Is a knowledge of the heavenly bodies, with regard to their magnitudes, motions, distances, &c. whether real or apparent; and of the natural causes on which their phenomena depend.

History of Astronomy.

The antiquity of this science may be gathered from what was spoken by the Deity at the time of creating the celestial luminaries, "Let them be for signs and seasons," &c.; whence it is thought probable that the human race never existed without some knowledge of astronomy among them. Indeed, besides the motives of mere curiosity, which of themselves may be supposed to have excited people to a contemplation of the glorious celestial canopy, as far as that was possible, it is easily to be seen that some parts of the science answer such essential purposes to mankind, that they could not possibly be dispensed with.

By some of the Jewish rabbins, Adam, in his state of innocence, is supposed to have been endowed with a knowledge of the nature, influence, and uses of the heavenly bodies; and Josephus ascribes to Seth and his posterity an extensive knowledge of astronomy. But whatever may be in this, the long lives of the Antediluvians certainly afforded such an excellent opportunity for observing the celestial bodies, that we cannot but suppose the science of astronomy to have been considerably advanced before the flood. Josephus says, that longevity was bestowed upon them for the very purpose of improving the sciences of geometry and astronomy. The latter could not be learned in less than 600 years: "for that period (says he) is the grand year." By which it is supposed he meant the period wherein the sun and moon came again into the same situation as they were in the beginning thereof, with regard to the nodes, apogee of the moon, &c. "This period (says Caffini), wherein we find no intimation in any monument of any other nation, is the finest period that ever was invented: for it brings out the solar year more exactly than that of Hipparchus and Ptolemy; and the lunar month within about one second of what is determined by modern astronomers. If the Antediluvians had such a period of 600 years, they must have known the motions of the sun and moon more exactly than their descendants knew them some ages after the flood."

On the building of the tower of Babel, Noah is supposed to have retired with his children born after the flood, to the north-eastern part of Asia, where his descendants peopled the vast empire of China. "This (says Dr Long) may perhaps account for the Chinese having so early cultivated the study of astronomy; their being so well settled in an admirable police, and continuing so many hundred years as they did in the worship of the true God." The vanity of that people indeed has prompted them to pretend a knowledge of astronomy almost as early as the flood itself. Some of the Jesuit missionaries have found traditional accounts among the Chinese, of their having been taught this science by their first emperor Fo-hi, supposed to be Noah; and Kempfer informs us, that this personage discovered the motions of the heavens, divided time into years and months, and invented the twelve signs into which they divide the zodiac, which they distinguish by the following... To the emperor Hong-ti, the grandson of Noah, they attribute the discovery of the pole-star, the invention of the mariner's compass, of a period of 60 years, and some kind of sphere. This extraordinary antiquity, however, is with good reason suspected, as is likewise their knowledge in the calculation of eclipses; of which Du Halde affirms us, that 36 are recorded by Confucius himself, who lived 551 years before Christ; and P. Trigault, who went to China in 1610, and read more than 100 volumes of their annals, says, "It is certain that the Chinese began to make astronomical observations soon after the flood; that they have observed a great number of eclipses, in which they have noted down the hour, day, month, and year, when they happened, but neither the duration nor the quantity; and that these eclipses have been made use of for regulating their chronology."

"But out of this abundance (says Dr Long), it is much to be regretted, that so very few of their observations have been particularized; for beside what has been mentioned above, we meet with no very ancient observations of the Chinese, except a winter solstice in the year 1111, and a summer solstice in the year 882, before Christ. Martini indeed speaks of a summer solstice 2342 years before that period. But M. Cassini, who calculated it, found that there must have been an error in the Chinese computation, of 500 years at least. An error of equal magnitude appears to have been committed in the conjunction of the five planets, which it is pretended they observed between the years 2513 and 2435 before Christ. In short, some have supposed, that none of these are real observations, but the result of bungling calculations; and it has been hinted, but surely on too slight a foundation, that even those good fathers themselves were greatly to be suspected. But let us come to things which are not contested.

"P. Gaubil informs us, that at least 120 years before Christ, the Chinese had determined by observation the number and extent of their constellations as they now stand; the situation of the fixed stars with respect to the equinoctial and solstitial points; and the obliquity of the ecliptic. He farther says, he cannot tell by what means it is that they foretell eclipses; but this is certain, that the theory by which they do predict them, was settled about the same time; and that they were acquainted with the true length of the solar year, the method of observing meridian altitudes of the sun by the shadow of a gnomon, and of learning from thence his declination and the height of the pole, long before. We learn, moreover, from the same missionary, that there are yet remaining among them some treatises of astronomy, which were written about 200 years before Christ; from which it appears, that the Chinese had known the daily motion of the sun and moon, and the times of the revolutions of the planets, many years before that period.

"We are informed by Du Halde, that, in the province of Honan, and city Teng-foang, which is nearly in the middle of China, there is a tower, on the top of which it is said that Tcheou-cong, the most skilful astronomer that ever China produced, made his observations. He lived 1200 years before Ptolemy, or more than 1000 years before Christ, and passed whole nights in observing the celestial bodies, and arranging them into constellations. He used a very large brass table placed perfectly horizontal, on which was fixed a long upright plate of the same metal, both of which were divided into degrees, &c. By these he marked the meridian altitudes; and from thence derived the times of the solstices, which were their principal epochs."

Dr Long represents the state of astronomy in China as at present very low; occasioned, he says, principally by the barbarous decree of one of their emperors*, to have all the books in the empire burnt, *See China, excepting such as related to agriculture and medicine. We are informed, however, by the Abbe Grosier, in his description of China, that astronomy is cultivated in Peking in the same manner as in most of the capital cities in Europe. A particular tribunal is established there, the jurisdiction of which extends to every thing relating to the observation of celestial phenomena. Its members are, an inspector; two presidents, one of them a Tartar and the other a Chinese; and a certain number of mandarins who perform the duty of assessors; but for near a century and a half the place of the Chinese president has been filled by an European. Since that time particular attention has been paid to the instruction of the astronomical pupils; and the presidents have always considered it as their duty to make them acquainted with the system and method of calculation made use of in Europe. Thus two-thirds of the astronomical pupils, maintained at the emperor's expense, in all about 200, have a tolerable notion of the state of the heavens, and understand calculation so well as to be able to compose ephemerides of sufficient exactness. The missionaries have never been the authors of any of these ephemerides: their employment is to revise the labours of the Chinese mathematicians, verify their calculations, and correct any errors into which they have fallen. The Portuguese mission still continues to furnish astronomers for the academy as it did at the first.

The astronomical tribunal is subordinate to that of ceremonies. When an eclipse is to be observed, information must be given to the emperor of the day and hour, the part of the heavens where it will be, &c. and this intelligence must be communicated some months before it happen: the eclipse must also be calculated for the longitude and latitude of the capital city of every every province of the empire. These observations, as well as the diagram which represents the eclipse, are preferred by the tribunal of ceremonies, and another called the calao, by whom it is transmitted to the different provinces and cities of the empire. Some days before the eclipse, the tribunal of ceremonies causes to be fixed up in a public place, in large characters, the hour and minute when the eclipse will commence; the quarter of the heavens in which it will be visible, with the other particulars relating to it. The mandarins are summoned to appear in state at the tribunal of astronomy, and to wait there for the moment in which the phenomenon will take place. Each of them carries in his hand a sheet of paper, containing a figure of the eclipse and every circumstance attending it. As soon as the observation begins to take place, they throw themselves on their knees, and knock their heads against the earth, and a horrid noise of drums and cymbals immediately commences throughout the whole city: a ceremony proceeding from an ancient superstitious notion, that by such noise they prevented the luminary from being devoured by the celestial dragon; and tho' this notion is now exploded in China, as well as everywhere else, such is the attachment of the people to ancient customs, that the ceremonial is still preserved.

While the mandarins thus remain prostrated in the court, others, stationed on the observatory, examine, with all the attention possible, the beginning, middle, and end of the eclipse, comparing what they observe with the figure and calculations given. They then write down their observations, affix their seal to them, and transmit them to the Emperor; who on his part has been no less assiduous to observe the eclipse with accuracy. A ceremonial of this kind is observed throughout the whole empire.

The Japanese, Siamese, and inhabitants of the Mogul's empire, have also, from time immemorial, been acquainted with astronomy; and the celebrated observatory at Benares, is a monument both of the ingenuity of the people and of their skill in this science.

Mr Bailly has been at great pains to investigate the progress of the Indians in astronomical knowledge, and gives a splendid account of their proficiency in the science, as well as of the antiquity of their observations. He has examined and compared four different astronomical tables of the Indian philosophers. 1. Of the Siamese, explained by M. le Gentil of the Academy of Sciences. 2. Those brought from India by M. le Gentil of the Academy of Sciences. 3. and 4. Two other manuscript tables found among the papers of the late M. de Lille. All of these tables have different epochs, and differ in form, being also constructed in different ways; yet they all evidently belong to the same astronomical system: the motions attributed to the sun and the moon are the same, and the different epochs are so well connected by the mean motions, as to demonstrate that they had only one, whence the others were derived by calculation. The meridians are all referred to that of Benares above-mentioned. The fundamental epoch of the Indian astronomy is a conjunction of the sun and moon, which took place at no less a distance of time than 3102 years before the Christian era. Mr Bailly informs us, that, according to our most accurate astronomical tables, a conjunction of the sun and moon actually did happen at that time. But though the bra-

mins pretend to have ascertained the places of the two luminaries at that time, it is impossible for us at this time to judge of the truth of their assertions, by reason of the unequal motion of the moon; which, as shall afterwards be more particularly taken notice of, now performs its revolution in a shorter time than formerly.

Our author informs us, that the Indians at present calculate eclipses by the mean motions of the sun and moon observed 5000 years ago; and with regard to the solar motion, their accuracy far exceeds that of the best Grecian astronomers. The lunar motions they had also settled by computing the space through which that luminary had passed in 1609,84 days, or somewhat more than 4383 years. They also make use of the cycle of 19 years attributed by the Greeks to Meton; and their theory of the planets is much better than that of Ptolemy, as they do not suppose the earth to be the centre of the celestial motions, and they believe that Mercury and Venus turn round the sun. Mr Bailly also informs us, that their astronomy agrees with the most modern discoveries of the decrease of the obliquity of the ecliptic, the acceleration of the motion of the equinoctial points, with many other particulars too tedious to enumerate in this place.

It appears also, that even the Americans were not unacquainted with astronomy, though they made use of the only of the solar, and not of the lunar motions, in their division of time. The Mexicans have had a strange predilection for the number 13. Their shortest periods consisted of 13 days; their cycle of 13 months, each containing 20 days; and their century of four periods of 13 years each. This excessive veneration for the number 13, according to Siguenza, arose from its being supposed the number of their greater gods. What is very surprising, though asserted as a fact by Abbé Clavigero, is, that having discovered the excess of a few hours in the solar above the civil year, they made use of intercalary days, to bring them to an equality: but with this difference in regard to the method established by Julius Caesar in the Roman calendar, that they did not interpose a day every four years, but 13 days (making use here even of this favourite number) every 52 years, which produces the same regulation of time.

Among those nations who first began to make any figure in ancient history, we find the Chaldeans and Egyptians most remarkable for their astronomical knowledge. Both of them pretended to an extravagant antiquity, and disputed the honour of having the first cultivators of the science. The Chaldeans boasted of their temple of Belus; and of Zoroaster whom they placed 5000 years before the destruction of Troy; the Egyptians boasted of their colleges of Priests, where astronomy was taught; and of the monument of Olymndyas, in which we are told was a golden circle 365 cubits in circumference and one cubit thick. The upper face was divided into 365 equal parts, answering to the days of the year; and on every division were written the name of the day, and the heliacal rising of the several stars for that day, with the prognostications from their rising, principally, as Long conjectures, for the weather.

The Chaldeans certainly began to make observations very soon after the confusion of languages; for when Alexander the Great took Babylon, Callisthenes, by his order, inquired after the astronomical observations recorded in that city, and obtained them for 1903 years back. Nothing, however, now remains of the Chaldean astronomy, excepting some periods of years which they had formed for the more ready computation of the heavenly bodies. But though they must have laboured under great disadvantages, for want of proper instruments, in those early ages, Gemina, as quoted by Petavius in his Uranologia, informs us, that they had determined, with tolerable exactness, the length both of a synodical and periodical month. They had also discovered, that the motion of the moon was not uniform, and even attempted to assign those parts of her orbit in which it was quicker or slower. Ptolemy also affirms us, that they were not unacquainted with the motion of the moon's nodes, and that of her apogee, supposing that the former made a complete revolution in $6285\frac{1}{7}$ days, or 18 years 15 days and 8 hours; which period, containing 223 complete lunations, is called the Chaldean Saros. The same author also gives us, from Hipparchus, several observations of lunar eclipses which had been made at Babylon about 720 years before Christ; but though he might very probably meet with many of a more ancient date, it was impossible to mention them particularly, on account of the imperfect state of the Chaldean chronology, which commenced only with the era of Nabonassar, 747 years before Christ. Aristotle likewise informs us, that they had many observations of the occultations of fixed stars and planets by the moon; and from hence, by a very natural and easy inference, they were led to conclude that the eclipses of the sun were occasioned also by the moon, especially as they constantly happened when the latter was in the same part of the heavens with the sun. They had also a considerable share in arranging the stars into constellations. Nor had the comets, by which astronomers in all ages have been so much perplexed, escaped their observation: for both Diodorus Siculus and Appollinus Myndius, in Seneca, inform us, that many of the Chaldeans held these to be lasting bodies, which have stated revolutions as well as the planets, but in orbits vastly more extensive; on which account they are only seen by us while near the earth, but disappear again when they go into the higher regions. Others of them were of opinion, that the comets were only meteors raised very high in the air, which blaze for a while, and disappear when the matter of which they consist is consumed or dispersed. Dialling was also known among them long before the Greeks were acquainted with any such thing.

It is evident, indeed, that the countries both of Chaldea and Egypt were exceedingly proper for astronomical observations, on account of the general purity and serenity of the air. The tower or temple of Belus, which was of an extraordinary height, with stairs winding round it up to the top, is supposed to have been an astronomical observatory; and the lofty pyramids of Egypt, whatever they were originally designed for, might possibly answer the same purpose. Indeed these very ancient monuments show the skill of this people in practical astronomy, as they are all situated with their four fronts exactly facing the cardinal points. Herodotus ascribes the Egyptian knowledge in astronomy to Seofris, whom Sir Isaac Newton makes contemporary with Solomon: but if this was the case, he could not be the instructor of the Egyptians in astronomical matters, since we find that Moses, who lived 500 years before Solomon, was skilled in all the wisdom of the Egyptians, in which we are undoubtedly to include astronomy.

From the testimony of some ancient authors, we learn that they believed the earth to be spherical, that they knew the moon was eclipsed by falling into its shadow, and that they made their observations with the greatest exactness. They even pretended to foretell the appearance of comets, as well as earthquakes and inundations; which extraordinary knowledge is likewise ascribed to the Chaldeans. They attempted to measure the magnitude of the earth and sun; but the methods they took to find out the latter were very erroneous. It does not indeed appear with certainty that they had any knowledge of the true system of the universe; and by the time of the Emperor Augustus, their astronomical knowledge was entirely lost.

From Chaldea the science of astronomy most probably passed into Phoenicia; though some are of opinion that the Phoenicians derived their knowledge of this science from the Egyptians. They seem, however, to have been the first who applied astronomy to the purposes of navigation; by which they became masters of the sea, and of almost all the commerce in the world. They became adventurous in their voyages, fleering their ships by one of the stars of the Little Bear; which being near the immovable point of the heavens called the Pole, is the most proper guide in navigation. Other nations made their observations by the Great Bear; which being too distant from the pole could not guide them in long voyages; and for this reason they never durst venture far from the coasts.

The first origin of astronomical knowledge among the Greeks is unknown. Sir Isaac Newton supposes that most of the constellations were invented about Greeks, the time of the Argonautic expedition; but Dr Long is of opinion that many of them must have been of a much older date; and that the shepherds, who were certainly the first observers, gave names to them according to their fancy; from whence the poets invented many of their fables. Several of the constellations are mentioned by Hesiod and Homer, the two most ancient writers among the Greeks, who lived about 870 years before Christ; Hesiod defining the farmer to regulate the time of sowing and harvest by the rising and setting of the Pleiades; and Homer informing us, that observations from the Pleiades, Orion, and Arcturus, were used in navigation. Their astronomical knowledge, however, was greatly improved by Thales the Milesian, who travelled into Egypt, and brought from thence the first principles of the science. He is said to have determined the height of the pyramids by measuring their shadows at the time the sun was 45 degrees high, and when of consequence the lengths of the shadows of objects are equal to their perpendicular heights. But his reputation was raised to the highest pitch among his countrymen, by the prediction of an eclipse, which happened just at the time that the armies of Alyattes king of Lydia, and Cyaxares the Mede, were about to engage; and being regarded as an evil omen by both parties, inclined them... to peace. To him Callimachus attributes the forming of the constellation of the Little Bear; the knowledge of which he certainly introduced into Greece. He also taught the true length of the year; determined the cosmical setting of the Pleiades in his time to have been 25 days after the autumnal equinox; divided the earth into five zones by means of the polar circles and tropics; taught the obliquity of the ecliptic; and showed that the equinoctial is cut by the meridians at right angles, all of which intersect each other at the poles. He is also said to have observed the exact time of the solstices, and from thence to have deduced the true length of the solar year; to have observed eclipses of the sun and moon; and to have taught that the moon had no light but what she borrowed from the sun.

According to Stanley, he also determined the diameter of the sun to be one-seventh part of his annual orbit.

"But (says Dr Long) these things should be received with caution. There are some reasons which might be assigned for supposing that the knowledge of Thales in these matters was much more circumscribed; and indeed it is not unreasonable to suppose, that that veneration for the ancients which leads authors to write profusely on the history of ancient times, may have induced them to ascribe full as much knowledge to them who lived in them as was really their due."

The successors of Thales, Anaximander, Anaximenes, and Anaxagoras, contributed considerably to the advancement of astronomy. The first is said to have invented or introduced the gnomon into Greece; to have observed the obliquity of the ecliptic; and taught that the earth was spherical, and the centre of the universe, and that the sun was not less than it. He is also said to have made the first globe, and to have set up a sun-dial at Lacedemon, which is the first we hear of among the Greeks; though some are of opinion that these pieces of knowledge were brought from Babylon by Pherecydes, a contemporary of Anaximander. Anaxagoras also predicted an eclipse which happened in the fifth year of the Peloponnesian war; and taught that the moon was habitable, consisting of hills, valleys, and waters, like the earth. His contemporary Pythagoras, however, greatly improved not only astronomy and mathematics, but every other branch of philosophy. He taught that the universe was composed of four elements, and that it had the sun in the centre; that the earth was round, and had antipodes; and that the moon reflected the rays of the sun; that the stars were worlds, containing earth, air, and ether; that the moon was inhabited like the earth; and that the comets were a kind of wandering stars, disappearing in the superior parts of their orbits, and becoming visible only in the lower parts of them. The white colour of the milky-way he ascribed to the brightness of a great number of small stars; and he supposed the distances of the moon and planets from the earth to be in certain harmonic proportions to one another. He is said also to have exhibited the oblique course of the sun in the ecliptic and the tropical circles, by means of an artificial sphere; and he first taught that the planet Venus is both the evening and morning star. This philosopher is said to have been taken prisoner by Cambyses, and thus to have become acquainted with all the mysteries of the Persian magi; after which he settled at Crotona in Italy, and founded the Italian sect.

About 440 years before the Christian era, Philolaus, a celebrated Pythagorean, asserted the annual motion of the earth round the sun; and soon after Hicetas, a Syracusan, taught its diurnal motion on its own axis. About this time also flourished Meton and Euctemon at Athens, who took an exact observation of the summer solstice 432 years before Christ; which is the oldest observation of the kind we have, excepting what is delivered by the Chinese. Meton is said to have composed a cycle of 19 years, which still bears his name; and he marked the risings and settings of the stars, and what seasons they pointed out: in all which he was assisted by his companion Euctemon. The science, however, was obscured by Plato and Aristotle, who embraced the system afterwards called the Ptolemaic, which places the earth in the centre of the universe.

Eudoxus the Cnidian was a contemporary with Aristotle, though considerably older, and is greatly celebrated on account of his skill in astronomy. He was the first who introduced geometry into the science, and he is supposed to be the inventor of many propositions attributed to Euclid. Having travelled into Egypt in the earlier part of his life, and obtained a recommendation from Agelaius to Nectanebus king of Egypt, he, by his means, got access to the priests, who had the knowledge of astronomy entirely among them, after which he taught in Asia and Italy. Seneca tells us that he brought the knowledge of the planetary motions from Egypt into Greece; and Archimedes, that he believed the diameter of the sun to be nine times that of the moon. He was also well acquainted with the method of drawing a sun-dial upon a plane; from whence it may be inferred that he understood the doctrine of the projection of the sphere; yet, notwithstanding what has been said concerning the observations of Eudoxus, it is not certain that his sphere was not taken from one much more ancient, ascribed to Chiron the Centaur. The reason given for this supposition is, that had the places of the stars been taken from his own observations, the constellations must have been half a sign farther advanced than they are said to be in his writings.

Soon after Eudoxus, Calippus flourished, whose system of the celestial sphere is mentioned by Aristotle; but he is better known from a period of 76 years, containing four corrected metonic periods, and which had its beginning at the summer solstice in the year 330, before Christ. But about this time, or rather earlier, the Greeks having begun to plant colonies in Italy, Gaul, and Egypt, these became acquainted with the Pythagorean system, and the notions of the ancient Druids concerning astronomy. Julius Caesar informs us, that the latter were skilled in this science; and that the Gauls in general were able sailors, which at that time they could not be without a competent knowledge of astronomy: and it is related of Pytheas, who lived at Marfelles in the time of Alexander the Great, that he observed the altitude of the sun at the summer solstice by means of a gnomon. He is also said to have travelled as far as Thule to settle the climates.

After the death of Alexander the Great, sciences State of flourished in Egypt more than in any other part of the world; and a famous school was set up at Alexandria, in Egypt under the auspices of Ptolemy Philadelphus, a prince educated in all kinds of learning, and the patron of all all those who cultivated them; and this school continued to be the seminary of all kinds of literature, till the invasion of the Saracens in 650. Timocharis and Aryllus, who first cultivated the astronomical science in this school, began to put it on a new footing; being much more careful in their observations, and exact in noting down the times when they were made, than their predecessors. Ptolemy assures us, that Hipparchus made use of their observations, by means of which he discovered that the stars had a motion in longitude of about one degree in an hundred years; and he cites many of their observations, the oldest of which is before the erection of this school, in the year 295, when the moon just touched the northern star in the forehead of the scorpion; and the last of them was in the 14th year of Philadelphus, when Venus hid the former star of the four in the left wing of Virgo.

From this time the study of astronomy continued greatly to advance. Aristarchus, who lived about 270 years before Christ, strenuously asserted the Pythagorean system, and gave a method of determining the distance of the sun by the moon's dichotomy. Eratosthenes, born at Cyrene in 271 B.C., determined the measure of a great circle of the earth by means of a gnomon. His reputation was so great, that he was invited from Athens to Alexandria by Ptolemy Euergetes, and made him keeper of the royal library at that place. At his instigation the same prince set up those armillae or spheres, which Hipparchus and Ptolemy the astronomers afterwards employed so successfully in observing the heavens. He also found the distance between the tropics to be eleven such parts as the whole meridian contains eighty-three. About the same time Berossus, a native of Chaldea, flourished at Athens. He is said to have brought many observations from Babylon, which are ascribed to the Greeks; while others contend, that the latter owe little or nothing of their astronomical knowledge to the Babylonians. The celebrated Archimedes, who next to Sir Isaac Newton holds the first place among mathematicians, was nothing inferior as an astronomer to what he was as a geometrician. He determined the distance of the moon from the earth, of Mercury from the moon, of Venus from Mercury, of the sun from Venus, of Mars from the sun, of Jupiter from Mars, and of Saturn from Jupiter; as likewise the distance of the fixed stars from the orbit of Saturn. That he made astronomical observations, is not to be doubted; and it appears from an epigram of the poet Claudian, that he invented a kind of planetarium, or orrery, to represent the phenomena and motions of the heavenly bodies.

Hipparchus was the first who applied himself to the study of every part of astronomy, his predecessors having chiefly considered the motions and magnitudes of the sun and moon. Ptolemy also informs us, that he first discovered the orbits of the planets to be eccentric, and on this hypothesis wrote a book against Eudoxus and Calippus. He gives many of his observations; and says, that by comparing one of his with another made by Aristarchus 145 years before, he was enabled to determine the length of the year with great precision. Hipparchus also first found out the anticipation of the moon's nodes, the eccentricity of her orbit, and that she moved slower in her apogee than in her perigee. He collected the accounts of such ancient eclipses as had been observed by the Chaldeans and Egyptians. He formed hypotheses concerning the celestial motions, and constructed tables of those of the sun and moon, and would have done the same with those of the other planets, if he could have found ancient observation sufficient for the purpose; but, these being wanting, he was obliged to content himself with collecting fit observations for that purpose, and endeavouring to form theories of the five planets. By comparing his own observations on the spica virginis with those of Timocharis at Alexandria made 100 years before, he discovered that the fixed stars changed their places, and had a slow motion of their own from west to east. He corrected the Calippic period, and pointed out some errors in the method laid down by Eratosthenes for measuring the circumference of the earth. By means of geometry, which was now greatly improved, he was enabled to attempt the calculation of the sun's distance in a more correct manner than any of his predecessors; but unhappily it required so much accuracy in observation as was found impracticable. His greatest work, however, was his catalogue of the fixed stars, which he was induced to attempt by the appearance of a new star. The catalogue is preserved by Ptolemy, and contains the longitudes and latitudes of 1022 stars, with their apparent magnitudes. He wrote also concerning the intervals between eclipses both solar and lunar, and is said to have calculated all that were to happen for no less than 600 years from his time.

Little progress was made in astronomy from the time of Hipparchus to that of Ptolemy, who flourished in the first century. The principles on which his system is built are indeed erroneous; but his work will always be valuable on account of the number of ancient observations it contains. It was first translated out of the Greek into Arabic in the year 827, and into Latin from the Arabic in 1230. The Greek original was unknown in Europe till the beginning of the 15th century, when it was brought from Constantinople, then taken by the Turks, by George a monk of Trapezond, who translated it into Latin. Various editions were afterwards published; but little or no improvement was made by the Greeks in this science.

During the long period from the year 800 to the beginning of the 14th century, the western parts of Asia-Europe were immersed in deep ignorance and barbarism. However, several learned men arose among the Arabians. The caliph Al Mamun was the first who introduced a taste for the sciences in his empire. His grandson Al Mamun, who ascended the throne in 814, was a great encourager of the sciences, and devoted much of his own time to the study of them. He made many astronomical observations himself, and determined the obliquity of the ecliptic to be $23^\circ 35'$. He employed many able mechanics in constructing proper instruments, which he made use of for his observations; and under his auspices a degree of the earth was measured a second time in the plain of Singar, on the border of the Red Sea. From this time astronomy was studiously cultivated by the Arabians; and Elements of Astronomy were written by Alferganus, who was partly contemporary with the caliph Al Mamun. But the most celebrated of all their astronomers is Albategnius. nianus, who lived about the year of Christ 880. He greatly reformed astronomy, by comparing his own observations with those of Ptolemy. Thus he calculated the motion of the sun's apogee from Ptolemy's time to his own; determined the precession of the equinoxes to be one degree in 70 years; and fixed the sun's greatest declination at $23^\circ 35'$. Finding that the tables of Ptolemy required much correction, he composed new ones of his own fitted to the meridian of Aracta, which were long held in estimation by the Arabians.

After his time, though several eminent astronomers appeared among the Saracens, none made any very valuable observations for several centuries, excepting Ibn Younis astronomer to the caliph of Egypt; who observed three eclipses with such care, that by means of them we are enabled to determine the quantity of the moon's acceleration since that time.

Other eminent Saracen astronomers were, Arzachel a Moor of Spain, who observed the obliquity of the eclipses, and constructed tables of sines, or half chords of double arcs, dividing the diameter into 300 parts; and Alhazen, his contemporary, who first showed the importance of the theory of refractions in astronomy; writing also upon the twilight, the height of the clouds, and the phenomenon of the horizontal moon.

Ulug Beg, a grandson of the famous Tartar prince Timur Beg, or Tamerlane, was a great proficient in practical astronomy. He is said to have had very large instruments for making his observations; particularly a quadrant as high as the church of Sancta Sophia at Constantinople, which is 180 Roman feet. He composed astronomical tables from his own observations for the meridian of Samarcan and his capital, so exact as to differ very little from those afterwards constructed by Tycho Brahe; but his principal work is his catalogue of the fixed stars, made from his own observations in the year of Christ 1437. The accuracy of his observations may be gathered from his determining the height of the pole at Samarcan to be $39^\circ 37' 23''$.

Besides these improvements, we are indebted to the Arabs for the present form of trigonometry. Menelaus, indeed, an eminent Greek astronomer who flourished about the year 90, had published three books of Spherics, in which he treated of the geometry necessary to astronomy, and which show great skill in the sciences; but his methods were very laborious, even after they had been improved and rendered more simple by Ptolemy; but Geber the Arabian, instead of the ancient method, proposed three or four theorems, which are the foundation of our modern trigonometry. The Arabs also made the practice still more simple, by using sines instead of the chords of double arcs. The arithmetical characters they had from the Indians.

During the greatest part of this time, almost all Europe continued ignorant not only of astronomy but of every other science. The emperor Frederick II. first began to encourage learning in 1230; restoring some universities, and founding a new one in Vienna. He also caused the works of Aristotle, and the Almagest or Astronomical Treatise of Ptolemy, to be translated into Latin; and from the translation of this book we may date the revival of astronomy in Europe. Two years after its publication, John de Sacro Bosco, or of Halifax, an Englishman, wrote his four books De Sphaera, which he compiled from Ptolemy, Albategnius, Alfer-

ganus, and other Arabian astronomers: this work was so much celebrated, that for 300 years it was preferred in the schools to every other; and has been thought worthy of several commentaries, particularly by Clavius in 1531. In 1240, Alphonso king of Castile caused the tables of Ptolemy to be corrected: for which purpose he assembled many persons skilled in astronomy, Christians, Jews, and Moors; by whom the tables called Alphonsine were composed, at the expense of 40,000, or according to others 400,000 ducats. About the same time Roger Bacon, an English monk, published many things relative to astronomy; particularly of the places of the fixed stars, solar rays, and lunar aspects. Vitello, a Polander, wrote a treatise on Optics about 1270, in which he showed the use of refractions in astronomy.

From this time to that of Purbach, who was born in 1423, few or no improvements were made in astro-nomy. He wrote a commentary on Ptolemy's Al-Purbach magest, some treatises on Arithmetic and Dialling, with tables for various climates. He not only used spheres and globes, but constructed them himself; and formed new tables of the fixed stars, reduced to the middle of that age. He composed also new tables of sines for every ten minutes, which Rigiomontanus afterwards extended to every single minute, making the whole fine 60, with 6 cyphers annexed. He likewise corrected the tables of the planets, making new equations to them, because the Alphonsine tables were very faulty in this respect. In his solar tables he placed the sun's apogee in the beginning of Cancer; but retained the obliquity of the ecliptic $23^\circ 35' 23''$, to which it had been reduced by the latest observations. He made new tables for computing eclipses, of which he observed some, and had just published a theory of the planets, when he died in 1461.

John Muller of Monteregio (Coningsberg), a town of Franconia, from whence he was called Regiomontanus, was the scholar and successor of Purbach. He completed the epitome of Ptolemy's Almagest which Purbach had begun; and after the death of the latter, went to Rome, where he made many astronomical observations. Having returned to Nuremberg in 1471, he was entertained by a wealthy citizen named Bernard Walther, who having a great love for astronomy, caused several instruments to be made under the direction of Regiomontanus, for observing the altitude of the sun and stars, and other celestial phenomena. Among these was an armillary astrolabe, like that which had been used by Hipparchus and Ptolemy at Alexandria, and with which many observations were made. He also made ephemerides for 30 years to come, showing the lunations, eclipses, &c. He wrote the Theory of the Planets and Comets, and a Treatise of Triangles yet in repute for several extraordinary cases. He is said to have been the first who introduced the use of tangents into trigonometry; and to have published in print (the art of printing having been lately invented) the works of many of the most celebrated ancient astronomers. After his death, which happened at Rome, Walther made a diligent search for all his instruments and papers which could be found; and continued his observations with the instruments he had till his death. The observations of both were collected by order of the senate of Nuremberg, and published there. The Motion of Venus and Mercury in respect of the Earth. there by John Schoner in 1544; afterwards by Snellius at the end of the observations made by the Landgrave of Hesse in 1618; and lastly, in 1666, with those of Tycho Brahe. Walther, however, as we are told by Snellius, found fault with his armilla, not being able to give any observation with certainty to less than ten minutes. He made use of a good clock, which also was a late invention in those days.

John Werner, a clergyman, succeeded Walther as astronomer at Nuremberg; having applied himself with great assiduity to the study of that science from his infancy. He observed the motion of the comet in 1500; and published several tracts, in which he handled many capital points of geometry, astronomy, and geography, in a masterly manner. He published a translation of Ptolemy's Geography, with a commentary, which is still extant. In this he first proposed the method of finding the longitude at sea by observing the moon's distance from the fixed stars; which is now so successfully put into practice. He also published many other treatises on mathematics and geography; but the most remarkable of all his treatises, are those concerning the motion of the eighth sphere or of the fixed stars, and a short theory of the same. In this he showed, by comparing his own observations of the stars regulus, spica virginis, and the bright star in the southern scale of the balance, made in 1514, with the places assigned to the same stars by Ptolemy, Alphonso, and others, that the motion of the fixed stars, now called the precession of the equinoctial points, is one degree ten minutes in 100 years, and not one degree only, as former astronomers had made it. He made the obliquity of the ecliptic 23° 28', and the first star of Aries 26° distant from the equinoctial point. He also constructed a planetarium representing the celestial motions according to the Ptolemaic hypotheses, and made a great number of meteorological observations with a view towards the prediction of the weather. The obliquity of the ecliptic was settled by Dominic Maria, the friend of Copernicus, at 23° 29', which is still held to be just.

The celebrated Nicolaus Copernicus next makes his appearance, and is undoubtedly the greatest reformer of the astronomical science. He was originally bred to the practice of medicine, and had obtained the degree of Doctor in that faculty; but having conceived a great regard for the mathematical sciences, especially astronomy, he travelled into Italy, where he for some time was taught by Dominic Maria, or rather affiliated him in his astronomical operations. On his return to his own country, being made one of the canons of the church, he applied himself with the utmost assiduity to the contemplation of the heavens, and to the study of the celestial motions. He soon perceived the deficiency of all the hypotheses by which it had been attempted to account for these motions; and for this reason he set himself to study the works of the ancients, with all of whom he also was dissatisfied excepting Pythagoras; who, as has been already related, placed the sun in the centre, and supposed all the planets, with the earth itself, to revolve round him. He informs us, that he began to entertain these notions about the year 1507; but not being satisfied with stating the general nature of his hypothesis, he became desirous of determining the several periodical revolutions of the planets, and hence of constructing tables of their motions which might be more agreeable to truth than those of Ptolemy and Alphonso. The observations he was enabled to make, however, must have been extremely inaccurate; as he tells us, that if with the instruments he made use of he should be able to come within ten minutes of the truth, he would rejoice no less than Pythagoras did when he discovered the proportion of the hypotenuse to the other two sides of a right-angled triangle. His work was completed in the year 1530; but he could not be prevailed upon to publish it till towards the end of his life, partly through diffidence, and partly through fear of the offence which might be taken at the singularity of the doctrines set forth in it. At last, overcome by the importunities of his friends, he suffered it to be published at their expense, and under the inspection of Schoner and Osiander, with a dedication to Pope Paul III., and a preface, in which it was attempted to palliate as much as possible the extraordinary innovations it contained. During the time of its publication, the author himself was attacked by a bloody flux, succeeded by a palsy; so that he received a copy only a few hours before his death, which happened on the 23rd of May 1543.

After the death of Copernicus, the astronomical science was greatly improved by Schoner, Nonius, Apian, and Gemma Frisius. Schoner survived Copernicus only four years; however, he greatly improved the methods of making celestial observations, reformed and explained the calendar, and published a treatise of cephography. Nonius had applied himself very early to the study of astronomy and navigation; but finding the instruments at that time in use excessively inaccurate, he applied himself to the invention of others which should be less liable to inconvenience. Thus he invented the astronomical quadrant, in which he divided the degrees into minutes by a number of concentric circles. The first of these was divided into 90 equal parts, the second into 89, the third into 88, and so on, as low as 46; and thus, as the index of the quadrant would always fall upon one or other of the divisions, or very near it, the minutes might be known by computation. He published many treatises on mathematical subjects, particularly one which detected the errors of Orontius, who had imagined that he could square the circle, double the cube, &c., by finding two mean proportions between two right lines. Apian's chief work was intitled The Cæsarean Astronomy; and was published at Ingolstadt in 1540, dedicated to the emperor Charles V. and his brother Ferdinand. In this he showed how to resolve astronomical problems by means of instruments, without either calculations or tables; to observe the places of the stars and planets by the astrolabe; and to foretell eclipses and describe the figures of them; the whole illustrated by proper diagrams. In his second book he describes the method of dividing an astronomical quadrant, and of using it properly. His treatise concludes with the observation of five comets. Gemma Frisius wrote a commentary on a work of Apian, intitled his Cephography, with many observations of eclipses. He invented also the astronomical ring, and several other instruments, which, though they could not boast of much exactness superior to others, were yet of considerable utility in taking observations at at sea; and he is also memorable for being the first who proposed a time-keeper for determining the longitude at sea.—George Joachim Rheticus was a scholar of Copernicus, to attend whose lectures he gave up his professorship of mathematics at Wittenberg. For the improvement of astronomical calculations, he began to construct a table of sines, tangents, and secants, for every minute and ten seconds of the quadrant. In this work he first showed the use of secants in trigonometry, and greatly enlarged the use of tangents, first invented by Regiomontanus; but he assigned for the radius a much larger number of places than had been done before, for the greater exactness of calculation. This great work he did not live to accomplish; but it was completed by his disciple Valentine Otho, and published at Heidelberg in 1594.

During this century, the list of astronomers was dignified by some very illustrious names. About the year 1561, William IV. Landgrave of Hesse Cassel, applied himself to the study of astronomy. With the assistance of Rothman and Burgius, the former an astrologer, the latter an excellent mathematical instrument maker, he erected an observatory on the top of his palace at Cassel, and furnished it with such instruments as were then in use, made in the best manner the artists of that age could execute. With these he made a great number of observations, which were by Hevelius preferred to those of Tycho Brahe, and which were published by Snellius in 1618. From these observations he determined the longitudes and latitudes of 400 stars, which he inserted in a catalogue where their places are rectified to the beginning of the year 1593.

Tycho Brahe began his observations about the same time with the Landgrave of Hesse, already mentioned. He observed the great conjunction of Saturn and Jupiter in 1563; and finding the instruments he could procure very inaccurate, he made a quadrant capable of showing single minutes, and likewise a sextant four cubits radius. In 1571, he discovered a new star in the chair of Cassiopeia; which induced him, like Hipparchus, to make a catalogue of the stars. This contained the places of 777 stars, rectified to the year 1600; but instead of the moon, which was used by the ancients to connect the places of the sun and stars, Tycho substituted Venus as having little or no parallax, and yet being like the moon visible both day and night. By the recommendation of the Landgrave of Hesse, he obtained from the king of Denmark the island of Hvenna, opposite to Copenhagen, where an Observatory Account of was built. The first stone of this building, afterwards Uraniburg called Uraniburg, was laid in the year 1576. It was his Observa- of a square form, one side of it being about 60 feet in length; and on the east and west sides were two round towers of 32 feet diameter each. The instruments were more large and solid than had ever been seen before by any astronomer. They consisted of quadrants, sextants, circles, semicircles, armilla both equatorial and zodiacal, parallactic rulers, rings, astrolabes, globes, clocks, and sun-dials. These instruments were divided as to show single minutes; and in some the arch might be read off to 10 seconds. Most of the divisions were diagonal; but he had one quadrant divided according to the method invented by Nonius; that is, by 47 concentric circles. The whole expense is said to have amounted to 200,000 crowns. The method of dividing by diagonals, which Tycho greatly admired, was the invention of Mr Richard Chancelor, an Englishman; Tycho, however, shows that it is not accurately true when straight lines are employed, and the circles at equal distances from each other; but that it may be corrected by making circular diagonals, which if continued would pass through the centre.

Tycho employed his time at Uraniburg to the best advantage; but falling into discredit on the death of the king, he was obliged to remove to Holstein, and at last found means to get himself introduced to the emperor, with whom he continued to his death. He is well known to have been the inventor of a system of astronomy, which bears his name; and which he vainly endeavoured to establish on the ruins of that of Copernicus: but the simplicity and evident conformity to the phenomena of nature, displayed in all parts of the Copernican system, soon got the better of the unnatural and complicated system of Tycho. His works, however, which are very numerous, discover him to have been a man of vast abilities. After his death the castle of Uraniburg quickly fell to decay, and indeed seems to have been purposely pulled down; for, in 1652, when Mr Huet went to Sweden, it was almost level with the ground, and few traces of the walls could be discerned. None of the neighbouring inhabitants had ever heard of the name of Tycho or Uraniburg, excepting one old man, whom Mr Huet found out with great difficulty, and who had been a servant in the family. All the discoveries of Purbach, Regiomontanus, and Tycho, were collected and published in the year 1621, by Longomontanus, who had been Tycho's favourite scholar.

While Tycho resided at Prague with the emperor, he invited thither John Kepler, afterwards so famous for his discoveries. Under the tuition of so great an astronomer, the latter quickly made an amazing progress. He found that his predecessors had erred in supposing the orbits of the planets to be circular, and their motions uniform: on the contrary, he perceived from his own observations, that they were elliptical, and their motions unequal, having the sun in one of the foci of their orbits; but that, however they varied in absolute velocity, a line drawn from the centre of the sun to the planet, and revolving with it, would always describe equal areas in equal times. He discovered, in the year 1618, that the squares of the periodical times are as the cubes of the distances of the planets; two laws which have been of the greatest importance to the advancement of astronomy. He seems to have had some notion of the extensive power of the principle of gravity: for he tells us, that gravity is a mutual power betwixt two bodies; that the moon and earth tend towards each other, and would meet in a point nearer the earth than the moon in the proportion of the superior magnitude of the former, were they not hindered by their projectile motions. He adds also, that the tides arise from the gravitation of the waters towards the moon; however, he did not adhere steadily to these principles, but afterwards substituted others as the causes of the planetary motions.

Contemporary with Kepler were Mr Edward Wright, and Napier baron of Merchiston. To the former we owe several very good meridional observations of the sun's fun's altitude, made with a quadrant of six feet radius, in the years 1594, 1595, and 1596; from which he greatly improved the theory of the sun's motion, and computed more exact tables of his declination than had been done by any person before. He published also, in 1599, an excellent Treatise, intitled, "Certain Errors in Navigation discovered and detected."

To the latter we are indebted for the knowledge of logarithms; a discovery, as was justly observed by Dr Halley, one of the most useful ever made in the art of numbering. John Bayer, a German, who lived about the same time, will ever be memorable for his work, intitled, Uranometria, which is a very complete celestial atlas, or a collection of all the constellations visible in Europe. To this he added a nomenclature, in which the stars in each constellation are marked with the letters of the Greek alphabet; and thus every star in the heavens may be referred to with the utmost precision and exactness. About the same time also, astronomy was cultivated by many other persons; abroad, by Maginus, Mercator, Maurolycus, Homelinus, Schultet, Stevin, &c.; and by Thomas and Leonard Digges, John Dee, and Robert Flood, in England: but none of them made any considerable improvement.

The beginning of the 17th century was distinguished not only by the discovery of logarithms, but by that of telescopes; a fort of instruments by which astronomy was brought to a degree of perfection utterly inconceivable by those who knew nothing of them. The question concerning the inventor is discussed under the article Optics; but whoever was entitled to this merit, it is certain that Galileo was the first who brought them to such perfection as to make any considerable discoveries in the celestial regions. With instruments of his own making, Galileo discovered the inequalities in the moon's surface, the satellites of Jupiter, and the ring of Saturn; though this last was unknown to him after he had seen it, and the view he got made him conclude that the planet had a threefold body, or that it was of an oblong shape like an olive. He discovered spots on the sun, by means of which he found out the revolution of that luminary on his axis; and he discovered also that the milky way and nebulae were full of small stars. It was not, however, till some time after these discoveries were made, that Galileo and others thought of applying the observations on Jupiter's satellites to the purpose of finding the longitude of places on the surface of the earth; and even after this was thought of, astronomers found it to difficult to construct tables of their motions, that it was not till after many observations had been made in distant places of the world, that Cassini was able to determine what positions of the satellites were most proper for finding out the longitude. At last he perceived that the entrance of the first satellite into the shadow of Jupiter, and the exit of it from the same, were the most proper for this purpose; that next to these the conjunction of the satellites with Jupiter, or with one another, may be made use of, especially when any two of them, moving in contrary directions, meet with each other; and lastly, that observations on the shadows of the satellites, which may be seen on the disk of Jupiter, are useful, as also the spots which are seen upon his face, and are carried along it with greater velocity than has hitherto been discovered in any of the other heavenly bodies.

While astronomers were thus busy in making new discoveries, the mathematicians in different countries were no less earnestly employed in constructing logarithmic tables to facilitate their calculations. Benjamin Ursinus, an excellent mathematician of Brandenburg, calculated much larger tables of logarithms than had been done by their noble inventor, and published them in 1625. They were improved by Henry Briggs, Savilian professor of Oxford; who by making unity the logarithm of ten, thus rendered them much more convenient for the purposes of calculation. Logarithmic tables of sines and tangents were also composed by Mr Briggs and Adrian Vlacq at Gouda, so that the business of calculation was now rendered nearly as easy as possible.

In 1633, Mr Horrox, a young astronomer of very extraordinary talents, discovered that Venus would pass over the disk of the sun on the 24th of November 1639. This event he announced only to one friend, a Mr Crabtree; and these two were the only persons in the world who observed this transit the first time it had ever been viewed by human eyes. Mr Horrox made many useful observations at the time; and had even formed a new theory of the moon, so ingenious as to attract the notice of Sir Isaac Newton: but the hopes of astronomers from the abilities of this excellent young man were blasted by his death in the beginning of January 1640.

About the year 1638 many learned men began to assemble at Paris in order to hold conferences on different scientific subjects, which was the first foundation of the Royal Academy of Sciences in that capital. This practice was introduced in France by Mericenus, Royal Society, and soon after at London by Oldenburg; which laid the foundation of the Royal Society there. About this time also the celebrated astronomer Hevelius flourished at Dantzig, building an observatory in his own house, and furnishing it with excellent instruments of his own construction; particularly octants and sextants of brass, of three and four feet radius, as well as telescopes, with which he constantly observed the spots and phases of the moon, and from which observations he afterwards compiled his excellent and beautiful work intitled Selenographia. This noble building, together with all the books and instruments it contained, was consumed by fire on the 26th of September 1679; but the memory, as well as the form and construction of the instruments, is preserved in a curious work of the ingenious inventor, intitled Machina Coelestis; though almost the whole impression of this book was involved in the same fate with the instruments it describes. The damage sustained on this occasion was estimated at 30,000 crowns.

The celebrated English mechanic Dr Hooke, who was contemporary with Hevelius, had in the mean time invented instruments with telescopic sights, which he preferred to those used by Hevelius so much, that a dispute commenced, which procured Hevelius a visit from Dr Halley. The latter had at that time taken a voyage to St Helena at the desire of the Royal Society, in order to observe and form a catalogue of the stars in the northern hemisphere. The result of his observations with Hevelius's instruments was, that three several observations on the Spica Virginis and Regulus differed only a few seconds from each other. They were the invention of Tycho Brahe, and are described described under the article Optics. At this visit Halley and Hevelius observed an occultation of Jupiter by the moon, and determined the diameter of the latter to be $36' 35''$.

In 1671, the Royal Observatory in Paris was founded, and the use of it assigned to Mr Cassini, after it had been furnished with instruments at a very great expense; and the observatory at Greenwich being likewise built five years after, Mr Flamsteed was appointed astronomer-royal. The observations in both these places, however, have been so numerous, that it is in vain to attempt any account of them.

Before the middle of the 17th century the construction of telescopes had been greatly improved, particularly by Fontana and Huygens. The latter constructed one of 12 feet, which is still preserved in the museum of the Royal Society at London. With this he observed the moon and planets for a long time, and discovered that Saturn was encompassed with a ring. The French, however, still outdid the English artists; and by means of telescopes of 200 and 300 feet focus, Mr Cassini was enabled to see all the five satellites of Saturn, his belts, and the shadows of Jupiter's satellites passing over his body. In 1666, Mr Azoult applied a micrometer to telescopes for the purpose of measuring the diameters of the planets, and small distances in the heavens; however, an instrument of this kind had been before invented by Mr Galcoigne, though it was but little known abroad.

Notwithstanding all these discoveries by means of telescopes, it was evident that they still continued in a very imperfect state, and their imperfections at the time appeared to be without remedy. One defect was the enormous length requisite to admit of any very considerable magnifying power; and another was the incorrectness of the image, arising from the aberration of the rays, as was then supposed, by the spherical figure of the glass. To obviate these inconveniences, Mercenius is said to have first proposed, in a letter to Descartes, the use of reflectors instead of lenses in the construction of telescopes; but this he did in such an obscure manner, that the latter laboured to persuade him of the falsehood of the principle on which his scheme was founded. In 1663, however, James Gregory of Aberdeen showed how such a telescope might be constructed. He showed also, that, in order to form a perfect image of an object in this manner, the figure of the speculum ought to be parabolic; but Sir Isaac Newton, who applied himself to the framing of telescopes of the reflecting kind, found it impracticable to grind them of the desired figure. Laying aside the idea of reflecting telescopes, therefore, he applied himself to the execution of a scheme formed by Descartes, viz. that of grinding lenses of the figure of one of the conic sections. In prosecuting this plan, he discovered, that the greatest errors to which telescopes were subject arose from the different refrangibility of the rays of light, for which he could not then find any remedy. He therefore returned to the scheme he had just abandoned; and, in the year 1672, presented to the Royal Society two reflectors which were constructed with spherical speculums, as he could not procure any other. The inconveniences arising from the different refrangibility of the rays of light, have since been in the fullest manner corrected by Mr Dolland, the excellency of whose achromatic telescopes are too well known to need any encomium.

About the beginning of the present century, the practical part of astronomy seemed to languish for want of proper instruments. Roemer, indeed, had invented some new ones, and Dr Hook had turned his attention towards this subject in a very particular manner; but either through want of skill in the art, or some other unfortunate circumstance, it happened that nothing effectual was done. But at the very time when this was the case with practical astronomy, the speculative part was carried in a manner to its utmost pitch by the labours of the immortal Newton, whose Principia gave an entire new face to the science. It was not, however, for many years relished by the foreign philosophers, though almost immediately adopted at home, and has continued ever since to spread its reputation farther and farther, so that now it is in a manner established all over the world. "But (says Dr Long) that, after Newton's system had for a long time been neglected, it should all at once be universally received and approved of, is not to be attributed to chance, or the caprice of fashion, as some who are ignorant of it are apt to think, and from thence to expect that some other system will hereafter take its place, and bury it in oblivion. The system of Newton, like that of Copernicus, is so agreeable to the phenomena of nature, and so well put together, that it may last as long as truth and reason endure, although time may perhaps bring the word attraction into disuse; and though it may no longer be thought inherent in matter, yet the laws of gravitation, as they are now called, and on which this system is founded, will never be forgotten."

It was also in Britain that the first improvements in astronomical instruments took place. The celebrated mechanic and watchmaker Graham, carried the accuracy of his instruments to a degree which surprised everyone. He also greatly improved the principles of watchwork, and made clocks to go with much greater regularity than before. The old eight feet mural arch at Greenwich was also constructed by him; as was a small equatorial sector for making observations out of the meridian; but he is chiefly remarkable for contriving the zenith sector of 24 feet radius, and afterwards one of 12½ feet, by which Dr Bradley discovered the aberration of the fixed stars. The reflecting telescope which had been invented by Gregory, and executed by Newton, was greatly improved by Mr Hadley, and a very complete and powerful instrument of that kind was presented to the Royal Society in 1719. The same gentleman has also immortalized his memory by the invention of the reflecting quadrant, which he presented to the Society in 1731, which is now in universal use at sea; and without which all improvements of the lunar theory would have been useless for determining the longitude, through the want of an instrument proper to make the observations with. It however appears, that an instrument, exactly similar to this in its principles, had been invented by Sir Isaac Newton; and a description of it, together with a drawing, given by the inventor to Dr Halley, when he was preparing for his voyage to discover the variation of the needle. needle in 1701. About the middle of this century, the constructing and dividing of large astronomical instruments was carried to a great degree of perfection by Mr John Bird; reflecting telescopes were equally improved by Mr Short, who first executed the divided object-glass micrometer. This had indeed been thought of by M. Louville, and several other persons long before; and a description of one nearly agreeing with that of Mr Short had been published in the Philosophical Transactions for 1753: but had it not been for the great skill of Mr Short in figuring and centering glasses of this kind, it is very probable the scheme might never have been executed. About this time also Mr Dollond brought refracting telescopes to such perfection, that they became superior to reflectors of equal length; though all of them are now excelled by those of Mr Herschel, whose Telescopic discoveries have been far more numerous and surprising than those of any other astronomer.

We shall close this history with a short account of the labours of the principal astronomers since the building the Royal Observatories at Paris, Greenwich, and the appointment of Mr Flamsteed to the office of astronomer royal. This gentleman not only made observations on the sun, moon, planets, and comets which appeared in his time, but on the fixed stars also, of which he gave a catalogue of 3000; many of them so small that they cannot be discerned without the help of a telescope: he also published new solar tables, and a theory of the moon according to Horrox. He published a very curious tract on the doctrine of the sphere, in which he showed how to construct eclipses of the sun and moon, as well as occultations of the fixed stars by the moon geometrically; and it was upon his observations that Halley's tables and Newton's theory of the moon were constructed. Mr Caffini also distinguished himself very considerably. He erected the gnomon, and drew the famous meridian line in the church of Petronia at Bologna. He enjoyed his office more than 40 years, making many observations on the sun, moon, planets, and comets, and greatly amended the elements of their motions; though the result of his labours was much inferior to Mr Flamsteed's. The office was continued in his family, and his grandson still enjoys it. Roemer, a celebrated Danish astronomer, first discovered the progressive motion of light by observing the eclipses of Jupiter, and read a dissertation upon it before the Royal Academy of Sciences at Paris in the year 1675. He was also the first who made use of a meridional telescope.

Mr Flamsteed was succeeded in 1719 by Dr Halley, "the greatest astronomer (says M. de la Lande) without contradiction in England;" and, adds Dr Long, "I believe he might have laid in the whole world." He had been sent, at the age of 21, by King Charles II. to the island of St Helena, in order to make a catalogue of the southern stars, which was published in 1679. In 1705, he published his Synopsis Astronomiae Cometicæ, in which, after immense calculation, he ventured to predict the return of one in 1758 or 1759. He also published many learned dissertations in the Philosophical Transactions concerning the use that might be made of the next transit of Venus in determining the distance of the sun from the earth. He was the first who discovered the acceleration of the moon, and gave a very ingenious method of finding her parallax by three observed phases of a solar eclipse. He composed tables of the sun, moon, and all the planets; and, in the nine years in which he was at Greenwich, made near 1500 observations of the moon: all which he compared with the tables, and noted the differences; and these, he thought, would return in about 18 years. He recommended the method of determining the longitude by means of the moon's distance from the sun and certain fixed stars. He was convinced of its superior excellence; and it has since been adopted by all the most eminent astronomers in Europe. It is at present the only sure guide to the mariner; and the great perfection to which it is now brought is much owing to the industry and exertions of Dr Maskelyne, the present astronomer-royal, to whom we are indebted for the publication of the Nautical Almanac, the Requisite Tables, and other works of the utmost service to practical astronomy.

In the mean time an attempt was made in France to measure a degree of the earth, which occasioned a very warm dispute concerning the figure of it. Cassini, from Picart's measure, concluded that the earth was an oblong spheroid; but Newton, from a consideration of the laws of gravity and the diurnal motion of the earth, had determined the figure of it to be an oblate spheroid, and flatted at the poles. To determine this point, Lewis XV. resolved to have two degrees of the meridian measured: one under, or very near the equator; and the other as near the pole as possible. For this purpose the Royal Academy of Sciences sent M. Maupertuis, Clairault, Camus, and Le Monier, to Lapland. They were accompanied by the Abbé Outhier, a correspondent of the same academy. They were joined by M. Celsius professor of anatomy at Upsal; and having set out from France in the spring of the year 1736, returned to it in 1737, after having fully accomplished their errand. On the southern expedition were dispatched M. Godin, Condamin, and Bouguer, to whom the king of Spain joined Don George Juan and Don Anthony de Ulloa, two very ingenious gentlemen and officers of the marine. They left Europe in 1735; and after enduring innumerable hardships and difficulties in the execution of their commission, returned to Europe at different times, and by different ways, in the years 1744, 1745, and 1746. The result of this arduous task was a confirmation of Newton's investigation. Picart's measure was revised by Cassini and de la Caille; and, after his errors were corrected, it was found to agree very well with the other two. On this occasion too it was discovered, that the attraction of the great mountains of Peru had an effect on the plumb-line of one of their largest instruments, drawing it seven or eight seconds from the true perpendicular.

Dr Halley, dying in 1742, was succeeded by Dr Bradley, who, though inferior as a mathematician, greatly exceeded him as a practical astronomer. He was the first who made observations with an accuracy sufficient to detect the lesser inequalities in the motions of the planets and fixed stars. Thus he discovered the aberration of light, the nutation of the earth's axis, and was able to make the lunar tables much more perfect than they had ever been. He also observed the places, and computed the elements of the comets which ap... peared in the years 1723, 1736, 1743, and 1757. He made new and most accurate tables of the motions of Jupiter's satellites, from his own observations and those of Dr Pound; and from a multitude of observations of the sun, moon, and stars, was enabled to give the most accurate table of mean refractions yet extant, as well as the best methods of computing the variations of those refractions arising from the different states of the air as indicated by the thermometer and barometer. In 1750, having procured a very large transit instrument made by Mr Bird, and a new mural quadrant of brass eight feet radius, he began to make observations with redoubled industry; so that between this time and his death, which happened in 1762, he made observations for settling the places of all the stars in the British catalogue, together with near 1500 places of the moon, much the greater part of which he compared with the tables of Mr Mayer.

In the meantime the French astronomers were assiduous in their endeavours to promote the science of astronomy. The theory of the moon, which had been given in a general way by Sir Isaac Newton, began to be particularly considered by Messrs Clairault, D'Alembert, Euler, Mayer, Simpson, and Walmly; tho' Clairault, Euler, and Mayer, distinguished themselves beyond any of the rest, and Mr Euler has been particularly happy in the arrangement of his tables for the ease and expedition of computation. He was excelled in exactness, however, by Mayer, who published his tables in the Gottingen Acts for 1753. In these the errors in longitude never exceeded two minutes; and having yet farther improved them, he sent a copy to the lords of the British admiralty in 1755; and it was this copy which Dr Bradley compared with his observations, as already mentioned. His last corrections of them were afterwards sent over by his widow; for which she and her children received a reward of L.3000.

Accurate tables for Jupiter's satellites were also composed by Mr Wargentin a most excellent Swedish astronomer, and published in the Upsal Acts in 1741; which have since been corrected by the author in such a manner as to render them greatly superior to any ever published before.

Amongst the many French astronomers who contributed to the advancement of the science, we are particularly indebted to M. de la Caille, for a most excellent set of solar tables, in which he has made allowances for the attractions of Jupiter, Venus, and the moon. In 1750 he went to the Cape of Good Hope, in order to make observations in concert with the most celebrated astronomers in Europe, for determining the parallax of the moon, as well as of the planet Mars, and from thence that of the sun; from whence it appeared that the parallax of the sun could not greatly exceed 10 seconds. Here he re-examined and adjusted the places of the southern stars with great accuracy, and measured a degree of the meridian at that place. In Italy the science was cultivated with the greatest assiduity by Signior Bianchini, father Bofcovich, Frisi, Manfredi, Zinotti, and many others; in Sweden by Wargentin already mentioned, Blingenstern, Mallet, and Planman; and in Germany, by Euler elder and younger, Mayer, Lambert, Grischow, &c. In the year 1760 all the learned societies in Europe began to prepare for observing the transit of Venus over the sun, foretold by Dr Halley upwards of 80 years before it happened, showing at the same time the important use which might be made of it. Unfortunately, however, for the cause of science, many of the astronomers sent out to observe this phenomenon were prevented by unavoidable accidents from reaching the places of their destination, and others were disappointed by the badness of the weather. It happened also, that the circumstances of the phenomenon were much less favourable for the purpose of determining the sun's parallax than had been expected by Dr Halley, owing to the faults of the tables he made use of; so that, notwithstanding all the labours of astronomers at that time, they were not able to determine the matter; and even after their observations in 1769, when the circumstances of the transit were more favourable, the parallax of the sun remained still uncertain.

Dr Bradley was succeeded in his office of astronomer-royal by Mr Bliss, Savilian professor of astronomy at Oxford; who being in a very declining state of health at the time of his accession to the office, did not enjoy it long. He was succeeded by the learned Nevil Maskelyne, D.D. the present astronomer-royal, whose name will be rendered immortal by his assiduity and success in bringing the lunar method of determining the longitude at sea into general practice.

Such was the general state of astronomy, when Mr Herschel's great discovery of augmenting the power of telescopes, beyond the most sanguine hopes of astronomers, opened at once a scene altogether unlooked for. By this indefatigable observer we are made acquainted with a new primary planet attended by two secondaries belonging to our solar system; so that the latter now appears to have double the bounds formerly assigned to it; this new planet being at least twice the distance of Saturn from the sun. In the still farther distant celestial regions, among the fixed stars, his observations are equally surprising; of which we shall only say with Dr Prickley *, "Mr Herschel's late discoveries in and beyond the bounds of the solar system, and Observations in the great views that he has given of the arrangement of the stars, their revolutions, and those of the immense systems into which they are formed, are peculiarly calculated to inspire an ardent desire of seeing so great a scene a little more unfolded. Such discoveries as these give us a higher idea of the value of our being, by raising our ideas of the system of which we are a part; and with this an earnest wish for the continuance of it."

Sect. I. Of the apparent Motions, Magnitudes, and Changes, in the celestial Bodies, as seen by the naked eye.

As the true motions of bodies at a great distance are to be gathered only from a careful observation of their apparent ones, it is absolutely necessary for those who want to become acquainted with the true motions of the heavenly bodies, to know perfectly the different changes which take place in the heavens as seen from this earth, the only place from which any observation can be made. By carefully attending to these, a little knowledge of optics will enable us to understand with great great certainty not only the true system of nature, but also what appearance the heavens would make to a spectator placed in any part of the visible creation.

The first and most obvious phenomenon is the daily rising of the sun in the east, and his setting in the west; after which the moon and stars appear, still keeping the same westerly course, till we lose sight of them altogether. This cannot be long taken notice of, before we must likewise perceive that neither the sun nor moon always rises exactly in the same point of the heavens. If we begin to observe the sun, for instance, in the beginning of March, we will find that he seems to rise almost every day sensibly more to the northward than he did the day before, to continue longer above the horizon, and to be more vertical at mid-day. This continues till towards the end of June, when he is observed to move backward in the same manner; and this retrograde motion continues to the end of December, or near it, when he begins again to move forwards, and so on.

The motion of the moon through the heavens, as well as her appearance at different times, is still more remarkable than that of the sun. When she first becomes visible at the time she is called the new moon, she appears in the western part of the heavens, and seems to be at no great distance from the sun himself. Every night she not only increases in size, but removes to a greater distance from the sun; till at last she appears in the eastern part of the horizon, just at the time the sun disappears in the western. After this she gradually moves farther and farther eastward, and therefore rises every night later and later, till at last she seems to approach the sun as nearly in the east as she did in the west, and rises only a little before him in the morning, as in the first part of her course she set in the west not long after him. All these different appearances are completed in the space of a month; after which they begin in the same order as before. They are not, however, at all times regular; for at some seasons of the year, particularly in harvest, the moon appears for several days to be stationary in the heavens, and to recede no farther from the sun, in consequence of which she rises for that time nearly at the same hour every night.

In contemplating the stars, it is observed that some of them have the singular property of neither rising in the east nor setting in the west; but seem to turn round one immovable point, near which is placed a single star called the pole, or pole-star. This point is more or less elevated according to the different parts of the earth from which we take our view. The inhabitants of Lapland, for instance, see it much more elevated above the horizon, or more vertical, than we do; we see it more vertical than it appears to the inhabitants of France and Spain; and they, again, see it more elevated than the inhabitants of Barbary. By continually travelling south, this star would at length seem depressed in the horizon, and another point would appear directly opposite to it, round which the stars in the southern part of the horizon would seem to turn. In this part of the heavens, however, there is no star so near the pole as there is in the northern part; neither is the number of stars in the southern part of the heavens so great as in the northern. Supposing us still to travel southward, the north-pole would then entirely disappear, and the whole hemisphere would appear to turn round a single point in the south, as the northern hemisphere appears to us to turn round the pole-star.

The general appearance of the heavens, therefore, is that of a vast concave sphere, turning round two points fixed in the north and south parts of it, once in 24 hours.

When we further consider the stars, we will find the greatest part of them to keep their places with respect to one another; that is, if we observe two stars having a certain apparent distance from each other this night, they will seem to have the same tomorrow, and every other succeeding night; but we will by no means observe them to have the same places either with respect to the sun or moon, as must be easily understood from what we have already said. Neither do all the stars in the heavens appear to be of this fixed kind: Some of them, on the contrary, change their places very remarkably with regard to the fixed stars, and with regard to one another. Of these, five were only observed formerly; but Mr Herschel has now discovered a sixth. They are distinguished by the appellation of planets, (from πλανεῖν, to err or wander); and called by the names of Mercury, Venus, Mars, Jupiter, Saturn, and the Georgian Sidus. The fixed stars are likewise distinguished from the planets by their continually exhibiting that appearance which is called the scintillation or twinkling of the stars. This is said to arise from the exceeding minuteness of their apparent diameter; so that the interposition of any little substance, of which there are many floating in the atmosphere, continually deprives us of the sight of them; but the interposing body soon changing its place, we again see the star, and thus the twinkling is produced.

Mercury is a small star, but emits a very bright white light; though, by reason of his always keeping near the sun, he is seldom to be seen; and when he does make his appearance, his motion towards the sun is so swift, that he can only be discerned for a short time. He appears a little after sunset, and again a little before sunrise.

Venus, the most beautiful star in the heavens, known by the names of the morning and evening star, likewise keeps near the sun, though she recedes from him almost double the distance of Mercury. She is never seen in the eastern quarter of the heavens when the sun is in the west; but always seems to attend him in the evening, or to give notice of his approach in the morning.

Mars is of a red fiery colour, and always gives a much duller light than Venus, though sometimes he equals her in size. He is not subject to the same limitation in his motions as Mercury or Venus; but appears sometimes very near the sun, and sometimes at a great distance from him; sometimes rising when the sun sets, or setting when he rises. Of this planet it is remarkable, that when he approaches any of the fixed stars, which all the planets frequently do, these stars change their colour, grow dim, and often become totally invisible, though at some little distance from the body of the planet: but Mr Herschel thinks this has been exaggerated by former astronomers.

Jupiter and Saturn likewise often appear at great distances from the sun. The former shines with a bright white light, and the latter with a pale faint one; and Besides the motions which we observe in all these planets, their apparent magnitudes are very different at different times. Every person must have observed that Venus, though she constantly appears with great splendor, is not always equally big; and this apparent difference of magnitude is so remarkable, that she appears no less than 32 times larger at some seasons than at others. This increase of magnitude is likewise very remarkable in Mars and Jupiter, but less so in Saturn and Mercury.

Though we have thus described the motions of the planets with respect to their apparent distances from the sun, they by no means appear to us to move regularly in the heavens, but, on the contrary, in the most complex and confused manner that can be imagined, sometimes going forward, sometimes backward, and sometimes seeming to be stationary. They all seem to describe looped curves; but it is not known when any of these curves would return into themselves, except that of Venus, which returns into itself every eighth year. On each side of the loops they appear stationary; in that part of each loop near the earth, retrograde; and in every other part of their path, direct.

These, however, are not the only moving bodies which are to be observed in the celestial regions. The six abovementioned are indeed the only ones which appear almost constantly, or disappear only at certain intervals, and then as certainly return. But there are others which appear at uncertain intervals, and with a very different aspect from the planets. These are very numerous, and no fewer than 450 are supposed to belong to our solar system. They are called Comets, from their having a long tail, somewhat resembling the appearance of hair. This, however, is not always the case; for some comets have appeared which were as well defined and as round as planets; but in general they have a luminous matter diffused around them, or projecting out from them, which to appearance very much resembles the Aurora Borealis. When these appear, they come in a direct line towards the sun, as if they were going to fall into his body; and after having disappeared for some time in consequence of their proximity to that luminary, they fly off again on the other side as fast as they came, projecting a tail much greater and brighter in their recesses from him than when they advanced towards him; but, getting daily at a farther distance from us in the heavens, they continually lose of their splendor, and at last totally disappear. Their apparent magnitude is very different: sometimes they appear only of the bigness of the fixed stars; at other times they will equal the diameter of Venus, and sometimes even of the sun or moon. So, in 1652, Hevelius observed a comet which seemed not inferior to the moon in size, though it had not so bright a splendor, but appeared with a pale and dim light, and had a dismal aspect. These bodies will also sometimes lose their splendor suddenly, while their apparent bulk remains unaltered. With respect to their apparent motions, they have all the inequalities of the planets; sometimes seeming to go forwards, sometimes backwards, and sometimes to be stationary.

Though the fixed stars are the only marks by which astronomers are enabled to judge of the courses of the moveable ones, and though they have never been observed to change their places; yet they seem not to be seemingly ended with the permanency even of the earth and destructible planets, but to be perishable or destructible by accident, and likewise generable by some natural cause. Several stars observed by the ancients are now no more to be seen, but are destroyed; and new ones have appeared, which were unknown to the ancients. Some of them have also disappeared for some time, and again become visible.

We are also assured from the observations of astronomers, that some stars have been observed which never were seen before, and for a certain time they have distinguished themselves by their superlative lustre; but afterwards decreasing, they vanished by degrees, and were no more to be seen. One of these stars being first seen and observed by Hipparchus, the chief of the ancient astronomers, set him upon composing a catalogue of the fixed stars, that by its posterity might learn whether any of the stars perish, and others are produced afresh.

After several ages, another new star appeared to Tycho Brahe and the astronomers who were contemporary with him; which put him on the same design with Hipparchus, namely, the making a catalogue of the fixed stars. Of this, and other stars which have appeared since that time, we have the following history by Dr Halley: "The first new star in the chair of Dr Halley's history of Cassiopeia, was not seen by Cornelius Gemma on the 8th of November 1572, who says, he that night considered that part of the heaven in a very serene sky, and saw it not; but that the next night, November 9, it appeared with a splendor surpassing all the fixed stars, and scarce less bright than Venus. This was not seen by Tycho Brahe before the 11th of the same month; but from thence he assures us that it gradually decreased and died away, so as in March 1574, after fifteen months, to be no longer visible; and at this day no signs of it remain. The place thereof in the sphere of fixed stars, by the accurate observations of the same Tycho, was 9° 17' a 1° * 9°, with 53° 45' north latitude.

"Such another star was seen and observed by the scholars of Kepler, to begin to appear on Sept. 30, A.D. anno 1604, which was not to be seen the day before; but it broke out at once with a lustre surpassing that of Jupiter; and like the former, it died away gradually, and in much about the same time disappeared totally, there remaining no footsteps thereof in January 1605. This was near the ecliptic, following the right leg of Serpentarius; and by the observations of Kepler and others, was in 7° 20° 20° a 1° * 9°, with north latitude 1° 56'. These two seem to be of a distinct species from the rest, and nothing like them has appeared since.

"But between them, viz. in the year 1596, we have the first account of the wonderful star in Collo Ceti, seen by David Fabricius on the third of August, A.D. as bright as a star of the third magnitude, which has been since found to appear and disappear periodically." THE SUN

with the GREAT SPOT in 1769.

Diagonal Scale of Miles.

Fig. 18.

Distance of the fourth Satellite from the third.

Earth Moon

Distance of the Moon from the Earth.

Fig. 16.

Fig. 17.

A.Bell Sculp. its period being precisely enough seven revolutions in six years, though it returns not always with the same lustre.

Nor is it ever totally extinguished, but may at all times be seen with a fix-feet cube. This was singular in its kind, till that in Collo Cygni was discovered. It precedes the first star of Aries $1^\circ 40'$, with $15^\circ 57'$ south latitude.

Another new star was first discovered by William Janfonius in the year 1600, in petto, or rather in eductione, Colli Cygni, which exceeded not the third magnitude. This having continued some years, became at length so small, as to be thought by some to have disappeared entirely; but in the years 1657, 1658, and 1659, it again arose to the third magnitude; tho' soon after it decayed by degrees to the fifth or sixth magnitude, and at this day is to be seen as such in $\delta$ $18^\circ 38'$ $\alpha$ $1^\text{ma} \gamma$, with $55^\circ 29'$ north latitude.

A fifth new star was first seen by Hevelius in the year 1670, on July $15^\text{th}$, as a star of the third magnitude, but by the beginning of October was scarce to be perceived by the naked eye. In April following it was again as bright as before, or rather greater than of the third magnitude, yet wholly disappeared about the middle of August. The next year, in March $1672$, it was seen again, but not exceeding the sixth magnitude; since when, it has been no further visible, though we have frequently sought for its return; its place is $9^\circ 3^\text{rd}$ $17'$ $\alpha$ $1^\text{ma} \gamma$, and has lat. north $47^\circ 28'$.

The fifth and last is that discovered by Mr G. Kirch in the year $1686$, and its period determined to be of $404\frac{1}{2}$ days; and though it rarely exceeds the fifth magnitude, yet it is very regular in its returns, as we found in the year $1714$. Since then we have watched, as the absence of the moon and clearness of the weather would permit, to catch the first beginning of its appearance in a fix feet tube, that, bearing a very great aperture, discovers most minute stars. And on June $15^\text{th}$, last, it was first perceived like one of the very least telecopic stars: but in the rest of that month and July, it gradually increased, so as to become in August visible to the naked eye; and so continued all the month of September. After that, it again died away by degrees; and on the eighth of December, at night, was scarce discernible by the tube; and, as near as could be guessed, equal to what it was at its first appearance on June $25^\text{th}$: so that this year it has been seen in all near six months, which is but little less than half its period; and the middle, and consequently the greatest brightness, falls about the $10^\text{th}$ of September.

Concerning the changes which happen among the fixed stars, Mr Montanere, professor of mathematics at Bononia, gave the following account, in a letter to the Royal Society, dated April $30^\text{th}$ $1670$. "There are now wanting in the heavens two stars of the second magnitude in the stern of the ship Argo, and its yard; Bayerus marked them with the letters $\beta$ and $\chi$. I and others observed them in the year $1664$, upon the occasion of the comet that appeared that year: when they disappeared first, I know not: only I am sure that in the year $1668$, upon the $10^\text{th}$ of April, there was not the least glimpse of them to be seen; and yet the rest about them, even of the third and fourth magnitudes, remained the same. I have observed many more changes among the fixed stars, even to the number of an hundred, though none of them are so great as those Apparent Motion, &c.

The late improvements in astronomy, and particularly those in the construction of telescopes, have now given astronomers an opportunity of observing the changes which take place among the stars with much greater accuracy than could be formerly done. In a paper in the 76th volume of the Philosophical Transf., Mr Pigot's actions, Mr Edward Pigot gives a dissertation on the remarks on stars suspected by the astronomers of last century to be changeable. For the greater accuracy in the investigation of his subject, he divides them into two classes; stars one containing those which are undoubtedly changeable, and the other those which are only suspected to be so. The former contains a list of 12 stars, from the first to the fourth magnitudes; including the new one which appeared in Cassiopeia in $1572$, and that in Serpentarius in $1604$: the other contains the names of 38 stars of all magnitudes, from the first to the seventh. He is of opinion, that the celebrated new star in Cassiopeia is a periodical one, and that it returns once in $150$ years. Mr Keill is of the same opinion: and Mr Pigot thinks, that its not being observed at the expiration of each period is no argument against the truth of that opinion; "since (says he), perhaps, as with most of the variables, it may at different periods have different degrees of lustre, so as sometimes only to increase to the ninth magnitude; and if this should be the case, its period is probably much shorter." For this reason, in September $1782$, he took a plan of the small stars near the place where it formerly appeared, but in four years had observed no alteration.

The star in the neck of the Whale had also been examined by Mr Pigot from the end of $1782$ to $1786$, to Ceti; but he never found it exceed the fifth magnitude; though Mr Goodricke had observed it on the $9^\text{th}$ of August to be of the second magnitude, and on the $3^\text{rd}$ of September the same year it was of the third magnitude. Mr Pigot deduced its period from its apparent equality with a small star in the neighbourhood, and thence found it to be $320$, $328$, and $337$ days.

The most remarkable of these changeable stars is that called Algol, in the head of Medusa. It had long been known to be variable; but its period was first ascertained by Mr Goodricke of York, who began to observe it in the beginning of $1783$. It changes continually from the first to the fourth magnitude; and the time taken up from its greatest diminution to its least is found, at a mean, to be $2$ days $20$ hours $49$ minutes and $3$ seconds. During four hours it gradually diminishes in lustre, which it recovers during the succeeding four hours; and in the remaining part of the period it invariably preserves its greatest lustre, and after the expiration of the term its diminution again commences. According to Mr Pigot, the degree of brightness of this star when at its minimum is variable in different periods, and he is of the same opinion with regard to its brightness when at its full; but whether these differences return regularly or not, has not been determined.

The $420$th of Mayer's catalogue, in Leo, has lately been shown to be variable by Mr Koch. Some years before $1782$, that gentleman perceived it undoubtedly smaller than the $419$th of the same catalogue. In February In February that year, it was of the same brightness with the 419th, that is, of the seventh magnitude. In April 1783, it was of the ninth magnitude; and in the same month 1784, it was of the tenth. Mr Pigot could never observe this star, though he frequently looked for it with a night-glass, and on the fifth of April 1785 with a three-feet achromatic transit instrument.

In 1704, Maraldi observed a variable star in Hydra, whose period he settled at about two years, though with considerable variations; but from the observations even of Maraldi, Mr Pigot concludes, that its period was then only 494 days; and from some others made by himself, he thinks that now it is only 487 days; so that since the time of Maraldi it has shortened seven days. The particulars relating to this star are as follow:

1. When at its full brightness it is of the fourth magnitude, and does not perceptibly change for a fortnight. 2. It is about six months in increasing from the tenth magnitude and returning to the same; so that it may be considered as invisible during that time. 3. It is considerably more quick, perhaps one half more, in its increase than in its decrease. 4. Though when at its full it may always be styled a star of the fourth magnitude, it does not constantly attain the same degree of brightness, but the differences are very small.

This star is the 30th of Hydra in Hevelius's catalogue, and is marked by him of the fifth magnitude.

The new star in Serpentarius, observed by Kepler, seems to have been of the same nature with that of Cassiopeia; and Mr Pigot therefore looks upon it also to be a periodical one, though, after taking a plan of the nearest stars in that part of the heavens, in the year 1782, he could, in four years time, perceive no alteration.

The variation of the star β Lyrae was discovered by Mr Goodricke above mentioned, who suspects its period to be six days nine hours; which coincides with the opinion of Mr Pigot.

The new star near the Swan's Head, observed by Don Anthelme in December 1669, soon became of the third magnitude, and disappeared in 1672. Mr Pigot has constantly looked for it since November 1781, but without success. He is of opinion, that had it only increased to the 10th or 11th magnitude, he would have seen it, having taken a plan of all the neighbouring small stars.

The next variable star in Mr Pigot's catalogue is the Antinoi, whose variation and period he discovered in 1785. From his corrected observations, he concludes that it continues at its greatest brightness 40 hours without decreasing; it is 66 hours after it begins to decrease before it comes to its full diminution; after which it continues stationary for 30 hours more, and then increases for 36 hours. In every period it seems to acquire its full brightness, and to be equally decreased.

The variable star in the Swan's Neck was observed for three years. The period of this star had been settled by Maraldi and Cassini at 405, and by M. Le Gentil at 405.3 days; but from a mean of the observations of Mr Pigot, it appears to be only 392. "Perhaps (says he) its period is irregular; to determine which several intervals of 15 years ought to be taken; and I am much inclined to believe that it will Apparent be found only 396 days 21 hours." The particulars Motion, &c., relating to this star are: 1. When at its full bright-ness it undergoes no perceptible change for a fortnight. 2. It is about three months and an half in increasing from the 11th magnitude to its full brightness, and the same in decreasing; for which reason it may be considered as invisible during six months. 3. It does not always attain the same degree of lustre, being sometimes of the fifth and sometimes of the seventh magnitude.

In 1660, G. Janssenius discovered a variable star in Swan's head, the break of the Swan, which was afterwards observed by different astronomers, and supposed to have a period of about ten years. The results of Mr Pigot's calculations from the observations of former astronomers are, 1. That it continues in full lustre for five years. 2. It decreases rapidly for two years. 3. It is invisible to the naked eye for four years. 4. It increases slowly during seven years. 5. All these changes are completed in 18 years. 6. It was at its minimum at the end of the year 1663. 7. It does not always increase to the same degree of brightness, being sometimes of the third, and at others only of the fifth magnitude.

"I am entirely ignorant (says Mr Pigot) whether it is subject to the same changes in this century, having not met with any series of observations on it; but if the above conjectures are right, it will be at its minimum in a very few years. Since November 1781 I have constantly seen it of the sixth magnitude. Sometimes I have suspected that it has decreased within these two last years, though in a very small degree."

The last star in Mr Pigot's first class is the Cephei, whose variation was discovered by Mr Goodricke. Its changes are very difficult to be seen, unless it is observed at the times of its greatest and least brightness. The result of the observations hitherto made upon it are, that its period consists of 5 days 8 hours 37 on a mean. The following observations relate to some stars of the second class.

1. Hevelius's 6th Cassiopeia was missing in 1782, Stars, the nor could Mr Pigot find it in 1783 and 1784. 2. ξ or 46th Andromeda, said to be variable, but the evidence is not convincing to Mr Pigot. 3. Flamsteed's 50, 52, τ Andromedæ, and Hevelius's 41 Andromedæ. The position and characters of these stars differ considerably in different catalogues, and some of them are said by Cassini to have disappeared and reappeared. Mr Pigot therefore gives their comparative brightness as observed in the years 1783, 1784, and 1785, during which time he does not mention any particular change. 4. Tycho's 20th Ceti. "This (says Mr Pigot) must be the star which Hevelius said had disappeared, being Tycho's second in the Whale's Belly. There can hardly be any doubt that it is the χ, misplaced by Tycho. This χ is of the fourth or fifth magnitude." 5. ε, or the 17th Eridani of Ptolemy and Ulug Beigh. Flamsteed says he could not see this star in 1691 and 1692; but in 1782, 1783, and 1784, Mr Pigot observed in that place one of the seventh magnitude, which appeared always of the same lustre. 6. Flamsteed's 41 Tauri was supposed by Cassini to be either a new or variable star; but Mr Pigot thinks there is no reason to be of that opinion. "That it is not not new (says he) is evident, since it is Ulug Beigh's 26th and Tycho's 43d.

7. A star about $2\frac{1}{2}$ north of 53 Eridani, and 47 Eridani. Cassini supposed the first of these stars to be a new one, and that it was not visible in 1664. He mentions another star thereabouts, which he also esteemed a new one.

8. γ Canis Majoris. Maraldi could not see this star in 1670; but in 1692 and 1693 it appeared of the fourth magnitude. Mr Pigot made frequent observations upon it from 1782 to 1786, but could perceive no variation.

9. α Gemini. "If any of these stars (says our author) have changed in brightness, it is probable the β. In 1783, 1784, and 1785, the β was undoubtedly brighter than α."

10. ε Leonis. According to Montanari, this star was hardly visible in 1693. In 1783, 1784, and 1785, it was of the fifth magnitude. By Tycho, Flamsteed, Mayer, Bradley, &c., it is marked of the fourth.

11. ι Leonis. This star is said to have disappeared before the year 1667; but according to Mr Pigot's observations, was constantly of the fifth or sixth magnitude since 1783.

12. 25 Leonis. In 1783 our author first perceived that this star was missing, and could not perceive it in 1784 and 1785, even with a transit instrument.

13. Bayer's i Leonis, or Tycho's 16 Leonis, was not visible in 1709, nor could it be seen in 1785. It is a different star from the i Leonis of the other catalogues, though Tycho's description of its place is the same.

14. ρ Ursae Majoris. This star is suspected to change in brightness, on account of its being marked by Tycho, the prince of Hesse, &c. of the second magnitude, while Hevelius, Bradley and others have marked it of the third. In 1786, and for three years before, it appeared as a bright star of the fourth magnitude.

15. ν Virginis. This is supposed to be variable, because Flamsteed, on the 27th of January 1680, could not see it; but he observed it in 1677, and some years afterwards. Mr Pigot observed it frequently in 1784 and 1785, and found it a star of the fifth magnitude without any perceptible change.

16. Bayer's star of the sixth magnitude 1° south of g Virginis. "This star (says Mr Pigot) is not in any of the nine catalogues that I have. Maraldi looked for it in vain; and in May 1785 I could not see the least appearance of it. It certainly was not of the eighth magnitude.

17. A star in the northern thigh of Virgo, marked by Riccioli of the sixth magnitude, could not be seen by Maraldi in 1793; nor was it of the ninth magnitude, if at all visible in 1785.

18. The 91 and 92 Virginis. In 1785, one of these stars, probably the 91, was missing: the remaining one is of the fifth or seventh magnitude.

19. α Draconis. Mr Pigot coincides in opinion with Mr Herschel, that this star is variable. Bradley, Flamsteed, &c. mark it of the second magnitude, but in 1786 it was only a bright fourth. It was frequently examined by Mr Pigot from the 4th of October 1782, but without any alteration being perceived.

20. Bayer's star in the west scale of Libra. Maraldi could not see this star, and it was likewise invisible to Mr Pigot in 1784 and 1785.

21. No 6 of Ptolemy and Ulug Beigh's unformed in Libra. This star is not mentioned in any other catalogues than the above. Mr Pigot frequently observed a little star of the seventh magnitude very near its place.

22. x Librae. This star is thought to be variable; but Mr Pigot is not of that opinion, though "certainly (says he) it is rather singular, that Hevelius, whose attention was directed to that part of the heavens to find Tycho's 11th, did not find the x; and the more so, as he has noticed two much smaller stars not far from it. During these three years I have found the x constantly of the fifth magnitude."

23. Tycho's 11th Librae. Mr Pigot is of opinion that no such star as this ever existed; and that it is no other than the x with an error of 2 degrees of longitude.

24. 33 Serpentis. This star was missing in 1784; nor could it be perceived with a night-glass in 1785.

25. A star marked by Bayer near i urse majoris. This star could not be seen by Cassini; nor was Mr Pigot able to discover it with a night-glass in 1782.

26. The ρ, or Ptolemy and Ulug Beigh's 14th Ophiuchi, or Flamsteed's 36th. Mr Pigot has no doubt that this is the star which is said to have disappeared before the year 1695; and it is evident that it was not seen by Hevelius. In 1784 and 1785, Mr Pigot found it of the fourth or fifth magnitude: but he is far from being certain of its having undergone any change, especially as it has a souther declination of 26 degrees; for which reason great attention must be paid to the state of the atmosphere.

27. Ptolemy's 13th and 18th Ophiuchi, fourth magnitude. Mr Pigot is of opinion that these stars are misplaced in the catalogues. The 18th of Ptolemy he thinks ought to be marked with a north latitude instead of a south, which would make it agree nearly with Flamsteed's 58th; and he is also of opinion that the 13th of Ptolemy is the 40th of Flamsteed.

28. ζ Sagittarii. Mr Herschel, as well as Mr Pigot, is of opinion, that this star has probably changed its magnitude, though the reason seems only to be the great disagreement concerning it among the different catalogues of stars.

29. θ Serpentis. This star, according to Mr Montanari, is of variable magnitude; but Mr Pigot never could perceive any alteration.

30. Tycho's 27th Capricorni was missing in Hevelius's time, and Mr Pigot could not find it with a transit instrument.

31. Tycho's 22d Andromedæ, and o Andromedæ. Mr Cassini informs us, that in his time the former had grown so small that it could scarcely be seen; and Mr Pigot, that no star was to be seen in its place in 1784 and 1785; but he is of opinion that Cassini may have mistaken the • Andromedæ for the 22d; for which reason he observed this star three years, but without any alteration in its brightness.

32. Tycho's 19th Aquarii. Hevelius says that this star was missing, and that Flamsteed could not see it with his naked eye in 1679. Mr Pigot could not see it in 1782; but is persuaded that it is the same with Flamsteed's Flamsteed's 55th marked f by Bayer, from which it is only a degree and an half distant. The 53rd of Flamstead, marked f in Ptolemy's catalogue, is a different star.

33. La Caille's 483 Aquarii was first discovered to be missing in 1778, and was not visible in 1783 and 1784.

Besides these there are several others certainly variable, but which cannot be seen in this country. There are some also suspected to be variable, but for which Mr Pigot thinks there is no reason. Mr Herschel also gives strong reasons for not laying great stress on all the observations by which new stars have been said to be discovered. Mr Pigot assures us from repeated experience, that even more than a single observation, if not particularized and compared with neighbouring stars, is very little to be depended upon; different streaks of the clouds, the state of the weather, &c., having often caused him error a whole magnitude in the brightness of a star.

As these changes to which the fixed stars are liable method of do not seem to be subject to any certain rule, Mr Wollaston has given an easy method of observing whether variations among them do take place in any part of the heavens or not, fixed stars, and that without much expense of instruments or waste of time, which are great objections to astronomical observations in general. His first idea was, that the work should be undertaken by astronomers in general; each taking a particular district of the heavens, and from time to time observing the right ascension and declination of every star in that space allotted to him, framing an exact map of it, and communicating their observations to one common place of information. This method, however, being too laborious, he next proposes the noting down at the time, or making a drawing of what one sees while they are observing. A drawing of this kind once made, would remain, and could be consulted on any future occasion; and if done at first with care, a transient review would discover whether any sensible change had taken place since it was last examined, which could not so well be done by catalogues or verbal description. For this purpose he recommends the following method: "To a night-glass, but of Dollond's construction, which magnifies about five times, and takes in about as many degrees of a great circle, I have added cross wires intersecting one another at an angle of 45 degrees. More wires may be crossed in other directions; but I apprehend these will be sufficient. This telescope I mount on a polar axis. One coarsely made, and without any divisions on its circle of declination, will answer the purpose, as there is no great occasion for accuracy in that respect; but as the heavenly bodies are more readily followed by an equatorial motion of the telescope, so their relative positions are much more easily discerned when they are looked at constantly as in the same direction. An horizontal motion, except in the meridian, would be apt to mislead the judgment. It is scarcely necessary to add, that the wires must stand so as for one to describe a parallel of the equator nearly; another will then be a horary circle, and the whole area will be divided into eight equal sectors.

Thus prepared, the telescope is to be pointed to a known star, which is to be brought into the centre or common intersection of all the wires. The relative positions of such other stars as appear within the field are apparent to be judged of by the eye; whether at \( \frac{1}{2} \), \( \frac{1}{4} \), or \( \frac{1}{8} \) from Motion, &c., the centre towards the circumference, or vice versa; &c.; and so with regard to the nearest wire respectively. These, as one sees them, are to be noted down with a black-lead pencil upon a large message card held in the hand, upon which a circle similarly divided is ready drawn. One of three inches diameter seems most convenient. The motion of the heavenly bodies in such a telescope is so slow, and the noting down of the stars so quickly done, that there is commonly full time for it without moving the telescope. When that is wanted, the principal star is easily brought back again into the centre of the field at pleasure, and the work resumed. After a little practice, it is astonishing how near one can come to the truth in this way; and tho' neither the right ascensions nor the declinations are laid down by it, nor the distances between the stars measured; yet their apparent situations being preserved in black and white, with the day and year, and hour, if thought necessary, written underneath, each card then becomes a register of the then appearance of the heavens; which is easily re-examined at any time with little more than a transient view; and which will yet show, on the first glance, if there should have happened in it any alteration of consequence."

Fig. 35 shows part of the corona borealis delineated in Plate LXVI, by making the stars \( \alpha \), \( \beta \), \( \gamma \), \( \delta \), \( \epsilon \), \( \zeta \), \( \eta \), \( \theta \), \( \iota \), \( \kappa \), \( \lambda \), \( \mu \), \( \nu \), \( \xi \), \( \pi \), \( \rho \), \( \sigma \), and \( \tau \) successively central; and these were joined with some of the stars of Bootes, for the sake of connecting the whole, and united into one map, as represented fig. 36.

In observing in this way, it is evident, that the places of such stars as happen to be under or very near any of the wires, are more to be depended upon than those which are in the intermediate spaces, especially if towards the edges of the fields; so also those which are nearest to the centre, because better defined, and more within the reach of one wire or another. For this reason, different stars of the same set must successively be made central, or brought towards one of the wires, where any suspicion arises of a mistake, in order to approach nearer to a certainty; but if the stand of the telescope be tolerably well adjusted and fixed, this is soon done.

In such a glass it is seldom that light sufficient for discerning the wires is wanting. When an illuminator is required, a piece of card or white pasteboard projecting on one side beyond the tube, and which may be brought forward occasionally, is better than any other. By cutting across a small segment of the object-glass, it throws a sufficient light down the tube though the candle be at a great distance, and one may lose sight of the false glare by drawing back the head, and moving the eye a little to one side, when the small stars will be seen as if no illuminator was there. See a delineation of the principal fixed stars, with the apparent path of the sun among them, on Plate LXIV. and LXV.

A very remarkable appearance in the heavens is that Galaxy, called the galaxy, or milky-way. This is a broad circle, milky-way sometimes double, but for the most part single, surrounding the whole celestial concave. It is of a whitish colour, somewhat resembling a faint aurora borealis; but Mr Brydone, in his journey to the top of mount Etna, found that phenomenon to make a glorious appearance. Fig. 19. The Moon in her mean libration with the spots according to Riccioli, Cassini &c.

Fig. 20.

Fig. 21. 1773 h.8.55 25 at h.11.8.

Fig. 22. Sept. 15. Hg. O.

Fig. 23. Nov. 13. 1773.

Fig. 22.

A Bell. Print. Wm. C. Sculptor fecit. pearance, being, as he expresses it, like a pure flame that shot across the heavens.

The only appearance, besides those already mentioned, which are very observable by the unassisted eye, are those unexpected obscurations of the sun and moon, commonly called eclipses. These are too well known, and attract the attention too much, to need any particular description. We have, however, accounts very well authenticated, of obscurations of the sun continuing for a much longer time than a common eclipse possibly can do, and likewise of the darkness being much greater than it usually is on such occasions; and that these accounts are probably true, we shall afterwards have occasion to observe.

Sect. II. Of the Appearances of the Celestial Bodies as seen through Telescopes.

1. The sun, though to human eyes so extremely bright and splendid, is yet frequently observed, even through a telescope of but very small powers, to have dark spots on its surface. These were entirely unknown before the invention of telescopes, though they are sometimes of sufficient magnitude to be discerned by the naked eye, only looking through a smoked glass to prevent the brightness of the luminary from destroying the sight. The spots are said to have been first discovered in the year 1611; and the honour of the discovery is disputed between Galileo and Scheiner, a German Jesuit at Ingolstadt. But whatever merit Scheiner might have in the priority of the discovery, it is certain that Galileo far exceeded him in accuracy, though the work of Scheiner has considerable merit, as containing observations selected from above 3000, made by himself. Since his time the subject has been carefully studied by all the astronomers in Europe; the result of whose observations, as given by Dr Long, is to the following purpose.

There is great variety in the magnitudes of the solar spots; the difference is chiefly in superficial extent of length and breadth; their depth or thickness is very small; some have been so large, as by computation to be capable of covering the continents of Asia and Africa; nay, the whole surface of the earth, or even five times its surface. The diameter of a spot, when near the middle of the disk, is measured by comparing the time it takes in passing over a cross hair in a telescope, with the time wherein the whole disk of the sun passes over the same hair; it may also be measured by the micrometer; and by either of these methods we may judge how many times the diameter of the spot is contained in the diameter of the sun. Spots are subject to increase and diminution of magnitude, and seldom continue long in the same state. They are of various shapes; most of them having a deep black nucleus surrounded by a dusky cloud, whereof the inner parts near the black are a little brighter than the outskirts. They change their shape, something in the manner that our clouds do; though not often so suddenly: thus, what is of a certain figure to-day, shall, to-morrow, or perhaps in a few hours, be of a different one; what is now but one spot, shall in a little time be broken into two or three; and sometimes two or three spots shall coalesce, and be united into one.

Dr Long, many years since, while he was viewing the image of the sun through a telescope cast upon white paper, saw one roundish spot, by elimination not much less than the diameter of our earth, break into two, which receded from one another with prodigious velocity. This observation was singular at the time; for though several writers had taken notice of this after it was done, none of them had been making any observation at the time it was actually doing.

The number of spots on the sun is very uncertain; sometimes there are a great many, sometimes very few; and sometimes none at all. Scheiner made observations on the sun from 1611 to 1629; and says he never found his disk quite free of spots, excepting a few days in December 1624. At other times he frequently saw 20, 30, and in the year 1625 he was able to count 50 spots on the sun at a time. In an interval afterwards of 20 years, from 1650 to 1670, scarce any spots were to be seen, and since that time some years have furnished a great number of spots, and others none at all; but since the beginning of the last century, not a year passed wherein some were not seen; and at present, says Mr Cassini, in his Elements d'Astronomie published in 1740, they are so frequent, that the sun is seldom without spots, and often shows a good number of them at a time.

From these phenomena, it is evident, that the spots are not endowed with any permanency, nor are they at all regular in their shape, magnitude, number, or in the time of their appearance or continuance. Hevelius observed one that arose and vanished in 16 or 17 hours; nor has any been observed to continue longer than 70 days, which was the duration of one in the year 1676; those spots that are formed gradually, are gradually dissolved; while those that arise suddenly, are for the most part suddenly dissolved. When a spot disappears, that part where it was generally becomes brighter than the rest of the sun, and continues so for several days; on the other hand, those bright parts (called faculae, as the others are called maculae) sometimes turn to spots.

The solar spots appear to have a motion which carries them across the sun's disk. Every spot, if it spots move, continues long enough without being dissolved, appears from west to enter the sun's disk on the east side, to go from thence with a velocity continually increasing till it has gone half its way; and then to move slower and slower, till it goes off at the west side, after which it disappears for about the same space of time that it spent in crossing the disk, and then enters upon the east side again, nearly in the same place, and crosses it in the same tract, and with the same unequal motion as before. This apparent inequality in the motion of the spots is purely optical, and is in such proportion as demonstrates them to be carried round equably or in a circle, the plane of which continued passes through or near the eye of a spectator upon the earth.

Besides the real changes of the spots already mentioned, there is another which is purely optical, and is owing to their being seen on a globe differently turned towards us. If we imagine the globe of the sun to have a number of circles drawn upon its surface, all passing through the poles, and cutting his equator at equal distances, these circles which we may call meridians, if they were visible, would appear to us at unequal distances, as in fig. 2. Now, suppose a spot were round, and so large as to reach from one meridian to another, it would appear round only at g, when it was in the middle of that half of the globe which is towards... Appearance of Celestial Bodies in length and breadth: in every other place it turns away from us, and appears narrower, though of the same length, the farther it is from the middle; and on its coming on at \(a\), and going off at \(b\), it appears as small as a thread, the thin edge being then all that we see.

The motion of the spots is in the order of the signs (the same way that all motions in the solar system, those of the comets alone excepted, are performed); and therefore, as the earth revolves round the sun the same way with the solar spots, one of these will appear to remain longer on the disk than it would otherwise do if the earth remained at rest. Thus, in fig. 3, let A B C D be the orbit of the earth, \(a b c d\) the equator of the sun; let \(a\) be a spot seen in the middle of the disk by a spectator upon the earth at A. The spot being carried round through \(b c d\), according to the order of the letters, will in about 25 days bring it again to \(a\); but during that interval, the earth will be got to B, and the middle of the disk at \(b\); so that about two days more will intervene before a spectator upon the earth at C will view it in the middle of the then apparent disk at \(c\). There are, however, but few instances of such returning spots; so that Scheiner, out of his multitude of observations, found only three or four of this kind.

As fig. 2 is an orthographic projection of meridians on the sun's disk, it may be thought that they would show the apparent diurnal motion of the spots; so that, for example, a spot which to-day at noon is in the meridian marked \(a\), would to-morrow at noon be in that marked \(b\), the next day in that marked \(c\), and so on: but Scheiner says, that, calling the sun's picture on paper through a telescope, the distance between the place of a spot at noon any given day and the place at noon the day immediately preceding, or the day immediately following, will be greater when the spot is near the circumference of the disk than according to the orthographic projection it ought to be. This deviation of spots he thought owing to the refraction of the glasses in the telescope being greater near the circumference than in the middle; and he was confirmed in this opinion, by finding, that if spots were observed by letting the sun shine through a small hole without a glass, upon white paper held at a good distance from the hole in a dark room, their places would then be every day according to the orthographic projection. But he found this method of observing the solar spots attended with great difficulties. Another proof that this deviation of the solar spots is occasioned by the different refraction of the glasses of the telescope, is deduced from the following experiment. Our author pierced with a needle 12 small holes at equal distances in a thin plate of brass; and placing the plate before the glasses of a short telescope, let the sun shine through, receiving 12 bright spots upon a white paper placed in such a manner that the light might fall perpendicularly upon it. Here also he found the distances between the spots near the outside greater than between those in the middle; whereas, when he received them upon paper without any glasses, the situation of the bright spots exactly corresponded to that of the small holes in the plate.

The face of the sun, when clear of spots, seen by the naked eye through a smoked or coloured glass, or appearance through a thin cloud, or the vapours near the horizon, appears all over equally luminous: but when viewed thro' the telescope, the glasses being smoked or coloured, besides the difference between the faculae and the other parts, the middle of the disk appears brighter than the outskirts; because the light is darted more directly towards us from the middle than from any other part, and the faculae appear more distinctly near the sides, as being on a darker ground than in the middle.

The phenomena of the solar spots, as delivered by Account of Scheiner and Hevelius, may be summed up in their phenomena by following particulars. 1. Every spot which hath a different nucleus, or considerably dark part, hath also an umbra, observers, or fainter shade, surrounding it. 2. The boundary betwixt the nucleus and umbra is always distinct and well defined. 3. The increase of a spot is gradual, the breadth of the nucleus and umbra dilating at the same time. 4. In like manner, the decrease of a spot is gradual, the breadth of the nucleus and umbra contracting at the same time. 5. The exterior boundary of the umbra never consists of sharp angles; but is always curvilinear, how irregular soever the outline of the nucleus may be. 6. The nucleus of a spot, whilst on the decrease, often changes its figure by the umbra encroaching irregularly upon it, inasmuch that in a small space of time new encroachments are discernible, whereby the boundary betwixt the nucleus and umbra is perpetually varying. 7. It often happens, by these encroachments, that the nucleus of a spot is divided into two or more nuclei. 8. The nuclei of the spots vanish sooner than the umbra. 9. Small umbrae are often seen without nuclei. 10. An umbra of any considerable size is seldom seen without a nucleus in the middle of it. 11. When a spot which consisted of a nucleus and umbra is about to disappear, if it is not succeeded by a facula, or spot brighter than the rest of the disk, the place where it was soon after not distinguishable from the rest.

In the Philosophical Transactions, Vol. LXIV. Dr Wilton, professor of astronomy at Glasgow, hath given a dissertation on the nature of the solar spots, and mentions the following appearances. 1. When the spot is about to disappear on the western edge of the sun's limb, the eastern part of the umbra first contracts, then vanishes, the nucleus and western part of the umbra remaining; then the nucleus gradually contracts and vanishes while the western part of the umbra remains. At last this disappears also; and if the spot remains long enough to become again visible, the eastern part of the umbra first becomes visible, then the nucleus; and when the spot approaches the middle of the disk, the nucleus appears environed by the umbra on all sides, as already mentioned. 2. When two spots lie very near to one another, the umbra is deficient on that side which lies next the other spot; and this will be the case, though a large spot should be contiguous to one much smaller; the umbra of the large spot will be totally wanting on that side next the small one. If there are little spots on each side of the large one, the umbra does not totally vanish; but appears flattened, or pressed in towards the nucleus on each side. When the little spots disappear, the umbra of the large one extends itself as usual. This circumstance, he observes, may sometimes prevent the disappearance of the umbra in the manner above- Appears abovementioned; so that the western umbra may disappear before the nucleus, if a small spot happens to break out on that side.

In the same volume, p. 337. Mr Wollaston observes, that the appearances mentioned by Dr Wilson are not constant. He positively affirms, that the facula or bright spots on the sun are often converted into dark ones. "I have many times (says he) observed, near the eastern limb, a bright facula just come on, which has the next day shown itself as a spot, though I do not recollect to have seen such a facula near the western one after a spot's disappearance. Yet, I believe, both these circumstances have been observed by others; and perhaps not only near the limbs. The circumstance of the facula being converted into spots I think I may be sure of. That there is generally (perhaps always) a mottled appearance over the face of the sun, when carefully attended to, I think I may be as certain. It is most visible towards the limbs, but I have undoubtedly seen it in the centre; yet I do not recollect to have observed this appearance, or indeed any spots, towards his poles. Once I saw, with a twelve-inch reflector, a spot burst to pieces while I was looking at it. I could not expect such an event, and therefore cannot be certain of the exact particulars; but the appearance, as it struck me at the time, was like that of a piece of ice when dashed on a frozen pond, which breaks to pieces and slides in various directions." He also acquaints us, that the nuclei of the spots are not always in the middle of the umbrae; and gives the figure of one seen November 13th 1773, which is a remarkable instance of the contrary. Mr Dunn, however, in his new Atlas of the Mundane System, gives some particulars very different from the above. "The face of the sun (says he) has frequently many large black spots, of various forms and dimensions, which move from east to west, and round the sun, according to some observations in 25 days, according to others in 26, and according to some in 27 days. The black or central part of each spot is in the middle of a great number of very small ones, which permit the light to pass between them. The small spots are scarce ever in contact with the central ones; but what is most remarkable, when the whole spot is near the limb of the sun, the surrounding small ones form nearly a straight line, and the central part projects a little over it, like Saturn in his ring."

The spots are by no means confined to one part of the sun's disk; though we have not heard of any being observed about his polar regions; and though their direction is from east to west, yet the paths they describe in their course over the disk are exceedingly different; sometimes being straight lines, sometimes curves, sometimes descending from the northern to the southern part of the disk, sometimes ascending from the southern to the northern, &c. This was observed by Mr Derham (Philos. Trans. No. 330) who hath given figures of the apparent paths of many different spots, wherein the months in which they appeared, and their particular progress each day, are marked.

Besides these spots, there are others which sometimes appear very round and black, travelling over the disk of the sun in a few hours. They are totally unlike the others, and will be shown to proceed from an interposition of the planets Mercury and Venus between the earth and the sun. Excepting the two kinds of spots above-mentioned, however, no kind of object is discoverable on the surface of the sun, but he appears like an immense ocean of elementary fire or light.

2. With the moon the case is very different. Many darkish spots appear in her to the naked eye; and, through a telescope, their number is prodigiously increased: she also appears very plainly to be more protuberant in the middle than at the edges, or to have the figure of a globe, and not a flat circle. When the moon is horned or gibbous, the one side appears very ragged and uneven, but the other always exactly defined and circular. The spots in the moon always keep their places exactly; never vanishing, or going from one side to the other, as those of the sun do. We sometimes see more or less of the northern and southern, and eastern and western part of the disk or face; but this is owing to what is called her libration, and will hereafter be explained.

The astronomers Florentius, Langrenus, John Hevelius of Dantzic, Grimaldus, Ricciolus, Cassini, and M. de la Hire, have drawn the face of the moon as she is seen through telescopes magnifying between 200 and 300 times. Particular care has been taken to note all the shining parts in her surface; and, for the better distinguishing them, each has been marked with a proper name. Langrenus and Ricciolus have divided the lunar regions among the philosophers, astronomers, and other eminent men; but Hevelius and others, fearing lest the philosophers should quarrel about the division of their lands, have endeavoured to spoil them of their property, by giving the names belonging to different countries, islands, and seas on earth, to different parts of the moon's surface, without regard to situation or figure. The names adopted by Riccioli, however, are those which are generally followed, as the names of Hipparchus, Tycho, Copernicus, &c. are more pleasing to astronomers than those of Africa, the Mediterranean Sea, Sicily, and Mount Aetna. On Plate LXIII. is a tolerably exact representation of the full moon in her mean libration, with the numbers to the principal spots according to Riccioli, Cassini, Mayer, &c. The asterisk refers to one of the volcanoes discovered by Dr Herschel, to be afterwards more particularly noticed. The names are as follow:

* Herschel's Volcano. 1 Grimaldus. 2 Galileus. 3 Aristarchus. 4 Keplerus. 5 Gassendus. 6 Shikardus. 7 Harpalus. 8 Heracleides. 9 Lansbergius. 10 Reinoldus. 11 Copernicus. 12 Helicon. 13 Capuanus. 14 Bullialdus. 15 Eratothenes. 16 Timocharis. 17 Plato. 18 Archimedes. 19 Insula Sinus Medii. 20 Pitatus. 21 Tycho. 22 Eudoxus. 23 Aristeoteles. 24 Manilius. 25 Menelaus. 26 Hermes. 27 Posidonius. 28 Dionysius. 29 Plinius. 30 Calharina Cyriillus. 31 Theophilus. 32 Pracaftorius. 33 Promontorium acutum. 34 Senforinus. 35 Messila. 36 Promontorium Somnii. 37 Proclus. 38 Cleomedes. 39 Snellius et Furnerins. 40 Petavius. We have already observed, that when the planet Mars approaches any of the fixed stars, they lose their light, and sometimes totally disappear before he seems to touch them; but it is not so with the moon; for though she very often comes in betwixt us and the stars, they preserve their lustre till immediately in seeming contact with her, when they suddenly disappear, and as suddenly re-appear on the opposite side. When Saturn, however, was hid by the moon in June 1662, Mr Dunn, who watched his appearance at the emersion, observed a kind of faint shadow to follow him for a little from the edge of the moon's disk. This appearance is represented fig. 88.

3. Mercury, when looked at through telescopes magnifying about 200 or 300 times, appears equally luminous throughout his whole surface, without the least dark spot. He appears indeed to have the same difference of phases with the moon, being sometimes horned, sometimes gibbous, and sometimes shining almost with a round face, though not entirely full, because his enlightened side is never turned directly towards us; but at all times perfectly well defined without any ragged edge, and perfectly bright.

4. Dr Long informs us, that the earliest account he had met with of any spots seen by means of the telescope on the disk of Venus was in a collection of letters printed at Paris in 1665, in one of which Mr Auzout relates his having received advice from Poland that Mr Burattini had, by means of large telescopes, seen spots upon the planet Venus similar to those upon the moon. In 1667, Cassini, in a letter to Mr Petit, mentions his having for a long time carefully observed Venus through an excellent telescope made by Campani, in order to know whether that planet revolved on its axis or not, as he had before found Jupiter and Mars to do. But though he then observed some spots upon her, he says, that even when the air was quiet and clear, they appeared faint, irregular, and not well defined; so that it was difficult to have such a distinct view of any of them as to be certain that it was the same spot which was seen again in any subsequent observation; and this difficulty was increased, in the first place, when Venus was in her inferior semicircle; because at that time she must be viewed through the thick vapours near the horizon; though otherwise it was most proper, on account of her being then nearest to us. In the second place, if we would observe her at some height above those vapours, it could only be for a short time; and, thirdly, when she is low in her inferior circle, and at that time nearest the earth, the enlightened part of her is too small to discover any motion in it. He was therefore of opinion, that he should succeed better in his observations when the planet was about its mean distance from us, showing about one-half of her enlightened hemisphere; at which time also he could observe her for a much longer time above the gross atmospherical vapours. His first appearance of success was October 14, 1666, at three-quarters past five in the evening; when he saw a bright spot (fig. 37.), but could not then view that appearance long enough to draw any inference concerning the planet's motion. He had no farther success till the 20th of April the following year; when, about a quarter of an hour before sunrise, he began again to perceive on the disk of Venus, now about half enlightened, a bright part near the section, distant from the southern horn a little more than a fourth part of the diameter of the disk, and near the eastern edge. He took notice also of a darkish oblong spot nearer to the northern than the southern horn: at sunrise the bright part was advanced further from the southern horn than when he first observed it; but though he was pleased to find that he had now a convincing proof of the planet's motion, he was surprised that the spots moved from south to north in the lower part of the disk, and from north to south in the upper part; a motion of which we have no example, except in the librations of the moon. This, however, was occasioned by the situation of the planet's axis. Cassini expected to have found the rotation of Venus similar to that of Jupiter and Mars, both of which have their axes perpendicular to their respective orbits, and turn round according to the order of the signs; so that, in each of them, the motion of the inferior half of their respective globes, or that part next the sun, is from east to west; in the superior half from west to east; but in Venus, whose axis is inclined 75 degrees towards her orbit, the coincidence is so near, that one-half of her disk appears to move from south to north, the other from north to south.

On the 21st of April, at sunrise, the bright part was a good way off the section, and about a fourth part of the diameter distant from the southern horn. When the sun was eight degrees five minutes high, it seemed to be got beyond the centre, and was cut through by the section. At the time the sun was seven degrees high, the section cut it in the middle, which showed its motion to have some inclination towards the centre.

May 9, a little before sunrise, the bright spot was seen near the centre, a little to the northward, with two obscure ones situated between the section and the circumference, at a distance from each other, equal to that of each of them from the nearest angular point or horn of the planet. The weather being at that time clear, he observed for an hour and a quarter the motion of the bright spot, which seemed to be exactly from south to north, without any sensible declination to east or west. A variation was at the same time perceived in the darkish spot too great to be ascribed to any optical cause. The bright spot was also seen on the 10th and 13th days of May before sunrise between the northern horn and the centre, and the same irregular change of darkish spots was taken notice of; but as the planet removed to a greater distance from the earth, it became more difficult to observe these appearances. The above phenomena are represented as they occurred, in fig. 37—43.

But though, from the appearances just now related, Cassini's M. Cassini was of opinion that Venus revolved on her conclusion axis, he was by no means so positive in this matter as concerning with regard to Mars and Jupiter. "The spots on the revolution of Venus (says he) I could attentively observe for a whole night, when the planets were in opposition to the sun: axis." I could see them return to the same situation, and consider their motion during some hours, and judge whether they were the same spots or not, and what time they took in turning round; but it was not the same with the spots of Venus; for they can be observed only for so short a time, that it is much more difficult to know with certainty when they return into the same situation. I can, however, supposing that the bright spot which I observed on Venus, and particularly this year, was the same, say that she finishes her motion, whether of rotation or libration, in less than a day; so that, in 23 days nearly, the spot comes into the same situation on nearly the same hour of the day, though not without some irregularity. Now (supposing the bright spot observed to be always the same) whether this motion is an entire turning round, or only a libration, is what I dare not positively affirm.

In 1669 Mr Cassini again observed Venus through a telescope, but could not then perceive any spots upon her surface; the reason of which Dr Hamel conjectures to have been the fluctuation of the vapours near the horizon, which prevented them from being visible. However, we hear nothing more of any spots being seen on her disk till the year 1726; when, on the 9th of February, Bianchini, with some of Campani's telescopes of 90 and 100 Roman palms, began to observe the planet at the altitude of 40° above the horizon, and continued his observations till, by the motion of several spots, he determined the position of her axis to be inclined as above mentioned, that the north pole pointed at a circle of latitude drawn through the 20th degree of Aquarius, elevated 15° or 20° above the orbit of Venus. He delineated also the figures of several spots which he supposed to be seas, and complimented the king of Portugal and some other great men by calling them by their names. Though none of Bianchini's observations were continued long enough to know whether the spots, at the end of the period assigned for the rotation of the planet, would have been in a different situation from what they were at the beginning of it; yet, from observations of two and of four days, he concluded the motion of the spots to be at the rate of 15° per day; at which advance the planet must turn round either once in 24 days or in 23 hours; but without farther observation it could not be determined which of the two was the period of revolution: for if an observer should at a particular hour, suppose seven in the evening, mark exactly the place of a spot, and at the same hour next evening find the spot advanced 15°, he would not be able to determine whether the spot had advanced only 15°, or had gone once quite round with the addition of 15° more in part of another rotation. Mr Bianchini, however, supposes Venus to revolve in 24 days eight hours; the principal proof adduced for which is an observation of three spots, A B C, being situated as in fig. 44, when they were viewed by himself and several persons of distinction for about an hour, during which they could not perceive any change of place. The planet being then hid behind the Barbarine palace, they could not have another view of her till three hours after, when the spots still appeared unmoved: "Now (says Mr Bianchini) if her rotation were so swift as to go round in 23 hours, in this second view, three hours after the former, the spots must have advanced near 50 degrees; so that the spot C would have been gone off at R, the spot B would have succeeded into the place of C, the spot A into the place of B, and there would have been no more but two spots, A and B, to have been seen."

Cassini, the son, in a memoir for 1732, denies the dispute between Bianchini to be certain. He says, that during the three hours interval, the spot C might be gone off the disk, and the spot B got into the place thereof; where, being near the edge, it would appear less than in the middle. That A, succeeding into the place of B, would appear larger than it had done near the edge, and that another spot might come into the place of A; and there were other spots besides these three on the globe of the planet, as appears by the figures of Bianchini himself, particularly one which would naturally come in the place of A. That if the rotation of Venus be supposed to be in 23 hours, it will agree with Bianchini's observations, as well as with those of his father; but that, on the other supposition, the latter must be entirely rejected as erroneous: and he concludes with telling us, that Venus had frequently been observed in the most favourable times by Mr Maraldi and himself with excellent telescopes of 80 and 100 feet focus, without being able to see any distinct spot upon her disk. "Perhaps," says Dr Long, "those seen by Bianchini had disappeared, or the air in France was not clear enough; which last might be the reason why his father could never see those spots in France which he had observed in Italy, even when he made use of the longest telescopes." Neither of these astronomers take notice of any indentings in the curve which divides the illuminated part from the dark in the disk of Venus, though in some views of that planet by Fontana and Ricciolus, the curve is indented; and it has from thence been concluded, that the surface of the planet is mountainous like that of the moon. This had also been supposed by Buratini, already mentioned; and a late writer has observed, that "when the air is in a good state for observation, mountains like those of the moon may be observed with a very powerful telescope."

Cassini, besides the discovery of the spots on the disk Cassini discovered of Venus by which he was enabled to ascertain her revolution on an axis, had also a view of her satellite or moon, of which he gives the following account.—"A.D. 1686, Aug. 28th, at 15 minutes after four in the morning, looking at Venus with a telescope of 34 feet, I saw, at the distance of one third of her diameter eastward, a luminous appearance, of a shape not well defined, that seemed to have the same phase with Venus, which was then gibbous on the western side. The diameter of this phenomenon was nearly equal to a fourth part of the diameter of Venus. I observed it attentively for a quarter of an hour, and having left off looking at it for four or five minutes, I saw it no more; but daylight was then advanced. I had seen a like phenomenon which resembled the phase of Venus, Jan. 25th, A.D. 1672, from 52 minutes after six in the morning to two minutes after seven, when the brightness of the twilight made it disappear. Venus was then horned, and this phenomenon, the diameter whereof was nearly a fourth part of the diameter of Venus, was of the same shape. It was distant from the southern horn of Venus, a diameter of the planet,..." on the western side. In these two observations, I was in doubt whether it was not a satellite of Venus of such a constitude as not to be very well fitted to reflect the light of the sun; and which, in magnitude, bore nearly the same proportion to Venus as the moon does to the earth, being at the same distance from the sun and the earth as Venus was, the phases whereof it resembled. Notwithstanding all the pains I took in looking for it after these two observations, and at divers other times, in order to complete so considerable a discovery, I was never able to see it. I therefore suspend my judgment of this phenomenon. If it should return often, there will be these two epochs, which, compared with other observations, may be of use to find out the periodical time of its return, if it can be reduced to any rule."

A similar observation was made by Mr Short on the 23d of October 1740, about sunrise. He used at this time a reflecting telescope of about 16.5 inches, which magnified between 50 and 60 times, with which he perceived a small star at about 10' distance from Venus, as measured by the micrometer; and, putting on a magnifying power of 240 times, he found the star put on the same appearance with the planet herself. Its diameter was somewhat less than a third of that of the primary, but its light was less vivid, though exceedingly sharp and well defined. The same appearance continued with a magnifying power of 140 times. A line, passing through the centre of Venus and it, made an angle of 18 or 20 degrees with the equator: he saw it several times that morning for about the space of an hour, after which he lost sight of it, and could never find it again.

From this time the satellite of Venus, though very frequently looked for by astronomers, could never be perceived, which made it generally believed that Cassini and Mr Short had been mistaken; but as the transits of the planet over the sun in 1761 and 1769 seemed to promise a greater certainty of finding it, the satellite was very carefully looked for by almost every one who had an opportunity of seeing the transit, but generally without success. Mr Baudouin at Paris had provided a telescope of 25 feet, in order to observe the passage of the planet over the sun, and to look for its satellite; but he did not succeed either at that time or in the months of April and May following. Mr Montaigne, however, one of the members of the Society of Limoges, had better success. On the 3d of May 1761, he perceived, about half an hour after nine at night, at the distance of 20' from Venus, a small crescent, with the horns pointing the same way as those of the planet; the diameter of the former being about one-fourth of that of the latter; and a line drawn from Venus to the satellite making an angle with the vertical of about 20° towards the south. But though he repeated this observation several times, some doubt remained whether it was not a small star. Next day he saw the same star at the same hour, distant from Venus about half a minute, or a minute more than before, and making with the vertical an angle of 10° below on the north side; so that the satellite seemed to have described an arc of about 30°, whereof Venus was the centre, and the radius 20'. The two following nights were hazy, so that Venus could only be seen; but on the 7th of May, at the same hour as before, he saw the satellite again above Venus, and on the north side, appearing at the distance of 25 or 26' upon a line which made an angle of about 45° with the vertical towards the right hand. The light of the satellite was always very weak, but it had the same phasis with its primary, whether viewed together with it in the field of his telescope or by itself. The telescope was nine feet long, and magnified an object between 40 and 50 times, but had no micrometer; so that the distances above mentioned are only from estimation.

Fig. 4 represents the three observations of Mr Montaigne. V is the planet Venus; ZN the vertical. EC, a parallel to the ecliptic, making then an angle with the vertical of 45°; the numbers 3, 4, 7, mark the situations of the satellite on the respective days. From the figure it appears that the points 3 and 7 would have been diametrically opposite, had the satellite gone 15° more round the point V at the last observation; so that in four days it went through 155°. Then, as 155° is to four days or 96 hours, so is 360° to a fourth number, which gives 9 days 7 hours for the whole length of the synodical revolution. Hence Mr Baudouin concluded, that the distance of this satellite was about 60 of the semidiameters of Venus from its surface; that its orbit cut the ecliptic nearly at right angles; had its ascending node in 22° of Virgo; and was in its greatest northern digression on the 7th at nine at night; and he supposed, that at the transit of the primary the satellite would be seen accompanying it. By a subsequent observation, however, on the 11th of May, he corrected his calculation of the periodical time of the satellite, which he now enlarged to 12 days; in consequence of which he found that it would not pass over the disk of the sun along with its primary, but go at the distance of above 20' from its southern limb; though if the time of its revolution should be 15 hours longer than 12 days, it might then pass over the sun after Venus was gone off. He imagined the reason why this satellite was so difficult to be observed might be, that one part of its globe was so difficultly crusted over with spots, or otherwise unfit to reflect the light of the sun. By comparing the periodical time of this satellite with that of our moon, he computed the quantity of matter in Venus to be nearly equal to that in our earth; in which case it must have considerable influence in changing the obliquity of the ecliptic, the latitudes and longitudes of stars, &c.

This is all the evidence which has yet been published concerning the existence of the satellite of Venus; as it does not appear that, during the transit of 1769, any of the observers had the good fortune to perceive it. In the Philosophical Transactions for 1761, Mr Hirst gives an account of his having observed an atmosphere round the planet Venus. The observations were made at Fort St George; and looking attentively at that part of the sun's disk where he expected the planet would enter, he plainly perceived a faint shade or penumbra; on which he called out to his two assistants, "Tis a coming!" and two or three seconds after, the first external contact took place, in the moment whereof all the three agreed; but he could not see the penumbra after the egress: and of the other two gentlemen, one had gone home, and the other lost the planet out of the field of his telescope. Mr Dunn at Chelsea saw a penumbra, or small diminution of light, light, that grew darker and darker for about five seconds before the internal contact preceding the egress; from whence he determines that Venus is surrounded with an atmosphere of about 50 geographical miles high. His observations, he tells us, were made with an excellent six-feet Newtonian reflector, with a magnifying power of 110, and of 220 times: he had a clear dark glass next his eye, and the sun's limb appeared well defined; but a very narrow watery penumbra appeared round Venus. The darkest part of the planet's phasis was at the distance of about a sixth part of her diameter from its edge; from which an imperfect light increased to the centre, and illuminated round about.

In the northern parts of Europe this penumbra could not be seen. Mr Wargentin, who communicated several observations of the first external contact, says, that he could not mark the time exactly, because of the undulation of the limb of the sun; but thought it very remarkable that, at the egress, the limb of Venus that was gone off the sun showed itself with a faint light during almost the whole time of emergence. Mr Bergman, who was then at the observatory at Upsal, begins his account at the time when three fourths of the disk of the planet was entered upon that of the sun; and he says, that the part which was not come upon the sun was visible, though dark, and surrounded by a crescent of faint light, as in fig. 7.; but this appearance was much more remarkable at the egress; for as soon as any part of the planet was got off the sun, that part was visible with a like crescent, but brighter, fig. 8. As more of the planetary disk went off that of the sun, however, that part of the crescent which was farther from the sun grew fainter, and vanished, until at last only the horns could be seen, as in fig. 9. The total ingress was not instantaneous: but, as two drops of water, when about to part, form a ligament between them; so there was a dark swelling stretched out between Venus and the sun, as in fig. 10.; and when this ligament broke, the planet appeared to have got about an eighth part of her diameter from the nearest limb of the sun, fig. 11.; he saw the like appearance at going off, but not so distinct, fig. 12. Mr Chappe likewise took notice, that the part of Venus which was not upon the sun was visible during part of the time of ingress and egress; that it was farther surrounded by a small luminous ring of a deep yellow near the place that appeared in the form of a crescent, which was much brighter at the going off than coming upon the sun; and that, during the whole time the disk of Venus was upon the sun, he saw nothing of it. The time of total ingress was instantaneous like a flash of lightning; but at the egress the limb of the sun began to be obscured three seconds before the interior contact.

Some of the French astronomers attributed this luminous ring round Venus to the inflection of the sun's rays, as they also do the light seen round the moon in solar eclipses; but Mr Chappe supposes it to have been owing to the sun enlightening more than one half of the planetary globe, though he owns this cause not to be altogether sufficient. Mr Fouchy, who observed the transit at La Muette in France, perceived, during the whole time, a kind of ring round Venus, brighter than the rest of the sun, which became fainter the farther it went from the planet, but appeared more vivid in proportion as the sun was clearer. Mr Ferner, who observed at the same place, confirms the testimony of Mr Fouchy. "During the whole time (says he) of my observing with the telescope, and the blue and green glases, I perceived a light round about Venus, which followed her like a luminous atmosphere more or less lively, according as the air was more or less clear. Its extent altered in the same manner; nor was it well terminated, throwing out, at it were, some feeble rays on all sides."

"I am not clear, (says Dr Long) as to the meaning of the luminous circle here mentioned, whether opinion on when the whole planet was upon the sun, they saw a thec obser- ring of light round it, distinct from the light of the vations, fun; or whether they mean only the light which surrounded that part of Venus that was not upon the fun." Mr Chappe takes this and other accounts of the observations made in France in this latter sense; and though he sometimes called the luminous part of the crescent that surrounded the part of the planet not upon the sun a ring, he explains himself that he did so, because at the coming upon the sun he perceived it at one side of the planet, and on the opposite side on its going off: for which reason he supposed that it surrounded it on all sides. See fig. 13. 14.

Much larger and more remarkable spots have been perceived on the disk of Mars than that of any other primary planet. They were first observed in 1666 by Cassini at Bologna with a telescope of Campani about feet long; and continuing to observe them for a month, he found they came into the same situation in 24 hours and 40 minutes. The planet was observed by some astronomers at Rome with longer telescopes made by Lautachio Divini; but they assigned to it a rotation in 13 hours only. This, however, was afterwards shown by Mr Cassini to have been a mistake, and to have arisen from their not distinguishing the opposite sides of the planet, which it seems have spots pretty much alike. He made further observations on the spots of this planet in 1670; from whence he drew an additional confirmation of the time the planet took to revolve. The spots were again observed in subsequent oppositions; particularly for several days in 1704 by Maraldi, who took notice that they were not always well defined, and that they not only changed their shape frequently in the space between two oppositions, but even in the space of a month. Some of them, however, continued of the same form long enough to ascertain the time of the planet's revolution. Among these there appeared this year an oblong spot, resembling one of the belts of Jupiter when broken. It did not reach quite round the body of the planet; but had, not far from the middle of it, a small protuberance towards the north, so well defined that he was thereby enabled to settle the period of its revolution at 24 hours 39 minutes; only one minute less than what Cassini had determined it to be. See Plate fig. 45.

The near approach of Mars to the earth in 1719, gave a much better opportunity of viewing him than had been obtained before; as he was then within 2 deg. of his perihelion, and at the same time in opposition to the sun. His apparent magnitude and brightness were thus so much increased, that he was by the vulgar taken for a new star. His appearance at that time, Appearance of Celestial Bodies, long belt that reached half way round, to the end of which another shorter belt was joined, forming an obtuse angle with the former, as in fig. 47. This angular point was observed on the 19th and 20th of August, at 11 hours 15 minutes, a little east of the middle of the disk; and 37 days after, on the 25th and 26th of September, returned to the same situation. This interval, divided by 36, the number of revolutions contained in it gives 24 hours 40 minutes for the period of one revolution; which was verified by another spot of a triangular shape, one angle whereof was towards the north pole, and the base towards the south, which on the 5th and 6th of August appeared as in fig. 48, and after 72 revolutions returned to the same situation on the 16th and 17th of October. The appearances of Mars, as delineated by Mr Hook, when viewed through a 36 feet telescope, are represented fig. 28. He appeared through this instrument as big as the full moon. Some of the belts of this planet are said to be parallel to his equator; but that seen by Maraldi was very much inclined to it.

Besides these dark spots, former astronomers took notice that a segment of his globe about the south pole exceeded the rest of his disk so much in brightness, that it appeared beyond them as if it were the segment of a larger globe. Maraldi informs us, that this bright spot had been taken notice of for 60 years, and was more permanent than the other spots on the planet. One part of it is brighter than the rest, and the least bright part is subject to great changes, and has sometimes disappeared.

A similar brightness about the north pole of Mars was also sometimes observed; and these observations are now confirmed by Mr Herschel, who hath viewed the planet with much better instruments, and much higher magnifying powers than any other astronomer ever was in possession of. His observations were made with a view to determine the figure of the planet, the position of his axis, &c. A very particular account of them is given in the 74th volume of the Philosophical Transactions, but which our limits will not allow us to insert. Fig. 49—72 show the particular appearances of Mars, as viewed on the days there marked. The magnifying powers used were sometimes as high as 932; and with this the fourth polar spot was found to be in diameter 41". Fig. 73 shows the connection of the other figures marked 64, 65, 66, 67, 68, 69, 70, which complete the whole equatorial succession of spots on the disk of the planet. The centre of the circle marked 65 is placed on the circumference of the inner circle, by making its distance from the circle marked 67 answer to the interval of time between the two observations, properly calculated and reduced to sidereal measure. The same is done with regard to the circles marked 66, 67, &c.; and it will be found by placing any one of these connected circles in such a manner as to have its contents in a similar situation with the figures in the single representation, which bears the same number, that there is a sufficient resemblance between them; though some allowance must undoubtedly be made for the distortions occasioned by this kind of projection.

With regard to the bright spots themselves, Mr Herschel informs us, that the poles of the planet are not exactly in the middle of them, though nearly so. From the appearance and disappearance (says he) of the bright north polar spot in the year 1781, we collect that the circle of its motion was at some considerable distance from the pole. By calculation, its latitude must have been about 76 or 77° north; for I find that, to the inhabitants of Mars, the declination of the disappearance, June 25th, 12 hours 15 minutes of our time was about 9° 50' south; and the spot must have been so far removed from the north pole as to fall a few degrees within the enlightened part of the disk to become visible to us. The south pole of Mars could not be many degrees from the centre of the large bright southern spot of the year 1781; though this spot was of such a magnitude as to cover all the polar regions farther than the 70th or 65th degree; and in that part which was on the meridian, July 3d, at 10 hours 54 minutes, perhaps a little farther.

From the appearances of the south polar spot in 1781, we may conclude that its centre was nearly polar. We find it continued visible all the time Mars revolved on his axis; and to present us generally with a pretty equal share of the luminous appearance, a spot which covered from 45 to 60° of a great circle on the globe of the planet, could not have any considerable polar distance. From the observations and calculations made concerning the poles of Mars, we may conclude, that his north pole must be directed towards some point of the heavens, between 9° 24' 35" of position and 7° 15' 3"; because the change of the situation of the poles the pole from left to right, which happened in the Mars time the planet passed from one place to the other, is a plain indication of its having gone through the node of its axis. Next, we may also conclude, that the node must be considerably nearer the latter point of the ecliptic than the former; for, whatever be the inclination of the axis, it will be seen under equal angles at equal distances from the node. But by a trigonometrical process of solving a few triangles, we soon discover both the inclination of the axis, and the place where it intersects the ecliptic at rectangles (which, for want of a better term, I have perhaps improperly called its node). Accordingly I find by calculation, that the node is in 17° 47' of Pisces, the north pole of Mars being directed towards that part of the heavens; and that the inclination of the axis to the ecliptic is 59° 40'. By further calculations we find that the pole of Mars on the 17th of April 1777, was then actually 81° 27' inclined to the ecliptic, and pointed towards the left as seen from the sun.

The inclination and situation of the node of the axis of Mars, with respect to the ecliptic, being found, may be thus reduced to the orbit of the planet himself. Let E C (fig. 74.) be a part of the ecliptic, O M part of the orbit of Mars, P E O a line drawn from P, the celestial pole of Mars, through E, that point which has been determined to be the place of the node of the axis of Mars in the ecliptic, and continued to O, where it intersects his orbit. Now, if according to Mr de La Lande, we put the node of the orbit of Mars for 1783 in 1° 17' 58", we have from the place of the node of the axis, that is, 11° 17' 47" to the place of the node of the orbit, an arch E N of 69° 11'. In the triangle N E O, right-angled... Appearances of the Celestial Bodies thro' the Telescopes.

At E, there is also given the angle ENO, according to the same author, 1° 51', which is the inclination of the orbit of Mars to the ecliptic. Hence we find the angle EON = 89° 5', and the side ON = 60° 12'. Again, when Mars is in the node of its orbit N, we have by calculation the angle PNE = 63° 7'; to which adding the angle ENO = 1° 51', we have PNO = 64° 58'; from which two angles, PON and PNO, with the distance ON, we obtain the inclination of the axis of Mars, and place of its node with respect to its own orbit; the inclination being 61° 18', and the place of the node of the axis 58° 31' preceding the intersection of the ecliptic with the orbit of Mars, or in our 19° 28' of Pisces."

Our author next proceeds to shew how the seasons in this planet may be calculated, &c. Which conjectures, though they belong properly to the next section, yet are so much connected with what has gone before, that we shall insert here what he says upon the subject.

"Being thus acquainted with what the inhabitants of Mars will call the obliquity of their ecliptic, and the situation of their equinoctial and solstitial points, we are furnished with the means of calculating the seasons on that planet, and may account, in a manner which I think highly probable, for the remarkable appearance about its polar regions.

"But first, it may not be improper to give an instance how to resolve any query concerning the Martial seasons. Thus, let it be required to compute the declination of the sun on Mars, June 25, 1781, at midnight of our time. If ψ, ξ, η, ζ, &c. (fig. 75.) represent the ecliptic of Mars, and ψ' the ecliptic of our planet, A, b, B the mutual intersection of the Martial and terrestrial ecliptics, then there is given the heliocentric longitude of Mars, ψ m = 9° 100' 30"; then taking away six signs, and ψ b or ψ a = 18° 10' 58", there remains b m = 18° 22' 32". From this arch, with the given inclination, 1° 51' of the orbits to each other, we have cosine of inclination to radius, as tangent of b m to tangent of BM = 18° 22' 33". And taking away Bψ = 1° 10' 29", which is the complement to ψ B (or ζ A, already shown to be 1° 28' 31"), there will remain ψ M = 21° 45', the place of Mars in its own orbit; that is, on the time above mentioned, the sun's longitude on Mars will be 6° 21' 45"; and the obliquity of the Martial ecliptic, 28° 42', being also given, we find, by the usual method, the sun's declination 6° 56' south.

"The analogy between Mars and the earth is perhaps by far the greatest in the whole solar system. Their diurnal motion is nearly the same; the obliquity of their respective ecliptics not very different: of all the superior planets, the distance of Mars from the sun is by far the nearest alike to that of the earth; nor will the length of the Martial year appear very different from what we enjoy, when compared to the surprising duration of the years of Jupiter, Saturn, and the Georgium Sidus. If then we find that the globe we inhabit has its polar regions frozen and covered with mountains of ice and snow that only partly melt when alternately exposed to the sun, I may well be permitted to surmise, that the same causes may probably have the same effect on the globe of Mars; that the bright polar spots are owing to the vivid reflection of light from frozen regions; and that the production of those spots is to be ascribed to their being exposed to the sun. In the year 1781, the south polar spot was extremely large, which we might well expect, as that pole had but lately been involved in a whole twelvemonth's darkness and absence of the sun; but in 1783, I found it considerably smaller than before, and it decreased continually from the 20th of May till about the middle of September, when it seemed to be at a stand. During this last period the south pole had already been above eight months enjoying the benefit of summer, and still continued to receive the sun-beams, though, towards the latter end, in such an oblique direction as to be but little benefited by them. On the other hand, in the year 1781, the north polar spot, which had then been its twelvemonth in the sunshine, and was but lately returning into darkness, appeared small, though undoubtedly increasing in size. Its not being visible in the year 1783, is no objection to these phenomena, being owing to the position of the axis, by which it was removed out of sight.

"That a planetary globe, such as Mars, turning on an axis, should be of a spheroidal form, will easily find form of admittance, when two familiar instances in Jupiter and Mars, the earth, as well as the known laws of gravitation and the centrifugal force of rotatory bodies, lead the way to the reception of such doctrines. So far from creating difficulties, or doubts, it will rather appear singular, that the spheroidal form of this planet has not already been noticed by former astronomers; and yet, reflecting on the general appearance of Mars, we soon find, that opportunities of making observations on its real form cannot be very frequent; for when it is near enough to view it to an advantage, we see it generally gibbous, and its appositions are so scarce, and of so short a duration, that in more than two years time, we have not above three or four weeks for such observations. Besides, astronomers being generally accustomed to see this planet distorted, the spheroidal form might easily be overlooked.

"September 25, 1783. At 9h 50m, the equatorial diameter of Mars measured 21" 53"; the polar diameter 21" 15", full measure; that is certainly not too small. This difference of the diameters was shewn, on the 28th of the same month, to Mr Wilson of Glasgow, who saw it perfectly well, so as to be convinced that it was not owing to any defect or distortion occasioned by the lens; and because I wished him to be satisfied of the reality of the appearance, I reminded him of several precautions; such as causing the planet to pass directly through the centre of the field of view, and judging of its figure when it was most distinct and best defined, &c. Next day, the difference between the two diameters was shown to Dr Blagden and Mr Aubert. The former not only saw it immediately, but thought the flattening almost as much as that of Jupiter. Mr Aubert also saw it very plainly, so as to entertain no manner of doubt about the appearance.

"September 30th, 10h 52m, the equatorial diameter was 22" 09", with a magnifying power of 278. By a second measure it was 22" 31", full large; the polar diameter, very exact, was 21" 26". On the 1st of October, at 10h 50m, the equatorial diameter measured 103 by the micrometer, and the polar 98; the values of the divisions in seconds and thirds not being well determined." determined, on account of some changes lately made in the focal length of the object metals of the telescope. On the 13th, the equatorial diameter was exactly 22" 35"; the polar diameter 21" 33". In a great number of succeeding observations, the same appearance occurred; but on account of the quick changes in the appearance of this planet, Mr Herschel thought proper to settle the proportion betwixt the equatorial and polar diameters from those which were made on the very day of the apposition, and which were also to be preferred on account of their being repeated with a very high power, and in a fine clear air, with two different instruments of an excellent quality. From these he determined the proportions to be as 103 to 98, or 1355 to 1272.

It has been commonly related by astronomers, that the atmosphere of this planet is possessed of such strong refractive powers, as to render the small fixed stars near which it passes invisible. Dr Smith relates an observation of Cassini, where a star in the water of Aquarius, at the distance of six minutes from the disk of Mars, became so faint before its occultation, that it could not be seen by the naked eye, nor with a three-feet telescope. This would indicate an atmosphere of a very extraordinary size and density; but the following observations of Mr Herschel seem to shew that it is of much smaller dimensions. "1783, Oct. 26th. There are two small stars preceding Mars, of different sizes; with 460 they appear both dusky red, and are pretty unequal; with 278 they appear considerably unequal. The distance from Mars of the nearest, which is also the largest, with 227 measured 3' 26" 25". Some time after, the same evening, the distance was 3' 8" 55", Mars being retrograde. Both of them were seen very distinctly. They were viewed with a new 20 feet reflector, and appeared very bright. October 27th, the small star is not quite so bright in proportion to the large one as it was last night, being a good deal nearer to Mars, which is now on the side of the small star; but when the planet was drawn aside, or out of view, it appeared as plainly as usual. The distance of the small star was 2' 5" 25". The largest of the two stars, (adds he), on which the above observations were made, cannot exceed the 12th, and the smallest the 13th or 14th magnitude; and I have no reason to suppose that they were any otherwise affected by the approach of Mars, than what the brightness of its superior light may account for. From other phenomena it appears, however, that this planet is not without a considerable atmosphere; for besides the permanent spots on its surface, I have often noticed occasional changes of partial bright belts, and also once a darkish one in a pretty high latitude; and these alterations we can hardly attribute to any other cause than the variable disposition of clouds and vapours floating in the atmosphere of the planet."

6. Jupiter has the same general appearance with Mars, only that the belts on his surface are much larger and more permanent. Their general appearance, as described by Dr Long, is represented fig. 76—79.; by Mr Dun, fig. 18.; by Mr Wollaston, fig. 21., 22., 23.; and by Mr Ferguson, fig. 153. But they are not to be seen but by an excellent telescope. They are said to have been first discovered by Fontana and two other Italians; but Cassini was the first who gave a good account of them. Their appearance is very variable, as sometimes only one, and at others no fewer than eight, may be perceived. They bodies thereof are generally parallel to one another, but not always so; Telescopes and their breadth is likewise variable, one belt having been observed to grow narrow, while another in its neighbourhood has increased in breadth, as if the one had flowed into the other: and in this case Dr Long observes, that a part of an oblique belt lay between them, as if to form a communication for this purpose. The time of their continuance is very uncertain, sometimes remaining unchanged for three months; at others, new belts have been formed in an hour or two. In some of these belts large black spots have appeared, which moved swiftly over the disk from east to west, times appearing and returned in a short time to the same place; from which the rotation of this planet about its axis has been determined. On the 9th of May 1664, Dr Hook, with a good 12 feet telescope, observed a small spot in the biggest of the three obscure belts of Jupiter; and observing it from time to time, found that in two hours it had moved from east to west about half the visible diameter of the planet. In 1665, Cassini observed a spot near the largest belt of Jupiter which is most frequently seen. It appeared round, and moved with the greatest velocity when in the middle, but appeared narrower, and moved slower, the nearer it was to the circumference. "These circumstances (says Dr Long), showed that the spot adhered to the body of Jupiter, and was carried round upon it. It continued thereon till the year following; long enough to determine the periodical time of Jupiter's rotation upon his axis to be nine hours 56 minutes." This principal, or ancient spot as it is called, is the largest, and of the longest continuance of any hitherto known, and has appeared and vanished no fewer than eight times between the years 1665 and 1708: from the year last mentioned it was invisible till 1713. The longest time of its continuing to be visible was three years; and the longest time of its disappearing was from 1708 to 1713; it seems to have some connection with the principal southern belt; for the spot has never been seen when that disappeared, though that belt has often been visible without the spot. Besides this ancient spot, Cassini, in the year 1699, saw one of less stability that did not continue of the same shape or dimensions, but broke into several small ones, whereof the revolution was but nine hours 51 minutes; and two other spots that revolved in nine hours 52 minutes and a half. The figure of Jupiter is evidently an oblate spheroid, the longest diameter of his disk being to the shortest as 13 to 12. His rotation is from west to east, like that of the sun, and the plane of his equator is very nearly coincident with that of his orbit; so that there can scarce be any difference of seasons in that planet. His rotation has been observed to be somewhat quicker in his aphelion than his perihelion.

The most remarkable circumstance attending that planet is his having four moons, which constantly revolve round him at different distances. See fig. 18. and 186. These are all supposed to move in ellipses; though the eccentricities of all of them are too small to be measured, excepting that of the fourth; and even this amounts to no more than 0.007 of its mean distance from the primary. The orbits of these planets were thought thought by Galileo to be in the same plane with that of their primary; but Mr Cassini has found that their orbits make a small angle with it; and as he did not find any difference in the place of their nodes, he concluded that they were all in the same place, and that their ascending nodes were in the middle of Aquarius. After observing them for more than 36 years, he found their greatest latitude, or deviation from the plane of Jupiter's orbit, to be $2^\circ 55'$. The first of these satellites revolves at the distance of 5.697 of Jupiter's semidiameters, or $1' 51''$ as measured by proper instruments; its periodical time is 1 d. 18 h. 27 m. 34 s.

The next satellite revolves at the distance of 9.017 semidiameters, or $2' 56''$, in 3 d. 13 h. 13 m. 42 s.; the third at the distance of 14.384 semidiameters, or $4' 42''$, in 7 d. 3 h. 42 m. 36 s.; and the fourth at the distance of 25.266, or $8' 16''$, in 16 d. 16 h. 32 m. 09 s.

Since the time of Cassini, it has been found that the nodes of Jupiter's satellites are not in the same place; and from the different points of view in which we have an opportunity of observing them from the earth, we see them sometimes apparently moving in straight lines, and at other times in elliptic curves. All of them, by reason of their immense distance, seem to keep near their primary, and their apparent motion is a kind of oscillation like that of a pendulum, going alternately from their greatest distance on one side to the greatest distance on the other, sometimes in a straight line, and sometimes in an elliptic curve. When a satellite is in its superior semicircle, or that half of its orbit which is more distant from the earth than Jupiter is, its motion appears to us direct, according to the order of the signs; but in its inferior semicircle, when it is nearer to us than Jupiter, its motion appears retrograde; and both these motions seem quicker the nearer the satellites are to the centre of the primary, slower the more distant they are, and at the greatest distance of all they appear for a short time to be stationary.

From this account of the system of Jupiter and his satellites, it is evident, that occultations of them must frequently happen by their going behind their primary, or by coming in betwixt us and it. The former takes place when they proceed towards the middle of their upper semicircle; the latter, when they pass thro' the same part of their inferior semicircle. Occultations of the former kind happen to the first and second satellite; at every revolution, the third very rarely escapes an occultation, but the fourth more frequently by reason of its greater distance. It is seldom that a satellite can be discovered upon the disk of Jupiter, even by the best telescopes, excepting at its first entrance, when by reason of its being more directly illuminated by the rays of the sun than the planet itself, it appears like a lucid spot upon it. Sometimes, however, a satellite in passing over the disk, appears like a dark spot, and is easily to be distinguished. This is supposed to be owing to spots on the body of these secondary planets; and it is remarkable, that the same satellite has been known to pass over the disk at one time as a dark spot, and at another so luminous that it could not be distinguished from Jupiter himself, except at its coming on and going off. To account for this, we must say, that either the spots are subject to change; or if they be permanent like those of our moon, that the satellites at different times turn different parts of their globes towards us. Possibly both these causes may contribute to produce the phenomena just mentioned. For these reasons also both the light and apparent magnitude of the satellites are variable; for the fewer spots there are upon that side which is turned towards us, the brighter it will appear; and as the bright side only can be seen, a satellite must appear larger the more of its bright side it turns towards the earth, and the less so the more it happens to be covered with spots. The fourth and apposite satellite, though generally the smallest, sometimes appears bigger than any of the rest; the third sometimes tude.

When the satellites pass through their inferior semicircles, they may cast a shadow upon their primary, times five, and thus cause an eclipse of the sun to his inhabitants if there are any; and in some situations this shadow disk of Jupiter may be observed going before or following the satellite. On the other hand, in passing through their superior semicircles, the satellites may be eclipsed in the same manner as our moon by passing through the shadow of Jupiter; and this is actually the case with moons the first, second, and third of these bodies; but the eclipsed in fourth, by reason of the largeness of its orbit, passes every revolution sometimes above or below the shadow, as is the case with our moon. The beginnings and endings of these eclipses are easily seen by a telescope when the earth is in a proper situation with regard to Jupiter and the sun; but when this or any other planet is in conjunction with the sun, the superior brightness of that luminary renders both it and the satellites invisible. From the time of its first appearing after a conjunction until near the apposition, only the immersions of the satellites into his shadow, or the beginnings of the eclipses, are visible; at the apposition, only the occultations of &c. of Jupiter's satellites, by going behind or coming before their primary, are observable; and from the apposition to the conjunction, only the immersions, or end of the eclipses, are to be seen. This is exactly true in the first satellite, of which we can never see an immersion with its immediately subsequent emersion; and it is but rarely that they can be both seen in the second; as in order to their being so, that satellite must be near one of its limits, at the same time that the planet is near his perihelion and quadrature with the sun. With regard to the third, when Jupiter is more than 46 degrees from conjunction with, or opposition to, the sun, both its immersions and immediately subsequent emissions are visible; as they likewise are in the fourth, when the distance of Jupiter from conjunction or apposition is 24 degrees. When Jupiter is in quadrature with the sun, the earth is farthest out of the line that passes through the centres of the sun and Jupiter, and therefore the shadow of the planet is then most exposed to our view; but even then the body of the planet will hide from us one side of that part of the shadow which is nearest to it, through which the first satellite passes; which is the reason that, though we see the entrance of that satellite into the shadow, or its coming out from thence, as the earth is situated on the east or west side thereof, we cannot see them both; whereas the other satellites going through the shadow at a greater distance from Jupiter, their ingress and egress are both visible.

7. Saturn, when viewed through a good telescope, makes a more remarkable appearance than any of the other planets. Galileo first discovered his uncommon shape, which he thought to be like two small globes, one on each side of a large one; and he published his discovery in a Latin sentence; the meaning of which was, that he had seen him appear with three bodies; though, in order to keep the discovery a secret, the letters were transposed. Having viewed him for two years, he was surprised to see him become quite round without these appendages, and then after some time to assume them as before. These adjoining globes were what are now called the ansae of his ring, the true shape of which was first discovered by Huygens about 40 years after Galileo, first with a telescope of 12 feet, and then with one of 23 feet, which magnified objects 100 times. From the discoveries made by him and other astronomers, it appears that this planet is surrounded by a broad thin ring, the edge of which reflects little or none of the sun's light to us, but the planes of the ring reflect the light in the same manner that the planet itself does; and if we suppose the diameter of Saturn to be divided into three equal parts, the diameter of the ring is about seven of these parts. The ring is detached from the body of Saturn in such a manner, that the distance between the innermost part of the ring and the body is equal to its breadth. If we had a view of the planet and his ring, with our eyes, perpendicular to one of the planes of the latter, we should see them as in fig. 8o.; but our eye is never so much elevated above either plane as to have the visual ray stand at right angles to it, nor indeed is it ever elevated more than about 30 degrees above it; so that the ring, being commonly viewed at an oblique angle, appears of an oval form, and through very good telescopes double, as represented fig. 18. and 153. Both the outward and inward rim is projected into an ellipsis, more or less oblong according to the different degrees of obliquity with which it is viewed. Sometimes our eye is in the plane of the ring, and then it becomes invisible, either because the outward edge is not fitted to reflect the sun's light, or more probably because it is too thin to be seen at such a distance. As the plane of this ring keeps always parallel to itself, that is, its situation in one part of the orbit is always parallel to that in any other part, it disappears twice in every revolution of the planet, that is, about once in 15 years; and he sometimes appears quite round for nine months together. At other times, the distance betwixt the body of the planet and the ring is very perceptible; insomuch that Mr Whiston tells us of Dr Clarke's father having seen a star through the opening, and supposed him to have been the only person who ever saw a sight so rare, as appearing, though certainly very large, appears very small to us. When Saturn appears round, if our eye be in the plane of the ring, it will appear as a dark line across the middle of the planet's disk; and if our eye be elevated above the plane of the ring, a shadowy belt will be visible, caused by the shadow of the ring as well as by the interposition of part of it betwixt the eye and the planet. The shadow of the ring is broadest when the sun is most elevated, but its obscure parts appear broadest when our eye is most elevated above the plane of it. When it appears double, the ring next the body of the planet appears brightest; when the ring appears of an elliptical form, the parts about the ends of the largest axis are called the ansae, as has been already mentioned. These, a little before and after the disappearing of the ring, are of unequal magnitude: the largest ansa is longer visible before the planet's round phase, and appears again sooner than the other. On the 1st of October 1714, the largest ansa was on the east side, and on the 12th on the west side of the disk of the planet, which makes it probable that the ring has a rotation round an axis; but whether or not this is the case with Saturn himself has not been discovered, on account of the deficiency of spots by which it might be determined. He has indeed two belts, discovered with very long telescopes, which appear parallel to that formed by the edge of the ring Saturn, above-mentioned; but these are rectilinear when the ring appears elliptic, as in fig. 81. and seem to be permanent. In 1683, however, Dom Cassini and Fatio perceived a bright streak upon Saturn, which was not permanent like the dark belts, but was visible one day and disappeared the next, when another came into view near the edge of his disk. This induced Cassini to suppose, that Saturn might have a rotation round his axis; but the distance of this planet is so great, that we can scarce hope to determine his revolution so accurately as that of the others. It will disappear in May 1789; the earth being about to pass from its northern side, which is enlightened, to the southern, which is obscure.

The astronomer-royal (Dr Maskelyne) informs us of this disappearance in 1789, and reappearance in 1790, in the following manner: "On May 3rd and August 26th 1789, the plane of Saturn's ring will pass through the earth; on October 11th it will pass through the sun; and January 29th 1790 it will again pass through the earth. Hence, and supposing with M. de la Lande that the ring is but just visible with the best telescopes in common use, when the sun is elevated 3' above its plane, or 3 days before the plane passes through the sun, and when the earth is elevated 2½' above the plane, or one day from the earth's passing it, the phenomena of disappearance and reappearance may be expected to take place as follows.

"May 2d 1789, Saturn's ring will disappear; the earth being about to pass from its northern side, which is enlightened, to its southern side, which is obscure.

"August 27th, the earth having repassed to the northern or enlightened side, the ring will reappear.

"October 8th, the ring will disappear; its plane being near passing through the sun, when it will change its enlightened side from the northern to the southern..." one; consequently the dark side will be then turned to the earth.

"January 30th 1790, the earth having passed from the northern or dark side of the ring, to the southern or enlightened one, the ring will become visible, to continue so until the year 1803."

In the diagram, fig. 159, are delineated the phases of the ring from its full appearance in the year 1782 to its disappearance in 1789, and its full reappearance 1796.

Saturn is still better attended than Jupiter (see figs. 18. and 186); having, besides the ring abovementioned, no fewer than five moons continually circulating round him. The first, at the distance of 2.097 semidiameters of his ring, and 4.893 of the planet itself, performs its revolution in 1 d. 21 h. 18 m. 27 s.; the second, at 2.686 semidiameters of the ring, and 6.268 of Saturn, revolves in 2 d. 17 h. 41 m. 22 s.; the third, at the distance of 8.754 semidiameters of Saturn, and 3.752 of the ring, in 4 d. 12 h. 25 m. 12 s.; the fourth, called the Huygenian satellite, at 8.668 semidiameters of the ring, and 20.295 of Saturn, revolves in 15 d. 22 h. 41 m. 12 s.; while the fifth, placed at the vast distance of 59.154 semidiameters of Saturn, or 25.348 of his ring, does not perform its revolution in less than 79 d. 7 h. 47 m. 12 s. The orbits of all these satellites, except the fifth, are nearly in the same plane, which makes an angle with the plane of Saturn's orbit of about 31°; and by reason of their being inclined at such large angles, they cannot pass either across their primary or behind it with respect to the earth, except when very near their nodes; so that eclipses of them happen much more seldom than of the satellites of Jupiter. There is, however, an account in the Philol. Transact. of an occultation of the fourth satellite behind the body of Saturn; and there is a curious account by Cassini, in the Memoirs of the Royal Academy for 1692, of a fixed star being covered by the fourth satellite, so that for 13 minutes they appeared both as one star. By reason of their extreme smallness, these satellites cannot be seen unless the air is very clear; and Dom. Cassini for several years observed the fifth satellite to grow less and less as it went through the eastern part of its orbit until it became quite invisible, while in the western part it gradually became more and more bright until it arrived at its greatest splendor.—"This phenomenon (says Dr Long) cannot be better accounted for than by supposing one half of the surface of this satellite to be unfit to reflect the light of the sun in sufficient quantity to make it visible, and that it turns round its axis nearly in the same time as it revolves round its primary; and that, by means of this rotation, and keeping always the same face toward Saturn, we upon the earth may, during one half of its periodical time, be able to see successively more and more of its bright side, and during the other half of its period have more and more of the spotted or dark side turned toward us. In the year 1703, this satellite unexpectedly became visible in all parts of its orbit through the very same telescopes that were before often made use of to view it in the eastern part without success: this shows the spots upon this satellite, like those upon Jupiter and some other of the primary planets, are not permanent, but subject to change."

With regard to the Georgium Sidus, still less is known than of Saturn. Its apparent magnitude is so small, that it can seldom be seen with the naked eye; and even with the telescope it appears but of a few seconds diameter. It is attended by two satellites at the proportional distances marked in fig. 82, according to the observations of Mr Herschel; but he had not an opportunity of observing them long enough to determine their periodical times with exactness; though he supposes the innermost to perform its revolution in about eight days and three quarters, the other in thirteen days and a half.

The Comets, viewed through a telescope, have a very different appearance from any of the planets. The nucleus, or star, seems much more dim. Sturmius tells us, that observing the comet of 1680 with a telescope, it appeared like a coal dimly glowing; or a rude mass of matter illuminated with a dusky fumid light, less sensible at the extremes than in the middle; and not at all like a star, which appears with a round disk and a vivid light.

Hevelius observed of the comet in 1661, that its body was of a yellowish colour, bright and conspicuous, but without any glittering light. In the middle was a dense ruddy nucleus, almost equal to Jupiter, encompassed with a much fainter, thinner matter.—Feb. 5th. The nucleus was somewhat bigger and brighter, of a gold colour, but its light more dusky than the rest of the stars; it appeared also divided into a number of parts.—Feb. 6th. The nuclei still appeared, though less than before. One of them on the left side of the lower part of the disk appeared to be much denser and brighter than the rest; its body round, and representing a little lucid star; the nuclei still encompassed with another kind of matter.—Feb. 10th. The nuclei more obscure and confused, but brighter at top than at bottom.—Feb. 13th. The head diminished much both in brightness and in magnitude.—March 2d. Its roundness a little impaired, and the edges lacerated.—March 8th. Its matter much dispersed; and no distinct nucleus at all appearing.

Wigellius, who saw through a telescope the comet of 1664, the moon, and a little cloud illuminated by the sun, at the same time, observed that the moon appeared of a continued luminous surface, but the comet very different, being perfectly like the little cloud enlightened by the sun's beams.

The comets, too, are to appearance surrounded with atmospheres of a prodigious size, often rising ten times higher than the nucleus. They have often likewise different phases, like the moon.

"The head of a comet (says Dr Long) to the eye, Dr Long's unassisted by glasses, appears sometimes like a cloudy star; sometimes shines with a dull light like that of the planet Saturn: some comets have been said to equal, some to exceed, stars of the first magnitude; some to have surpassed Jupiter, and even Venus; and to have cast a shadow as Venus sometimes does.

"The head of a comet, seen through a good telescope, appears to consist of a solid globe, and an atmosphere that surrounds it. The solid part is frequently called the nucleus; which through a telescope is easily distinguished from the atmosphere or hairy appearance.

"A comet is generally attended with a blaze or tail, whereby it is distinguished from a star or planet." as it is also by its motion. Sometimes the tail only of a comet has been visible at a place where the head has been all the while under the horizon; such an appearance is called a beam.

"The nucleus of the comet of 1618 is said, a few days after coming into view, to have broken into three or four parts of irregular figures. One observer compares them to so many burning coals; and says they changed their situation while he was looking at them, as when a person stirs a fire; and a few days after were broken into a great number of smaller pieces. Another account of the same is, that on the 1st and 4th of December, the nucleus appeared to be a round, solid, and luminous body, of a dusky lead colour, larger than any star of the first magnitude. On the 8th of the same month it was broken into three or four parts of irregular figures; and on the 20th was changed into a cluter of small stars.

"As the tail of a comet is owing to the heat of the sun, it grows larger as the comet approaches near to, and shortens as it recedes from, that luminary. If the tail of a comet were to continue of the same length, it would appear longer or shorter according to the different views of the spectator; for if his eye be in a line drawn through the middle of the tail lengthwise, or nearly so, the tail will not be distinguished from the rest of the atmosphere, but the whole will appear round; if the eye be a little out of that line, the tail will appear short as in fig. 83; and it is called a bearded comet when the tail hangs down towards the horizon, as in that figure. If the tail of a comet be viewed side-wise, the whole length of it is seen. It is obvious to remark, that the nearer the eye is to the tail, the greater will be the apparent length thereof.

"The tails of comets often appear bent, as in figs. 84 and 85, owing to the resistance of the aether; which, though extremely small, may have a sensible effect on so thin a vapour as the tails consist of. This bending is seen only when the earth is not in the plane of the orbit of the comet continued. When that plane passes thro' the eye of the spectator, the tail appears straight, as in fig. 86, 87.

"Longomontanus mentions a comet, that, in 1618, Dec. 10th, had a tail above 100 degrees in length; which shows that it must then have been very near the earth. The tail of a comet will at the same time appear of different lengths in different places, according as the air in one place is clearer than in another. It need not be mentioned, that in the same place, the difference in the eyes of the spectators will be the cause of their disagreeing in their estimate of the length of the tail of a comet.

"Hevelius is very particular in telling us, that he observed the comet of 1665 to cast a shadow upon the tail; for in the middle thereof there appeared a dark line. It is somewhat surprising, that Hooke should be positive in affirming, on the contrary, that the place where the shadow of that comet should have been, if there had been any shadow, was brighter than any other part of the tail. He was of opinion that comets have some light of their own: His observations were made in a hurry; he owns they were short and transitory. Hevelius's were made with so much care, that there is more reason to depend upon them. Dom. Cassini observed, in the tail of the comet of 1680, a darkness in the middle; and the like was taken notice of by a curious observer in that of 1744.

"There are three comets, viz. of 1680, 1744, and 1759, that deserve to have a farther account given of them. The comet of 1680 was remarkable for its near approach to the sun; so near, that in its perihelion it was not above a sixth part of the diameter of that luminary from the surface thereof. Fig. 85, ta., the comet of Newton's Principia, represents so much of the trajectory of this comet as it passed through while it was visible to the inhabitants of our earth, in going from and returning to its perihelion. It shows also the tail, as it appeared on the days mentioned in the figure. The tail, like that of other comets, increased in length and brightness as it came nearer to the sun; and grew shorter and fainter as it went farther from him and from the earth, till that and the comet were too far off to be any longer visible.

"The comet of 1744 was first seen at Lausanne in Switzerland, Dec. 13th, 1743, N. S. From that time it increased in brightness and magnitude as it was coming nearer to the sun. The diameter of it, when at the distance of the sun from us, measured about one minute; which brings it out equal to three times the diameter of the earth. It came so near Mercury, that, if its attraction had been proportional to its magnitude, it was thought probable it would have disturbed the motion of that planet. Mr Betts of Oxford, however, from some observations made there, and at Lord Macclesfield's observatory at Sherburn, found, that when the comet was at its least distance from Mercury, and almost twice as near the sun as that planet was, it was still distant from him a fifth part of the distance of the sun from the earth; and could therefore have no effect upon the planet's motions. He judged the comet to be at least equal in magnitude to the earth. He says, that in the evening of Jan. 23, this comet appeared exceedingly distinct and bright, and the diameter of its nucleus nearly equal to that of Jupiter. Its tail extended above 16 degrees from its body; and was in length, supposing the sun's parallax 10", no less than 23 millions of miles. Dr Bevis, in the month of May 1744, made four observations of Mercury, and found the places of that planet, calculated from correct tables, differed so little from the places observed, as to show that the comet had no influence upon Mercury's motion.

"The nucleus, which had before been always round, on the 10th of February appeared oblong in the direction of the tail, and seemed divided into two parts by a black stroke in the middle. One of the parts had a sort of beard brighter than the tail; this beard was surrounded by two unequal dark strokes, that separated the beard from the hair of the comet. The odd phenomena disappeared the next day, and nothing was seen but irregular obscure spaces like smoke in the middle of the tail; and the head resumed its natural form. Feb. 15th, the tail was divided into two branches; the eastern part about seven or eight degrees long, the western 24. On the 23d, the tail began to be bent; it showed no tail till it was as near to the sun as the orbit of Mars; the tail grew longer as it approached nearer the sun; and at its greatest length was computed to equal a third part of the distance of the earth from the sun. Fig. 84, is a view of this comet, taken by... by an observer at Cambridge. I remember that in viewing it I thought the tail seemed to sparkle, or vibrate luminous particles. Hevelius mentions the like in other comets; and that their tails lengthen and shorten while we are viewing. This is probably owing to the motion of our air.

"The comet of 1759 did not make any considerable appearance by reason of the unfavourable situation of the earth all the time its tail might otherwise have been conspicuous; the comet being then too near the sun to be seen by us; but deserves our particular consideration, as it was the first that ever had its return foretold." See the following Section.

Hevelius gives pictures of comets of various shapes; as they are described by historians to have been like a sword, a buckler, a tun, &c. These are drawn by fancy only, from the description in words. He gives, however, also pictures of some comets, engraved by his own hand from the views he had of them through a very long and excellent telescope. In these we find changes in the nucleus and the atmosphere of the same comet. The nucleus of the comet of 1661, which in one observation appeared as one round body, as it is represented in fig. 87, in subsequent views seemed to consist of several smaller ones separated from one another, as in fig. 86. The atmosphere surrounding the nucleus, at different times, varied in the extent thereof; as did also the tail in length and breadth. The nuclei of other comets, as has already been observed, have sometimes phases like the moon. Those of 1744 and 1769 had both this kind of appearance. See fig. 34.

The fixed stars, when viewed through the best telescopes, appear not at all magnified, but rather diminished in bulk; by reason, as is thought by some, that the telescope takes off that twinkling appearance they make to the naked eye; but by others more probably, that the telescope tube excludes a quantity of the rays of light, which are not only emitted from the particular stars themselves, but by many thousands more, which falling upon our eye-lids and the aerial particles about us, are reflected into our eyes so strongly as to excite vibrations, not only on those points of the retina where the images of the stars are formed, but also in other points at the same distance round about. This without the telescope makes us imagine the stars to be much bigger than when we see them only by a few rays coming directly from them, so as to enter our eyes without being intermixed with others. The number of stars appears increased prodigiously through the telescope; so stars have been counted in the constellation called Pleiades, and no fewer than 2000 in that of Orion. The late improvements of Mr Herschel, however, have shown the number of stars to be exceedingly beyond even what the discoveries of former astronomers would induce us to suppose. He has also shown, that many which to the eye, or through ordinary glasses, appear single, do in fact consist of two or more stars; and that the galaxy or milky-way owes its light entirely to multitudes of small stars placed so close, that the naked eye, or even ordinary telescopes, cannot discover them.

He has shown also, that the nebulæ, or small whitish specks, discoverable by telescopes in various parts of the heavens, are owing to the same cause. Former astro-

nomers could only reckon 103; but Dr Herschel has discovered upwards of 1250. He has also discovered a species of them, which he calls planetary nebulae, on account of their brightness and shining with a well-defined disk, being also capable of being magnified more than the fixed stars.

Sect. III. Conclusions from the foregoing Appearances.

The conjectures which have been formed concerning the nature of the celestial bodies are so numerous, that a recital of them would fill a volume; while at the same time many of them are so ridiculous, that absurdity itself would seem almost to have been exhausted on this subject.

1. As a specimen of what were the opinions of the ancients concerning the nature of the sun, it may suffice to mention, that Anaximander and Anaximenes held, that there was a circle of fire all along the heavens, which they called the circle of the sun; between the earth and this fiery circle was placed another circle of some opaque matter, in which there was a hole like the mouth of a German flute. Through this hole the light was transmitted, and appeared to the inhabitants of this earth as a round and distinct body of fire. The eclipses of the sun were occasioned by stopping this hole.

We must not, however, imagine, that the opinions of all the ancients were equally absurd with those of Anaximander and Anaximenes. Many of them had more just notions, though very imperfect and obscure. Anaxagoras held the sun to be a fiery globe of some solid substance, bigger than Peloponnesus; and many of the moderns have adopted this notion, only increasing the magnitude of the globe prodigiously. Sir Isaac Newton has proposed it as a query, Whether the sun and fixed stars are not great Earths made vehemently hot, whose parts are kept from fuming away by the vast weight and density of their superincumbent atmospheres, and whose heat is preserved by the prodigious action and reaction of their parts upon one another? But though Sir Isaac has proposed this as a query, and taken the existence of a solar atmosphere for granted, there have yet been no proofs adduced in favour of that opinion besides those of analogy and probability.

There is, however, an appearance in the heavens termed the semita luminosa, or zodiacal light, which is now generally supposed to be owing to the sun's atmosphere. This was first discovered by Dom. Cassini in 1683. It is something like the milky-way, a faint twilight, or light, the tail of a comet, thin enough to let stars be seen through it, and seems to surround the sun in the form of a lens, the plane whereof is nearly coincident with that of the sun's equator. It is seen stretched along the zodiac, and accompanies the sun in his annual motion through the twelve signs. Each end terminates in an angle of about 21°; the extent of it in length from either of the angular points varies from 50 to 100°; it reaches beyond the orbit of Venus, but not so far as that of the earth. The breadth of it near the horizon is also various; from 12 almost to 30°; near the fun, where it may reasonably be supposed to be broadest, it cannot be seen. This light is weakest in the morning and strongest at night; disappearing in full moon. as it is also by its motion. Sometimes the tail only of a comet has been visible at a place where the head has been all the while under the horizon; such an appearance is called a beam.

"The nucleus of the comet of 1618 is said, a few days after coming into view, to have broken into three or four parts of irregular figures. One observer compares them to so many burning coals; and says they changed their situation while he was looking at them, as when a person stirs a fire; and a few days after were broken into a great number of smaller pieces. Another account of the same is, that on the 1st and 4th of December, the nucleus appeared to be a round, solid, and luminous body, of a dusky lead colour, larger than any star of the first magnitude. On the 8th of the same month it was broken into three or four parts of irregular figures; and on the 20th was changed into a cluter of small stars.

"As the tail of a comet is owing to the heat of the sun, it grows larger as the comet approaches near to, and shortens as it recedes from, that luminary. If the tail of a comet were to continue of the same length, it would appear longer or shorter according to the different views of the spectator; for if his eye be in a line drawn through the middle of the tail lengthwise, or nearly so, the tail will not be distinguished from the rest of the atmosphere, but the whole will appear round; if the eye be a little out of that line, the tail will appear short as in fig. 83.; and it is called a bearded comet when the tail hangs down towards the horizon, as in that figure. If the tail of a comet be viewed side-wise, the whole length of it is seen. It is obvious to remark, that the nearer the eye is to the tail, the greater will be the apparent length thereof.

"The tails of comets often appear bent, as in fig. 84 and 85, owing to the resistance of the ether; which, though extremely small, may have a sensible effect on so thin a vapour as the tails consist of. This bending is seen only when the earth is not in the plane of the orbit of the comet continued. When that plane passes thro' the eye of the spectator, the tail appears straight, as in fig. 86, 87.

"Longomontanus mentions a comet, that, in 1618, Dec. 10th, had a tail above 100 degrees in length; which shows that it must then have been very near the earth. The tail of a comet will at the same time appear of different lengths in different places, according as the air in one place is clearer than in another. It need not be mentioned, that in the same place, the difference in the eyes of the spectators will be the cause of their disagreeing in their estimate of the length of the tail of a comet.

"Hevelius is very particular in telling us, that he observed the comet of 1665 to cast a shadow upon the tail; for in the middle thereof there appeared a dark line. It is somewhat surprising, that Hooke should be positive in affirming, on the contrary, that the place where the shadow of that comet should have been, if there had been any shadow, was brighter than any other part of the tail. He was of opinion that comets have some light of their own: His observations were made in a hurry; he owns they were short and transitory. Hevelius's were made with so much care, that there is more reason to depend upon them. Dom. Cassini, observed, in the tail of the comet of 1689, a darkness in the middle; and the like was taken notice of by a curious observer in that of 1744.

"There are three comets, viz. of 1680, 1744, and 1759, that deserve to have a farther account given of them. The comet of 1680 was remarkable for its near approach to the sun; to near, that in its perihelion it was not above a sixth part of the diameter of that luminary from the surface thereof. Fig. 85, taken from Newton's Principia, represents so much of the trajectory of this comet as it passed through while it was visible to the inhabitants of our earth, in going from and returning to its perihelion. It shows also the tail, as it appeared on the days mentioned in the figure. The tail, like that of other comets, increased in length and brightness as it came nearer to the sun; and grew shorter and fainter as it went farther from him and from the earth, till that and the comet were too far off to be any longer visible.

"The comet of 1744 was first seen at Lausanne in Switzerland, Dec. 13th, 1743, N. S. From that time it increased in brightness and magnitude as it was coming nearer to the sun. The diameter of it, when at the distance of the sun from us, measured about one minute; which brings it out equal to three times the diameter of the earth. It came so near Mercury, that, if its attraction had been proportionable to its magnitude, it was thought probable it would have disturbed the motion of that planet. Mr Betts of Oxford, however, from some observations made there, and at Lord Macclesfield's observatory at Sherburn, found, that when the comet was at its least distance from Mercury, and almost twice as near the sun as that planet was, it was still distant from him a fifth part of the distance of the sun from the earth; and could therefore have no effect upon the planet's motions. He judged the comet to be at least equal in magnitude to the earth. He says, that in the evening of Jan. 23, this comet appeared exceedingly distinct and bright, and the diameter of its nucleus nearly equal to that of Jupiter. Its tail extended above 16 degrees from its body; and was in length, supposing the sun's parallax 10", no less than 23 millions of miles. Dr Bevis, in the month of May 1744, made four observations of Mercury, and found the places of that planet, calculated from correct tables, differed so little from the places observed, as to show that the comet had no influence upon Mercury's motion.

"The nucleus, which had before been always round, on the 10th of February appeared oblong in the direction of the tail, and seemed divided into two parts by a black stroke in the middle. One of the parts had a sort of beard brighter than the tail; this beard was surrounded by two unequal dark strokes, that separated the beard from the hair of the comet. The odd phenomena disappeared the next day, and nothing was seen but irregular obscure spaces like smoke in the middle of the tail; and the head resumed its natural form. Feb. 15th, the tail was divided into two branches; the eastern part about seven or eight degrees long, the western 24. On the 23d, the tail began to be bent; it showed no tail till it was as near to the sun as the orbit of Mars; the tail grew longer as it approached nearer the sun; and at its greatest length was computed to equal a third part of the distance of the earth from the sun. Fig. 84, is a view of this comet, taken by J. by an observer at Cambridge. I remember that in viewing it I thought the tail seemed to sparkle, or vibrate luminous particles. Hevelius mentions the like in other comets; and that their tails lengthen and shorten while we are viewing. This is probably owing to the motion of our air.

"The comet of 1759 did not make any considerable appearance by reason of the unfavourable situation of the earth all the time its tail might otherwise have been conspicuous; the comet being then too near the sun to be seen by us; but deserves our particular consideration, as it was the first that ever had its return foretold." See the following Section.

Hevelius gives pictures of comets of various shapes; as they are described by historians to have been like a sword, a buckler, a tun, &c. These are drawn by fancy only, from the description in words. He gives, however, also pictures of some comets, engraved by his own hand from the views he had of them through a very long and excellent telescope. In these we find changes in the nucleus and the atmosphere of the same comet. The nucleus of the comet of 1661, which in one observation appeared as one round body, as it is represented in fig. 87, in subsequent views seemed to consist of several smaller ones separated from one another, as in fig. 86. The atmosphere surrounding the nucleus, at different times, varied in the extent thereof; as did also the tail in length and breadth. The nuclei of other comets, as has already been observed, have sometimes phases like the moon. Those of 1744 and 1769 had both this kind of appearance. See fig. 34.

The fixed stars, when viewed through the best telescopes, appear not at all magnified, but rather diminished in bulk; by reason, as is thought by some, that the telescope takes off that twinkling appearance they make to the naked eye; but by others more probably, that the telescope tube excludes a quantity of the rays of light, which are not only emitted from the particular stars themselves, but by many thousands more, which falling upon our eye-lids and the aerial particles about us, are reflected into our eyes so strongly as to excite vibrations, not only on those points of the retina where the images of the stars are formed, but also in other points at the same distance round about. This without the telescope makes us imagine the stars to be much bigger than when we see them only by a few rays coming directly from them, so as to enter our eyes without being intermixed with others. The number of stars appears increased prodigiously through the telescope; 70 stars have been counted in the constellation called Pleiades, and no fewer than 2000 in that of Orion. The late improvements of Mr Herschel, however, have shown the number of stars to be exceedingly beyond even what the discoveries of former astronomers would induce us to suppose. He has also shown, that many which to the eye, or through ordinary glasses, appear single, do in fact consist of two or more stars; and that the galaxy or milky-way owes its light entirely to multitudes of small stars placed so close, that the naked eye, or even ordinary telescopes, cannot discover them.

He has shown also, that the nebulae, or small whitish specks, discoverable by telescopes in various parts of the heavens, are owing to the same cause. Former astronomers could only reckon 103; but Dr Herschel has discovered upwards of 1250. He has also discovered a from the species of them, which he calls planetary nebulae, on account of their brightness and shining with a well-defined disk, being also capable of being magnified more than the fixed stars.

Sect. III. Conclusions from the foregoing Appearances.

The conjectures which have been formed concerning the nature of the celestial bodies are so numerous, that a recital of them would fill a volume; while at the same time many of them are so ridiculous, that absurdity itself would seem almost to have been exhausted on this subject.

1. As a specimen of what were the opinions of the ancients concerning the nature of the sun, it may suffice to mention, that Anaximander and Anaximenes held, that there was a circle of fire all along the heavens, which they called the circle of the sun; between the earth and this fiery circle was placed another circle of some opaque matter, in which there was a hole like the mouth of a German flute. Through this hole the light was transmitted, and appeared to the inhabitants of this earth as a round and distinct body of fire. The eclipses of the sun were occasioned by stopping this hole.

We must not, however, imagine, that the opinions of all the ancients were equally absurd with those of Anaximander and Anaximenes. Many of them had more just notions, though very imperfect and obscure. Anaxagoras held the sun to be a fiery globe of some solid substance, bigger than Peloponnesus; and many of the moderns have adopted this notion, only increasing the magnitude of the globe prodigiously. Sir Isaac Newton has proposed it as a query, Whether the sun and fixed stars are not great Earths made vehemently hot, whose parts are kept from fuming away by the vast weight and density of their superincumbent atmospheres, and whose heat is preserved by the prodigious action and reaction of their parts upon one another? But though Sir Isaac has proposed this as a query, and taken the existence of a solar atmosphere for granted, there have yet been no proofs adduced in favour of that opinion besides those of analogy and probability.

There is, however, an appearance in the heavens termed the semita luminum, or zodiacal light, which is now generally supposed to be owing to the sun's atmosphere. This was first discovered by Dom Cassini in 1683. It is something like the milky-way, a faint twilight, or light, the tail of a comet, thin enough to let stars be seen through it, and seems to surround the sun in the form of a lens, the plane whereof is nearly coincident with that of the sun's equator. It is seen stretched along the zodiac, and accompanies the sun in his annual motion through the twelve signs. Each end terminates in an angle of about 21°; the extent of it in length from either of the angular points varies from 50 to 100°; it reaches beyond the orbit of Venus, but not so far as that of the earth. The breadth of it near the horizon is also various; from 12 almost to 30° near the sun, where it may reasonably be supposed to be broadest; it cannot be seen. This light is weakest in the morning and strongest at night; disappearing in full moon. Conclusions moonlight or in strong twilight, and therefore is not at all visible about midsummer in places so near either of the poles as to have their twilight all the night long, but may be seen in those places in the middle of winter both morning and evening, as it may in places under and near the equator all the year round. In north latitude it is most conspicuous after the evening twilight about the latter end of February, and before the morning twilight in the beginning of October; for at those times it stands most erect above the horizon, and is therefore clearest from the thick vapours of the twilight. Besides the difference of real extension of this light in length and breadth at different times, it is diminished by the nearness of any other light in the sky; not to mention that the extent of it will be differently determined by different spectators according to the goodness of their eyes.

Cassini, inquiring into the cause of this light, says first, that it might be owing to a great number of small planets surrounding the sun within the orbit of Venus; but soon rejects this for what he thinks a more probable solution, viz. that as by the rotation of the sun some parts are thrown up on his surface, whereof spots and nebulosities are formed; so the great rapidity wherewith the equatorial parts are moved, may throw out to a considerable distance a number of particles of a much finer texture, of sufficient density to reflect light: now, that this light was caused by an emanation from the sun, similar to that of the spots, he thought probable from the following observation: That after the year 1683, when this light began to grow weaker, no spots appeared upon the sun; whereas, in the preceding years, they were frequently seen there; and that the great inequality in the intervals between the times of the appearances of the solar spots has some analogy to the irregular returns of weakness and strength in this light, in like circumstances of the constitution of the air, and of the darkness of the sky. Cassini was of opinion that this light in the zodiac, as it is subject to great increase at one time and diminution at another, may sometimes become quite imperceptible; and thought this was the case in the years 1665, 1672, and 1681, when he saw nothing of it, though he surveyed with great attention those parts of the heaven where, according to his theory, it must have appeared if it had been as visible then as it was in others. He cites also passages out of several authors both ancient and modern, which make it probable that it had been seen both in former and latter ages, but without being sufficiently attended to, or its nature inquired into. It had been taken for the tail of a comet, part of the twilight, or a meteor of short continuance; and he was fully convinced of its having appeared formerly, from a passage in an English book of Mr Childrey's, printed in 1661. This passage is as follows:

"There is something more that we would recommend to the observation of the mathematicians, namely, that in the month of February, and a little before and after it (as I have observed for several years), about six o'clock in the evening, when the twilight has entirely left the horizon, a path of light tending from the twilight towards the Pleiades, and touching them as it were, presented itself very plainly to my view. This path is to be seen when the weather is clear, but best of all when the moon does not shine." The same conclusion appearance is taken notice of in Gregory's astronomy, from the foregoing and there expressly attributed to the sun's atmosphere.

With regard to the solar spots, Dr Long informs us, that "they do not change their places upon the sun, but adhere to his surface, or float in his atmosphere, very near his body; and if there be 20 spots Dr Long's or faculae upon him at a time, they all keep in the same opinion of the situation with respect to one another; and, as long as the solar spots last, are carried round together in the same manner: by the motion of the spots therefore we learn, what we should not otherwise have known, that the sun is a globe, and has a rotation about his axis." Notwithstanding this he tells us afterwards, "The spots, generally speaking, may be said to adhere to the sun, or to be so near him as to be carried round upon him uniformly; nevertheless, sometimes, though rarely, a spot has been seen to move with a velocity a little different from the rest; spots that were in different parallels have appeared to be carried along, not keeping always the same distance, but approaching nearer to each other; and when two spots moved in the same parallel, the hindmost has been observed to overtake and pass by the other. The revolution of spots near the equator of the sun is shorter than of those that are more distant from it."

The apparent change of shape in the spots, as they approach the circumference of the disk, according to our author, is likewise a proof of the sun's rotation round his axis, and that they either adhere to the surface of the luminary, or are carried round his atmosphere very near his surface.

"The rotation of the sun (says Dr Long) being known, we may consider his axis and poles, and their situation; as also his equator, or a circle imagined to be drawn upon that luminous globe equally distant from his poles; we may also imagine lesser circles drawn thereon, parallel to his equator.

"The rotation of the sun is according to the order of the signs; that is, any point on the surface of that vast globe turns round so as to look successively at Aries, Taurus, Gemini, &c. which is also the way that all the primary planets are carried round him, though each of them in a plane a little different from that of the rest. We must likewise observe, that the plane of the sun's equator produced, does not coincide with the heliocentric orbit of any of the planets, but cuts every one of them at a small angle: it is nearest to coincidence with the orbit of Venus.

"The sun being a globe at a great distance from us, we always see nearly one half of that globe at a time; but the visible half is continually changing, by the rotation of the sun, and the revolution of the earth in her orbit. To speak accurately, we do not see visible part quite half the sun's globe at a time; we want so much of the sun's of it as the sun's apparent diameter amounts to, which, at his mean distance, is about 32 minutes; so much is invisible. The diameter of the invisible part of the sun greater than that of the visible part; for this reason a spot may be about two hours longer invisible than visible.

"The time between the entrance of a spot upon the disk and its exit therefrom, gives us nearly half the apparent period of the sun's rotation, which is usually in about 13 days; a spot that, after passing the disk and disappearing, returns again, gives the whole time, Conclusions time, but not with precision; because the spot may perhaps not keep all the while exactly in the same place, but have some floating motion of its own upon the surface of the sun. Dom. Cassini, taking notice that several spots had often appeared in the same parallel, thought that some particular places of the sun might be more disposed than others to supply the matter of these spots; and so, that they would not move far from the place of their origin, just as the smoke of mount Etna, if it could be seen from the sun, would appear always to return to the same place of the disk of the earth once every 24 hours, very nearly; sometimes a little sooner, sometimes a little later, according as the smoke was driven by the wind from the place of its eruption. In consequence of this supposition, he compared several large intervals between the appearances of spots carried in the same parallel, which he judged to be returns of the same spots arising out of the same place on the surface of the sun, and found that 27 days 22 hours and 20 seconds was a common measure of those intervals very nearly; this, therefore, he thought the most proper period to be taken for an apparent revolution of the solar spots, and consequently of the sun himself as seen from the earth. These observations were made in April and May, nearly in the same time of the year, and therefore are not much affected by the inequality of the earth's motion. The same period is confirmed by Dom. Cassini.

The time of the apparent revolution of a spot being known, the true time of its going round upon the sun may be thus found: In fig. 3, the arc AC, which in the month of May, the earth goes through in her orbit in 27 days 12 hours and 20 minutes, is $26^\circ 22'$; the arc ac being equal to AC: the apparent revolution of a spot is the whole circle a b c d, or $360^\circ$, with the addition of the arc a c of $26^\circ 22'$, which makes $386^\circ 22'$: then say, as $386.22$ is to $27d.12h.20m.$, so is $360^\circ$ to $25d.15h.16m.$; the true time of the rotation of the sun, as it would be seen from a fixed star.

The angle of intersection of the sun's equator with the ecliptic is but small, according to Scheiner being never more than $8^\circ$, nor less than $6^\circ$; for which reason he settled it at $7^\circ$, though Cassini makes it $7^\circ$. This plane continued cuts the ecliptic in two opposite points, which are called the sun's nodes, being $8^\circ$ of N. and $8^\circ$ of S.; and two points in the ecliptic, $90^\circ$ from the nodes, may be called the limits. These are $8^\circ$ of X and $8^\circ$ of W. When the earth is in either of these nodes, the equator of the sun, if visible, would appear as a straight line; and, by reason of the vast distance of the sun from us, all his parallels would likewise appear as straight lines; but in every other situation of the earth, the equator and parallels of the sun would, if visible, appear as ellipses growing wider the farther the earth is from the nodes, and widest of all when the earth is in one of her limits.

"In the present age (says Dr Long), on the 18th of May, the earth is in the $8^\circ$ of S., one of the nodes of the sun, and consequently the sun's equator and parallels, if visible, would appear as straight lines, fig. 92. From that time the sun's equator, and every parallel, begin to appear as half of an ellipse convex, or swelling towards the south, and growing wider every day to the 20th of August, where it is at the widest, as in fig. 93, the earth being then in the $8^\circ$ of X, one of the limits. Immediately after, the apparent curvature of the sun's equator and parallels continually decreases from the 19th of November, when they again appear as straight lines, the earth being then in the other node."

From that time the equator of the sun and parallels become elliptical, convex towards the north; their curvature continually increasing to the 15th of February, when the earth is arrived at the other limit; and their curvature then decreases continually to the 18th of May, when they again appear as straight lines. Every spot is carried round the sun in his equator, or in a parallel; therefore the apparent motion of the spots upon the sun is rectilinear every year in May and November, at all other times elliptical." See fig. 16, 17, where the paths of some solar spots are delineated by Mr Dunn, in a manner seemingly inconsistent with what is just now delivered from Dr Long. From a farther consideration of the nature of the paths described by the solar spots, the Doctor concludes that their appearance may be retarded about four hours by the unequal motion of the earth in its orbit.

The nature and formation of the solar spots have been the subject of much speculation and conjecture. Some have thought that the sun is an opaque body, mountainous and uneven, as our earth is, covered all over spots with a fiery and luminous fluid; that this fluid is subject to ebbing and flowing, after the manner of our tides, so as sometimes to leave uncovered the tops of rocks or hills, which appear like black spots; and that the nebulosities about them are caused by a kind of froth. Others have imagined, that the fluid which sends us so much light and heat, contains a nucleus or solid globe, wherein are several volcanoes, that, like Etna or Vesuvius, from time to time cast up quantities of bituminous matter to the surface of the sun, and form those spots which are seen thereon; and that as this matter is gradually consumed by the luminous fluid, the spots disappear for a time, but are seen to rise again in the same places when those volcanoes cast up new matter. A third opinion is, that the sun consists of a fiery luminous fluid, wherein are immersed several opaque bodies of irregular shapes; and that these bodies, by the rapid motion of the sun, are sometimes buoyed or raised up to the surface, where they form the appearance of spots, which seem to change their shapes according as different sides of them are presented to the view. A fourth opinion is, that the sun consists of a fluid in continual agitation; that, by the rapid motion of this fluid, some parts more gross than the rest are carried up to the surface of the luminary, like the scum of melted metal rising up to the top in a furnace; that these scums, as they are differently agitated by the motion of the fluid, form themselves into those spots we see on the solar disk; and, besides the optical changes already mentioned, grow larger, are diminished in their apparent magnitude, recede a little from, or approach nearer to, each other, and are at last entirely dissipated by the continual rapid motion of the fluid, or are otherwise consumed or absorbed.

In the 64th volume of the Philosophical Transactions, Dr Wilson advances a new opinion concerning the solar spots, viz. that they are hollows in the surface of the luminary. "All the foregoing appearances (says he), when taken together, and when duly considered, seem to prove in the most convincing manner..." Conclusions that the nucleus of this spot (December 1769) was considerably beneath the level of the sun's spherical surface.

"The next thing which I took into consideration was, to think of some means whereby I could form an estimate of its depth. At the time of the observation I had, on December 12th, remarked that the breadth of the side of the umbra next the limb was about 14"; but, for determining the point in question, it was also requisite to know the inclination of the shelving side of the umbra to the sun's spherical surface. And here it occurred, that, in the case of a large spot, this would in some measure be deduced from observation. For, at the time when the side of the umbra is just hid, or begins first to come in view, it is evident, that a line joining the eye and its observed edge, or uppermost limit, coincides with the plane of its declivity. By measuring therefore the distance of the edge from the limb, when this change takes place, and by representing it by a projection, the inclination or declivity may in some measure be ascertained. For in fig. 27, let LLDK be a portion of the sun's limb, and ABCD a section of the spot, SL the sun's semidiameter, LG the observed distance from the limb, when the side of the umbra changes; then will the plane of the umbra CD coincide with the line EDG drawn perpendicular to SL at the point G. Let FH be a tangent to the limb at the point D, and join SD.

"Since GL, the verified sine of the angle LSD, is given by observation, that angle is given, which by the figure is equal to FDE or GDH; which angle is therefore given, and is the angle of inclination of the plane of the umbra to the sun's spherical surface. In the small triangle therefore CMD, which may be considered as rectangular, the angle MDC is given, and the side DC equal to AB is given nearly by observation; therefore the side MC is given, which may be regarded as the depth of the nucleus without any material error.

"I had not an opportunity, in the course of the foregoing observations, to measure the distance GL, not having seen the spot at the time when either of the sides of the umbra changed. It is, however, certain, that when the spot came upon the disk for the second time, this change happened some time in the night between the 11th and 12th of December, and I judge that the distance of the plane of the umbra, when in a line with the eye, must have been about 1' 55" from the sun's eastern limb; from which we may safely conclude, that the nucleus of the spot was, at that time, not less than a semidiameter of the earth below the level of the sun's spherical surface, and made the bottom of an amazing cavity, from the surface downwards, whose other dimensions were of much greater extent."

Having thus demonstrated that the solar spots are vast cavities in the sun, the Doctor next proceeds to offer some queries and conjectures concerning the nature of the sun himself, and to answer some objections to his hypothesis. He begins with asking, Whether it is not reasonable to think, that the vast body of the sun is made up of two kinds of matter very different in their qualities; that by far the greatest part is solid and dark; and that this dark globe is encompassed with a thin covering of that resplendent substance, from which the sun would seem to derive the whole of his vivifying heat and energy?—This, if granted, will afford a satisfactory solution of the appearance of spots; for, if any part of this resplendent substance shall by any means be displaced, the dark globe must necessarily appear; the bottom of the cavity corresponding to the nucleus, and the shelving sides to the umbra. The shining substance, he thinks, may be displaced by the action of some elastic vapour generated within the substance of the dark globe. This vapour, swelling into such a volume as to reach up to the surface of the luminous matter, would thereby throw it aside in all directions; and as we cannot expect any regularity in the production of such a vapour, the irregular appearance and disappearance of the spots is by that means accounted for; as the reflux of the luminous matter must always occasion the dark nucleus gradually to decrease, till at last it becomes indistinguishable from the rest of the surface.

Here an objection occurs, viz. That, on this supposition, the nucleus of a spot whilst on the decrease should always appear nearly circular, by the gradual descent of the luminous matter from all sides to cover it. But to this the Doctor replies, that in all probability the surface of the dark globe is very uneven and mountainous, which prevents the regular reflux of the shining matter. This, he thinks, is rendered very probable by the enormous mountains and cavities which are observed in the moon; and why, says he, may there not be the same on the surface of the sun? He thinks his hypothesis also confirmed by the dividing of the nucleus into several parts, which might arise from the luminous matter flowing in different channels in the bottom of the hollow.—The appearance of the umbra after the nucleus is gone, he thinks, may be owing to a cavity remaining in the luminous matter, tho' the dark globe is entirely covered.

As to a motion of the spots, distinct from what they are supposed to receive from the rotation of the sun round his axis, he says he never could observe any, except what might be attributed to the enlargement or diminution of them when in the neighbourhood of one another. "But (says he) what would farther contribute towards forming a judgment of this kind, is the apparent alteration of the relative place, which must arise from the motion across the disk on a spherical surface; a circumstance which I am uncertain if it has been sufficiently attended to."

The abovementioned hypothesis, the Doctor thinks, is further confirmed by the disappearance of the umbrae on the sides of spots contiguous to one another; as the action of the elastic vapour must necessarily drive the luminous matter away from each, and thus as it were accumulate it between them, so that no umbra can be perceived. As to the luminous matter itself, he conjectures, that it cannot be any very ponderous fluid, but that it rather resembles a dense fog which broods on the surface of the sun's dark body. His general conclusion we shall give in his own words.

"According to the view of things given in the foregoing queries, there would seem to be something very extraordinary in the dark and unignited state of the great internal globe of the sun. Does not this seem to indicate that the luminous matter that encompasses it derives not its splendor from any intensity of heat? For, if this were the case, would not the parts underneath, Conclusions beneath, which would be perpetually in contact with that glowing matter, be heated to such a degree as to become luminous and bright? At the same time it must be confessed, that although the internal globe was in reality much ignited, yet when any part of it forming the nucleus of a spot is exposed to our view, and is seen in competition with a substance of such amazing splendor, it is no wonder that an inferior degree of light should, in these cases, be unperceivable.

In order to obtain some knowledge of this point, I think an experiment might be tried, if we had an opportunity of a very large spot, by making a contrivance in the eye-piece of a telescope, whereby an observer could look at the nucleus alone with the naked eye, without being in danger of light coming from any other part of the sun. In this case, if the observer found no greater splendor than what might be expected from a planet very near the sun, and illumined by as much of his surface as corresponds to the spot's umbra, we might reasonably conclude, that the solar matter, at the depth of the nucleus, is in reality not ignited. But from the nature of the thing, doth there seem any necessity for thinking that there prevails such a raging and fervent heat as many have imagined? It is proper here to attend to the distinction between this shining matter of the sun and the rays of light which proceed from it. It may perhaps be thought, that the reaction of the rays upon the matter, at their emission, may be productive of a violent degree of heat. But whoever would urge this argument in favour of the sun being intensely heated, as arising from the nature of the thing, ought to consider that all polished bodies are less and less disposed to be heated by the action of the rays of light, in proportion as their surfaces are more polished, and as their powers of reflection are brought to a greater degree of perfection. And is there not a strong analogy between the reaction of light upon matter in cafes where it is reflected, and in cafes where it is emitted?

To this account of the solar spots, some objections have been made, particularly by Mr Wollaston, in the Philosophical Transactions, and M. de la Lande in the Memoirs of the Academy of Sciences; and to these Dr Wilson replied in the Philosophical Transactions for 1783, to the following purpose.

First of all (says he) it has been urged, as an objection of great weight, that the absence of the umbra on one side, where spots are near the limb, is not always constant; and of this I was sufficiently aware, having stated three cases from my own observation, when I did not perceive this change to take place. The reverend Francis Wollaston is the only person who, in the Philosophical Transactions, has bestowed any remarks on my publication; and though he acknowledges that the umbra generally changes in the manner I have determined, yet he expresses a difficulty as to my conclusion, on account of this circumstance not obtaining universally. Under similar expressions, M. de la Lande produces from his own observations, which appear to have been long continued, only three cases of the same kind, and four more from the ancient observations of M. Cassini and De la Hire. In regard to these last, I am not sure if such obsolete ones ought to be referred to in a question of the present kind. These excellent observers, entertaining no thought that any thing of moment depended upon a nice attention to the form of the spots, might easily overlook conclusions less obvious circumstances, especially when they were found near the limb. We may add farther, that even when they were so situated, they retain the umbra at both ends; and that whole side of it which lies farthest from the centre of the disk and these parts in the aggregate, they might sometimes mistake for the umbra as not deficient in any particular place. But, even admitting the anomaly we at present consider to be much more frequent than can be contended for, still such cases can only be brought as so many exceptions to the general law or uniformity of appearance, from which the condition of by far the greatest number of spots is most undeniably deduced. The utmost, therefore, that can be alleged is, that some few spots differ from all the rest, or from the multitude; and are not, like them, excavations in the sun. But notwithstanding these few instances where the umbra is not found to change, when we consider how perfectly all spots resemble one another in their most striking features, there naturally rises some presumption for all under that description we have given, partaking of one common nature; and for this only difference in the phenomena depending upon something, which does not necessarily imply a complete generical distinction. It comes therefore to be inquired, how far spots, which when near the middle of the disk appear equal and similar in all things, may yet differ from one another as excavations, or as possessing the third dimension of depth? and how far the peculiar circumstances by which they may disagree, can contribute to make some resist this change of the umbra when near the limb much more than others?

In order to this, suppose two spots which occupy a space upon the sun corresponding to the equal arches GD, fig. 94, and let GM, DM, be drawn so as to coincide with the plane of the excavation in such a case. The breadth of the nucleus being commonly equal to that of the surrounding umbra, if the base MD of the triangle GDM conceived rectilineal, be divided in L, so as ML : LD :: MD : DG; and if through L be drawn LS parallel to DG, then will DGS be the section of two spots having this condition; and which, as to size, would, when far away from the limb, be equal in all apparent measures; tho' very unequal in the third dimension HE, or depth of the nucleus SL, and also in the inclination DGM of their sides parallel to the spherical surface of the sun. Now it is manifest from the construction of the figure, that the distances AB, AK, from the limb A, when the sides GS of the umbra disappear, must depend very much on the latter of these two circumstances; and when, according as the angle of inclination DGM is smaller, the respective spot will go nearer to the limb than the other, before the side of the umbra GS vanishes. But these very exceptions to the general phenomena which we are at present examining, are of this kind; and may perhaps, from what has been now shown, proceed wholly from the shallowness and the very gradual shelving of some few spots which break out in certain tracts of the sun's body, over which the luminous matter lies very thinly mantled.

In order to avoid circumlocution, we may call that side of the umbra which lies nearest the limb the nearest umbra, and the side opposite the farthest umbra. Conclusions and to enter more particularly into the consideration now before us, let us suppose a spot of 45" over all, with its nucleus and umbra equally broad; then will the depth of the nucleus, and the apparent breadth of the nearest umbra, when the plane of the farthest comes to coincide with the visual ray, be expressed as in the following examples, where the apparent femidiameter of the sun is supposed to be 16", and his parallax 8.5".

| Farthest umbra | Depth of nucleus in English miles and seconds | Apparent breadth of nearest umbra | |----------------|---------------------------------------------|----------------------------------| | I. 1/2" | 4.54" | 8.58 | | II. 0 | 3.99 | 6.02 | | III. 0 | 2.09 | 4.13 | | IV. 0 | 1.44 | 2.87 |

"Now because in every aspect of a spot the real breadth of either the farthest or nearest umbra must be to the projected or apparent breadth as radius to the sine of the angle which this respective plane makes with the visual ray, it follows, that at any time before the spot comes so near the limb as is expressed in the above examples, the apparent breadth of the nearest and farthest umbra cannot differ so much as by the quantity there set down for the apparent breadth of the nearest when the other is supposed to vanish. Regarding, therefore, the farthest and nearest umbra of the spot in Case IV. as two neighbouring visible objects which turn narrower by degrees as the spot goes towards the limb, we should undoubtedly judge that they contract as to fence alike; since, so long as the farthest could be perceived, the other cannot appear to exceed it by a quantity that we could distinguish; and by the time the former coincides with the visual ray, the extreme narrowness to the limb would prevent our forming any certain judgment of either.

From this last example, therefore, it appears manifest, that a spot answering to the description and conditions therein mentioned, or one a little more shallow, would approach the limb, and finally go off the disk, without that peculiar change of the umbra on one side which is so obvious on common occasions, notwithstanding it were an excavation whose nucleus or bottom is so many miles below the level of the surface. In the four cases above stated, the distance of the remotest part of the nucleus from the sun's limb, when the visual ray coming from it is just interrupted by the lip of the excavation, or, in other words, the distance of the nucleus from the limb when it was totally hid, was also computed. These distances are as follows:

Case I. - 16.93" Case III. - 4.70" II. - 8.90" IV. - 2.70"

And it is remarkable, from the two last, how very near the limb a shallow spot of not more than 40" in diameter may come, before the nucleus wholly disappears."

After describing the method in which these computations were made, the Doctor proceeds thus: "Perhaps it may be urged, that very shallow spots ought always to be known from the rest, and discover themselves, by a surrounding umbra, very narrow, compared to the extent of the nucleus; but we know far too little of the qualities of the luminous matter, and of the proximate causes of the spots to say anything at all upon a point of this kind. The breadth of the Conclusions umbra is, as assumed in the computations, about equal from the that of the nucleus, though sometimes it varies more foregoing or less; but how far these relative dimensions indicate Appearances, or shallowness, must be expounded only by observation, and not by any vague and imperfect notions of the nature and constitution of the sun.

The mention of a pit, or hollow or excavation several thousands of miles deep, reaching to that extent down through a luminous matter to darker regions, is ready to strike the imagination in a manner unfavourable to a just conception of the nature of the solar spots as now described. Upon first thoughts it may look strange how the sides and bottom of such vast abysses can remain so very long in sight, whilst, by the sun's rotation, they are made to present themselves more and more obliquely to our view. But when it is considered how extremely inconsiderable their greatest depth is compared to the diameter of the sun, and how very wide and shelving they are, all difficulties of this sort will be entirely removed." Unless, however, we duly attend to these proportions, our notions upon the subject must be very erroneous; and it seems the more necessary to offer this caution, as this very thing is inaccurately represented in fig. 9. belonging to the Memoir under review, and in a way that may lead to mistakes. Instead of exhibiting a spot as depressed below the surface of the sun one hundredth part of his femidiameter, the section of it is there determined by two lines drawn from the circumference, and meeting in a point at the prodigious distance of one fifth of the femidiameter below. Any reader, therefore, who pleases, by turning to fig. 95, may see how very small a portion of the sun's body is made up of the luminous matter when supposed every where 3967 English miles deep. A is a section of a spot 50" diameter, situated in the deepest part of this refulgent substance.

What has now been insisted on at so much length concerning the shallowness and more gradual shelving of some few spots, will also apply to another objection which M. de la Lande views in a strong light.

Here we find quoted the great spot in 1719, seen by M. Cassini; and, for the second time, that of June 1703, seen by M. de la Hire; both which, on their arrival at the limb, are said to have made an indentation or dark notch in the disk; and this phenomenon is mentioned as absolutely incompatible with spots being below the surface.

It is most true, that if we look for any thing like this when the plane which coincides with the external boundary of the spots passes through the eye, the way that M. de la Lande considers the matter, it must be very large indeed before the disk could be perceived deficient by any dark segment. But may not a spot, even no larger than M. Cassini's, considered as an excavation, make, in a manner very different from this, something like a notch; for, by the way, this phenomenon is not in the Mem. Acad. nor anywhere else that I know of, described with any sort of precision.—M. Cassini's great spot, by which we understood the nucleus, was of 30"; and supposing the umbra equally broad, its diameter over all must have been 1' 30". It would therefore occupy an extent upon the sun's surface of 5° 22' fully. Now, suppose a circular space of that size upon the sun distinguished from the surrounding... Conclusions ing lustre by such a failure of light as is peculiar to some spots, and suppose that it just touches the limb, it would still subtend an angle of more than 4°. This being the case, might not a dulky shade, more or less remarkable according to the darkness of the umbra, commencing at the limb, and reaching inwards upon the disk, or, in other words, a notch, be perceived? Had M. Cassini's spot been a very shallow excavation, it appears by Case IV. that when viewed in this aspect, some small part of the nucleus might have yet been visible; and might have contributed, along with the shade of the farthest umbra, and the still broader and deeper shade of the two ends of the umbra, to mark out the indentation.

"Should it be said, that these notches are always distinct and jet-black impressions on the disk, of an obvious breadth, and originating entirely from the opaque nucleus conceived as something prominent above the general surface, this can be shown inconsistent with some circumstances we find accidentally mentioned in the case of M. de la Hire's spot; for of this great one it was said, that when only 8° distant from the limb, the nucleus was seen as a very narrow line. This was on June 3d 1703, at six o'clock in the morning. Now, forasmuch as at that time its alleged elevation must have been to its apparent subtenue very nearly as radius to cosine of that arch of the sun's circumference whose versed sine was the 8° of distance from the limb, it is impossible that its breadth could have increased sensibly in its further progress towards the limb; and how any obvious black notch could be produced by the elevation contended for in this case is not conceivable.

"I do not imagine, therefore, that the phenomena of notches in the disk, so inconsiderable and dubious as these seem to be, are by any means a proof of projecting nuclei, or that they are not reconcilable to spots being depressions on the sun. A large shallow excavation, with the sloping sides or umbra darker than the common, may, as has been shown, be more or less perceptible at the limb; and what perhaps is a further confirmation of this, and seems to evince that such a concurrence of circumstances is necessary, is, that sometimes even large spots make no indentation. M. Cassini, in Mem. Acad. Tom. X. p. 581, describes the great spot of 1676, which he saw at its entrance with a telescope of 35 feet, as an obscure line parallel to the limb; but nowhere mentions that it made a notch in it.

"Though we now and then see the surrounding umbra darker than at other times; yet when spots are deep, and the umbrae but little dusky, it is indeed impossible that we should see anything of them, even though large, very near the limb; for here even the nucleus, which lies buried, cannot in the least contribute to the effect, as it may do a little before its state of evanescence, when spots are very shallow. Accordingly, cases of this kind are perfectly agreeable to experience.

"In reasoning concerning the nature of the spots, and particularly about their third dimension, the only arguments which are admissible, and which carry with them a perfect conviction, are those grounded upon the principles of optical projection. If, for example, the far greater number of them be excavations some thousands of miles deep, certain changes of the umbra would be observable when near the limb, as has been shown at so much length. Were they very shallow, or quite superficial, both sides of the umbra would as to senescence contract alike in their progress toward the limb; for if, in case 4th above stated, the spot had been supposed superficial, the apparent breadth of the side of the umbra next the centre of the disk would have made them only 1.62°, and that of the side opposite 1.27°. Now, the whole of either of these quantities, and much more their difference, would be quite insensible. Again, if the nucleus extended much above the common level, whilst the surrounding umbra was superficial, we should behold the manifest indications of this by such an opaque body when seen very obliquely being projected across the farthest side of the umbra, and by hiding the whole or part of it before the time it would otherwise disappear. According to this or that condition of the spot, such things must infallibly obtain by the known laws of vision; and hence arguments resting upon such principles may be denominated optical ones. On the other hand, when spots are contemplated near the middle of the disk, a great variety of changes are observed in them, which depend not upon position, but upon certain physical causes producing real alterations in their form and dimensions. It is plain, that arguments derived from the consideration of such changes, and which, on that account, may be called physical arguments, can assist us but little in investigating their third dimensions; and, from the nature of the thing, must be liable to great uncertainty. The author of the Memoire, in p. 511, &c. takes new ground, and proceeds with a number of objections depending upon that sort of reasoning which we have last defined. I must take notice, that a certain distinction has been here overlooked, which in my paper I have endeavoured to point out. Presuming upon our great ignorance of many things which doubtless affect deeply the constitution of that wonderful body the sun, I offered in Part II. an account of the production, changes, and decay, of the spots, considered as excavations, in the most loose and problematical manner; stating every thing on this head in the form of queries.—Hence I would remark, that whatever inconsistencies are imagined in the account I have delivered Part II., though such may be justly chargeable on certain principles there assumed, yet they ought not to be stated as presumptions against the spots being really excavations or depressions in the luminous matter of the sun. This opinion must rest entirely upon the evidence held forth in the first part of the paper, whatever be the fate of the account laid down in the second. It does not enter there as an hypothesis, but as a matter of fact previously established by optical arguments; and from optical arguments alone can there arise even any just presumptions against it.

"It remains now only to make a few strictures up-Remarks on M. de la Lande's theory of the solar spots, humbly submitting them to the consideration of the reader. The theory of M. de la Lande's import of it is, 'that the spots as phenomena arise from the solar dark bodies like rocks, which by an alternate flux and reflux of the liquid igneous matter of the sun, sometimes raise their heads above the general surface. That part of the opaque rock, which at any time thus stands above, gives the appearance of the nucleus, whilst those parts,' Conclusions from the foregoing Appearances.

Parts, which in each ly only a little under the igneous matter, appear to us as the surrounding umbra.

"In the first place it may be remarked, that the whole proceeds upon mere supposition. This indeed the author himself very readily acknowledges. Though therefore it could not be disputed by arguments derived from observation, yet conjecture of any kind, if equally plausible, might fitly be employed to set aside its credit. Without entering into any tedious dissection, however, we shall confine ourselves to such particulars as appertain to the more obvious characters of the spots, and which also seem to be irreconcilable with the theory; and first of all with regard to the distinguishing features of the umbra.

"M. Cassini, Mem. Acad. tom. x. p. 582. Pl. VII. and M. de la Hire, Mem. Acad. 1703, p. 16, and I may add all other observers, and all good representations of the spots, bear testimony to the exterior boundary of the umbra being always well defined, and to the umbra itself being less and less shady the nearer it comes to the nucleus. Now it may be asked, how this could possibly be, according to M. de la Lande's theory? If the umbra be occasioned by our seeing parts of the opaque rock which lie a little under the surface of the igneous matter, should it not always be darkest next the nucleus? and, from the nucleus outward, should it not wax more and more bright, and at last lose itself in the general lustre of the sun's surface, and not terminate all at once in the darkest shade, as in fact it does? These few incongruities, which meet as it were in the very threshold of the theory, are so very palpable, that of themselves they raise unsurmountable doubts. For, generally speaking, the umbra immediately contiguous to the nucleus, instead of being very dark, as it ought to be, from our seeing the immersed parts of the opaque rock through a thin stratum of the igneous matter, is on the contrary very nearly of the same splendor as the external surface.

"Concerning the nucleus, or that part of the opaque rock which stands above the surface of the sun, M. de la Lande produces no optical arguments in support of this third dimension or height. Neither does he say anything particular as to the degree of elevation above the surface. But from what has been already hinted in the course of this paper, it appears, that if this were any thing sensible, it ought to be discovered by phenomena very opposite to those which we have found to be so general.

"Again, a flux and reflux of the igneous matter, so considerable as sometimes to produce a great number of spots all over the middle zone, might affect the apparent diameter of the sun, making that which passes through his equator less than the polar one, by the retreat of the igneous matter towards those regions where no spots ever appear. But as a difference of this kind, of nearly one thousandth part of the whole, would be perceivable, as we learn from M. de la Lande's own observations, compared with those of Mr Short in Histoire Acad. 1760, p. 123, it would seem, that the theory had this difficulty also to combat. Further, when among spots very near one another some are observed to be increasing whilst others are diminishing, how is it possible this can be the effect of such a supposed flux and reflux? This last inconsistency is mentioned by the author himself, who endeavours to avoid it by making a new demand upon the general fund of hypothesis, deriving from thence such qualities of the igneous matter as the case seems to require; and such must be the method of proceeding in all systems merely theoretical. But it is unnecessary to pursue at more length illusive speculations of this kind, especially as we lie under a conviction founded on fact, of the theory being utterly erroneous. It hardly differs in any respect from that proposed by M. de la Hire, and a little amended by the writer of the Histoire de l'Academie for 1707, p. 111. Views very much of the same kind were even entertained by some long ago as the days of Scheiner, as we find mentioned by that indefatigable author in his Rosa Ursina, p. 746."

Concerning the moon, it is allowed on all hands, Great inequality that there are prodigious inequalities on her surface, qualities on this is proved by looking at her through a telescope, the surface at any other time than when she is full; for then there is no regular line bounding light and darkness; but the confines of these parts appear as it were toothed and cut with innumerable notches and breaks; and even in the dark part, near the borders of the lucid surface, there are seen some small spaces enlightened by the sun's beams. Upon the fourth day after new moon, there may be perceived some shining points like rocks or small islands within the dark body of the moon; but not far from the confines of light and darkness there are observed other little spaces which join to the enlightened surface, but run out into the dark side, which by degrees change their figure, till at last they come wholly within the illuminated face, and have no dark parts round them at all. Afterwards many more shining spaces are observed to arise by degrees, and to appear within the dark side of the moon, which before they drew near to the confines of light and darkness were invisible, being without any light, and totally immersed in the shadow. The contrary is observed in the decreasing phases, where the lucid spaces which joined the illuminated surface by degrees recede from it, and, after they are quite separated from the confines of light and darkness, remain for some time visible, till at last they also disappear. Now it is impossible that this should be the case, unless these shining points were higher than the rest of the surface, so that the light of the sun may reach them.

Not content with perceiving the bare existence of these lunar mountains, astronomers have endeavoured measuring to measure their height in the following manner. Let the lunar EGD be the hemisphere of the moon illuminated by the sun, ECD the diameter of the circle bounding light and darkness, and A the top of a hill within the dark part when it first begins to be illuminated. Observe with a telescope the proportion of the right line AE, or the distance of the point A from the lucid surface to the diameter of the moon ED; and because in this case the ray of light ES touches the globe of the moon, AEC will be a right angle by 16th prop. of Euclid's third book; and therefore in the triangle AEC having the two sides AE and EC, we can find out the third side AC; from which subduing BC or EC, there will remain AB the height of the mountain. Riccioli affirms, that upon the fourth day after new moon he has observed the top of the hill called St Catherine's to be illuminated, and that it was distant from the confines of the lucid surface about a sixteenth part of Conclusions of the moon's diameter. Therefore, if \( CE = 8 \), \( AE \) will be 1, and \( AC^2 = CE^2 + AE^2 \) by prop. 47, of Euclid's first book. Now, the square of \( CE \) being 64, and the square of \( AE \) being 1, the square of \( AC \) will be 65, whose square root is 8.062, which expresses the length of \( AC \). From which deducing \( BC = 8 \), there will remain \( AB = 0.062 \). So that \( CB \) or \( CE \) is therefore to \( AB \) as 8 is to 0.062, that is, as 8000 is to 62.

If the diameter of the moon therefore was known, the height of this mountain would also be known. This demonstration is taken from Dr Keil, who supposes the semidiameter of the moon to be 1182 miles; according to which, the mountain must be somewhat more than nine miles of perpendicular height; but astronomers having now determined the moon's semidiameter to be only 1090 miles, the height of the mountain will be nearly 8½ miles.

In the former edition of this work, we could not help making some remarks on the improbability that the mountains of the moon, a planet so much inferior in size to the earth, should exceed in such vast proportion the highest of our mountains, which are computed at little more than one-third of the height just mentioned. Our remark is now confirmed by the observations of Mr Herschel. After explaining the method used by Galileo, Hevelius, &c., for measuring the lunar mountains, he tells us, that the former takes the distance of the top of a lunar mountain from the line that divides the illuminated part of the disk from that which is in the shade to be equal to one-twentieth of the moon's diameter; but Hevelius makes it only one twenty-sixth. When we calculate the height of such a mountain, therefore, it will be found, according to Galileo, almost 5½ miles; and according to Hevelius 3½ miles, admitting the moon's diameter to be 2180 miles. Mr Ferguson, however, says (Astronomy explained, § 252.), that some of her mountains, by comparing their height with her diameter, are found to be three times higher than the highest hills on earth; and Keil, in his Astronomical Lectures, has calculated the height of St Catharine's hill, according to the observations of Ricciolus, and finds it nine miles. Having premised these accounts, Mr Herschel explains his method of taking the height of a lunar mountain from observations made when the moon was not in her quadrature, as the method laid down by Hevelius answers only to that particular case: for in all others the projection must appear shorter than it really is. "Let \( SLM \), says he, or \( slm \), (fig. 96.) be a line drawn from the sun to the mountain, touching the moon at \( L \) or \( l \), and the mountain at \( M \) or \( m \). Then, to an observer at \( E \) or \( e \), the lines \( LM \), \( lm \), will not appear of the same length, though the mountain should be of an equal height; for \( LM \) will be projected into \( on \), and \( lm \) into \( ON \). But these are the quantities that are taken by the micrometer when we observe a mountain to project from the line of illumination. From the observed quantity \( on \), when the moon is not in her quadrature, to find \( LM \), we have the following analogy. The triangles \( oOL \), \( rML \), are similar; therefore \( LO : LO :: LR : LM \), or \( \frac{LO \times on}{LO} = LM \); but \( LO \) is the radius of the moon, and \( LR \) or \( on \) is the observed distance of the mountain's projection; and \( LO \) is the sine of the angle \( ROL = oLS \); which we may take to be the distance of the sun from the moon without any material error, and which therefore from the foregoing we may find at any given time from an ephemeris.

"The telescope used in these observations was a Newtonian reflector of six feet eight inches focal length, to which a micrometer was adapted, consisting of two parallel hairs, one of which was moveable by means of a fine screw. The value of the parts shown by the index was determined by a trigonometrical observation of a known object at a known distance, and was verified by several trials. The power was always 222, excepting where another is expressly mentioned; and this was also determined by experiment, which frequently differs from theory on account of some small errors in the data, hardly to be avoided. The moon having sufficient light, an aperture of no more than four inches was made use of; and, says Mr Herschel, "I believe, that, for distinctness of vision, this instrument is perhaps equal to any that ever was made."

With this instrument he observed a prominence, which he calls a rock, situated near the Lacus Niger of Hevelius, and found that it projected 41.56". To reduce this into miles, put \( R \) for the semidiameter of the moon in seconds, as given by the nautical almanack at the time of observation, and \( Q \) for the observed quantity, also in seconds and centesimal; then it will be in general, \( R : 1090 :: Q : \frac{1090Q}{R} = on \) in miles.

Thus it is found, that 41.56" is 46.79 miles. The distance of the sun from the moon at that time was, by the nautical almanack, about 93° 57'"; the sine of which to the radius 1 is .9985, &c. and \( \frac{on}{L} \) in this case is \( LM = 46.85 \) miles. Then, by Hevelius's method, the perpendicular height of the rock is found to be about one mile. At the same time, a great many rocks, situated about the middle of the disk, projected from 25.92" to 26.56"; which gives \( on \) about 29.3 miles: so that these rocks are all less than half a mile high.

These observations were made on the 13th of November 1779. On the 13th of January 1780, examining the mountains of the moon, he found that there was not one of them fairly placed on level ground, which is very necessary for an exact measurement of the projection: for if there should be a declivity on the moon before the mountains, or a tract of hills placed so as to cast a shadow upon that part before them which would otherwise be illuminated, the projection would appear too large; and, on the contrary, should there be a rising ground before them, it would appear too little.

Proceeding in this cautious manner, Mr Herschel measured the height of many of the lunar prominences, and draws at last the following conclusions.—"From these observations I believe it is evident, that the height of the lunar mountains in general is greatly over-rated; and that, when we have excepted a few, the generality do not exceed half a mile in their perpendicular elevation. It is not so easy to find any certain mountain exactly in the same situation it has been measured in before; therefore some little difference must be expected in these measures. Hitherto I have not had an opportunity of particularly observing the three mountains mentioned by Hevelius; nor that which Conclusions which Ricciolus found to project a sixteenth part of from the moon's diameter. If Keill had calculated the height of this last mentioned hill according to the theorem I have given, he would have found (supposing the observation to have been made, as he says, on the fourth day after new moon) that its perpendicular height could not well be less than between 11 and 12 miles. I shall not fail to take the first opportunity of observing these four, and every other mountain of any eminence; and if other persons, who are furnished with good telescopes and micrometers, would take the quantity of the projection of the lunar mountains, I make no doubt but that we should be nearly as well acquainted with their heights as we are with the elevation of our own. One caution I would beg leave to mention to those who may use the excellent 3 feet refractors of Mr Dollond. The admirable quantity of light, which on most occasions is so desirable, will probably give the measure of the projection somewhat larger than the true, if not guarded against by proper limitations placed before the object-glass. I have taken no notice of any allowance to be made for the refraction: a ray of light must suffer in passing through the atmosphere of the moon, when it illuminates the top of the mountain, whereby its apparent height will be lessened, as we are too little acquainted with that atmosphere to take it into consideration. It is also to be observed, that this would equally affect the conclusions of Hevelius, and therefore the difference in our inferences would still remain the same.

In the continuation of his observations, Mr Herschel informs us that he had measured the height of one of the mountains which had been measured by Hevelius. "Antitaurus (says he), the mountain measured by Hevelius, was badly situated; because Mount Monsbus and its neighbouring hills cast a deep shadow, which may be mistaken for the natural convexity of the moon. A good, full, but just measure, 25.105"; in miles, 29.27; therefore LM 31.7 miles, and the perpendicular height not quite half a mile. As great exactness was desired in this observation, it was repeated with very nearly the same result. Several other mountains were measured by the same method; and all his observations concurred in making the height of the lunar mountains much less than what former astronomers had done. Mount Lipulus was found to be near two-thirds of a mile; one of the Apennine mountains between Lacus Trasimenus and Pontus Euxinus measured a mile and a quarter; Mons Armenia, near Taurus, two thirds of a mile; Mons Leucopatera three quarters of a mile. Mons Sacer projected 45.625"; but (says he) I am almost certain that there are two very considerable cavities or places where the ground descends below the level of the convexity, just before these mountains; so that these measures must of course be a good deal too large: but supposing them to be just, it follows, that \( \alpha \) is 50.193 miles, LM = 64 miles, and the perpendicular height above \( \frac{1}{2} \) miles.

As the moon has on its surface mountains and valleys in common with the earth, some modern astronomers have discovered a still greater similarity, viz. that some of these are really volcanoes, emitting fire as those on earth do. An appearance of this kind was discovered some years ago by Don Ulloa in an eclipse of the sun. It was a small bright spot like a star near the margin of the moon, and which he at that time supposed to have been a hole with the sun's light shining through it. Succeeding observations, however, have induced astronomers to attribute appearances of this kind to the eruption of volcanic fire; and Mr Herschel has particularly observed several eruptions of the lunar volcanoes, the last of which he gives an account of in the Phil. Trans. for 1787, "April 19. 10 h. 30' sidereal time. I perceive (says he) three volcanoes in different places of the dark part of the new moon. Two of them are either already nearly extinct, or otherwise in a state of going to break out; which perhaps may be decided next lunation. The third shows an actual eruption of fire or luminous matter. I measured the distance of the crater from the northern limb of the moon, and found it 3° 57' 3"; its light is much brighter than the nucleus of the comet which M. Mechain discovered at Paris the 10th of this month.

"April 20. 10 h. 2' sidereal time. The volcano burns with greater violence than last night. Its diameter cannot be less than 3", by comparing it with that of the Georgian planet: as Jupiter was near at hand, I turned the telescope to his third satellite, and estimated the diameter of the burning part of the volcano to be equal to at least twice that of the satellite; whence we may compute that the shining or burning matter must be above three miles in diameter. It is of an irregular round figure, and very sharply defined on the edges. The other two volcanoes are much farther towards the centre of the moon, and resemble large, pretty faint nebulæ, that are gradually much brighter in the middle; but no well-defined luminous spot can be discerned in them. These three spots are plainly to be distinguished from the rest of the marks upon the moon; for the reflection of the sun's rays from the earth is, in its present situation, sufficiently bright, with a ten-feet reflector, to show the moon's spots, even the darkest of them; nor did I perceive any similar phenomena last lunation, though I then viewed the same places with the same instrument.

"The appearance of what I have called the actual fire, or eruption of a volcano, exactly resembled a small piece of burning charcoal when it is covered by a very thin coat of white ashes, which frequently adhere to it when it has been some time ignited; and it had a degree of brightness about as strong as that with which such a coal would be seen to glow in faint daylight. All the adjacent parts of the volcanic mountain seemed to be faintly illuminated by the eruption, and were gradually more obscure as they lay at a greater distance from the crater. This eruption resembled much that which I saw on the 4th of May in the year 1783, but differed considerably in magnitude and brightness; for the volcano of the year 1783, though much brighter than that which is now burning, was not nearly so large in the dimensions of its eruption: the former seen in the telescope resembled a star of the fourth magnitude as it appears to the naked eye; this, on the contrary, shows a visible disk of luminous matter very different from the sparkling brightness of star-light."

Concerning the nature of the moon's substance there have been many conjectures formed. Some have imagined, Conclusions gained, that, besides the light reflected from the sun, the moon hath also some obscure light of her own, by which she would be visible without being illuminated by the sun-beams. In proof of this it is urged, that during the time of even total eclipses the moon is still visible, appearing of a dull red colour, as if obscured by a great deal of smoke. In reply to this it hath been advanced, that this is not always the case; the moon sometimes disappearing totally in the time of an eclipse, so as not to be discernible by the best glasses, while little stars of the fifth and sixth magnitudes were distinctly seen as usual. This phenomenon was observed by Kepler twice, in the years 1580 and 1583; and by Hevelius in 1620. Ricciolus and other Jesuits at Bologna, and many people throughout Holland, observed the same on April 14, 1642; yet at Venice and Vienna she was all the time conspicuous. In the year 1703, Dec. 23, there was another total obscuration. At Arles, she appeared of a yellowish brown; at Avignon, ruddy and transparent, as if the sun had shone through her; at Marseilles, one part was reddish and the other very dusky; and at length, though in a clear sky, she totally disappeared. The general reason for her appearance at all during the time of eclipses shall be given afterwards: but as for these particular phenomena, they have not yet, as far as we know, been satisfactorily accounted for.

Different conjectures have also been formed concerning the spots on the moon's surface. Some philosophers have been so taken with the beauty of the brightest places observed in her disk, that they have imagined them to be rocks of diamonds; and others have compared them to pearls and precious stones. Dr. Keill and the greatest part of astronomers now are of opinion, that these are only the tops of mountains, which by reason of their elevation are more capable of reflecting the sun's light than others which are lower. The darkish spots, he says, cannot be seas, nor any thing of a liquid substance; because, when examined by the telescope, they appear to consist of an infinity of caverns and empty pits, whose shadows fall within them, which can never be the case with seas, or any liquid substance: but, even within these spots, brighter places are also to be observed; which, according to his hypothesis, ought to be the points of rocks standing up within the cavities. Dr. Long, however, is of opinion that several of the dark spots on the moon are really water. May not the lunar seas and lakes (says he) have islands in them, wherein there may be pits and caverns? And if some of these dark parts be brighter than others, may not that be owing to the seas and lakes being of different depths, and to their having rocks in some places and flats in others?

It has also been urged, that if all the dark spots observed on the moon's surface were really the shadows of mountains, or of the sides of deep pits, they could not possibly be so permanent as they are found to be; but would vary according to the position of the moon with regard to the sun, as we find shadows on earth are varied according as the earth is turned towards or from the sun. Accordingly it is pretended, that variable spots are actually discovered on the moon's disk, and that the direction of these is always opposite to the sun. Hence they are found among those parts which are soonest illuminated in the increasing moon, and in the decreasing moon lose their light sooner than the intermediate ones; running round, and appearing sometimes longer, and sometimes shorter. The permanent appearance of dark spots, therefore, it is said, must be some matter which is not fitted for reflecting the rays of the sun so much as the bright parts do; and this property, we know by experience, belongs to water rather than land; whence these philosophers conclude, that the moon, as well as our earth, is made up of land and seas.

It has been a matter of dispute whether the moon has any atmosphere or not. The following arguments have been urged by those who take the negative side:

1. The moon constantly appears with the same brightness when there are no clouds in our atmosphere; which could not be the case if she were surrounded with an atmosphere like ours, so variable in its density, and so frequently obscured by clouds and vapours.

2. In an appearance of the moon to a star, when the comes so near that part of her atmosphere is interposed between our eye and the star, refraction would cause the latter seem to change its place, so that the moon would appear to touch it later than by her own motion she would do.

3. Some philosophers are of opinion, that because there are no seas or lakes in the moon, there is therefore no atmosphere, as there is no water to be raised up in vapours.

All these arguments, however, have been answered by other astronomers in the following manner. 1. It is denied that the moon appears always with the same brightness, even when our atmosphere appears equally clear. Hevelius relates, that he has several times found in skies perfectly clear, when even stars of the fifth and seventh magnitude were visible, that at the same altitude of the moon, and the same elongation from the earth, and with one and the same telescope, the moon and its maculae do not appear equally lucid, clear, and conspicuous at all times; but are much brighter and more distinct at some times than at others. From the circumstances of this observation, say they, it is evident that the reason of this phenomenon is neither in our air, in the tube, in the moon, nor in the spectator's eye; but must be looked for in something existing about the moon. An additional argument is drawn from the different appearances of the moon already mentioned in total eclipses, which are supposed to be owing to the different conditions of the lunar atmosphere.

To the second argument Dr. Long replies, that Sir Isaac Newton has shown (Princip. prop. 37, cor. 5.), that the weight of any body upon the moon is but a third part of what the weight of the same would be upon the earth: now the expansion of the air is reciprocally as the weight that compresses it: the air, light is not therefore, surrounding the moon, being pressed together by a weight, or being attracted towards the centre the moon's of the moon by a force equal only to one-third of that which attracts our air towards the centre of the earth, it thence follows, that the lunar atmosphere is only one-third as dense as that of the earth, which is too little to produce any sensible refraction of the starlight. Other astronomers have contended that such refraction was sometimes very apparent. Mr. Cassini says that he frequently observed Saturn, Jupiter, and the fixed stars, to have their circular figure changed into an elliptical one, when they approached either to the moon's dark or illuminated limb, though they... Conclusions own, that, in other occultations, no such change could be observed. With regard to the fixed stars, indeed, it has been urged, that, granting the moon to have an atmosphere of the same nature and quantity as ours, no such effect as a gradual diminution of light ought to take place; at least, that we could by no means be capable of perceiving it. Our atmosphere is found to be so rare at the height of 44 miles as to be incapable of refracting the rays of light. This height is the 180th part of the earth's diameter; but since clouds are never observed higher than four miles, we must conclude that the vaporous or obscure part is only one 180th. The mean apparent diameter of the moon is $3' 29''$, or 188 seconds; therefore the obscure parts of her atmosphere, when viewed from the earth, must subtend an angle of less than one second; which space is passed over by the moon in less than two seconds of time. It can therefore hardly be expected that observation should generally determine whether the supposed obscuration takes place or not.

The third argument is necessarily inconclusive, because we know not whether there is any water in the moon or not; nor, though this could be demonstrated, would it follow that the lunar atmosphere answers no other purpose than the raising of water into vapour. There is, however, a strong argument in favour of the existence of a lunar atmosphere, taken from the appearance of a luminous ring round the moon in the time of solar eclipses. In the eclipse of May 1, 1766, Captain Stanyan, from Bern in Switzerland, writes, that "the sun was totally darkened there for the space of four minutes and a half; that a fixed star and planet appeared very bright; that his getting out of the eclipse was preceded by a blood-red streak of light from his left limb, which continued not longer than six or seven seconds of time; then part of the sun's disk appeared, all on a sudden, brighter than Venus was ever seen in the night; and in that very instant gave light and shadow to things as strong as moon-light uses to do." The publisher of this account observes, that the red streak of light preceding the emergence of the sun's body, is a proof that the moon has an atmosphere; and its short continuance of five or six seconds shows that its height is not more than the five or six hundredth part of her diameter.

Fatio, who observed the same eclipse at Geneva, tells us, that "there was seen during the whole time of the total immersion, a whiteness which seemed to break out from behind the moon, and to encompass her on all sides equally; this whiteness was not well defined on its outward side, and the breadth of it was not a twelfth part of the diameter of the moon. The planet appeared very black, and her disk very well defined within the whiteness which encompassed it about, and was of the same colour as that of a white crown or halo of about four or five degrees in diameter, which accompanied it, and had the moon for its centre. A little after the sun had begun to appear again, the whiteness, and the crown which had encompassed the moon, did entirely vanish." "I must add (says Dr Long), that this description is a little perplexed, either through the fault of the author or of the translator; for I suppose Fatio wrote in French; however, it plainly appears by it that the moon's atmosphere was visible, surrounded by a light of larger extent, which I think must be that luminous appearance (the zodiacal Conclusions light) mentioned from Caffini." Flamstead, who published this account, takes notice, that, according to these observations, the altitude of the moon's atmosphere cannot be well supposed less than 180 geographical miles; and that probably this atmosphere was never discovered before this eclipse, by reason of the smallness of the refraction, and the want of proper observations.

An account of the same eclipse, as it appeared at Zurich, is given by Dr Scheuchzer, in the following words: "We had an eclipse of the sun, which was both total and annular: total, because the whole sun was covered by the moon; annular, not what is properly so called, but by refraction; for there appeared round the moon a bright shining, which was owing to the rays of the sun refracted through the atmosphere of the moon.

Dom. Caffini, from a number of accounts sent him from different parts, says, that in all those places where it was total, during the time of total darkness, there was seen round the moon a crown or broad circle of pale light, the breadth whereof was about a 12th part of the moon's diameter: that at Montpelier, where the observers were particularly attentive to see if they could distinguish the zodiacal light already mentioned, they took notice of a paler light of a larger extent, which surrounded the crown of light before mentioned, and spread itself on each side of it, to the distance of four degrees. He then mentions Kepler's opinion, that the crown of light which appears round the moon during the total darkness in an eclipse of the sun, is caused by some celestial matter surrounding the moon, of sufficient density to receive the rays of the sun and send them to us; and that the moon may have an atmosphere similar to that of our earth, which may reflect the sun's light.

A total eclipse of the sun was observed on the 22nd of April O.S. in the year 1715, by Dr Halley at London, and by M. Louville of the Academy of Sciences at Paris. Dr Halley relates, that "when the first part of the sun remained on his east side, it grew very faint, and was easily supportable to the naked eye even through the telescope, for above a minute of time before the total darkness; whereas, on the contrary, the eye could not endure the splendour of the emerging beams through the telescope even from the first moment. To this, two causes perhaps concurred: the one, that the pupil of the eye did necessarily dilate itself during the darkness, which before had been much contracted by looking on the sun: the other, that the eastern parts of the moon, having been heated with a day near as long as 30 of ours, must of necessity have that part of its atmosphere replete with vapours raised by the so long continued action of the sun; and, by consequence, it was more dense near the moon's surface, and more capable of obstructing the sun's beams; whereas at the same time the western edge of the moon had suffered as long a night, during which there might fall in dew all the vapours that were raised in the preceding long day; and for that reason, that that part of its atmosphere might be seen much more pure and transparent."

"About two minutes before the total immersion, the remaining part of the sun was reduced to a very fine horn, whose extremities seemed to lose their acuteness," Conclusionsness, and to become round like stars; and for the space of about a quarter of a minute a small piece of the southern horn of the eclipse seemed to be cut off from the rest by a good interval, and appeared like an oblong star rounded at both ends: which appearance would proceed from no other cause but the inequalities of the moon's surface; there being some elevated parts thereof near the moon's southern pole, by whose interpolation part of that exceedingly fine filament of light was intercepted. A few seconds before the sun was totally hid, there discovered itself round the moon a luminous ring, about a digit, or perhaps a tenth part of the moon's diameter in breadth. It was of a pale whiteness, or rather of a pearl colour, seeming to me a little tinged with the colour of the iris, and to be concentric with the moon; whence I concluded it the moon's atmosphere. But the great height of it, far exceeding that of our earth's atmosphere, and the observations of some who found the breadth of the ring to increase on the west side of the moon as the emersion approached, together with the contrary sentiments of those whose judgments I shall always revere, make me less confident, especially in a matter to which I gave not all the attention requisite.

Whatever it was, this ring appeared much brighter and whiter near the body of the moon than at a distance from it; and its outward circumference, which was ill defined, seemed terminated only by the extreme rarity of the matter of which it was composed, and in all respects resembled the appearance of an enlightened atmosphere seen from far: but whether it belonged to the sun or moon, I shall not pretend to determine.

During the whole time of the total eclipse, I kept my telescope constantly fixed on the moon, in order to observe what might occur in this uncommon appearance; and I saw perpetual flashes or coruscations of light, which seemed for a moment to dart out from behind the moon, now here, now there, on all sides, but more especially on the western side, a little before the emersion; and about two or three seconds before it, on the same western side, where the sun was just coming out, a long and very narrow streak of dusky but strong red light seemed to colour the dark edge of the moon, though nothing like it had been seen immediately after the immersion. But this instantly vanished after the appearance of the sun, as did also the aforementioned luminous ring.

Mr Louville relates, that a luminous ring of a silver colour appeared round the moon as soon as the sun was entirely covered by her disk, and disappeared the moment he recovered his light; that this ring was brightest near the moon, and grew gradually fainter towards its outer circumference, where it was, however, defined; that it was not equally bright all over, but had several breaks in it; but he makes no doubt of its being occasioned by the moon's atmosphere, and thinks that the breaks in it were occasioned by the mountains of the moon: he says also, that this ring had the moon, and not the sun, for its centre, during the whole time of its appearance. Another proof brought by him of the moon having an atmosphere is, that, towards the end of the total darkness, there was seen on that side of the moon on which the sun was going to appear, a piece of a circle, of a lively red, which might be owing to the red rays that are least refrangible being transmitted through the moon's atmosphere in the greatest quantity: and that he might be assured this from the redness did not proceed from the glasses of his telescope, he took care to bring the red part into the middle of his glasses.

He lays great stress on the streaks of light which he saw dart instantaneously from different places of the moon during the time of total darkness, and chiefly near the eastern edge of the disk: these he takes to be lightning, such as a spectator would see flashing from moon, the dark hemisphere of the earth, if he were placed upon the moon, and saw the earth come between himself and the sun. "Now (says Dr Long) it is highly probable, that if a man had, at any time, a view of that half of the earth where it is night, he would see lightning in some part of it or other." Louville farther observes, that the most mountainous countries are most liable to tempests; and that mountains being more frequent in the moon, and higher than on earth, thunder and lightning must be more frequent there than elsewhere; and that the eastern side of the moon would be most subject to thunder and lightning, those parts having been heated by the sun for half the month immediately preceding. It must here be observed, that Halley, in mentioning these flashes, says they seemed to come from behind the moon; and Louville, though he says they came sometimes from one part and sometimes from another, owns, that he himself only saw them near the eastern part of the disk; and that, not knowing at that time what it was that he saw, he did not take notice whether the same appearance was to be seen on other parts of the moon or not. He tells us, however, of an English astronomer, who presented the Royal Society with a draught of what he saw in the moon at the time of this eclipse; from which Louville seems to conclude that lightnings had been observed by that astronomer near the centre of the moon's disk. "Now (says Dr Long) thunder and lightning would be a demonstration of the moon having an atmosphere similar to ours, wherein vapours and exhalations may be supported, and furnish materials for clouds, storms, and tempests. But the strongest proof brought by Louville of the moon having an atmosphere is this, that as soon as the eclipse began, those parts of the sun which were going to be hid by the moon grew sensibly palish as the former came near them, suffering beforehand a kind of imperfect eclipse or diminution of light: this could be owing to nothing else but the atmosphere of the moon, the eastern part whereof going before her reached the sun before the moon did. As to the great height of the lunar atmosphere, which from the breadth of the luminous ring being about a whole digit would upon calculation come out 180 miles, above three times as high as the atmosphere accounted of the earth, Louville thinks that no objection; for, since, if the moon were surrounded with an atmosphere of the same nature with that which encompasses the earth, the gravitation thereof towards the moon would be but one third of that of our atmosphere towards the earth; and consequently its expansion would make the height of it three times as great from the moon as is the height of our atmosphere from the earth."

The same luminous ring has been observed in other total eclipses, and even in such as are annular, though without the luminous streaks or flashes of lightning. Conclusions have mentioned; it is even taken notice of by Plutarch: however, some members of the academy at Paris have endeavoured to account for both these phenomena without having recourse to a lunar atmosphere; and for this purpose they made the following experiments.

The image of the sun coming through a small hole into a darkened room, was received upon a circle of wood or metal of a diameter a good deal larger than that of the sun's image; then the shadow of this opaque circle was cast upon white paper, and there appeared round it, on the paper, a luminous circle such as that which surrounds the moon. The like experiment being made with a globe of wood, and with another of stone not polished, the shadows of both these casts upon paper were surrounded with a palish light, most bright near the shadows, and gradually more diluted at a distance from them. They observe also, that the ring round the moon was seen in the eclipse of 1706 by Wurzelbaur, who call her shadow upon white paper. The same appearance was observed on holding an opaque globe in the sun, so as to cover his whole body from the eye; for, looking at it through a smoked glass, in order to prevent the eye from being hurt by the glare of light it would otherwise be exposed to, the globe appeared surrounded with a light resembling that round the moon in a total eclipse of the sun.

Thus they solve the phenomenon of the ring seen round the moon by the inflection, or diffraction as they call it, of the solar rays passing near an opaque substance. As for the small streaks of light abovementioned, and which are supposed to be lightning, they explain these by an hypothesis concerning the cavities of the moon themselves; which they consider as concave mirrors reflecting the light of the sun nearly to the same point; and as these are continually changing their situation with great velocity by the moon's motion from the sun, the light which any one of them tends to our eye is seen but for a moment. This, however, will not account for the flashes, if any such there are, seen near the centre of the disk, though it does, in no very satisfactory manner, account for those at the edges.

It has already been observed, that the occultations of the fixed stars and planets by the moon, in general happen without any kind of refraction of their light by the lunar atmosphere. The contrary, however, has sometimes been observed, and the stars have been seen manifestly to change their shape and colour on going behind the moon's disk. An instance of this happened on the 28th of June N.S. in the year 1715, when an occultation of Venus by the moon happened in the daytime. Some astronomers in France observing this with a telescope, saw Venus change colour for about a minute before she was hid by the moon; and the same change of colour was observed immediately after her emergence from behind the disk. At both times the edge of the disk of Venus that was nearest the moon appeared reddish, and that which was most distant of a bluish colour. These appearances, however, which might have been taken for proofs of a lunar atmosphere, were supposed to be owing to the observers having directed the axis of their telescopes towards the moon. This would necessarily cause any planet or star near the edge of the moon's disk to be seen through those parts of the glasses which are near their circumference, and consequently to appear coloured. This was evidently the case from other observations of an occultation of Jupiter by the moon the same year, when no such appearance of refraction could be perceived while he was kept in the middle of the telescope. Maraldi also informs us, that he had observed before this two other occultations of Venus and one of Jupiter; and was always attentive to see whether those planets changed their figure or colour either upon the approach of the moon to cover them, or at their first coming again into sight; but never could perceive any such thing. Nor could he, in a great number of occultations of the fixed stars, perceive the smallest apparent change in any of them, excepting once that a fixed star seemed to increase its distance a little from the moon as it was going to be covered by her; but this, he suspected, might be owing to his telescope being directed so as to have the star seen too far from the middle of its aperture. He concludes, therefore, that the moon has no atmosphere; and he remarks, that at Montpelier, perhaps because the air is clearer there than at London, the luminous ring round the moon appeared much larger than at London; that it was very white near the moon, and, gradually decreasing in brightness, formed round her a circular area of about eight degrees in diameter. If, says he, this light was caused by the atmosphere of the moon, of what a prodigious extent must that atmosphere be?

Before we enter upon any further speculations concerning the celestial bodies, we shall here take some notice of the doctrine of a plurality of worlds; to which we are naturally led by the question, Whether the moon is inhabited or not? This is an hypothesis of very ancient date, and which in modern times has been revived in such a manner as now to be almost adopted as an undoubted truth. Plutarch, Diogenes, Laertius, and Stobæus, inform us, that this doctrine was embraced by several of the ancient Greek philosophers; from which authors Gregory has given us extracts in the Preface to his Astronomy. "Among the moderns (says Dr Long), Huygens has written a treatise, which he calls Cosmothecos, or A view of the world, worth perusing. One thing, however, I must find fault with; that, in peopling the planets with reasonable creatures, he inflicts upon their being in all points exactly similar to the human race, as to the shape of their bodies and the endowments of their minds: this is too confined a thought; for we cannot but acknowledge that infinite Power and Wisdom is able to form rational beings of various kinds, not only in shape and figure different from the human, but endowed also with faculties and senses very different; such as in our present state we can have no idea of." With regard to the probability of the doctrine itself, the Doctor expresses himself in the following manner: "That the earth and all the creatures thereon were created to be subservient to the use of man, we may believe upon the authority of the sacred writer, Psalm viii., but that the stars and planets were formed only to bespeak the canopy of heaven with their glimmering, which does not furnish us with the twentieth part of the light the moon gives, I think is not at all probable: this is contrary to the observation made by the best philosophers, that nature is magnificent in all her designs, but frugal in the execution of them. It is commonly said, that

Conclusions nature does nothing in vain; now by Nature, in a found sense, must be underfed the present order and disposition of things according to the will of the supreme Being.'

Objections have been made to this possibility of this hypothesis from the different degrees of heat and light which the planets receive from the sun, according to their various distances from him. On Venus, for instance, the heat must be more than double what it is with us, and on Mercury upwards of ten times as great; so that were our earth brought near the sun as Mercury, every drop of body would immediately be evaporated into steam, and every combustible solid set on fire; while, on the other hand, were we removed to the distance of the superior planets, such as the Georgium Sidus, Saturn, or even Jupiter, there is the highest probability that our liquids would all be congealed into ice, at the same time that the climate would be utterly insupportable by such creatures as are. Objections of the same kind are drawn from the small quantity of light which falls upon the more distant planets, which it is thought would be insufficient for the purpose of living and rational creatures. Such arguments as these, however, are by no means conclusive; for, as Dr Long justly observes, "we are sure, that if the all-wise supreme Being hath placed animals upon the respective planets in manner similar, they may be applied, together with a clear conception of the ideas of fitness and order that form the prototypes in the mind of that Great Being who directs their motions. These considerations show the absurdity of attempting to explain the final causes of events we see; but they by no means require that we should neglect them if judges where we have reason to believe that we understand the phenomena, and have sufficient experience to be assured that we discern the principal, or at least one of the principal, purposes to which things may have been designed. Thus it is scarcely to be imagined that we can err in concluding, that the eyes, ears, legs, wings, and other parts of animals, were made for the purposes of seeing, hearing, walking, flying, and so forth. Neither can we avoid inferring, that the Power who constructed living creatures with mouths, teeth, and organs to digest and assimilate food for their nutriment, did likewise form other organized bodies, which we call vegetables, for the express purpose of affording that food. It is needles to multiply instances. We cannot avoid seeing them every moment; and their effect is so striking, from the that we are insensibly forced from analogy to allow the existence of a final cause in all cases, whether we are able to discover it or not.

"On this ground, an inquiry into the final cause of the planetary bodies offers itself to our consideration. The earth is shown to be a planet in circumstances similar to the other five; we know its mates very similar to the other five; we know it in the heavens, and by analogy it may be concluded, that the others are also habitable worlds; though, from their different proportions of heat, it is credible that beings of our make and temperature could not live upon them. However, even that can scarcely be affirmed of all the plants; for the warmest climate on the planet Mars is not colder than many parts of Norway or Lapland are in the spring or autumn. Jupiter, Saturn, and the Georgium Sidus, it must be granted, are colder than any of the inhabited parts of our globe. The greatest heat on the planet Venus exceeds the heat on the island of St Thomas on the coast of Guinea, or Sumatra in the East Indies, about as much as the heat in those places exceeds that of the Orkney islands, or the city of Stockholm in Sweden: therefore, at 60° north latitude on that planet, if its axis were perpendicular to the plane of its orbit, the heat would not exceed the greatest heat on the earth; and of course vegetation like ours might be there carried on, and animals of the species on earth might subsist. If Mercury's axis be supposed to have a like position, a circle of about 2° diameter round each pole would enjoy the same temperature as the warmer regions of the earth, though in its hottest climate, water would continually boil, and oil inflammable substances would be parched up, destroyed, or converted into vapor. But it is not at all necessary that the planets should be peopled with animals like those on the earth; the Creator has doubtless adapted the inhabitants of each to their situation.

"From the observations that have been just made, Comparing a better idea may be formed of the proportions of heat on the planes than can be conveyed by numbers. It is right of the mediator, however, to remote from our purpose to compare the light of the superior planets with that of our day; from whence it will appear, that they are by no means in a state of darkness, notwithstanding their great distance from the sun. This might be investigated by several methods; as by the sun's light admitted into a dark chamber, and received on paper with different degrees of obliquity, or by a greater number of candles brought into a room with the purpose of illuminating it with various degrees of light, or by various optical methods that need not here be described. It will be sufficient for the illustration of the subject, to compare their different proportions of light with that of a moonshine night at the time of full.

"When the moon is visible in the daytime, its light is so nearly equal to that of the lighter thin clouds, that it is with difficulty distinguished among them. Its light continues the same during the night, but the absence of the sun suffering the pupil of the eye to dilate itself, it becomes more conspicuous. It therefore follows, that if every part of the sky were equally... Conclusions equally luminous with the moon's disk; the light would be the same as if in the day-time it were covered with the thin clouds abovementioned. This day-light is consequently in proportion to that of the moon as the whole surface of the sky or visible hemisphere is to the surface of the moon; that is to say, nearly as 90,000 to 1. The light of the Georgium Sidus being to that of the earth as 0.276 to 100, will be equal to the effect of 248 full moons. Jupiter's day will equal the light of 3,330 moons; and that of Mars will require 38,700, a number so great that they would almost touch one another. It is even probable, that the comets in the most distant parts of their orbits enjoy a degree of light much exceeding moonshine.

Of all the celestial bodies, comets have given rise to the greatest number of speculations and conjectures. Their strange appearance has in all ages been a matter of terror to the vulgar, who uniformly have looked upon them to be evil omens and forerunners of war, pestilence, &c. Others, less superstitious, supposed them to be meteors raised in the higher regions of the air. But we find that some part of the modern doctrine concerning them had been received into the ancient Italic and Pythagorean schools; for they held them to be so far from the nature of planets, that they had their periodical times of appearing; that they were out of sight for a long time, while they were carried aloft at an immense distance from the earth, but became visible when they descended into the lower regions of the air, when they were nearer to us.

These opinions were probably brought from Egypt, from whence the Greeks borrowed great part of their learning. However, it seems not to have been generally received; for Aristotle, who mentions it, asserted that the heavens were unchangeable, and not liable to generation or corruption. Comets, therefore, which he believed to be generated when they first made their appearance, and destroyed when they vanished from our sight, he maintained could not be heavenly bodies, but rather meteors or exhalations raised into the upper regions of the atmosphere, where they blazed out for a while, and disappeared when the matter of which they were formed was consumed. Seneca, who lived in the first century, mentions Apollonius of Myndus, a very careful observer of natural causes, to have been of the same sentiments with the most ancient Greek philosophers with regard to comets. He himself had seen two; one in the reign of Claudius, the other in that of Nero; besides another which he saw while a boy, before the death of Augustus. He plainly intimates, that he thought them above the moon; and argues strongly against those who supposed them to be meteors, or held other absurd opinions concerning them; declaring his belief that they were not fires suddenly kindled, but the eternal productions of nature. He points out also the only way to come at a certainty on this subject, viz. by collecting a number of observations concerning their appearance, in order to discover whether they return periodically or not.

For this purpose (says he) one age is not sufficient; but the time will come when the nature of comets and their magnitudes will be demonstrated, and the routes they take, so different from the planets, explained. Posterity will then wonder that the preceding ages should be ignorant of matters so plain and easy to be known."

For a long time this prediction of Seneca seemed very unlikely to be fulfilled. The great authority which Aristotle maintained for many ages, determined them to be nothing but meteors casually lighted up in the air; though they were manifestly at a great height, not only above the clouds, but subject to the diurnal revolution of the earth. In the dark and superstitious ages, they were held to be the forerunners of every kind of calamity, and were supposed to have different degrees of malignity according to the shape they assumed; from whence also they were differently denominated. Thus, some were said to be bearded, some hairy; some to represent a beam, sword, or species of spear; others a target, &c.; whereas modern astronomers acknowledge only one species of comets, and account for their different appearances from their different situations from the sun and earth.

It was not till some time after people began to throw off the fetters of superstition and ignorance which had so long held them, that any rational hypothesis was formed concerning comets. Kepler, in other respects a very great genius, indulged the most extravagant conjectures, not only concerning comets, but the whole system of nature in general. The planets he imagined to be huge animals who swam round the sun by means of certain fins acting upon the ethereal fluid, as those of fishes do on the water; and agreeable to this notion, he imagined the comets to be monstrous and uncommon animals generated in the celestial spaces; and he explained how the air engendered them by an animal faculty. A yet more ridiculous opinion, if possible, was that of John Bodin, a learned man of France in the 16th century. He maintained that comets "are spirits, which having lived on the earth innumerable ages, and being at last arrived on the confines of death, celebrate their last triumph, or are recalled to the firmament like shining stars!" This is followed by famine, plague, &c. because the cities and people destroy the governors and chiefs who appease the wrath of God." This opinion he says he borrowed from the philosopher Democritus, who imagined them to be the souls of famous heroes; but that being irreconcilable with Bodin's Christian sentiments, he was obliged to suppose them to be a kind of genii, or spirits subject to death, like those so much mentioned in the Mahometan fables. Others, again, have denied even the existence of comets, and maintained that they were only false appearances occasioned by the refraction or reflection of light.

The first rational conjecture we meet with is that of James Bernouilli, an Italian astronomer, who imagined them to be the satellites of some very distant planet, which was invisible to us on account of its distance, as were also the satellites, unless when in a certain part of their course.

Tycho Brahe was the first who restored the comets to their true rank in the creation. Before his time, several comets had been observed with tolerable exactness by Regiomontanus, Appian, Fabricius, and others; yet they all thought them below the moon. But Tycho, being provided with much better instruments, set himself with great diligence to observe the famous... Conclusions famous comet of 1577; and from many careful observations, deduced that it had no sensible diurnal parallax; and therefore was not only far above the regions of our atmosphere, but much higher than the moon.

But though few have come so near the earth as to have any diurnal parallax, all of them have what may be called an annual parallax; that is, the revolution of the earth in her orbit causes their apparent motion to be very different from what it would be if viewed from the sun; and this shows them to be much nearer than the fixed stars, which have no such parallax. Kepler, the disciple of Tycho, notwithstanding his ridiculous conjecture already mentioned, was very attentive to the motions of the comets, and found that they did not move in straight lines, as had been supposed. He showed that their paths were concave towards the sun, and supposed them to move in parabolic trajectories.

Their true motion, however, was only discovered from the observations made by Sir Isaac Newton on determined the great comet of 1680. This defended almost perpendicularly towards the sun with a prodigious velocity; ascending again with the same velocity retarded, as it had been before accelerated. It was seen in the morning by a great number of astronomers in different parts of Europe, from the 4th to the 25th of November, in its way toward the sun; and in the evening from the 12th of December to the 9th of March following. The many exact observations made on this comet enabled Sir Isaac Newton to determine that they are a kind of planets which move in very eccentric ellipses; and this opinion is now looked upon to be certainly established. It was opposed, however, by M. de la Hire, and some other French philosophers; and it is evident that the whole dispute now turned on mere practical observation. If the return of any comet could be predicted, and its periodical time calculated like that of a planet, then the doctrine might be concluded certainly true, but not otherwise. Dr Halley therefore set himself to collect all the observations he could on comets; and afterwards calculated the periodical times of 24 of them, on a supposition of their being parabolas; but afterwards found that they agreed better with the supposition of their motion being performed in very eccentric elliptical orbits. On this he calculated a table of their elements; from which it was manifest that they were not comprehended in the zodiac, some of them making an angle of upwards of 80° with the ecliptic.

By computations founded on these elements, the Doctor concluded that the comet of 1682 was the same which had appeared in 1607 and 1531; that it had a period of 75 or 76 years; and he ventured to foretell that it would return about the year 1758. The comet which appeared in 1661 was supposed to be the same with that of 1532, and to have a period of 129 years; and from the equality of periods, and similitude of appearances, it was concluded that the great comet of 1680 was the same which had appeared in 1106 in the time of Henry I. and the consulate of Lampadius and Oracles about the year 531, and in the year 44 B.C. before Julius Caesar was murdered; and thence concluded that its period was 575 years. Mr Dunthorne, however, has endeavoured to show from a MS. in Pembroke-hall library, that the comet of 1106 could not be the same with that of 1680; but M. de la Lande thinks the four appearances related by Dr Halley stronger proofs than a single observation, which might be very faulty.

Since the time of Dr Halley, other astronomers have calculated the elements of 25 other comets; all of which, excepting one of three which appeared in 1759, and which differs but little from that of 1531, 1607, and 1682, and is therefore accounted the same, differ very much from each other; so that we cannot help concluding them all to be different, and that the number of these bodies is very great. "It is not, however, unlikely (says Dr Long), from the immense interval between the orbit of Saturn and the nearest fixed stars, that many of them have not descended into the planetary regions since they have been looked upon as their celestial bodies, and observed accordingly; besides, it seldom happens that a body may finish its whole period without being observed by us, on account of the unfavourable situation of the earth in her orbit when the comet is in its perihelion. Thus, if the comet be either behind or before the sun, or nearly so, it must be above our horizon in the day-time, and consequently invisible, except the sun should at that time be in a total eclipse; for then the comet might be seen near the sun, as well as the stars and planets are; and this case is said to have happened; for Seneca relates from Posidonius, that a comet was seen when the sun was eclipsed, which had before been invisible by being near that luminary."

A greater number of comets are seen in the hemisphere towards the sun than in the opposite; the reason are seen in which will easily appear from fig. 97, wherein 'S' the hemisphere represents the sun, 'E' the earth, 'A B C D' the sphere towards the fixed stars; and because comets either do not reflect light enough to be visible, or emit tails conspicuous enough to attract our notice, till they come within the planetary regions, commonly a good way within the sphere of Jupiter, let K L M N be a sphere concentric to the sun, at such a distance from him, that no comet can be seen by us till it come within that distance; through E draw the plane B D perpendicular to S E, which will divide the sphere K L M N into two hemispheres, one of which, B C D, is towards the sun, the other, D A B, opposite. Now it is manifest, that the spherical portion L M N, which is in the hemisphere B C D towards the sun, is larger than the portion N K L in the hemisphere opposite to him; and consequently a greater number of comets will appear in the hemisphere B C D than in that marked D A B.

Though the orbits of all comets are very eccentric ellipses, there are vast differences among them; for excepting Mercury, there are no great differences among tricities of them either as to the eccentricity of their orbits, or the inclination of their planes; but the planes of some comets of comets are almost perpendicular to others, and some of their ellipses are much wider than others. The narrowest ellipse of any comet hitherto observed was that of 1680. There is also a much greater inequality in the motion of the comets than of the planets; the velocity of the former being incomparably greater in their perihelion than in their aphelion; but the planets are but very little accelerated.

Astronomers are now generally agreed, that comets are opaque bodies, enlightened by the sun. Hevelius, concerning a large work, wherein he gives the opinion of various places. Conclusions our authors on the subject, mentions some who were of the same sentiments with himself, that comets were so far transparent as to let the light of the sun pass through them, which formed their tails. Sir Isaac Newton was of opinion, that they are quite opaque; and in confirmation of this, he observes, that if a comet be seen in two parts of its orbit, at equal distances from the earth, but at unequal distances from the sun, it always shines brightest in that nearest the sun. They are of very different magnitudes, which may be conjectured from their apparent diameter and brightness. Thus the head of a comet, when of the same brightness and apparent diameter with Saturn, may be supposed to be nearly about the same magnitude with that planet; though this must be attended with some uncertainty, as we know not whether the heads of comets reflect the sun's light in the same manner the planets do. Their distance may be known from their parallax, in the manner related in a subsequent section.

In this manner he found the distance of the comet diameters, or 1577 to be about 210 semidiameters of the earth, &c. of some or about 840,000 miles distant from us, its apparent diameter being seven minutes; whence he concluded, that the true diameter of the comet was to that of the earth as 3 to 14. "But (says Dr Long) it was the hemisphere of the comet which was then measured." Hevelius, from the parallax and apparent diameter of the head of the comet in 1652, computed its diameter to be to that of the earth as 52 to 100. By the same method he found the diameter of the head of the comet of 1664 to be at one time 12 semidiameters of the earth, and at another not much more than 5. "That the head of a comet must appear less the farther it is from the earth (says Dr Long) is obvious; but besides this apparent change, there is also a real one in the dimensions of the head of the same comet; for, when near the sun, the atmosphere is diminished by the heat raising more of it into the tail; whereas, at a greater distance, the tail is diminished and the head enlarged." Hevelius computed the diameter of the nucleus of the comets of 1661 and 1665 to be only about a tenth part of that of the earth; and Cylatus makes the true diameter of the comet of 1618 to be about the same size. Some comets, however, from their apparent magnitude and distance, have been supposed much larger than the moon, or even equal in magnitude to some of the primary planets; and some have imagined, that by an interposition of these bodies between the earth and sun, we might account for those darknesses which cannot be derived from any interposition of the moon. Such are those mentioned by Herodotus, l. 7. c. 37. and l. 9. c. 10.; likewise the eclipse mentioned by Dion, which happened a little before the death of Augustus; and it is observable that Seneca saw a comet that year. Some have even attempted to account in this manner for the darkness which happened at our Saviour's crucifixion; and indeed it is certain, that were a comet in its perigee to come between the earth and sun, and to be moving the same way with the earth, it must cause a darkness much more intense, as well as of more considerable duration, than what could take place in any lunar eclipse.

Various conjectures have been formed respecting the tails of comets; though it is acknowledged by all, that they depend on the sun somehow or other; and for this plain reason, that they are always turned from him; but in what manner this is accomplished, from the we cannot easily determine. Apian, Tycho-Brahe, and others, thought the tail was formed by the sun's rays transmitted through the nucleus of the comet, which they fancied transparent, and was there refracted as in a lens of glass, so as to form a beam of light behind the comet; but this cannot be the case, as well because the figure of a comet's tail does not answer to such a refraction, as that such refracted light would not be seen by a spectator placed sideways to it, unless it fell upon some substance sufficiently dense to cause a reflection. Des Cartes and his followers were of opinion, that the tail of a comet was owing to the refraction of its head; but if this were the case, the planets and principal fixed stars must have tails also; for the rays from them pass through the same medium as the light from the comets. Sir Isaac Newton was of opinion, that the tail of a comet is a very thin vapour which the head sends out by reason of its heat; that it ascends from the sun just as smoke does from the earth; that as the ascent of smoke is caused by the rarefaction of the air wherein it is entangled, causing such air to ascend and carry the smoke up with it; so the sun's rays acting upon the coma or atmosphere of the comet, do by rarefaction and refraction heat the same; that this heated atmosphere heats, and by heating rarefies, the ether that is involved therein; and that the specific gravity with which such ether tends to the sun, is so diminished by its rarefaction, that it will now ascend from him by its relative lightness, and carry with it the reflecting particles whereof the tail is composed. Tho' the immensely large tails of some comets seem to require a great quantity of matter to produce them, this is no objection to the foregoing solution: for every day's experience shows what a great quantity of smoke is produced from a very little wood or coal; and Newton has demonstrated, that a cubic inch of air equally rarefied with that at the distance of a semidiameter from the earth's surface, would fill all the planetary regions to the orbit of Saturn and beyond. Mairan entertained a very different opinion. He supposed the tails of the comets to be formed out of the luminous matter whereof the sun's atmosphere consists. This he supposes to extend as far as the orbit of the earth, and to furnish matter for the aurora borealis. M. de la Lande is for joining the two last opinions together. Part of the matter which forms the tails of comets he supposes to arise from their own atmosphere rarified by heat and pushed forward by the force of the light streaming from the sun; and also that a comet passing through the sun's atmosphere is drenched therein, and carries away some of it. Mr Rowning objects to Newton's account, that it can hardly be supposed the thin vapour of the tail should go before the more solid body of the comet, when the motion thereof is sometimes so extremely swift, as that of some of the comets is said to be after the rate, as Sir Isaac Newton calculated the motion of the comet of 1680 to be, of no less than 880,000 miles an hour. He therefore supposes the atmosphere of the comet to extend every way round it as far as the tail reaches; and that the part of it which makes the tail is distinguished from the rest, so as to fall thick upon that part of the atmosphere which goes before the comet in its progress along its elliptic orbit. The greatest objection to this is the immense magnitude of the atmospheres; as it must now be supposed to account for the vast lengths of the tails of some comets, which have been said to measure above 80 millions of miles.

The many discoveries which, since the time of Newton, Halley, and other celebrated mathematicians, have been made in electricity, having brought in a new element unknown to former ages, and which shows a vast power through every part of the creation with which we are acquainted, it became natural to imagine that it must extend also into those higher regions which are altogether inaccessible to man. The similarity of the tails of comets to the aurora borealis, which is commonly looked upon to be an electrical phenomenon, therefore suggested an opinion at present far from being generally disbelieved, that the tails of comets are streams of electric matter. An hypothesis of this kind was published by Dr Hamilton of Dublin in a small treatise, intitled, Conjectures on the Nature of the Aurora Borealis, and on the Tails of Comets. His hypothesis is, that the comets are of use to bring back the electric fluid to the planets, which is continually discharged from the higher regions of their atmospheres. Having given at length the abovementioned opinion of Sir Isaac, "We find (says he) in this account, that Sir Isaac attributes the ascent of comets tails to their being rarer and lighter, and moving round the sun more swiftly, than the solar atmosphere, with which he supposes them to be surrounded whilst in the neighbourhood of the sun; he says also, that whatever position (in respect to each other) the head and tail of a comet then receive, they will keep the same afterwards most freely; and in another place he observes, 'That the celestial spaces must be entirely void of any power of resisting, since not only the solid bodies of the planets and comets, but even the exceeding thin vapours of which comets tails are formed, move thro' those spaces with immense velocity, and yet with the greatest freedom.' I cannot help thinking that this account is liable to many difficulties and objections, and that it seems not very consistent with itself or with the phenomena.

"I do not know that we have any proof of the existence of a solar atmosphere of any considerable extent, nor are we anywhere taught how to guess at the limits of it. It is evident that the existence of such an atmosphere cannot be proved merely by the ascent of comets tails from the sun, as that phenomenon may possibly arise from some other cause. However, let us suppose for the present, that the ascent of comets tails is owing to an atmosphere surrounding the sun, and see how the effects arising from thence will agree with the phenomena. When a comet comes into the solar atmosphere, and is then descending almost directly to the sun, if the vapours which compose the tail are raised up from it by the superior density and weight of that atmosphere, they must rise into those parts that the comet has left, and therefore at that time they may appear in a direction opposite to the sun. But as soon as the comet comes near the sun, and moves in a direction nearly at right angles with the direction of its tail, the vapours which then arise, partaking of the great velocity of the comet, and being specifically lighter than the medium in which they move, and being vastly expanded through it, must necessarily suffer a resistance immensely greater than what the small and dense body from the comet meets with, and consequently cannot possibly keep up with it, but must be left behind, or, as it were, driven backwards by the resistance of that medium into a line directed towards the parts which the comet has left, and therefore can no longer appear in a direction opposite to the sun. And, in like manner, when a comet passes its perihelion, and begins to ascend from the sun, it certainly ought to appear ever after with its tail behind it, or in a direction pointed towards the sun; for if the tail of the comet be specifically lighter than the medium in which it moves with so great velocity, it must be just as impossible it should move foremost, as it is that a torch moved swiftly through the air should project its flame and smoke before it. Since therefore we find that the tail of a comet, even when it is ascending from the sun, moves foremost, and appears in a direction nearly opposite to the sun, I think we must conclude that the comet and its tail do not move in a medium heavier and denser than the matter of which the tail consists, and consequently that the constant ascent of the tail from the sun must be owing to some other cause. For that the solar atmosphere should have density and weight sufficient to raise up the vapours of a comet from the sun, and yet not be able to give any sensible resistance to these vapours in their rapid progress through it, are two things inconsistent with each other: And therefore, since the tail of a comet is found to move as freely as the body does, we ought rather to conclude, that the celestial spaces are void of all resisting matter, than that they are filled with a solar atmosphere, be it ever so rare.

"But there is, I think, a further consideration, which will show that the received opinion, as to the ascent of comets tails, is not agreeable to the phenomena, and may at the same time lead us to some knowledge of the matter of which these tails consist; which I suspect is of a very different nature from what it has been hitherto supposed to be. Sir Isaac says, the vapours, of which the tail of a comet consists, grow hot by reflecting the rays of the sun, and thereby warm and rarefy the medium which surrounds them; which must therefore ascend from the sun, and carry with it the reflecting particles of which the tail is formed; for he always speaks of the tail as shining by reflected light. But one would rather imagine, from the phenomena, that the matter which forms a comet's tail has not the least sensible power of reflecting the rays of light. For it appears from Sir Isaac's observation which I have quoted already, that the light of the smallest stars, coming to us through the immense thicknesses of a comet's tail, does not suffer the least diminution. And yet, if the tail can reflect the light of the sun to copiously as it must do if its great splendour be owing to such reflection, it must undoubtedly have the same effect on the light of the stars; that is, it must reflect back the light which comes from the stars behind it, and by so doing must intercept them from our sight, considering its vast thickness, and how exceedingly slender a ray is that comes from a small star; or if it did not intercept their whole light, it must at least increase their twinkling. But we do not find that it has even this small effect; for those stars that appear through the tail are not observed to twinkle more than others." Conclusions others in their neighbourhood. Since therefore this fact is supported by observations, what can be a plainer proof that the matter of a comet's tail has no power of reflecting the rays of light? and consequently that it must be a self-shining substance. But the same thing will further appear, from considering that bodies reflect and refract light by one and the same power; and therefore if comets tails want the power of refracting the rays of light, they must also want the power of reflecting them. Now, that they want this refracting power appears from hence: If that great column of transparent matter which forms a comet's tail, and moves either in a vacuum or in some medium of a different density from its own, had any power of refracting a ray of light coming through it from a star to us, that ray must be turned far out of its way in passing over the great distance between the comet and the earth; and therefore we should very sensibly perceive the smallest refraction that the light of the stars might suffer in passing through a comet's tail. The consequence of such a refraction must be very remarkable: the stars that lie near the tail would, in some cases, appear double; for they would appear in their proper places by their direct rays, and we should see their images behind the tail, by means of their rays which it might refract to our eyes; and those stars that were really behind the tail would disappear in some situations, their rays being turned aside from us by refraction. In short, it is easy to imagine what strange alterations would be made in the apparent places of the fixed stars by the tails of comets, if they had a power of refracting their light, which could not fail to be taken notice of if any such ever happened. But since astronomers have not mentioned any such apparent changes of place among the stars, I take it for granted that the stars seen through all parts of a comet's tail appear in their proper places, and with their usual colours; and consequently I infer, that the rays of light suffer no refraction in passing through a comet's tail. And thence I conclude (as before), that the matter of a comet's tail has not the power of refracting or reflecting the rays of light, and must therefore be a lucid or self-shining substance.

But whatever probability the Doctor's conjecture concerning the materials whereof the tails are formed may have in it, his criticism on Sir Isaac Newton's account of them seems not to be just: for that great philosopher supposes the comets to have an atmosphere peculiar to themselves; and consequently, in their nearest approaches to the sun, both comet and atmosphere are immersed in the atmosphere of that luminary. In this case, the atmosphere of the comet being prodigiously heated on the side next to the sun, and consequently the equilibrium in it broken, the denser parts will continually pour in from the regions farthest from the sun; for the same reason, the more rarefied part which is before will continually fly off opposite to the sun, being displaced by that which comes from behind; for tho' we must suppose the comet and its atmosphere to be heated on all sides to an extreme degree, yet still that part which is farthest from the sun will be left hot, and consequently more dense, than what is nearest to his body. The consequence of this is, that there must be a constant stream of dense atmosphere descending towards the sun, and another stream of rarefied vapours and atmosphere ascending on the contrary side; just as in a common fire, there is a constant stream of dense from the air descending, which pushes up another of rarefied foregoing air, flame, and smoke. The refraction of the solar atmosphere may indeed be very well supposed to occasion the curvature observable in the tails of comets, and their being better defined in the fore part than behind; and this appearance we think Dr Hamilton's hypothesis is incapable of solving. We grant, that ton's hypothesis is the utmost probability that the tails of comets are streams of electric matter; but they who advance a theory of any kind ought to solve every phenomenon, otherwise their theory is insufficient. It was incumbent on Dr Hamilton, therefore, to have explained how this stream of electric matter comes to be bent into a curve; and also why it is better defined and brighter on the outer side of the arch than on the inner. This, indeed, he attempts in the following manner: "But that this curvature was not owing to any resisting matter appears from hence, that the tail must be bent into a curve, though it met with no resistance; for it could not be a right line, unless all its particles were projected in parallel directions, and with the same velocity, and unless the comet moved uniformly in a right line. But the comet moves in a curve, and each part of the tail is projected in a direction opposite to the sun, and at the same time partakes of the motion of the comet; so that the different parts of the tail must move on lines which diverge from each other; and a line drawn from the head of a comet to the extremity of the tail, will be parallel to a line drawn from the sun to the place where the comet was when that part of the tail began to ascend, as Sir Isaac observes; and so all the chords or lines drawn from the head of the comet to the intermediate parts of the tail, will be respectively parallel to lines drawn from the sun to the places where the comet was when these parts of the tail began to ascend. And therefore, since these chords of the tail will be of different lengths, and parallel to different lines, they must make different angles with a great circle passing through the sun and comet; and consequently a line passing through their extremities will be a curve.

"It is observed, that the convex side of the tail which is turned from the sun is better defined, and shines a little brighter, than the concave side. Sir Isaac accounts for this, by saying, that the vapour on the convex side is fresher (that is, has ascended later) than that on the concave side; and yet I cannot see how the particles on the convex side can be thought to have ascended later than those on the concave side which may be nearer to the head of the comet. I think it rather looks as if the tail, in its rapid motion, met with some slight resistance just sufficient to cause a small condensation in that side of it which moves foremost, and which would occasion it to appear a little brighter and better defined than the other side; which slight resistance may arise from that subtile ether which is supposed to be dispersed through the celestial regions, or from this very electric matter dispersed in the same manner, if it be different from the ether."

On the last part of this observation we must remark, that though a slight resistance in the ethereal medium would have served Sir Isaac Newton's turn, it will by no means serve Dr Hamilton's; for though a stream of water Astronomy

Diffusions water or air may be easily destroyed or broken by the distance, yet a stream of electric matter seems to set every obstacle at defiance. If a sharp needle is placed on the conductor of an electric machine, and the machine set in motion, we will perceive a small stream of electric matter issuing from the point; but though we blow against this stream of fire with the utmost violence, it is impossible either to move it, or to brighten it on the side against which we blow. If the celestial spaces then are full of a subtle ether capable of thus affecting a stream of electric matter, we may be sure that it also will resist very violently; and we are then as much difficulted to account for the projectile motion continuing amidst such violent resistance; for if the ether resists the tail of the comet, it is impossible to prove that it doth not resist the head also.

This objection may appear to some to be but weakly founded, as we perceive the electric fluid to be endowed with such extreme subtilty, and to yield to the impulsion of solid bodies with such facility, that we easily imagine it to be of a very passive nature in all cases. But it is certain, that this fluid only shows itself passive where it passes from one body into another, which it seems very much inclined to do of itself. It will also be found, on proper examination of all the phenomena, that the only way we can manage the electric fluid at all is by allowing it to direct its own motions. In all cases where we ourselves attempt to assume the government of it, it shows itself the most unruly and stubborn being in nature. But these things come more properly under the article Electricity, where they are fully considered. Here it is sufficient to observe, that a stream of electric matter resists air, and from the phenomena of electric repulsion we are sure that one stream of electric matter resists another: from which we may be also certain, that if a stream of electric matter moves in an aerial fluid, such fluid will resist it; and we can only judge of the degree of resistance it meets with in the heavens from what we observe on earth. Here we see the most violent blast of air has no effect upon a stream of electric fluid; in the celestial regions, either air or some other fluid has an effect upon it according to Dr Hamilton. The resistance of that fluid, therefore, must be greater than that of the most violent blast of air we can imagine.

As to the Doctor's method of accounting for the curvature of the comet's tail, it might do very well on Sir Isaac Newton's principles, but cannot do so on his. There is no comparison between the celerity with which rarefied vapour ascends in our atmosphere, and that whereby the electric fluid is discharged. The velocity of the latter seems to equal that of light; of consequence, supposing the velocity of the comet to be equal to that of the earth in its annual course, and its tail equal in length to the distance of the sun from the earth, the curvature of the tail could only be to a straight line as the velocity of the comet in its orbit is to the velocity of light, which, according to the calculations of Dr Bradley, is as 10,201 to 1. The apparent curvature of such a comet's tail, therefore, would at this rate only be \( \frac{1}{10,201} \) part of its visible length, and thus would always be imperceptible to us. The velocity of comets is indeed sometimes inconceivably great. Mr Bredone observed one at Palermo, in July 1770, which in 24 hours described an arch in the heavens upwards of 50 degrees in length; according to which he supposes, that if it was as far distant as the sun, it must have moved at the rate of upwards of 60 millions of miles in a day. But this comet was attended with no appearance, so that we cannot be certain whether the curvature of the tails of these bodies corresponds with their velocity or not.

The near approach of some comets to the sun subjects them to intense and inconceivable degrees of heat. Newton calculated that the heat of the comet of 1680 must have been near 2000 times as great as that of red-hot iron. The calculation is founded upon this principle, that the heat of the sun falling upon any body at different distances is reciprocally as the squares of those distances; but it may be observed, that the effect of the heat of the sun upon all bodies near our earth depends very much on the constitution of those bodies, and of the air that surrounds them. "The comet in question (says Dr Long) certainly acquired a prodigious heat; but I cannot think it came up to what the calculation makes it: the effect of the strongest burning-glass that has ever been made use of was the vitrification of most bodies placed in its focus. What would be the effect of a still greater heat we can only conjecture; it would perhaps to disunite the parts as to make them fly off every way in atoms. This comet, according to Halley, in passing thro' its southern node, came within the length of the sun's semidiameter of the orbit of the earth. Had the earth then been in the part of her orbit nearest to that node, their mutual gravitation must have caused a change in the plane of the orbit of the earth, and in the length of our year: he adds, that if so large a body, with so rapid a motion as that of this comet, were to strike against the earth, a thing by no means impossible, the shock might reduce this beautiful frame to its original chaos."

We must not conclude this account without observing, that Whiston, who, from Flamsteed's measure of its apparent diameter, concluded the nucleus of the comet to be about ten times as big as the moon, or equal to a fourth part of the earth, attributes the universal deluge in the time of Noah to the near approach thereof. His opinion was, that the earth passing thro' the atmosphere of the comet, attracted therefrom great part of the water of the flood; that the nearness of the comet raised a great tide in the subterraneous waters, so that the outer crust of the earth was changed from a spherical to an oval figure; that this could not be done without making fissures and cracks in it, thro' which the waters forced themselves, by the hollow of the earth being changed into a less capacious form; that along with the water thus squeezed up on the surface of the earth, much slime or mud would rise; which, together with the grossest part of the comet's atmosphere, would, after the subsiding of the water, partly into the fissures and partly into the lower parts of the earth to form the sea, cover all over, to a considerable depth, the antediluvian earth. Thus he accounts for trees and bones of animals being found at very great depths in the earth. He also held that, before the fall, the earth revolved round the sun in the plane of the ecliptic, keeping always the same points of its surface towards the same fixed stars. By this means, as every meridian would come to the sun but once in every revolution, a day and a year were then the same: but Conclusions that a comet striking obliquely upon some part of the earth gave it the diurnal rotation; that the antediluvian year consisted of 360 days; but that the additional matter deposited upon the earth from the atmosphere of the comet at the flood, so retarded the revolution thereof round the sun, that it is not now performed in less than 365 days and about a quarter. The same comet he thought would probably, coming near the earth when heated in an immense degree in its perihelion, be the instrumental cause of that great catastrophe, the general conflagration, foretold in the sacred writings, and from ancient tradition.

These conjectures lead us to speak somewhat more particularly concerning the nature of comets, and the purposes they may possibly answer in the creation. Hevelius, in order to account for the various appearances of the nucleus already related, supposed that they were composed of several masses compacted together with a transparent fluid interspersed, but the apparent changes in the nucleus may be only on the surface: comets may be subject to spots as the planets are; and the vastly different degrees of heat they go through may occasion great and sudden changes, not only in their surfaces, but even in their internal frame and texture. Newton places all these apparent changes to the atmosphere that environs them; which must be very dense near the surface, and have clouds floating therein. It was his opinion, that the changes mentioned may all be in the clouds, not in the nucleus. This last indeed he looked upon to be a body of extreme solidity, in order to sustain such an intense heat as the comets are sometimes destined to undergo; and that, notwithstanding their running out into the immense regions of space, where they were exposed to the most intense degrees of cold, they would hardly be cooled again on their return to the sun. Indeed, according to his calculation, the comet of 1680 must be forever in a state of violent ignition. He hath computed that a globe of red-hot iron of the same dimensions with the earth, would scarce be cool in 50,000 years. If then the comet be supposed to cool 100 times faster than red-hot iron, as its heat was 2000 times greater, it must require upwards of a million of years to cool it. In the short period of 575 years, therefore, its heat will be in a manner scarcely diminished; and, of consequence, in its next and every succeeding revolution, it must acquire an increase of heat: so that, since the creation, having received a proportional addition in every succeeding revolution, it must now be in a state of ignition very little inferior to that of the sun itself. Sir Isaac Newton hath farther concluded, that this comet must be considerably retarded in every succeeding revolution by the atmosphere of the sun within which it enters; and this must continually come nearer and nearer his body, till at last it falls into it. This, he thinks, may be one use of the comets, to furnish fuel for the sun, which otherwise would be in danger of wasting from the continual emission of its light.

He adds, that for the conservation of the water and moisture of the planets, comets seem absolutely requisite; from whole condensed vapors and exhalation all the moisture which is spent in vegetation and putrefaction, and turned into dry earth, &c., may be resupplied and recruited; for all vegetables grow and Conclusions increase wholly from fluids; and again, as to their foregoing greatest part, turn by putrefaction into earth; an earthly appearance being perpetually precipitated to the bottom of ces putrefying liquors. Hence the quantity of dry earth must continually increase, and the moisture of the globe decrease, and be quite evaporated, if it have not a continual supply from some part or other of the universe. "And I suspect (adds our great author), that the spirit, which makes the finest, subtlest, and best part of our air, and which is absolutely requisite for the life and being of all things, comes principally from the comets."

Mr Brydone observes, that the comets without tails seem to be of a very different species from those which have tails: To the latter, he says, they appear to bear a much less resemblance than they do even to planets. He tells us, that comets with tails have seldom been visible but on their recesses from the sun: that none of them are kindled up, and receive their alarming appearance, in their near approach to this glorious luminary; but that those without tails are seldom or never seen without but on their way to the sun; and he does not recollect tails, any whose return has been tolerably well ascertained.

"I remember indeed (says he), a few years ago, a small one, that was said to have been discovered by a telescope after it had passed the sun, but never more became visible to the naked eye. This assertion is easily made, and nobody can contradict it; but it does not at all appear probable that it should have been so much less luminous after it had passed the sun than before it approached him: and I will own to you, when I have heard that the return of these comets had escaped the eyes of the most acute astronomers, I have been tempted to think that they did not return at all, but were absorbed in the body of the sun, which their violent motion towards him seemed to indicate." He then attempts to account for the continual emission of the sun's light without waste, by supposing that there are numberless bodies throughout the universe that are attracted into the body of the sun, which serve to supply the waste of light, and which for some time remain obscure and occasion spots on his surface, till at last they are perfectly dissolved and become bright like the rest. This hypothesis may account for the dark spots becoming as bright, or even brighter, than the rest of the disk, but will by no means account for the brighter spots becoming dark. Of this comet too, Mr Brydone remarks, that it was evidently surrounded by an atmosphere which refracted the light of the fixed stars, and seemed to cause them change their places as the comet came near them.

A very strange opinion we find set forth in a book intitled "Observations and Conjectures on the Nature, Hypothesis, and Properties of Light, and on the Theory of Comets, by William Cole." This gentleman supposes that the comets belong to no particular system; but were originally projected in such directions as would successively expose them to the attraction of different centres, and thus they would describe various curves of the parabolic and the hyperbolic kind. This treatise is written in answer to some objections thrown out in Mr Brydone's Tour, against the motions of the comets by means of the two forces of gravitation and projection, which The appearance of Saturn as emerging from behind the dark limb of the Moon 10 June 1762 at 4h 22m apparent time at Chelsea.

A.Bell Pin. Ital. Sculptor fecit. Conclusions which were thought sufficient for that purpose by Sir Isaac Newton; of which we shall treat as fully as our limits will allow in the next section.

The analogy between the periodical times of the planets and their distances from the sun, discovered by Kepler, takes place also in the comets. In consequence of this, the mean distance of a comet from the sun may be found by comparing its period with the time of the earth's revolution round the sun. Thus the period of the comet that appeared in 1531, 1607, 1682, and 1759, being about 76 years, its mean distance from the sun may be found by this proportion:

As 1, the square of one year, the earth's periodical time, is to 5776 the square of 76, the comet's periodical time; so is 1,000,000, the cube of 100 the earth's mean distance from the sun, to 5,776,000,000, the cube of the comet's mean distance. The cube root of this last number is 1794; the mean distance itself in such parts as the mean distance of the earth from the sun contains 100. If the perihelion distance of this comet, 58, be taken from 358 double the mean distance, we shall have the aphelion distance, 3530, of such parts as the distance of the earth contains 100; which is a little more than 35 times the distance of the earth from the sun. By a like method, the aphelion distance of the comet of 1680 comes out 138 times the mean distance of the earth from the sun, supposing its period to be 575 years: so that this comet, in its aphelion, goes more than 14 times the distance from the sun that Saturn does. Euler computes the orbit of this comet from three of Flamsteed's observations taken near together, compared with a fourth taken at some distance from the other three; and from thence concludes the period to be a little more than 170 years. "It seems something surprising (says Dr Long), that, from the same observations which were used by Newton and Halley, he should bring out a period so very different from what those great men have determined: but it is the less to be wondered at, if we consider how small a portion of the comet's orbit lay between the most distant places used in this computation, or indeed that could be had for that purpose; so small, that the form of the ellipsis cannot be found with precision by this method, except the comet's places were more exactly verified than is possible to be done; and that he does not pretend to confirm his determination of the period by pointing out and comparing together any former appearances of this comet; a method which Newton recommended as the only one whereby the periodical times and transverse diameters of the orbits of the comets can be determined with accuracy."

The period of the comet in 1744 is much longer than even that of 1680. Mr Betts, in attempting to compute the transverse axis of its orbit, found it come out so near infinite, that, though the orbit showed itself in this manner to be a very long one, he found it impossible to calculate it without some observations made after its perihelion. Halley, after he had finished his table of comets, found such a similitude in the elements of those of 1531, 1607, and 1682, that he was induced to believe them to be returns of the same comet in an elliptic orbit: but as there was such a difference in their periodical times and inclinations of their orbits as seemed to make against this opinion; and as the observations of the first of them in 1531 by Appian, and the second in 1607 by Kepler, were not exact enough to determine so nice a point when he first published his synopsis in 1705; he only mentioned this as a thing probable, and recommended it to posterity to watch for an appearance of the same in 1758. Afterwards, looking over the catalogue of ancient comets, and finding three others at equal intervals with those now mentioned, he grew more positive in his opinion; and knowing a method of calculating with ease a motion in an elliptic orbit, how eccentricsoever it might be, instead of the parabolic orbit which he had given for the comet of 1682, he set about adapting the plan of that orbit to an ellipsis of a given space and magnitude, having the sun in one of its foci, so as totally with the observations of that comet made by Flamsteed with great accuracy, by the help of a very large sextant. He likewise corrected the places of the comet of 1531 from Appian, and those of the comet 1607 from Kepler and Longomontanus, by rectifying the places of the stars they had made use of, and found those places agree as well with the motion in such an ellipsis as could be expected from the manner of observing of these astronomers, and the imperfections of their instruments. The greatest objection to this theory was some difference in the inclination of the orbits, and that there was above a year's difference between the two periods. The comet of 1531 was in Why the its perihelion August 24.; that of 1607, October 16.; and that of 1682, September 4.: so that the first return of these periods was more than 76, the latter not quite 75 years. To obviate this, he reminds his readers of an observation made by him of the periodical revolution of Saturn having at one time been about 13 days longer than at another time; occasioned, as he supposed, by the near approach of Saturn and Jupiter, and the mutual attraction and gravitation of the two planets; and observes, that in the summer of the year 1681, the comet in its descent was for some time too near Jupiter, that its gravitation towards that planet was one-fiftieth part of its gravitation towards the sun. This, he concluded, would cause a change in the inclination of its orbit, and also in the velocity of its motion: for by continuing longer near the planet Jupiter on the side most remote from the sun, its velocity would be more increased by the joint forces of both those bodies, than it would be diminished by them acting contrary-wise, when on the side next the sun where its motion was swiftest. The projectile motion being thus increased, its orbit would be enlarged, and its period lengthened; so that he thought it probable it would not return till after a longer period than 76 years, about the end of the year 1758 or beginning of 1759.

As Halley expressed his opinion modestly, though clearly enough, that this comet would appear again about the end of 1758, or the beginning of the following year, M. de la Lande pretends he must have been at a loss to know whether the period he foretold would have been of 75 or of 76 years; that he did not give a decisive prediction, as if it had been the result of calculation; and that, by considering the affair in so loose a manner as Halley did, there was a good deal of room for objecting to his reasoning. After these reflections, he is very large in his commendation of the performance of Clairault; who, he says, not only Conclusions only calculated strictly the effect of the attraction of Jupiter in 1681 and 1683, when the comet was again near Jupiter, but did not neglect the attraction of that planet when the comet was most distant; that he considered the uninterrupted attractions of Jupiter and Saturn upon the sun and upon the comet, but chiefly the attraction of Jupiter upon the sun, whereby that luminary was a little displaced, and gave different elements to the orbit of the comet. By this method he found the comet would be in its perihelion about the middle of April; but that, on account of some small quantities necessarily neglected in the method of approximation made use of by him, Mr Clairault desired to be indulged one month; and that the comet came just 30 days before the time he had fixed for its appearance.

That comets may have their motion disturbed by the planets, especially by the two largest, Jupiter and Saturn, appears by an instance just now mentioned. They may also affect one another by their mutual gravitation when out of the planetary regions; but of this we can take no account, nor can we estimate the resistance of the ether through which they pass; and yet both these causes may have some influence on the inclination of their orbits and the length of their periods.

Thus much concerning the bodies of which our solar system is composed. But the conjectures of astronomers have reached even beyond its boundaries: they have supposed every one of the innumerable multitude of fixed stars to be a sun attended by planets and comets, each of which is an habitable world like our own; so that the universe may in some measure be represented by fig. 161, where several adjacent systems are marked. The strongest argument for this hypothesis is, that they cannot be magnified by a telescope on account of their extreme distance; whence we must conclude that they shine by their own light, and are therefore as many suns; each of which we may suppose to be equal, if not superior, in lustre and magnitude to our own. They are not supposed to be at equal distances from us, but to be more remote in proportion to their apparent smallness. This supposition is necessary to prevent any interference of their planets; and thus there may be as great a distance between a star of the first magnitude and one of the second apparently close to it, as between the earth and the fixed stars first mentioned.

Those who take the contrary side of the question affirm, that the disappearance of some of the fixed stars is a demonstration that they cannot be suns, as it would be to the highest degree absurd to think that God would create a sun which might disappear of a sudden, and leave its planets and their inhabitants in endless night. Yet this opinion we find adopted by Dr Keil, who tell us, "It is no ways improbable that these stars lost their brightness by a prodigious number of spots which entirely covered and overwhelmed them. In what dismal condition must their planets remain, who have nothing but the dim and twinkling light of the fixed stars to enlighten them?" Others, however, have made suppositions more agreeable to our notions of the benevolent character of the Deity. Sir Isaac Newton thinks that the sudden blaze of some stars may have been occasioned by the falling of a comet into them, by which means they would be enabled to emit a prodigious light for a little time, after which from the foregoing they would gradually return to their former state. Others have thought that the variable ones, which disappear for a time, were planets, which were only visible during some part of their course. But this, their apparent immobility, notwithstanding their decrease of lustre, will not allow us to think. Some have imagined, that one side of them might be naturally much darker than the other, and when by the revolution of the star upon its axis the dark side was turned towards us, the star became invisible, and, for the same reason, after some interval, returned its former lustre. Mr Maupeurtuis, in his dissertation on the figures of the celestial bodies (p. 61—63), is of opinion, that some stars, by their prodigious quick rotations on their axes, may not only assume the figures of oblate spheroids, but that, by the great centrifugal force arising from such rotations, they may become of the figures of mill-stones, or be reduced to flat circular planes, so thin as to be quite invisible when their edges are turned towards us; as Saturn's ring is in such positions. But when very eccentric planets or comets go round any flat star, in orbits much inclined to its equator, the attraction of the planets or comets in their perihelions must alter the inclination of the axis of that star; on which account it will appear more or less large and luminous, as its broad side is more or less turned towards us. And thus he imagines we may account for the apparent changes of magnitude and lustre in those stars, and likewise for their appearing and disappearing.

Lastly, Mr Dunn (Phil. Trans. Vol. LII.) in a dissertation concerning the apparent increase of magnitude in the heavenly bodies when they approach the horizon, conjectures that the interposition of some gross atmosphere may solve the phenomena both of nebulous and new stars. "The phenomena of nebulous and new stars (says he) have engaged the attention of curious astronomers; but none that I know of have given any reason for the appearance of nebulous stars. Possibly what has been before advanced may also be applicable for investigating reasons for those strange appearances in the remote parts of the universe. From many instances which might be produced concerning the nature and properties of lights and illuminations on the earth's surface, concerning the nature and properties of the earth's atmosphere, and concerning the atmospheres and illuminations of comets, we may safely conclude, that the atmospheres of comets and of our earth are more gross in their nature than the ethereal medium which is generally diffused throughout the solar system; possibly a more aqueous vapour in the one than the other, makes the difference. Now, as the atmospheres of comets and of planets in our solar system are more gross than the ether which is generally diffused throughout our solar system, why may not the ethereal medium diffused throughout those other solar systems (whose centres are their respective fixed stars) be more gross than the ethereal medium diffused throughout our solar system? This indeed is an hypothesis, but such an one as agrees exactly with nature. For these nebulous stars appear so much like comets, both to the naked eye and through telescopes, that the one cannot always, by any difference of their extraneous light, be known from the other. Such orbs of gross ether re- Conclusions affecting light more copiously, or like the atmospheres from the comets, may help us to judge of the magnitudes of the orbs illuminated by those remote lights, when all other means seem to fail. The appearance of new stars, and disappearance of others, possibly may be occasioned by the interposition of such an ethereal medium, within their respective orbs, as either admits light to pass freely, or wholly absorbs it at certain times, whilst light is constantly pursuing its journey through the vast regions of space.

In the Philosophical Transactions for 1783, however, Mr Michell, in proposing a method to determine the distance, magnitude, &c. of the fixed stars by the diminution of the velocity of their light, should any such thing be discovered, makes such suppositions as seem totally incongruous with what has been just now advanced. "The very great number of stars (says Mr Michell) that have been discovered to be double, triple, &c., particularly by Mr Herschel, if we apply the doctrine of chances, as I have heretofore done in my Inquiry into the probable parallax, &c. of the fixed stars, published in the Philosophical Transactions for the year 1767, cannot leave a doubt with any one who is properly acquainted with the force of those arguments, that by far the greatest part, if not all of them, are systems of stars so near each other, as probably to be liable to be affected sensibly by their mutual gravitation; and it is therefore not unlikely, that the periods of the revolutions of some of these about their principals (the smaller ones being, upon this hypothesis, to be considered as satellites to the others) may some time or other be discovered." Having then shown in what manner the magnitude of a fixed star, if its density were known, would affect the velocity of its light, he concludes at last, that "if the semidiameter of a sphere of the same density with the sun were to exceed his in the proportion of 500 to 1, a body falling from an infinite height towards it (or moving in a parabolic curve at its surface) would have acquired a greater velocity than that of light; and consequently, supposing light to be attracted by the same force in proportion to its vis inertiae with other bodies, all light emitted from such a body would be made to return towards it by its own proper gravity. But if the semidiameter of a sphere, of the same density with the sun, was of any other size less than 497 times that of the sun, though the velocity of light emitted by such a body would never be wholly destroyed, yet it would always suffer some diminution, more or less according to the magnitude of the sphere. The same effects would likewise take place if the semidiameters were different from those already mentioned, provided the density was greater or less in the duplicate ratio of those semidiameters inversely.

After proceeding in his calculations, in order to find the diameter and distance of any star, he proceeds thus: "According to Mr Bouguer the brightness of the sun exceeds that of a wax candle in no less a proportion than that of 8000 to 1. If therefore the brightness of any of the fixed stars should not exceed that of our common candles, which, as being something less luminous than wax, we will suppose in round numbers to be only one ten thousandth part as bright as the sun, such a star would not be visible at more than one hundredth part of the distance at which it would be seen if it were as bright as the sun. Now, because the sun would fill, I apprehend, appear as bright and luminous as the star Sirius, if removed to from the 400,000 times his present distance, such a body, if no brighter than our common candles, would only appear equally luminous with that star at 4000 times the distance of the sun; and we might then be able, with the best telescopes, to distinguish some sensible apparent diameter of it: but the apparent diameters of the stars of lesser magnitudes would still be too small to be distinguishable even with our best telescopes, unless they were yet a good deal less luminous; which may possibly, however, be the case with some of them: for though we have indeed very slight grounds to go upon with regard to the specific brightness of the fixed stars, compared with that of the sun at present, and can therefore form only very uncertain and random conjectures concerning it; yet from the infinite variety which we find in the works of the creation, it is not unreasonable to suspect, that very possibly some of the fixed stars may have so little natural brightness in proportion to their magnitude, as to admit of their diameters having some sensible apparent size when they shall come to be more carefully examined, and with larger and better telescopes than have been hitherto in common use.

"With regard to the sun, we know that his whole Luminous surface is extremely luminous, a very small and temporary interruption sometimes, from a few spots, excepted. This universal and excessive brightness of the whole surface is probably owing to an atmosphere, which being luminous throughout, and in some measure also transparent, the light proceeding from a considerable depth of it all arrives at the eye, in the same manner as the light of a great number of candles would do if they were placed one behind another, and their flames were sufficiently transparent to permit the light of the more distant ones to pass through those that were nearer without interruption.

"How far the same constitution may take place in the fixed stars we do not know: probably, however, it may still do so in many; but there are some appearances, with regard to a few of them, which seem to make it probable that it does not do so universally. Now, if I am right in supposing the light of the sun to proceed from a luminous atmosphere which must necessarily diffuse itself equally over the whole surface, and I think there can be very little doubt that this is really the case, this constitution cannot well take place in those stars which are in some degree periodically more and less luminous, such as that in Collo Ceti, &c. It is also not very improbable, that there is some difference from that of the sun in the constitution of those stars which have sometimes appeared and disappeared, of which that in the constellation of Cassiopeia is a notable instance. And if these conjectures are well founded which have been formed by some philosophers concerning stars of this kind, that they are not wholly luminous, or at least not constantly so, but that all, or by far the greatest part of their surfaces, is subject to considerable changes, sometimes becoming luminous, at others extinguished; it is amongst stars of this sort that we are most likely to meet with instances of a sensible apparent diameter, their light being much more likely not to be so great in proportion as that of the sun, which if removed to 400,000 times his present distance, would still appear, I apprehend, as bright as Sirius, as I have observed above; whereas it is hardly to be expected, with any telescope whatsoever, that we should ever be able to distinguish a well-defined disk of any body of the same size with the sun at much more than 10,000 times his present distance.

Hence the greatest distance at which it would be possible to distinguish any sensible apparent diameter of a body as dense as the sun, cannot well greatly exceed five hundred times ten thousand; that is, five million times the distance of the sun; for if the diameter of such a body was not less than 500 times that of the sun, its light, as has been shown above, could never arrive at us."

Mr Herschel, improving on Mr Michell's idea of the fixed stars being collected into groups, and fitted by his own observations with the extraordinary telecopic powers already mentioned, has suggested a theory concerning the constitution of the universe entirely new and singular. It had been the opinion of former astronomers, that our sun, besides occupying the centre of the system which properly belongs to him, occupied also the centre of the universe: but Mr Herschel is of a very different opinion. "Hitherto (says he) the sidereal heavens have, not inadequately for the purpose designed, been represented by the concave surface of a sphere, in the centre of which the eye of the observer might be supposed to be placed. It is true, the various magnitudes of the fixed stars even then plainly suggested to us, and would have better suited, the idea of an expanded firmament of three dimensions; but the observations upon which I am now going to enter, still farther illustrate and enforce the necessity of considering the heavens in this point of view. In future therefore we shall look upon those regions into which we may now penetrate by means of such large telescopes, (A) as a naturalist regards a rich extent of ground or chain of mountains, containing strata variously inclined and directed, as well as consisting of very different materials. A surface of a globe or map therefore will but ill delineate the interior parts of the heavens."

With the powerful telescope mentioned in the note, Mr Herschel first began to survey the Via Lactea, and found that it completely resolved the whitish appearance into stars, which the telescopes he formerly used had not light enough to do. The portion he first observed was that about the hand and club of Orion; and found therein an astonishing multitude of stars, whose number he endeavoured to estimate by counting many fields (B), and computing from a mean of these how many might be contained in a given portion of the milky-way. In the most vacant place to be met with in that neighbourhood he found 63 stars; other six fields contained 110, 60, 70, 90, 70, and 74 stars; a mean of all which gave 79 for the number of stars to each field: and thus he found, that by allowing 15 minutes for the diameter of his field of view, a belt of 15 degrees long and two broad, which he had often seen pass before his telescope in an hour's time, could contain less than 50,000 stars, large enough to be distinctly numbered; besides which, he suspected twice as many more, which could be seen only now and then by faint glimpses for want of sufficient light.

The success he had with the milky-way soon induced him to turn his telescope to the nebulous parts of the heavens, of which an accurate list had been published in the Connaissance des Temps for 1783 and 1784. Most of these yielded to a Newtonian reflector of 20 feet focal distance and 12 inches aperture; which plainly discovered them to be composed of stars, or at least to contain stars, and to show every other indication of consisting of them entirely. "The nebulæ (says he) are arranged into strata, and run on to a great length; and some of them I have been able to pursue, and to guess pretty well at their form and direction. It is probable enough that they may surround the whole starry sphere of the heavens, not unlike the milky-way, which undoubtedly is nothing but a stratum of fixed stars: And as this latter immense starry bed is not of equal breadth or lustre in every part, nor runs on in one straight direction, but is curved, and even divided into two streams along a very considerable portion of it; we may likewise expect the greatest variety in the strata of the clusters of stars and nebulæ. One of these nebulæous beds is so rich, that, in passing through a section of it in the time of only 36 minutes, I have detected no less than 31 nebulæ, all distinctly visible upon a fine blue sky." Their situation and shape, as well as condition, seem to denote the greatest variety imaginable. In another stratum, or perhaps a different branch of the former, I have often seen double and treble nebulæ variously arranged; large ones with small seeming attendants; narrow, but much extended lucid nebulæ or bright dashes; some of the shape of a fan, resembling an electric brush issuing from a lucid point; others of the cometic shape, with a seeming nucleus in the centre, or like cloudy stars, surrounded with a nebulæous atmosphere: a different sort again contain a nebulosity of the milky kind, like that wonderful inexplicable phenomenon about ο Orionis; while others shine with a fainter mottled kind of light, which denotes their being resolvable into stars.

"It is very probable that the great stratum called why the milky-way, is that in which the sun is placed, though milky-way perhaps not in the very centre of its thickness. We appear to gather this from the appearance of the galaxy, which seems to encompass the whole heavens, as it certainly must do if the sun is within the same. For suppose a number of stars arranged between two parallel planes, indefinitely extended every way, but at a given considerable distance from one another, and calling this a sidereal stratum, an eye placed somewhere within it will see all the stars in the direction of the planes of the stratum projected into a great circle, which will appear lucid on account of the accumulation of the stars, while the rest of the heavens at the sides will only seem to be scattered over with constellations, more or less.

(A) Mr Herschel's observations, on which this theory is founded, were made with a Newtonian reflector of 20 feet focal length, and an aperture of 18 inches.

(B) By this word we are to understand the apparent space in the heavens he could see at once through his telescope. Conclusions less crowded according to the distance of the planes or from the foregoing Apparations.

Thus in fig. 102, an eye at S within the stratum \(a b\), will see the stars in the direction of its length \(ab\), or height \(ed\), with all those in the intermediate situation, projected into the lucid circle ABCD; while those in the sides \(m e, n w\), will be seen scattered over the remaining part of the heavens at MVNW.

If the eye were placed somewhere without the stratum, at no very great distance, the appearance of the flares within it would assume the form of one of the lesser circles of the sphere, which would be more or less contracted to the distance of the eye; and if this distance were exceedingly increased, the whole stratum might at last be drawn together into a lucid spot of any shape, according to the position, length, and height of the stratum.

Let us now suppose, that a branch or smaller stratum should run out from the former in a certain direction, and let it also be contained between two parallel planes extended indefinitely onwards, but so that the eye may be placed in the great stratum somewhere before the separation, and not far from the place where the strata are still united; then will this second stratum not be projected into a bright circle like the former, but will be seen as a lucid branch proceeding from the first, and returning to it again at a certain distance less than a semicircle. Thus, in the same figure, the stars in the small stratum \(p g\) will be projected into a bright arch at PRRP, which, after its separation from the circle CBD, unites with it again at P.

What has been instanced in parallel planes may easily be applied to strata irregularly bounded, and running in various directions; for their projection will of consequence vary according to the quantities of the variations in the strata and the distance of the eye from the same. And thus any kind of curvatures, as well as various degrees of brightness, may be produced in the projections.

From appearances, then, as I observed before, we may infer, that the sun is most likely placed in one of the great strata of the fixed stars, and very probably not far from the place where some smaller stratum branches out from it. Such a supposition will satisfactorily, and with great simplicity, account for all the phenomena of the milky way; which, according to this hypothesis, is no other than the appearance of the projection of the stars contained in this stratum and its secondary branch. As a farther inducement to look on the galaxy in this point of view, let it be considered, that we can no longer doubt of its whitish appearance arising from the mixed lustre of the numberless stars that compose it. Now, should we suppose it to be an irregular ring of stars, in the centre nearly of which we must then suppose the sun to be placed, it will appear not a little extraordinary, that the sun, being a fixed star like those which compose this imagined ring, should just be in the centre of such a multitude of celestial bodies, without any apparent reason for this singular distinction; whereas, on our supposition, every star in this stratum, not very near the termination of its length or height, will be so placed as also to have its own galaxy, with only such variations in the form and lustre of it as may arise from the particular situation of each star.

Various methods may be taken to come to a knowledge of the sun's place in the sidereal stratum, one of which I have already begun to put in practice: I call it gauging the heavens, or the star-gauge. It consists in repeatedly taking the number of stars in ten Mercier's fields of view of my reflector very near each other; and by adding their sums, and cutting off one decimal on the right, a mean of the contents of the heavens in all the parts which are thus gauged are obtained. Thus it appears, that the number of stars increases very much as we approach the milky way; for in the parallel from 92 to 94 degrees north polar distance, and right ascension 15 h. 10', the star-gauge runs up from 9.4 stars in the field to 13.6 in about an hour and a half; whereas in the parallel from 78 to 80 degrees north polar distance, and R.A. 11, 12, 13, and 14 hours, it very seldom rises above 4. We are, however, to remember, that, with different instruments, the account of the gauges will be very different, especially on our supposition of the sun in a stratum of stars.

For let \(a b\), fig. 98, be the stratum, and suppose the small circle \(g b l k\) to represent the space into which, by the light and power of a given telescope, we are enabled to penetrate, and let GHI.K be the extent of another portion which we are enabled to visit by means of a larger aperture and power; it is evident, that the gauges with the latter instrument will differ very much in their account of stars contained at MN and at KG or LH, when with the former they will hardly be affected with the change from \(m n\) to \(k g\) or \(l k\).

The situation of the sun in the sidereal stratum will be found by considering in what manner the star-gauge agrees with the length of a ray involving in several directions about an assumed point, and cut off by the bounds of the stratum. Thus, in fig. 99, let S be the place of an observer; \(S r r r, S r r r\), lines in the plane \(r S r, r S r\), drawn from S within the stratum to one of the boundaries here represented by the plane AB. Then, since neither the situation of S nor the form of the limiting surface AB is known, we are to assume a point, and apply to it lines proportional to the several gauges that have been obtained, and at such angles from each other as they may point out: then will the termination of these lines delineate the boundary of the stratum, and consequently manifest the situation of the sun within the same.

In my late observations on nebulae, I soon found, Observations that I generally detected them in certain directions rather than in others: that the spaces preceding them were generally quite deprived of their stars, so as often to afford many fields without a single star in it: that the nebula generally appeared some time after among stars of a certain considerable size, and but seldom among very small stars: that when I came to one nebula, I generally found several more in the neighbourhood: that afterwards a considerable time passed before I came to another parcel. These events being often repeated in different altitudes of my instrument, and some of them at considerable distances from each other, it occurred to me that the intermediate spaces between the sweeps might also contain nebulae; and finding this to hold good more than once, I ventured to give notice. notice to my assistant at the clock, that 'I found myself on nebulous ground.' But how far these circumstances of vacant places preceding and following the nebulous strata, and their being as it were contained in a bed of stars sparingly scattered between them, may hold good in more distant portions of the heavens, and which I have not been yet able to visit in any regular manner, I ought by no means to hazard a conjecture.

I may venture, however, to add a few particulars about the direction of some of the capital strata or their branches. The well-known nebula of Cancer, visible to the naked eye, is probably one belonging to a certain stratum, in which I suppose it to be placed as to lie nearest to us. This stratum I shall call that of Cancer. It runs from Cancri towards the south, over the 67th nebula of the Connaissance des Temps, which is a very beautiful and pretty much compressed cluster of stars, easily to be seen by any good telescope; and in which I have observed above 200 stars at once in the field of view of my great reflector with a power of 157. This cluster appearing so plainly with any good common telescope, and being so near to the one which may be seen with the naked eye, denotes it to be probably the next in distance to that within the quartile formed by γ, η, δ. From the 67th nebula, the stratum of Cancer proceeds towards the head of Hydra; but I have not yet had time to trace it farther than the equator.

Another stratum, which perhaps approaches nearer to the solar system than any of the rest, and whose situation is nearly at right angles with the great sidereal stratum in which the sun is placed, is that of Coma Berenices, as I shall call it. I suppose the Coma itself to be one of the clusters in it, and that on account of its nearness it appears to be so scattered. It has many capital nebulæ very near it; and in all probability this stratum runs out a very considerable way. It may perhaps even make the circuit of the heavens, though very likely not in one of the great circles of the sphere; for unless it should chance to intersect the great sidereal stratum of the milky way before mentioned, in the very place in which the sun is stationed, such an appearance would hardly be produced. However, if the stratum of Coma Berenices should extend so far as I apprehend it may, the direction of it towards the north lies probably, with some windings, through the Great Bear onwards to Cassiopeia, thence through the Girdle of Andromeda and the Northern Fish, proceeding towards Cetus; while towards the south it passes through the Virgin, probably on to the tail of Hydra and Centaurus."

By a continued series of observations, Mr Herschel became confirmed in his notions; and in a succeeding paper * has given a sketch of his opinions concerning the interior construction of the heavens.—"That the milky way (says he) is a most extensive stratum of stars of various sizes, admits no longer of the least doubt; and that our sun is one of the heavenly bodies belonging to it is as evident. I have now viewed and gauged this shining zone in almost every direction, and find it composed of shining stars, whose number, by the account of those gauges, constantly increases and decreases in proportion to its apparent brightness to the naked eye. But in order to develope the ideas of the universe that have been suggested by my late observations, it will be best to take the subject from a point of view at a considerable distance both of space and time.

Let us then suppose numberless stars of various sizes scattered over an indefinite portion of space, in such a manner as to be almost equally distributed throughout the whole. The laws of attraction, which no doubt extend to the remotest regions of the fixed stars, will operate in such a manner as most probably to produce the following remarkable effects.

"I. It will frequently happen, that a star, being the star considerably larger than its neighbouring ones, will attract them more than they will be attracted by others that are immediately around them; by which means they will be in time, as it were, condensed about a centre; or, in other words, form themselves into a cluster of stars of almost a globular figure, more or less regularly so according to the size and original distance of the surrounding stars. The perturbations of these mutual attractions must undoubtedly be very intricate, as we may easily comprehend, by considering what Sir Isaac Newton has said, Princip. lib. i. prob. 38, et seq.; but in order to apply this great author's reasoning of bodies moving in ellipses to such as are here for a while supposed to have no other motion than what their mutual gravity has imparted to them, we must suppose the conjugate axes of these ellipses indefinitely diminished, whereby the ellipses will become straight lines.

II. The next case, which will happen almost as frequently as the former, is where a few stars, though not superior in size to the rest, may chance to be rather nearer each other than the surrounding ones; for here also will be formed a prevailing attraction in the combined centre of gravity of them all, which will occasion the neighbouring stars to draw together; not, indeed, so as to form a regular globular figure, but, however, in such a manner as to be condensed towards the common centre of gravity of the whole irregular clutter. And this construction admits of the utmost variety of shapes, according to the number and situation of the stars which first gave rise to the condensation of the rest.

III. From the composition and repeated conjunction of both the foregoing forms, a third may be derived, when many large stars, or combined small ones, are situated in long extended regular or crooked rows, hooks, or branches; for they will also draw the surrounding ones so as to produce figures of condensed stars—coarsely similar to the former, which gave rise to these condensations.

IV. We may likewise admit of still more extensive combinations; when, at the same time that a cluster of stars is forming in one part of space, there may be another collecting in a different, but perhaps not far distant, quarter, which may occasion a mutual approach towards their common centre of gravity.

V. In the last place, as a natural consequence of vacancies, the former cases, there will be great cavities or vacuums occupied by the retreat of the stars towards the various centres which attract them; so that, upon the whole, there is evidently a field of the greatest variety open for the mutual and combined attractions of the heavenly bodies to exert themselves in.

From this theoretical view of the heavens, which Conclusions has been taken from a point not less distant in time than in space, we will now retreat to our own retired station, in one of the planets attending a star in its great combination with numberless others; and in order to investigate what will be the appearances from this contracted situation, let us begin with the naked eye. The stars of the first magnitude, being in all probability the nearest, will furnish us with a step to begin our scale. Setting off, therefore, with the distance of Sirius or Arcturus, for instance, as unity, we will at present suppose, that those of the second magnitude are at double, those of the third at treble, the distance, &c. Taking it for granted, then, that a star of the seventh magnitude (the smallest supposed visible with the naked eye) is about seven times as far as one of the first, it follows, that an observer who is inclosed in a globular cluster of stars, and not far from the centre, will never be able with the naked eye to see to the end of it; for since, according to the above estimations, he can only extend his view to about seven times the distance of Sirius, it cannot be expected that his eyes should reach the borders of a cluster which has perhaps not less than 50 stars in depth everywhere around him. The whole universe to him, therefore, will be comprised in a set of constellations richly ornamented with scattered stars of all sizes: Or, if the united brightness of a neighbouring cluster of stars should, in a remarkable clear night, reach his sight, it will put on the appearance of a small, faint, whitish, nebulous cloud, not to be perceived without the greatest attention. Let us suppose him placed in a much extended stratum, or branching cluster of millions of stars, such as may fall under the third form of nebula already considered. Here also the heavens will not only be richly scattered over with brilliant constellations, but a shining zone or milky way will be perceived to surround the whole sphere of the heavens, owing to the combined light of those stars which are too small, that is, too remote to be seen. Our observer's sight will be so confined, that he will imagine this single collection of stars, though he does not even perceive the thousandth part of them, to be the whole contents of the heavens. Allowing him now the use of a common telescope, he begins to suspect that all the milkeness of the bright path which surrounds the sphere may be owing to stars. He perceives a few clusters of them in various parts of the heavens, and finds also that there are a kind of nebulons patches: but still his views are not extended to reach so far as to the end of the stratum in which he is situated; so that he looks upon these patches as belonging to that system which to him seems to comprehend every celestial object. He now increases his power of vision; and, applying himself to a close observation, finds that the milky way is indeed no other than a collection of very small stars. He perceives, that those objects which had been called nebulons, are evidently nothing but clusters of stars. Their number increases upon him; and when he resolves one nebula into stars, he discovers ten new ones which he cannot resolve. He then forms the idea of immense strata of fixed stars, of clusters of stars and of nebulons; till, going on with such interesting observations, he now perceives, that all these appearances must naturally arise from the confined situation in which we are placed. Confined it may justly be called, Conclusions though in no less a space than what appeared before to the foregoing, be the whole region of the fixed stars, but which now has assumed the shape of a crookedly branching nebula; not indeed one of the least, but perhaps very far from being the most considerable, of those numberless clusters that enter into the construction of the heavens."

Our author now proceeds to show that this theoretical view of the heavens is perfectly consistent with facts, and seems to be confirmed by a series of observations. Many hundreds of nebulons of the first and second forms are to be seen in the heavens; and their places, he says, will hereafter be pointed out; many of the third on nebulons, four described, and instances of the fourth related; a few of the cavities mentioned in the fifth particularized, though many more have been already observed: so that, "upon the whole (says he), I believe it will be found, that the foregoing theoretical view, with all its consequential appearances, as seen by an eye inclosed in one of the nebulons, is no other than a drawing from nature, wherein the features of the original have been closely copied: and I hope the resemblance will not be called a bad one, when it shall be considered how very limited must be the pencil of an inhabitant of so small and retired a portion of an indefinite system in attempting the picture of so unbounded an extent."

Mr Herschel next presents us with a long table of star-gauges, or accounts of the number of stars at once in the field of his telescope, which go as high as 588; after which he proposes the following

**Problem.**

"The stars being supposed nearly equally scattered, and their number, in a field of view of a known angular diameter, being given, to determine the length of the visual ray.

Here, the arrangement of the stars not being fixed upon, we must endeavour to find which way they may be placed so as to fill a given space most equally. Suppose a rectangular cone cut into frustums by many equidistant planes perpendicular to the axis; then, if one star be placed at the vertex and another in the axis at the first intersection, six stars may be set around it so as to be equally distant from one another and from the central star. These positions being carried on in the same manner, we shall have every star within the cone surrounded by eight others at an equal distance from that star taken as a centre. Fig. 100 contains four sections of such a cone distinguished by alternate shades; which will be sufficient to explain what sort of arrangement I would point out.

The series of the number of stars contained in the several sections will be 1, 7, 19, 37, 61, 91, &c., which, continued to n terms, the sum of it, by the differential method, will be \( na + \frac{n-1}{2} d' + \frac{n-1}{2} d'' + \frac{n-2}{2} d''' + \ldots \), where \( a \) is the first term, \( d', d'', d''' \), &c., the first, second, and third differences. Then, since \( a = 1 \), \( d' = 6 \), \( d'' = 6 \), \( d''' = 0 \), the sum of the series will be \( n^3 \).

Let \( S \) be the given number of stars; \( r \) the diameter of the base of the field of view; and \( B \) the diameter of the great rectangular cone; and by trigonometry we shall have..." have \( B = \frac{\text{Radius}}{\text{Tang. } \frac{1}{2} \text{ field}} \). Now, since the field of view of a telescope is a cone, we shall have its solidity to that of the great cone of the stars formed by the above construction, as the square of the diameter of the base of the field of view to the square of the diameter of the great cone, the height of both being the same; and the stars in each cone being in the ratio of the solidity, as being equally scattered, we have \( n = \sqrt{B^2 S} \); and the length of the visual ray \( = n - 1 \), which was to be determined.

Another solution of this problem, on the supposition of another arrangement of stars, is given; but Mr Herschel prefers the former.

From the data now laid down, Mr Herschel next endeavours to prove that the earth is the planet of a star belonging to a compound nebula of the third form. "I shall now (says he) proceed to show, that the stupendous sidereal system we inhabit, this extensive stratum, and its secondary branch, consisting of many millions of stars, is in all probability a detached nebula. In order to go upon grounds that seem to me to be capable of great certainty, they being no less than an actual survey of the boundaries of our sidereal system, which I have plainly perceived as far as I have yet gone round it, everywhere terminated, and in most places very narrowly too, it will be proper to show the length of my founding line, if I may so call it, that it may appear whether it was sufficiently long for the purpose.

"In the most crowded parts of the milky-way, I have had fields of view that contained no fewer than 588 stars, and these were continued for many minutes; so that in one quarter of an hour's time there passed no less than 116,000 stars through the field of view of my telescope. Now, if we compute the length of the visual ray, by putting \( S = 588 \), and the diameter of the field of view 15 minutes, we shall find

\[ n = \sqrt{B^2 S} = 498; \]

so that it appears the length of what I have called my Sounding Line, or \( n - 1 \), was not probably less than 497 times the distance of Sirius from the sun.

"It may seem inaccurate that we should found an argument on the stars being equally scattered, when, in all probability, there may not be any two of them in the heavens whose mutual distance shall be equal to that of any other two given stars; but it should be considered, that when we take all the stars collectively, there will be a mean distance which may be assumed as the general one; and an argument founded on such a supposition will have in its favour the greatest probability of not being far short of truth. And here I must observe, that the difference between a crowded place and a cluster (none of the latter being put into the gauge table), may easily be perceived by the arrangement as well as the size and mutual distance of the stars; for in a clutter they are generally not only resembling each other pretty nearly in size, but a certain uniformity of distance also takes place: they are more and more accumulated towards the centre, and put on all the appearances which we should naturally expect from a number of them collected into a group at a certain distance from us. On the other hand, the rich parts of the milky-way, as well as those in the distant broad parts of the stratum, consist of a mixture of stars of all possible sizes, that are seemingly placed without any particular apparent order. Perhaps we might recollect, that a greater condensation towards the centre of our system than towards the borders of it should be taken into consideration; but with a nebula of the third form, containing such various and extensive combinations as I have found to take place in ours, this circumstance, which in one of the first form would be of considerable moment, may, I think, be safely neglected.

"If some other high gauge be selected from the table, such as 472 or 344, the length of the visual ray will be found 461 and 415. And although in consequence of what has been said, a certain degree of doubt may be left about the arrangement and scattering of the stars, yet when it is recollected, that in those parts of the milky-way, where these high gauges were taken, the stars were neither so small nor so crowded as they must have been, on a supposition of a much farther continuance of them, when certainly a milky or nebulons appearance must have come on, I need not fear to have over-rated the extent of my visual ray; and indeed every thing that can be said to shorten it will only contract the limits of our nebula, as it has in most places been of sufficient length to go far beyond the bounds of it. Thus in the sides of our stratum, opposite to our situation in it, where the gauges often run below 5, our nebula cannot extend to 100 times our nebula, the distance of Sirius; and the same telescope which could show 588 stars in a field of view of 15 minutes, must certainly have presented me also with the stars in these situations, had they been there. If we should answer this by observing, that they might be at too great a distance to be perceived, it will be allowing that there must at least be a vacancy amounting to the length of a visual ray, not short of 400 times the distance of Sirius; and this is amply sufficient to make our nebula a detached one. It is true, that it would not be consistent confidently to affirm that we were on an island, unless we had found ourselves everywhere bounded by the ocean; and therefore I shall go no farther than the gauges will authorise: but considering the little depth of the stratum in all those places which have been actually gauged, to which must be added all the intermediate parts that have been viewed and found to be much like the rest, there is but little room to expect a connection between our nebula and any of the neighbouring ones. A telescope, with a much larger aperture than my present one, grasping together a greater quantity of light, and thereby enabling us to see farther into space, will be the surest means of completing and establishing the arguments that have been used; for if our nebula is not absolutely a detached one, I am firmly persuaded that an instrument may be made large enough to discover the places where the stars continue onwards. A very bright milky nebulosity must there undoubtedly come on, since the stars in a field of view will increase in the ratio of \( n^3 \) greater than that of the cube of the visual ray. Thus, if 588 stars in a given field of view are to be seen by a ray of 497 times the distance of Sirius, when this is lengthened to 1000, which is but little more than double the former, the number of stars in the same field of view will Conclusions will be no less than 4774; for when the visual ray \( r \) is given, the number of stars \( S \) will be \( \frac{n^3}{3} \); where \( n = r + 1 \); and a telescope with a threefold power of extending into space, or with a ray of 1500, which I think may easily be constructed, will give us 16,096 stars. Nor would these be so close, but that a good power applied to such an instrument might easily distinguish them; for they need not, if arranged in regular squares, approach nearer to each other than 6°.27; but the milky nebulosity I have mentioned, would be produced by the numberless stars beyond them, which, in one respect, the visual ray might also be said to reach. To make this appear, we must return to the naked eye; which, as we have before estimated can only see the stars of the seventh magnitude so as to distinguish them; but it is nevertheless very evident, that the united lustre of millions of stars, such as I suppose the nebula in Andromeda to be, will reach our sight in the shape of a very small faint nebulosity; since the nebula of which I speak may easily be seen in a fine evening. In the same manner, my present telescope, as I have argued, has not only a visual ray that will reach the stars at 497 times the distance of Sirius, so as to distinguish them, and probably much farther, but also a power of showing the united lustre of the accumulated stars that compose a milky nebulosity at a distance far exceeding the former limits; so that from these considerations it appears again highly probable, that my present telescope not showing such a nebulosity in the milky-way, goes already far beyond its extent; and consequently much more would an instrument, such as I have mentioned, remove all doubt on the subject, both by showing the stars in the continuation of the stratum, and by exposing a very strong milky nebulosity beyond them, that could no longer be mistaken for the dark ground of the heavens.

To these arguments, which rest on the firm basis of a series of observation, we may add the following considerations drawn from analogy. Among the great number of nebulae which I have now already seen, amounting to more than 900, there are many which in all probability are equally extensive with that which we inhabit; and yet they are all separated from each other by very considerable intervals. Some, indeed, there are that seem to be double and treble; and though with most of these it may be that they are at a very great distance from each other, yet we allow that some such conjunctions really are to be found; nor is this what we mean to exclude: But then these compound, or double nebulae, which are those of the third and fourth forms, still make a detached link in the great chain. It is also to be supposed, that there may be some thinly scattered solitary stars between the large interstices of nebulae; which being situated so as to be nearly equally attracted by the several clusters when they were forming, remain unassociated: and though we cannot expect to see these stars on account of their vast distance, yet we may well presume that their number cannot be very considerable in comparison to those that are already drawn into systems; which conjecture is also abundantly confirmed in situations where the nebulae are near enough to have their stars visible; for they are all insulated, and generally to be seen upon a very clear and pure ground, without any star near them that might be thought to belong to them. And though I have often seen them in beds of stars, yet from the size of these latter we may be certain, that they were much nearer to us than those nebulae, and belonged undoubtedly to our own system.

Having thus determined that the visible system of nature, by us called the universe, consisting of all the celestial bodies, and many more than can be seen by the naked eye, is only a group of stars or suns with their planets, constituting one of those patches called a nebula, and perhaps not one ten thousandth part of what is really the universe, Mr Herschel goes on to delineate the figure of this vast nebula, which he is of opinion may now be done; and for this purpose he gives a table, calculating the distance of the stars which form its extreme boundaries, or the length of the visual ray in different parts, by the number of stars contained in the field of his telescope at different times, according to the principles already laid down. He does not, however, as yet attempt the whole nebula, but of a particular section, represented fig. 160. "I have taken one (says he) which passes through the poles of our system, and is at rectangles to the conjunction of the branches, which I have called its length. The name of poles seemed to me not improperly applied to those points which are 90 degrees distant from a circle passing along the milky-way; and the north pole is here supposed to be situated in right ascension 186°, and polar distance (that is from the pole commonly so called) 58°. The section is one which makes an angle of 35 degrees with our equator, crossing it in 124° and 304° degrees. A celestial globe, adjusted to the latitude of 55° north, and having \( \sigma \) Ceti near the meridian, will have the plane of this section pointed out by the horizon." The visual rays are to be projected on the plane of the horizon of the latitude just mentioned, which may be done accurately enough by a globe adjusted in the manner directed. The stars in the border, which are marked larger than the rest, are those pointed out by the gauges. The intermediate parts are filled up by smaller stars, arranged in straight lines between the gauged ones. From this figure, which I hope is not a very inaccurate one, we may see that our nebula, as we observed before, is of the third form; that is, a very extensive, branching, compound congeries of many millions of stars, which most probably owes its origin to many remarkably large, as well as pretty closely scattered small stars, that may have drawn together the rest. Now, to have some idea of the wonderful extent of this system, I must observe, that this section of it is drawn upon a scale where the distance of Sirius is no more than the 80th part of an inch; so that probably all the stars, which in the finest nights we are able to distinguish with the naked eye, may be comprehended within a sphere drawn round the large star near the middle, representing our situation in the nebula of less than half a quarter of an inch radius."

Mr Herschel now proceeds to offer some further thoughts on the origin of the nebulous strata of the heavens; in doing which he gives some hints concerning the antiquity of them. "If it were possible (says he) to distinguish between the parts of an indefinitely extended whole, the nebula we inhabit might be said..." Conclusions to be one that has fewer marks of antiquity than any from the foregoing appearances.

To explain this idea perhaps more clearly, we should recollect, that the condensation of clusters of flares has been ascribed to a gradual approach; and whoever reflects on the number of ages that must have passed before some of the clusters that are to be found in my intended catalogue of them could be so far condensed as we find them at present, will not wonder if I ascribe a certain air of youth and vigour to many very regularly scattered regions of our sidereal stratum.

There are, moreover, many places in it which, if we may judge from appearances, there is the greatest reason to believe that the stars are drawing towards secondary centres, and will in time separate into clusters, so as to occasion many subdivisions. Hence we may surmise, that when a nebulous stratum consists chiefly of nebulae of the first and second forms, it probably owes its origin to what may be called the decay of a great compound nebula of the third form; and that the subdivisions which happened to it in length of time, occasioned all the small nebulae which sprung from it to lie in a certain range, according as they were detached from the primary one. In like manner, our system, after numbers of ages, may very possibly become divided, so as to give rise to a stratum of two or three hundred nebulae; for it would not be difficult to point out so many beginning or gathering clusters in it.

This throws a considerable light upon that remarkable collection of many hundreds of nebulae which are to be seen in what I have called the nebulous stratum in Coma Berenices. It appears from the extended and branching figure of our nebula, that there is room for the decomposed small nebulae of a large reduced former great one to approach nearer to us in the sides than in any other parts. Nay, possibly there might originally be another very large joining branch, which in time became separated by the condensation of the stars; and this may be the reason of the little remaining breadth of our system in that very place; for the nebulae of the stratum of the Coma are brightest and most crowded just opposite to our situation, or in the pole of our system. As soon as this idea was suggested, I tried also the opposite pole; where accordingly I have met with a great number of nebulae, though under a much more scattered form.

Some parts of our system indeed seem already to have sustained greater ravages of time than others: for instance, in the body of the Scorpion is an opening or hole, which is probably owing to this cause. It is at least four degrees broad; but its height I have not yet ascertained. It is remarkable, that the 80 Nebuleuses sans Etoiles of the Connaissance des Temps, which is one of the richest and most compressed clusters of small stars I remember to have seen, is situated just on the west border of it, and would almost authorise a suspicion that the stars of which it is composed were collected from that place, and had left the vacancy. What adds not a little to this surmise is, that the same phenomenon is once more repeated with the fourth cluster of the Connaissance des Temps; which is also on the western border of another vacancy, and has moreover a small miniature cluster, or easily resolvable nebula, of about 2½ minutes in diameter north, following it at no very great distance.

There is a remarkable purity or clearness in the heavens when we look out of our stratum at the sides; conclusions that is, towards Leo, Virgo, and Coma Berenices on the one hand, and towards Cetus on the other; whereas the ground of the heavens becomes troubled as we approach towards the length or height of it. These troubled appearances are easily to be explained by ascribing them to some of the distant straggling stars that yield hardly light enough to be distinguished. And I have indeed often experienced this to be the case, by examining these troubled spots for a long while together, when at last I generally perceived the stars which occasioned them. But when we look towards the poles of our system, where the visual ray does not graze along the side, the straggling stars will of course be very few in number; and therefore the ground of the heavens will assume that purity which I have always observed to take place in those regions.

Thus, then, according to Mr Herschel, the universe consists of nebulae, or innumerable collections of innumerable stars, each individual of which is a sun not only equal, but much superior to ours: at least if the words of Mr Nicholson have any weight; for he tells us, that "each individual sun is destined to give light to hundreds of worlds that revolve about it, but which can no more be seen by us, on account of their great distance, than the solar planets can be seen from the fixed stars." Yet (continues he), as in this unexplored, and perhaps unexplorable, abyss of space, it is no necessary condition that the planets should be of the same magnitudes as those belonging to our system, it is not impossible but that planetary bodies may be discovered among the double and triple stars.

Though in the above extracts from Mr Herschel's papers, the words condensation, clusters, &c. of stars frequently occur, we are by no means from thence to imagine that any of the celestial bodies in our nebula, are nearer to one another than we are to Sirius, whose distance is supposed not to be less than 400,000 times that of the sun from us, or 38 millions of millions of miles. The whole extent of the nebula being in some places near 500 times as great, must be such, that the light of a star placed at its extreme boundary, supposing it to fly with the velocity of 12 millions of miles every minute, must have taken near 3000 years to reach us. Mr Herschel, however, is by no means of opinion, that our nebula is the most considerable in the universe. "As we are used (says he) to call the appearance of the heavens, where it is surrounded with a bright zone, the milky-way, it may not be amiss to point out some other very remarkable nebulae, which cannot well be less, but are probably much larger, than our own system; and being also extended, the inhabitants of the planets that attend the stars which compose them, must likewise perceive the same phenomena: for which reason they may also be called milky-ways, by way of distinction.

My opinion of their size is grounded on the following observations: There are many round nebulae of the first form, of about five or six minutes in diameter, of the stars of which I can see very distinctly; and on nebulae comparing them with the visual ray calculated from some of my long gauges, I suppose by the appearance of the small stars in those gauges, that the centres of these round nebulae may be 600 times the distance of Sirius. Sirius from us." He then goes on to tell us, that the stars in such nebulae are probably twice as much condensed as those of our system; otherwise the centre of it would not be less than 6000 times the distance of Sirius from us; and that it is possibly much under-rated by supposing it only 600 times the distance of that star.

"Some of these round nebulae (says Mr Herschel), have others near them, perfectly similar in form, colour, and the distribution of stars, but of only half the diameter: and the stars in them seem to be doubly crowded, and only at about half the distance from each other. They are indeed so small, as not to be visible without the utmost attention. I suppose these miniature nebulae to be at double the distance of the first. An instance equally remarkable and instructive is a case where, in the neighbourhood of two such nebulae as have been mentioned, I met with a third similar, resolvable, but much smaller and fainter nebula. The stars of it are no longer to be perceived; but a resemblance of colour with the former two, and its diminished size and light, may well permit us to place it at full twice the distance of the second, or about four or five times the distance of the first. And yet the nebulosity is not of the milky kind; nor is it so much as difficultly resolvable or colourless. Now in a few of the extended nebulae, the light changes gradually, so as from the resolvable to approach to the milky kind; which appears to me an indication, that the milky light of nebulae is owing to their much greater distance. A nebula, therefore, whose light is perfectly milky, cannot well be supposed to be at less than six or eight thousand times the distance of Sirius; and though the numbers here assumed are not to be taken otherwise than as very coarse estimates, yet an extended nebula, which in an oblique situation, where it is possibly foreshortened by one half, two-thirds, or three-fourths of its length, subtends a degree or more in diameter, cannot be otherwise than of a wonderful magnitude, and may well outvie our milky-way in grandeur."

Mr Herschel next proceeds to give an account of several remarkable nebulae, and then concludes thus:

"Now, what great length of time must be required to produce these effects (the formation of nebulae) may easily be conceived, when, in all probability, our whole system, of about 800 stars in diameter, if it were seen at such a distance that one end of it might assume the resolvable nebulosity, would not, at the other end, present us with the irresolvable, much less with the colourless and milky sort of nebulosities." Great indeed must be the length of time requisite for such distant bodies to form combinations by the laws of attraction, since, according to the distances he has assumed, the light of some of his nebulae must be thirty-five or forty-eight thousand years in arriving from them to us. It would be worth while then to enquire, whether attraction is a virtue propagated in time or not; or whether it moves quicker or slower than light?

In the course of Mr Herschel's observations and inquiries concerning the structure of the heavens, an objection occurred, that if the different systems were formed by the mutual attractions of the stars, the whole would be in danger of destruction by the falling of them one upon another. A sufficient answer to this, he thinks, is, that if we can really prove the system of the universe to be what he has said, there is no doubt but that the great Author of it has amply provided for the preservation of the whole, though it should not appear to us in what manner this is effected. Several circumstances, however, he is of opinion, manifestly tend to a general preservation; as, in the first place, the indefinite extent of the sidereal heavens; which must produce a balance that will effectually secure all the great parts of the whole from approaching to each other. "There remains then (says he) only to see how the particular stars belonging to separate clusters are prevented from rushing on to their centres of attraction." This he supposes may be done by projectile forces; "the admission of which will prove such a barrier against the seeming destructive power of attraction, as to secure from it all the stars belonging to a cluster, if not for ever, at least for millions of ages. Besides, we ought perhaps to look upon such clusters, and the destruction of a star now and then in some thousands of ages, as the very means by which the whole is preserved and renewed. These clusters may be the laboratories of the universe, wherein the most salutary remedies for the decay of the whole are prepared."

In speaking of the planetary nebulae, by which name he distinguishes those spots that are all over equally luminous, he says, "If we should suppose them to be nebulae, single stars with large diameters, we shall find it difficult to account for their not being brighter, unless we should admit that the intrinsic light of some stars may be very much inferior to that of the generality; which, however, can hardly be imagined to extend to such a degree. We might suppose them to be comets about their aphelion, if the brightness, as well as magnitude of their diameters, did not oppose this idea: so that, after all, we can hardly find any hypothesis so probable as that of their being nebulae; but then they must consist of stars that are compressed and accumulated in the highest degree. If it were not perhaps too hazardous to pursue a former surmise of a renewal in what I figuratively called the Laboratories of the Universe, the stars forming these extraordinary nebulae, by some decay or waste of nature being no longer fit for their former purposes, and having their projectile forces, if any such they had, retarded in each other's atmosphere, may rush at last together; and, either in succession or by one general tremendous shock, unite into a new body. Perhaps the extraordinary and sudden blaze of a new star in Cassiopeia's chair, in 1572, might possibly be of such a nature. If a little attention to these bodies should prove that, having no annual parallax, they belong most probably to the class of nebulae, they may then be expected to keep their station better than any one of the stars belonging to our system, on account of their being probably at a very great distance."

Having thus at length got through the conjectures and theories concerning the nature and situations of the fixed heavenly bodies, we must now proceed to consider those projectile forces which are supposed necessary for the preservation of the system of Nature, and to prevent the stars from falling upon one another more frequently than they do. It was first suspected by Dr Halley, that many of the stars which we call fixed, are really in motion, though that motion is either so slow Conclusions in itself; or their distance is so great, that it can scarce be perceptible in half a century. It is, however, now confirmed by astronomical observations, that Arcturus, Sirius, Aldebaran, Procyon, Castor, Rigel, Altair, and many others, are actually in motion: which consideration, with the length of time necessary to show any change of place in bodies at such extreme distance, with the lateness of any observations on this head, would lead us (says Mr Herschel) to suppose that there is not one fixed star in the heavens:" but "many other reasons (adds he) will render this so obvious, that there can hardly remain a doubt of the general motion of all the starry systems; consequently of the solar one among the rest.

"I might begin with principles drawn from the theory of attraction, which evidently oppose every idea of absolute rest in any one of the stars, when once it is known that some of them are in motion: for the change that must arise by such motion, in the value of a power which acts inversely as the squares of the distances, must be felt in all the neighbouring stars; and if these be influenced by the motion of the former, they will again affect those that are next to them, and so on, till all are in motion. Now, as we know several stars in divers parts of the heavens do actually change their place, it will follow, that the motion of our solar system is not a mere hypothesis. And what will give additional weight to this consideration is, that we have the greatest reason to suppose most of those very stars which have been observed to move, to be such as are nearest to us; and therefore their influence on our situation would alone prove a powerful argument in favour of the proper motion of the sun had it been originally at rest."

After enumerating a great many changes, which, from his own observation, have happened among the fixed stars, and of which we have already given an account, "Does it not seem natural (says he), that these observations should cause a strong suspicion that most probably every star in the heaven is more or less in motion?" And though we have no reason to think that the disappearance of some stars, or new appearance of others, nor indeed that the frequent changes in the magnitude of so many of them, are owing to their change of distance from us by proper motions, which could not occasion these phenomena without being inconceivably quick; yet we may well suppose, that motion is some way or other concerned in producing these effects. A slow motion, for instance, in an orbit round some large opaque body, where the star which is lost or diminished in magnitude might undergo occasional occultations, would account for some of those changes; while others might perhaps be owing to the periodical return of some large spots on that side of the surface which is alternately turned towards us by the rotatory motion of the star. The idea, also, of a body much flattened by a quick rotation, and having a motion similar to the moon's orbit by a change of the place of its nodes, whereby more of the luminous surface would one time be exposed to us than another, tends to the same end: for we cannot help thinking with Mr de la Lande (Mem. 1776), that the same force which gave such rotations, would probably also produce motions of a different kind by a translation of the centre. Now, if the proper motion of the stars in general be once admitted, who can refuse to allow that our sun, with all its conclusions planets and comets, that is, the solar system, is no less from the foregoing appearances liable to such a general agitation as we find to obtain among the rest of the celestial bodies?

"Admitting this for granted, the greatest difficulty will be, how to discern the proper motion of the sun among so many other and variously compounded motions of the stars. This is an arduous task indeed; but I shall point out a method of detecting the direction of the motion and quantity of the supposed proper motion of the sun by a few geometrical deductions; and at the same time show, by an application of them to some known facts, that we have already some reason to guess which way the solar system is probably tending its course.

"Suppose the sun to be at S, fig. 101, the fixed stars to be dispersed in all possible directions and distances around, at s₁, s₂, s₃, etc. Now, setting aside the proper motion of the stars, let us first consider what will be the consequence of a proper motion in the sun, and let it move in a direction from A towards B. Suppose it now arrived at C: here, by a mere inspection of the figure, it will be evident, that the stars s₁, s₂, s₃, which were before seen at a₁a₂a₃, will now, by the motion of the sun from S to C, appear to have gone in a contrary direction, and be seen at b₁b₂b₃; that is to say, every star will appear more or less to have receded from the point B, in the order of the letters ab₁, ab₂, ab₃. The converse of this proposition is equally true; for if the stars should all appear to have had a retrograde motion with respect to the point B, it is plain, on supposition of their being at rest, the sun must have a direct motion towards the point B, to occasion all these appearances. From a due consideration of what has been said, we may draw the following inferences:

"1. The greatest, or total systematical parallax of the fixed stars (fig. 103), will fall upon those that are in the line D E, at rectangles to the direction A B of the sun's motion.

"2. The partial systematical parallax of every other star s₁, s₂, s₃, not in the line D E, will be to the total parallax as the sine of the angle BS₁a₁, being the star's distance from that point towards which the sun moves, to radius.

"3. The parallax of stars at different distances will be inversely as these distances; that is, one half at double the distance, one third at three times, and so on; for the subtense SC remaining the same, and the parallactic angle being very small, we may admit the angle S₁C to be inversely as the side S₁s₁, which is the star's distance.

"4. Every star at rest, to a system in motion, will appear to move in a direction contrary to that which the system has. Hence it follows, that if the solar system be carried towards any star situated in the ecliptic, every star, whose angular distance in antecedentia (reckoned upon the ecliptic from the star towards which the system moves) is less than 180 degrees, will decrease in longitude; and that, on the contrary, every star, whose distance from the same star (reckoned upon the ecliptic, but in consequentia) is less than 180 degrees, will increase in longitude in both cases, without alteration of latitude.

"The immense regions of the fixed stars may be considered as an infinitely expanded globe, having the..."

View of the proportional Magnitudes of the Planetary Orbits.

Orbit of the Georgium Sidus.

Proportional Magnitude of the Primary Planets:

Saturn

Jupiter

Georgium Sidus

Mars

Earth

Venus

Mercury

Apparent Magnitude of the Sun seen from each Planet:

From Mercury

From Venus

From Earth

From Mars

From Jupiter

From Saturn

From Georgium Sidus

NB. The proportional Magnitude of the Sun, with respect to the figures of the Planets here given, is represented by the Circle of Saturn's Orbit marked γ. Conclusions solar system for its centre. The most proper method therefore of finding out the direction of the motion of the sun is, to divide our observations on the systematical parallax of the fixed stars into three principal zones. These, for the convenience of fixed instruments, may be assumed so as to let them pass around the equator and the solstitial colures, every one being at rectangles to the other two, according to the three dimensions of solids." Our author, then, having informed us that observations on double stars are most proper for ascertaining this point, gives an account of three zones he has marked out for this purpose; the equatorial zone, containing 150 double stars; that of the equinoctial colure, extending 10 degrees of a great circle on each side, as far as it is visible on our hemisphere, which will contain about 70 double stars; and that of the solstitial colure, including 120, besides a zone of the ecliptic containing a great many double stars which may undergo occultations by the moon. It is of the same extent, and includes about 120 double stars.

To apply this theory, it is necessary, in the first place, to observe, that the rules of philosophy direct us to refer all phenomena to as few and simple principles as are sufficient to explain them. Astronomers, therefore, having already observed what they call a proper motion in several of the fixed stars, and which may be supposed common to them all, ought to resolve it, as far as possible, into a single and real motion of the solar system, as far as that will answer the known facts; and only to attribute to the proper motion of each particular star the deviations from the general law which the stars seem to follow.

Dr Maskelyne informs us, that the proper motions in right ascension of Sirius, Castor, Procyon, Pollux, Regulus, Arcturus, and α Aquilae, are as follow—c°.63; c°.28; c°.80; c°.93; c°.41; c°.40; and + c°.57. Two of them, Sirius and Arcturus, have also a change of declination; viz. 1°.20 and 2°.01; both southward. Let now fig. 104 represent an equatorial zone with the abovementioned stars referred to it, according to their respective right ascensions, having the solar system in its centre. Assume the direction A B from a point somewhere not far from the 77th degree of right ascension to its opposite 257th degree, and suppose the sun to move in that direction from S towards E, then will that one motion answer that of all the stars together; for if the supposition be true, Arcturus, Regulus, Pollux, Procyon, Castor, and Sirius, should appear to decrease in right ascension, while α Aquilae, on the contrary, should appear to increase. Moreover, suppose the sun to ascend at the same time, in the same direction, towards some point in the northern hemisphere, for instance towards the constellation Hercules; then will also the observed change of declination of Sirius and Arcturus be resolved into the single motion of the system. Many difficulties indeed yet remain; such as the correspondence of the exact quantity of motion observed in each star, with what will be assigned to it by this hypothesis. But it is to be remembered, that the very different and still unknown distances of the fixed stars must, for a good while yet, leave us in the dark as to the strict application of the theory; and that any deviation from it may easily be accounted for from the still unknown real proper motion of the stars; for if the solar system have in reality the motion now ascribed to it, then what astronomers have already observed concerning the change of place of the stars, and have called their proper motion, will become only an apparent motion; and future observation must still point out, by the deviations from the general law, which the stars will follow in those apparent motions, what may be their real proper motions, as well as relative distances. "But (says Mr Herschel) lest I should be censured for admitting so new and capital a motion upon too slight a foundation, I must observe, that the concurrence of those seven principal stars cannot but give some value to an hypothesis that will simplify the celestial motions in general. We know that the sun, at the distance of a fixed star, would appear like one of them; and from analogy we conclude the fixed stars to be suns. Now, since the apparent motions of those seven stars may be accounted for, either by supposing them to move in the manner they appear to do, or else by supposing the sun alone to have a motion in a direction somehow not far from that which I have assigned to it, I think we are no more authorized to suppose the sun at rest than we should be to deny the diurnal motion of the earth; excepting in this respect, that the proofs of the latter are very numerous, whereas the former rest only on a few, though capital testimonies."

The following table, taken from De la Lande, of Change of the change of right ascension and declination of twelve right ascension and declination of twelve stars, is brought as an additional proof of this doctrine.

| Names of Stars | Change of R. A. | Change of Declin. | |----------------|----------------|------------------| | Arcturus | - 1' 11'' | 1' 53'' | | Sirius | - 37 | 52 | | β Cygni | - 3 | 49 | | Procyon | - 33 | 47 | | ε Cygni | + 20 | 34 | | γ Arietis | - 14 | 29 | | γ Gemini | - 8 | 24 | | Aldebaran | + 3 | 18 | | β Gemini | - 48 | 16 | | γ Piscium | + 53 | 7 | | α Aquilae | + 32 | 4 | | α Gemini | - 24 | 1 |

Fig. 105 represents them projected on the plane of the equator. They are all in the northern hemisphere except Sirius, which must be supposed to be viewed in the concave part of the opposite half of the globe, while the rest are drawn on the convex surface. Regulus being added to that number, and Castor being double, we have 14 stars; and every star's motion except Regulus, being assigned in declination as well as right ascension, we have no fewer than 27 given motions to account for. Now, by assuming a point somewhere near α Hercules, and supposing the sun to have the stars a proper motion towards that part of the heavens, we shall account for 22 of these motions. For β Cygni, α Aquilae, ε Cygni, γ Piscium, γ Arietis, and Aldebaran, ought, upon the supposed motion of the sun, to have an apparent progression according to the hour-circle XVIII, XIX, XX, &c., or to increase in right ascension; while Arcturus, Regulus, the two stars of α Geminorum, Pollux, Procyon, Sirius, and γ Geminorum, should apparently go back in the order XVI, XV, XIV, &c., of the hour-circle, so as to decrease in right ascension. But according to De la Lande's table, excepting ε Cygni and γ Arietis, all their motions... Conclusions from the foregoing Appearances.

Some circumstances in the quantity of these motions also deserve our notice. In the first place, Arcturus and Sirius being the largest of the stars, and therefore probably the nearest, ought to have the greatest apparent motion both in right ascension and declination; which is agreeable to observation, as appears from the table. 2. With regard to the right ascension only, Arcturus being better situated to show its motion, ought to have it much greater; as we find it actually has. Aldebaran, both badly situated and considerably smaller, ought, according to the same rule, to show but little motion, &c.; all of which is conformable to the table. A very striking agreement with the hypothesis may also be observed in Castor and Pollux, both of which are pretty well situated; and accordingly we find that Pollux, for the size of the star, shows as much motion in right ascension as we could expect; though it is remarkable that Castor, though equally well placed, shows no more than half the motion by the table. This is seemingly contrary to the hypothesis; but it must be remembered, that Castor is a double star, and the two of which it consists are nearly equal to each other in lustre; so that, as we can allow only half the light to each, there is a strong presumption of their being at twice the distance of Pollux, which agrees very well with observation. It might also be observed, that we should be involved in great difficulty by supposing the motion of Castor really to be in the star: for how extraordinary must be the concurrence, that two stars, viz. those that make up this apparently single one, should both have a proper motion so exactly alike, that in all our observations hitherto, we have not found them disagree a single second either in right ascension, or in declination for 50 years together?

In a postscript to this paper on the motion of the solar system, Mr Herschel brings several additional confirmations of his hypothesis from the works of Mr Mayer. These contain a catalogue of the places of 80 stars observed by Mr Mayer in 1756, and whose places he compared with those of the same stars given by Roemer in 1706. From the goodness of the instrument with which Mr Roemer made his observations, Mr Mayer gives it as his opinion, that where the disagreement in the place of a star is but small, it may be attributed to the imperfection of the instrument, but that when it amounts to 10" or 15", it is a very probable indication of motion in such a star; and he adds, that when the disagreement is so much as in some stars which he names (among which is Fomalhaut, where the difference is 21" in 50 years), he has not the least doubt of a proper motion. The following tables are extracted from Mr Mayer's work; one contains the stars whose motion agrees with Mr Herschel's hypothesis; the other those that disagree with it, and whose phenomena must therefore be either ascribed to a proper motion of the stars themselves, or to some other more hidden cause.

Table I.

| Names of Stars | Motion in R.A. | Motion of Declin. | |----------------|---------------|------------------| | β Cete | + 32 | - | | α Arietis | + 10 | - | | δ Ceti | + 15 | - | | α Ceti | + 16 | - | | α Persei | + 16 | - | | η Pleiadium | - | - | | γ Eridani | + 14 | - | | ε Tauri | - | - | | α Auriga | + 11 | - | | β Orionis | inf. | inf. | | β Tauri | - | - | | ξ Hydra | - | - | | ε Leporis | - | - | | ε Ursae Majoris| - | - | | α Serpentis | inf. | - | | γ Draconis | + 12 | - | | α Lyrae | inf. | - | | γ Aquila | - | - | | γ Capricorn | + 19 | - | | ε Pegasi | - | - | | δ Capricorn | + 24 | - | | α Aquar. | + 13 | - | | α Orionis | inf. | - | | μ Geminorum | - | - | | ε Navis | - | - | | β Cancri | - | - | | ε Ursae Majoris| - | - | | ε Pegasi | - | - | | Fomalhaut | + 21 | - | | β Pegasi | + 12 | - | | α Androm. | - | - | | β Cassiopeia | + 34 | - |

Table II.

| Names of Stars | Motion in R.A. | Motion in Declin. | |----------------|---------------|------------------| | Polaris | - | + 13 | | γ Ceti | - | + 14 | | β Persei | - | + 10 | | ε Leporis | - | + 11 | | μ Geminor. | - | + 15 | | ε Canis Major. | - | + 10 | | ξ Hydra | - | + 24 | | α Hydrae | - | + 13 | | β Herculis | + 14 | - | | γ Cygni | - | + 13 | | ε Pegasi | - | + 14 | | ε Pegasi | - | + 20 |

"From the first table, (says Mr Herschel), we gather that the principal stars, Lucida Lyrae, Capella, α Orionis, Rigel, Fomalhaut, α Serpentarii, α Aquarii, α Arietis, α Persei, α Andromedae, α Tauri, α Ceti, and 20 more of the most distinguished of the second and third rank of the stars, agree with our proposed solar motion; when, on the contrary, the second table contains but a few stars, and not a single one of the first magnitude among them to oppose it. It is also remarkable, that many stars of the first table agree both in right ascension and declination with the supposition of a solar motion; whereas there is not one among those of the second table which opposes it in both directions. This seems to indicate, that the solar motion, in some..." Conclusions of them at least, has counteracted, and thereby destroyed from the foregoing Appearances.

The effect of their own proper motion in one direction, so as to render it insensible; otherwise it would appear improbable, that eight stars out of twelve, contained in the latter table, should only have a motion at rectangles, or in opposition to any one given direction. The same may also be said of 19 stars of the former table, that only agree with the solar motion one way, and are as to lense at rest in the other direction; but these singularities will not be near so remarkable when we have the motion of the sun to compound with their own proper motions.

The motions of α Lyrae and α Ursae Majoris towards the north, are placed in the first table: to understand the reason of which, it will be necessary to point out the general law by which the apparent declinations of the stars at present under consideration are governed. Let an arch of 90° be applied to a sphere representing the fixed stars, so as always to pass through the apex of the solar motion: then, while one end of it is drawn along the equator, the other will describe on the spherical surface a curve which will pass through the pole of the equator, and return into itself at the apex. This curve, not taken notice of by other authors, Mr Herschel calls a spherical conchoid, from the manner in which it is generated. The law then is, that all the stars in the northern hemisphere, situated within the nodated part of the conchoid, will seem to go to the north by the motion of the solar system towards its apex, the rest will appear to go southwards. A similar curve is to be delineated in the southern hemisphere. Mr Herschel then shows a method of finding whether any star whose place in the heavens is known, will fall without or within the conchoid; after which he accounts for the want of sensible motion in α Lyrae and α Orionis in right ascension, and of Rigel both in right ascension and declination, in the following manner:

"These stars are so bright, that we may reasonably suppose them to be among those that are nearest to us: and if they had any considerable motion, it would most likely have been discovered, since the variations of Sirius, Arcturus, Procyon, Castor, and Pollux, &c., have not escaped our notice. Now, from the same principle of the motion of the solar system, by which we have accounted for the apparent motion of the latter stars, we may account for the apparent rest of the former. Those two bright stars, α Lyrae and α Orionis, are placed so near the direction of the assigned solar motion, that from the application of the second theorem (p. 236), their motion ought to be insensible in right ascension, and not very considerable in declination; all which is confirmed by observations. With respect to Rigel and α Serpentarii, admitting them both as stars large enough to have shown a proper motion, were their situation otherwise than it is, we find that they also should be apparently at rest in right ascension; and Rigel, having southern declination, and being a less considerable star than α Orionis, which shows but 1′′ motion towards the south in 50 years, its apparent motion in declination may on that account be also too small to become visible."

Our author concludes with a remarkable passage from Mayer, to the following purpose, viz. "If it be possible that the sun has any proper motion of his own, the stars in that part of the heavens towards which he moves, must appear to open and recede from each other, while on the other hand, those on the opposite side will seem to contract their distances, and come nearer each other." Now (says Mr Herschel), if we recollect what has been said of the motion of the stars, we find that those, towards which I suppose the solar system to move, do really recede from each other: for instance, Arcturus from α Lyrae, α Aquilae and α Aquarii from α Serpentarii and α Ursae Majoris; and, on the contrary, those in the opposite part of the heavens do really come nearer each other, as Sirius to Aldebaran, Procyon to α Arietis, Castor, Pollux, Regulus, &c. to α Ceti, α Persei, α Andromedae, &c. It must be added however, that we cannot expect immediately to perceive any effects of this motion, excepting in such stars as are nearest to us. But as we have at present no other method of judging of the relative distance of the fixed stars than from their apparent brightness, those that are most likely on that account to be affected by a parallax arising from the motion of the solar system, are the very stars which have been pointed out from Mayer's own table."

With regard to the quantity of motion in the solar system, or the velocity with which the sun and planets change their places in absolute space, Mr Herschel proposes only a few distant hints. "From the annual parallax of the fixed stars (says he) which from my own observations I find much less than it has hitherto been thought to be, we may certainly admit, that the diameter of the earth's orbit, at the distance of Sirius or Arcturus, would not nearly subtend an angle of one second; but the apparent motion of Arcturus, if owing to a translation of the solar system, amounts to no less than 2′′.7 a-year, as will appear if we compound the two motions of 1′′.1″ in right ascension, and 1′′.55″ in declination into one single motion, and reduce it into an annual quantity. Hence we may, in a general way, estimate, that the solar motion can certainly not be less than that which the earth has in her annual orbit."

Sect. IV. Of the different Systems by which the Celestial Phenomena have been accounted for.

In treating of the various systems which have been invented in different ages, we do not mean to give an account of the various absurdities that have been broached by individuals on this subject; but shall confine ourselves to those systems which have been of considerable note, and been generally followed for a number of years. Concerning the opinions of the very first astronomers about the system of nature, we are necessarily as ignorant as we are of those astronomers themselves. Whatever opinions are handed down to us, must be of a vastly later date than the introduction of astronomy among mankind. If we may hazard a conjecture, however, we are inclined to think that the first opinions on this subject were much more just than those that were held afterwards for many ages. We are told that Pythagoras maintained the motion of the earth, which is now universally believed, but that time appears to have been the opinion of only a few detached individuals of Greece. As the Greeks borrowed many things from the Egyptians, and Pythagoras had travelled into Egypt and Phoenice, it is probable he might receive an account of this hypothesis from thence: but whether he did so or not, we have... Of the different systems by which the celestial phenomena have been accounted for, there is no means of knowing, neither is it of any importance whether he did or not. Certain it is, however, that this opinion did not prevail in his days, nor for many ages after. In the 3rd century after Christ, the very name of the Pythagorean hypothesis was superseded by a system erected by the famous geographer and astronomer Claudius Ptolemy. This system, which commonly goes by the name of the Ptolemaic, he seems not to have originally invented, but adopted as the prevailing one of that age; and perhaps made it somewhat more consistent than it was before. He supposed the earth at rest in the centre of the universe. Round the earth, and nearest to it of all the heavenly bodies, the moon performed its monthly revolutions. Next to the moon was placed the planet Mercury; then Venus; and above that the Sun, Mars, Jupiter, and Saturn, in their proper orbits; then the sphere of the fixed stars; above these, two spheres of what he called crystalline heavens; above these was the primum mobile, which, by turning round once in 24 hours, by some unaccountable means or other, carried all the rest along with it. This primum mobile was encompassed by the empyrean heaven, which was of a cubic form, and the seat of angels and blessed spirits. Besides the motions of all the heavens round the earth once in 24 hours, each planet was supposed to have a particular motion of its own; the moon, for instance, once in a month, performed an additional revolution, the sun in a year, &c. See Fig. 150.

It is easy to see, that, on this supposition, the confused motions of the planets already described could never be accounted for. Had they circulated uniformly round the earth, their apparent motion ought always to have been equal and uniform, without appearing either stationary or retrograde in any part of their courses. In consequence of this objection, Ptolemy was obliged to invent a great number of circles, interfering with each other, which he called epicycles and eccentrics. These proved a ready and effectual salvo for all the defects of his system; as, whenever a planet was deviating from the course it ought on his plan to have followed, it was then only moving in an epicycle or an eccentric, and would in due time fall into its proper path. As to the natural causes by which the planets were directed to move in these epicycles and eccentrics, it is no wonder that he found himself much at a loss, and was obliged to have recourse to divine power for an explanation, or, in other words, to own that his system was unintelligible.

This system continued to be in vogue till the beginning of the 16th century, when Nicolaus Copernicus, a native of Thorn (a city of regal Prussia), and a man of great abilities, began to try whether a more satisfactory manner of accounting for the apparent motions of the heavenly bodies could not be obtained than was afforded by the Ptolemaic hypothesis. He had recourse to every author upon the subject, to see whether any had been more consistent in explaining the irregular motions of the stars than the mathematical schools; but he received no satisfaction, till he found first from Cicero, that Nicetas the Syracusan had maintained the motion of the earth; and next from Plutarch, that others of the ancients had been of the same opinion. From the small hints he could obtain from the ancients, Copernicus then deduced a most complete system, capable of solving every phenomenon in a satisfactory manner. From him this system hath ever afterwards been called the Copernican, and represented fig. 152. Here the sun which is supposed to be in the centre; next him revolves the celestial planet Mercury; then Venus; next, the Earth, with the Moon; beyond these, Mars, Jupiter, and Saturn; and far beyond the orbit of Saturn, he supposed the fixed stars to be placed, which formed the boundaries of the visible creation.

Though this hypothesis afforded the only natural and satisfactory solution of the phenomena which so much perplexed Ptolemy's system, it met with great opposition at first; which is not to be wondered at, considering the age in which he lived. Even the famous astronomer Tycho Brahe could never attest to the earth's motion, which was the foundation of Copernicus's scheme. He therefore invented another system, whereby he avoided the ascribing of motion to the earth, and at the same time got clear of the difficulties with which Ptolemy was embarrassed. In this system, the earth was supposed the centre of the orbits of the sun and moon; but the sun was supposed to be the centre of the orbits of the five planets; so that the sun with all the planets were by Tycho Brahe supposed to turn round the earth, in order to save the motion of the earth round its axis once in 24 hours. This system was never much followed, the superiority of the Copernican scheme being evident at first sight.

The system of Copernicus coming soon into universal credit, philosophers began to inquire into the causes of the planetary motions; and here, without entering upon what has been advanced by detached individuals, we shall content ourselves with giving an account of the three famous systems, the Cartesian, the Newtonian, and what is sometimes called the Mechanical system.

Des Cartes, the founder of that system which since his time has been called the Cartesian, flourished about the beginning of the 17th century. His system seems to have been borrowed from the philosophers Democritus and Epicurus; who held, that everything was formed by a particular motion of very minute bodies called atoms, which could not be divided into smaller parts. But though the philosophy of Des Cartes resembled that of the Corporealists in accounting for all the phenomena of nature merely from matter and motion; he differed from them in supposing the original parts of matter capable of being broken. To this property his Materia Subtilis owes its origin. To each of his atoms, or rather small masses of matter, Des Cartes attributed a motion on its axis, and likewise maintained that there was a general motion of the whole matter of the universe round like a vortex or whirlpool. From this complicated motion, those particles, which were of an angular form, would have their angles broke off; and the fragments which were broke off being smaller than the particles from which they were abraded, behaved to form a matter of a more subtile kind than that made of large particles; and as there was no end of the attrition, different kinds of matter of all degrees of fineness would be produced. The finest forts, he thought, would naturally separate themselves from the rest, and be accumulated in particular places. The finest of all would therefore be collected in the sun, which was the centre of the universe, whose vortex was the whole ethereal matter in the cre- Of the different systems by which the celestial phenomena have been accounted for.

As all the planets were immersed in this vortex, they behoved to be carried round by it, in different times, proportioned to their distances; those which were nearest the sun circulating the most quickly; and those farther off more slowly; as those parts of a vortex which are farthest removed from the centre are observed to circulate more slowly than those which are nearest. Besides this general vortex of the sun, each of the planets had a particular vortex of their own by which their secondary planets were carried round, and any other body that happened to come within reach of it would likewise be carried away.

It is easy to see, from this short account of Des Cartes's system, that the whole of it was a mere petitio principii: for had he been required to prove the existence of his materia subtilis, he must undoubtedly have failed in the attempt; and hence, though his hypothesis was for some time followed for want of a better, yet it gave way to that of Newton almost as soon as it was propounded.

The general view of the solar system given by this celebrated philosopher, is not different from what has been laid down in the foregoing sections. The sun is placed in or near the centre; about whom the six planets, to which a seventh, the Georgium Sidus, is now added, continually move with different degrees of velocity, and at different distances. The first and nearest to the sun is Mercury, next Venus, then the Earth and Moon: beyond these is Mars; after him, Jupiter; then Saturn; and last of all, at least as far as discoveries have hitherto reached, the Georgium Sidus. Four of these primary planets, as they are called, are attended by moons or satellites, as well as the earth. These are, Venus*, Jupiter, Saturn, and the Georgium Sidus: of whom the first has only one; the second, four; the third, five; and the fourth two, though probably there may be more yet undiscovered by reason of their smallness or distance.

Though these planets uniformly and at all times respect the sun as the centre of their motion, yet they do not always preserve the same distance from him; neither do they all move in the same plane, though every one of them revolves in an orbit whose plane if extended would pass through the sun's centre. The line in which the plane of any of the planetary orbits crosses the orbit of the earth is called the line of its nodes, and the points of intersection are the nodes themselves. Each of them moves in an orbit somewhat elliptical; and thus sometimes approaches nearer, and at others recedes farther from, the sun than before. This deviation from a circle is called the eccentricity of the orbit; the point where it is farthest distant from the sun is called its aphelion; and where nearest, the perihelion. The eccentricities of the different planets, however, are very different. In Saturn the proportion of the greatest distance to the least is something less than 9 to 8, but much nearer to this than to 9; in Jupiter, it is something greater than that of 11 to 10; in Mars, it exceeds the proportion of 6 to 5; in the earth, it is only in the proportion of about 30 to 29; in Venus still less, being only as 70 to 69; but in Mercury it is much greater than in any of the rest, being little less than that of 3 to 2. The apheles of all the planets are not situated on the same side of the sun, but in the positions shown fig. 106.; though these positions are also variable, as shall be afterwards more fully explained. The eccentricity of the Georgium Sidus is not yet determined, though it is supposed to be less than that of the rest. All of them revolve from west to east; and the most remote is the longest of finishing its course round the sun.

Each of the planets moves in its orbit round the sun in such a manner, that the line drawn from the sun to the planet, by accompanying the planet in its spaces in equal times, will describe about the sun equal spaces in round the equal times. There is also a certain relation between the greater axes of these ellipses and the times in which the planets perform their revolutions through them, which may be expressed in the following manner: Let the period of one planet be expressed by the letter A, the greater axis of its orbit by D; let the period of another planet be denoted by B, and the greater axis of this planet's orbit by E. Then if C be taken to bear the same proportion to B as B bears to A; likewise if F be taken to bear the same proportion to E as E bears to D, and G taken to bear the same proportion likewise to F as E bears to D; then A shall bear the same proportion to C as D bears to G.

§ 1. Of Centripetal Powers in general.

Before we attempt to give any particular explanation of the causes producing the planetary motions, it will be necessary to premise something of Sir Isaac Newton's doctrine of centripetal forces, as upon that depends his doctrine of gravitation, and of the whole celestial system. The first effect of these powers is, to cause any body projected in a straight line deviate from it, and describe an incurved one, which shall always be bent towards the centre to which the body is supposed to have a tendency. It is not, however, necessary that the moving body should approach the centre; it may even recede farther from it, notwithstanding its being drawn by it; but this property uniformly belongs to it, that the line in which it moves will be continually concave towards the centre to which the power is directed.

Let A (fig. 107.) be the centre of a force. Let a body in B be moving in the direction of the straight line BC, in which line it would continue to move if undisturbed; but being attracted by the centripetal force towards A, the body must necessarily depart from this line BC; and being drawn into the curve line BD, must pass between the lines AB and BC. It is evident, therefore, that the body in B being gradually turned off from the straight line BC, it will at first be convex towards that line, and concave towards A. And that the curve will always continue to have this concavity towards A, may thus appear: In the line BC, near to B, take any point, as E, from which the line ETF may be so drawn as to touch the curve line BD in some point, as F. Now, when the body is come to F, if the centripetal power were immediately to be suspended, the body would no longer continue to move in a curve line, but, being left to itself, would forthwith reassume a straight course, and that straight course would be in the line FG; for that line is in the direction of the body's motion of the point F. But the centripetal force continuing its energy, the body will be gradually drawn from this line FG so as to keep in the Of Centripetal Powers.

the line FD, and make that line, near the point F, to be concave towards the point A; and in this manner the body may be followed in its course throughout the line BD, and every part of that line be shown to be concave towards the point A.

Again, the point A (fig. 108.) being the centre of a centripetal force, let a body at B set out in the direction of the straight line BC perpendicular to the line AB. It will be easily conceived, that there is no other point in the line BC so near to A as the point B; that AB is the shortest of all the lines which can be drawn from A to any part of the line BC; all others, as AD or AE, being longer than AB. Hence it follows, that the body setting out from it, if it moved in the line BC, would recede more and more from the point A. Now, as the operation of a centripetal force is to draw a body towards the centre of that force, if such a force act upon a resting body, it must necessarily put that body into motion as to cause it move towards the centre of the force; if the body were of itself moving towards that centre, it would accelerate that motion, and cause it to move faster down; but if the body were in such a motion that it would of itself recede from the centre, it is not necessary that the action of a centripetal power should make it immediately approach the centre from which it would otherwise have receded; the centripetal force is not without effect if it cause the body to recede more slowly from that centre than otherwise it would have done. Thus, the smallest centripetal power, if it act on the body, will force it out of the line BC, and cause it to pass in a bent line between BC and the point A, as has been already explained. When the body, for instance, has advanced to the line AD, the effect of the centripetal force discovers itself by having removed the body out of the line BC, and brought it to cross the line AD somewhere between A and D, suppose at F. Now, AD being longer than AB, AF may also be longer than AB. The centripetal power may indeed be so strong, that AF shall be shorter than AB; or it may be so evenly balanced with the progressive motion of the body that AF and AB shall be just equal; in which case the body would describe a circle about the centre A; this centre of the force being also the centre of the circle.

If now the body, instead of setting out in the line BC perpendicular to AB, had set out in another line BG more inclined towards the line AB, moving in the curve line BH; then, as the body, if it were to continue its motion in the line BG, would for some time approach the centre A, the centripetal force would cause it to make greater advances toward that centre: But if the body were to set out in the line BI, reclined the other way from the perpendicular BC, and were to be drawn by the centripetal force into the curve line BK; the body, notwithstanding any centripetal force, would for some time recede from the centre; since some part at least of the curve line BK, lies between the line BI and the perpendicular BC.

Let us next suppose a centripetal power directed toward the point A (Fig. 109.), to act on a body in B, which is moving in the direction of the straight line BC, the line BC reclining off from AB. If from A the straight lines AD, AE, AF, are drawn to the line QB prolonged beyond B to G, it appears that AD is inclined to the line GC more obliquely than AB, Of Centripetal AE more obliquely than AD, and AF than AE; or, to speak more correctly, the angle under ADG is less than that under ABG, that under AEG is less than ADG, and AFG less than AEG. Now suppose the body to move in the curve line BHIK, it is likewise evident that the line BHIK being concave towards A and convex towards BC, it is more and more turned off from that line; so that in the point H, the line AK will be more obliquely inclined to the curve line BHIK than the same line AHD is inclined to BC at the point D; at the point I the inclination of the line AI to the curve line will be more different from the inclination of the same line AIE to the line BC at the point IE; and in the points K and F the difference of inclination will be still greater; and in both, the inclination at the curve will be less oblique than at the straight line BC. But the straight line AB is less obliquely inclined to BG than AD is inclined towards DG; therefore, although the line AH be less obliquely inclined towards the curve HB than the same line AHD is inclined towards DG, yet it is possible, that the inclination at H may be more oblique than the inclination at B. The inclination at H may indeed be less oblique than the other, or they may be both the same. This depends upon the degree of strength wherewith the centripetal force exerts itself during the passage of the body from B to H; and in like manner the inclinations at I and K depend entirely on the degree of strength wherewith the centripetal force acts on the body in its passage, from H to K: if the centripetal force be weak enough, the lines AH and AI drawn from the centre A to the body at H and at I, shall be more obliquely inclined to the curve than the line AB is inclined towards BG. The centripetal force may be of such a strength as to render all these inclinations equal; or if stronger, the inclinations at I and K will be less oblique than at B; and Sir Isaac Newton has particularly shown, that if the centripetal power decreases after a certain manner without the increase of distance, a body may describe such a curve line, that all the lines drawn from the centre to the body shall be equally inclined to that curve line.

We must farther remark, that if the centripetal power, while the body increases its distance from the centre, retain sufficient strength to make the lines round a length less oblique to the curve; then, if this diminution of the obliquity continue, till at last the line drawn from the centre to the body shall cease to be obliquely inclined to the curve, and become perpendicular thereto; from this instant the body shall no longer recede from the centre, but in its following motion shall again descend, and describe a curve in all respects like that which it has described already, provided the centripetal power, every where at the same distance from the body, acts with the same strength. This return of the body may be proved by the following proposition: That if the body in any place, suppose at I, were to be stopped, and thrown directly backward with the velocity wherewith it was moving forward in that point I, then the body, by the action of the centripetal force upon it, would move back again over the path IHB, in which it had before advanced. The truth of this proposition may be illustrated in the following manner. Suppose, in fig. 110, that a body were carried after the following manner through the bent figure ABCDEF, composed of the straight lines AB, BC, CD, DE, EF; let the body then first be supposed to receive an impulse at some point within the concavity of the figure, as G. Now, as this body, when once moving in the straight line AB, will continue to move on in this line as long as it shall be left to itself; but being disturbed at the point B by the impulse given it, it will be turned out of this line AB into some other straight line, wherein it will afterwards continue to move as long as it shall be left to itself; therefore, let this impulse have strength sufficient to turn the body into the line BC; then let the body move on undisturbed from B to C; but at C let it receive another impulse pointed also towards G, and of sufficient strength to turn the body into the line CD; at D let a third impulse turn it into the line DE; and at E let another turn it into EF. Now, if the body, while moving on in the line EF, be stopped and turned back again with the same velocity with which it was moving forward, then by the repetition of the former impulse at E, the body will be turned into the line ED, and move in it from E to D with the same velocity as that wherewith it was moving forward in this line: then by a repetition of the impulse at D, when the body shall have returned to that point, it will be turned into the line DC; and by the repetition of the former impulses at C and at B, the body will be brought back again into the line BA, with the velocity wherewith it first moved in that line.

To illustrate this still farther, let DE and FE be continued beyond E. In DE thus continued, take at pleasure the length EH, and let HI be so drawn as to be equidistant from the line GE; then, from the second law of motion, it follows, that after the impulse on the body on E, it will move through the space EI in the same time it would have employed in moving from E to H with the velocity it had in the line DE. In FE prolonged, take EK equal to EI and draw KL equidistant from GE. Then, because the body is thrown back in the line FE with the same velocity with which it went forward in that line, if, when the body was returned to E, it were permitted to go straight on, it would pass through EK in the same time as it took up in passing through EI, when it went forward in the line EF. But if, at the body's return to the point E, such an impulse directed toward the point D were to be given as was sufficient to turn it into the line DE, it is plain that this impulse must be equal to that which originally turned the body out of the line DE into EF; and that the velocity with which the body will return into the line ED is the same as that wherewith it moved before through this line from D to E. Because EK is equal to EI, and KL and HI being each equidistant from GE, are by consequence equidistant from each other; it follows, that the two triangular figures IHEH and KEL are altogether like and equal to each other. EK there- Of Centripetal Powers.

Astronomy, thereby greatly enriching that science, we might make a transition from this figure, composed of a number of straight lines, to a figure of one continued curvature, and from a number of separate impulses repeated at distinct intervals to a continued centripetal force, and show, that because what has been here advanced holds universally true whatever be the number of straight lines whereof the curve figure ACF is composed, and however frequently the impulses at the angles of this figure are repeated; therefore the same will still remain true although this figure should be converted into one of a continued curvature; and these distinct impulses should be changed into a continual centripetal force.

This being allowed, suppose the body in K to have the line AK no longer obliquely inclined to its motion. In this case, if the body be turned back in the manner we have been considering, it must be directed back perpendicularly to AK; but if it had proceeded forward, it would likewise have moved in a direction perpendicular to AK; consequently, whether it move from this point K backward or forward, it must describe the same kind of course. Therefore, since by being turned back it will go over again the line LHB, if it be permitted to go forward, the line KL, which it shall describe, will be altogether similar to the line KHB.

In like manner we may determine the nature of the motion, if the line wherein the body sets out be inclined, as in fig. 111., down toward the line BA drawn between the body and the centre. If the centripetal power so much increases in strength as the body approaches, that it can bend the path in which the body moves to that degree as to cause all the lines, AH, AI, AK, to remain no less oblique to the motion of the body than AB is oblique to BC, the body shall continually more and more approach the centre: But if the centripetal power increases in so much less a degree as to permit the line drawn from the centre to the body, as it accompanies the body in its motion, at length to become more and more erect to the curve wherein the body moves, and in the end, suppose at K, to become perpendicular to it; from that time the body shall rise again. This is evident from what has been said above; because, for the very same reason, here also the body will proceed from the point K to describe a line altogether similar to that in which it has moved from B to K. Thus it happens as in the pendulum, which, all the time it approaches a perpendicular position towards the horizon, descends more and more; but as soon as it is come into that situation, it immediately rises again by the same degrees as it descended before: so here the body more and more approaches the centre all the time it is moving from B to K; but thenceforward it rises from the centre again by the same degrees as it approached before.

If, as in fig. 112., the line BC be perpendicular to AB; then, as has already been observed, the centripetal power may be so balanced with the progressive motion of the body, that it may keep moving round the centre A constantly at the same distance; as the body does when whirled about any point to which it is tied by a string. If the centripetal power be too weak to produce this effect, the motion of the body will presently become oblique to the line drawn from itself to the centre; but if it be stronger, the body of Centripetal Power must constantly keep moving in a curve to which a line drawn from it to the body is perpendicular.

If the centripetal power change with the change of distance, in such a manner that the body, after its motion has become oblique to the line drawn from itself to the centre, shall again become perpendicular thereto; then the body shall, in its subsequent motion, return again to the distance of AB, and from that distance take a course similar to the former: and thus, if the body move in a space void of all resistance, which has been all along supposed, it will continue in a perpetual motion about the centre, descending and ascending from it alternately. If the body, setting out from B (fig. 113.) in the line BC perpendicular to AB, describe the line BDE, which in D shall be oblique to the line AD, but in E shall again become erect to AE, drawn from the body in E to the centre A; then from this point E the body shall describe the line EFG entirely similar to BDE, and at G shall be at the same distance as it was at B; and the line AG shall be erect to the body's motion. Therefore the body shall proceed to describe from G the line GHI altogether similar to the line GFE, and at I it will have the same distance from the centre as it had at E; and also have the line AI erect to its motion: so that its subsequent motion must be in the line IKL similar to IKG, and the distance AL equal to AG. Thus the body will go on in a perpetual round without ceasing, alternately enlarging and contracting its distance from the centre.

If it so happen that the point E fall upon the line BA, continued beyond A; then the point G will fall upon B, I on E, and L also on B; so that the body will in this case describe a simple curve line round the centre A, like the line BDEF in fig. 114., in which it will revolve from B to E, and from E to B, without end. If AE in fig. 113. should happen to be perpendicular to AB, in this case also a simple line will be described; for the point G will fall on the line BA prolonged beyond A; the point I on the line AE prolonged beyond A; and the point L on B; so that the body will describe a line like the curve line BEGI in fig. 115., in which the opposite points B and G are equally distant from A; and the opposite points E and I, are also equally distant from the same point A. In other cases the body will have a course of a more complicated nature.

Thus it must be apparent how a body, while it is constantly attracted towards a centre, may notwithstanding by its progressive motion keep itself from falling down to the centre, describing about it an endless circuit, sometimes approaching and sometimes receding from it. Hitherto, however, we have supposed, that the centripetal power is everywhere of equal strength at the same distance from the centre; and this is indeed the case with that power which keeps the planets in their orbits; but a body may be kept on in a perpetual circuit round a centre, although the centripetal power be kept moving in any curve line whatever, that shall have its concavity turned everywhere towards the centre of the force. To illustrate this, we shall in the first place propose the case of a body moving the incurvated figure ABCDE (fig. 116.), which is composed of the straight lines, AB, BC, CD, DE, and Of Centripetal Powers; the motion being carried on in the following manner. Let the body first move in the line AB with any uniform velocity. When it is arrived at the point B, let it receive an impulse directed towards any point F taken within the figure; and let the impulse be of such a strength as to turn the body out of the line AB into the line BC: The body after this impulse, while left to itself, will continue moving in the line BC. At C let the body receive another impulse directed towards the same point F, of such a strength as to turn it from the line CB into CD. At D, let the body, by another impulse, directed likewise to the point F, be turned out of the line CD into DE. At E, let another impulse, directed likewise toward the point F, turn the body from the line DE into EA: and thus the body will, by means of these impulses, be carried thro' the whole figure ABCDE.

Again, when the body is come to the point A, if it there receive another impulse directed like the rest to the point F, and of such a degree of strength as to turn it into the line AB, wherein it first moved; the body will then return into this line with the same velocity it had originally. To understand this, let AB be prolonged beyond B at pleasure, suppose to G; and from G let GH be drawn; which, if produced, should always continue equidistant from BF, i.e., let GH be drawn parallel to BF, in the time, then, in which the body would have moved from B to G, had it not received a new impulse in B; by the means of that impulse it will have acquired a velocity which will carry it from B to H. After the same manner, if CI be taken equal to BH, and IK be drawn parallel to CF, the body will have moved from C to K, with the velocity which it has in the line CD; in the same time it would have employed in moving from C to I with the velocity it had in the line BC. Therefore, since CI and BH are equal, the body will move through CK in the same time as it would have taken up in moving from B to G with the velocity whereewith it moved through the line AB. Again, DL being taken equal to CK, and LM drawn parallel to DF, the body will, for the same reason as before, move through DM with the velocity which it has in the line DE, in the same time it would employ in moving through BG with its original velocity. Lastly, if EN be taken equal to DM, and NO be drawn parallel to EF; likewise, if AP be taken equal to EO, and PQ be drawn parallel to AF; then the body, with the velocity wherewith it returns into the line AB, will pass thro' AQ in the time it would have employed in passing through BG with its original velocity. Now as all this follows directly from what has been delivered concerning oblique impulses impressed upon bodies in motion; so we must here observe farther, that it can be proved by geometry, that AQ will always be equal to BG; which being granted, it follows, that the body has returned into the line AB with the same velocity which it had when it first moved in that line; for the velocity with which it returns into the line AB, will carry it over the line AQ in the same time as would have been taken up in its passing over an equal line BG with the original velocity.

The conclusion naturally deduced from the above reasoning is, that by means of a centripetal and projectile force, a body may be carried round any fixed point in a curve figure which shall be concave towards Of Centripetal Powers, as that marked ABC, fig. 117, and when it is receded to that point from whence it set out, it shall recover again the velocity with which it departed from that point. It is not indeed always necessary that it should return again into its first course, for the curve may be any curvilinear direction, if the body set out from any point B in the direction BE, and moved through the line by BCD till it returned to B; here the body would not means of enter again into the line BCD, because the two parts centripetal BD and BC of the curve line make an angle at the force point B; so that the centripetal power, which at the point B would turn the body from the line BF into the curve, will not be able to turn it into the line BC from the direction in which it returns to the point B. A forcible impulse must be given the body in the point B to produce that effect. If, at the point B, whence the body sets out, the curve line return into itself, as in fig. 117, then the body, upon its arrival again at B, may return into its former course, and thus make an endless circuit about the centre.

The force requisite to carry a body in any curve line Calculation proposed, is to be deduced from the curvature which the figure has in any part of it. Sir Isaac Newton has laid down the following proposition as a foundation for discovering this, viz. that if a line be drawn from some fixed point to the body, and remaining by one extreme united to that point, it be carried round along with the body; then if the power whereby the body is kept in its course be always pointed to this fixed point as a centre, this line will move over equal spaces in equal portions of time. Suppose a body were moving through the curve line ABCD (fig. 120.), and passed over the arches AB, BC, CD in equal portions of time; then if a point, as E, can be found, from whence the line EA being drawn to the body in accompanying it in its motion, it shall make the spaces EAB, EBC, and ECD, over which it passes, equal where the times are equal; then is the body kept in this line by a power always pointed to E as a centre.

To prove this, suppose a body let out from the point A, fig. 121. to move in the straight line AB; and after it had moved for some time in that line, it were to receive that impulse directed to some point, as C. Let it receive that impulse at D, and thereby be turned into the line DE; and let the body, after this impulse, take the same time in passing from D to E that is employed in passing from A to D. Then the straight lines CA, CD, and CE being drawn, the triangular spaces CAD and CDE are proved to be equal in the following manner. Let EF be drawn parallel to CD. Then, it follows, from the second law of motion, that since the body was moving in the line AB when it received the impulse in the direction DC, it will have moved after that impulse through the line DE in the same time as it would have moved through DF, provided it had received no disturbance in D. But the time of the body's moving from D to E is supposed to be equal to the time of its moving through AD; therefore the time which the body would have employed in moving through DF, had it not been disturbed in D, is equal to the time wherein it moved through AD; consequently DF is equal in length to AD; for if the body had gone on to move through the line AB without interruption, it would have moved through all the parts of it with the same velocity, and have passed over equal parts of that line in equal portions of time. Now CF being drawn, since AD and DF are equal, the triangular space CDF is equal to the triangular space CAD. Further, the line EF being parallel to CD, it follows from the 37th proposition of Euclid's first book, that the triangle CED is equal to the triangle CFD; therefore the triangle CED is equal to the triangle CAD.

In like manner, if the body receive at E another impulse directed toward the point C, and be turned by that impulse into the line EG; if it move afterwards from E to G in the same space of time as was taken up by its motion from D to E, or from A to D; then CG being drawn, the triangle CEG is equal to CDE. A third impulse at G, directed as the two former to C, whereby the body shall be turned into the line GH, will have also the like effect with the rest. If the body move over GH in the same time as it took up in moving over EG, the triangle CGH will be equal to the triangle CEG. Lastly, if the body at H be turned by a fresh impulse directed toward C into the line HI, and at I by another impulse directed also to C be turned into the line IK; and if the body move over each of the lines HI and IK in the same time as it employed in moving over each of the preceding lines AD, DE, EG, and GH: then each of the triangles CHI and CIK will be equal to each of the preceding. Likewise, as the time in which the body moves over ADE is equal to the time of its moving over EGH, and to the time of its moving over HIK; the space CADE will be equal to the space CEGH and to the space CHIK. In the same manner, as the time in which the body moved over ADEG is equal to the time of its moving over GHIK, so the space CADEG will be equal to the space CGHIK. From this principle Sir Isaac Newton demonstrates the above mentioned proposition, by making the transition from this incurvated figure composed of straight lines, to a figure of continued curvature; and by showing, that since equal spaces are described in equal times, in this present figure composed of straight lines, the same relation between the spaces described and the times of their description will also have place in a figure of one continued curvature. He also deduces from this proposition the reverse of it; and proves, that whenever equal spaces are continually described, the body is acted upon by a centripetal force directed to the centre at which the spaces terminate.

Having thus endeavoured to illustrate the fundamental principle of the Newtonian philosophy, at least as far as it regards the motion of the planets and heavenly bodies, we shall now proceed to the more particular application of it. The first thing undertaken by Sir Isaac in order to explain those motions, is to demonstrate, that in the celestial spaces there is no sensible matter. That the heavenly bodies suffer no sensible resistance from any matter of this kind, is evident from the agreement between astronomical observations in all ages with regard to the time in which the planets have been found to perform their revolutions. Descartes, however, was of opinion, that the planets might be kept in their courses by means of a fluid matter, which continually circulating round, should carry the planets along with it; and there is one appearance which seems to favour this opinion, viz. that the sun turns round his axis the same way the planets move; the earth also turns round its axis the same way as the moon turns round the earth; and the planet Jupiter turns round his axis the same way that his satellites revolve round him. It might therefore be supposed, that if the whole planetary region were filled with a fluid matter, the sun, by turning round on his own axis, might communicate motion first to that part of the fluid which was contiguous, and by degrees propagate the like motion to the parts more remote. After the same manner the earth might communicate motion to this fluid to a degree sufficient to carry round the moon; and Jupiter might communicate the like to the distance of its satellites. This system has been particularly examined by Sir Isaac Newton; who finds, that the velocities with which the parts of this fluid should move in different distances from the centre of motion, will not agree with the motions observed in the different planets; for instance, that the time of one entire circulation of the fluid wherein Jupiter should swim, would bear a greater proportion to the time of one entire circulation of the fluid where the earth is, than the period of Jupiter bears to that of the earth. He proves also, that the planet cannot circulate in such a fluid, so as to keep long in the same course, unless the planet and the contiguous fluid are of the same density, and the planet be carried along with the same velocity as the fluid. There is also another remark made on this motion by Sir Isaac, viz. that some vivifying force will be continually necessary at the centre of the motion. The sun, in particular, by communicating motion to the ambient fluid, will lose from itself as much motion as it communicates to the fluid, unless some acting principle reside in the sun to renew its motion continually. If the fluid were infinite, this gradual loss of motion would continue till the whole should stop; and if the fluid were limited, this loss of motion would continue till there would remain no swifter a revolution in the sun than in the outermost part of the fluid, so that the whole would turn together about the axis of the sun like one solid globe. We must likewise observe, that as the planets do not move in perfect circles round the sun, there is a greater distance between their orbits in some places than others. For instance, the distance between the orbit of Mars and Venus is near half as great again in some part of their course as in others. Now here the fluid in which the earth should swim, must move with a less rapid motion where there is this greater interval between the contiguous orbits; but, on the contrary, where the space is straitest, the earth moves more slowly than where it is widest.

Again, if our globe of earth swam in a fluid of equal density with the earth itself, that is, in a fluid more dense than water, all bodies put in motion hereupon the earth's surface must suffer a great resistance by it; whereas Sir Isaac Newton has made it evident, by experiments, that bodies, falling perpendicularly through the air, suffer only about a hundred and sixtieth part of the resistance from it that they meet with in water. These experiments are applied by Sir Isaac yet farther to the general question concerning the absolute plenitude of space. He objects against the filling of all space with a subtle fluid, after the manner of Des Cartes, That all bodies must be immeasurably resisted by it. And lest it should be thought that this objection might be evaded, by ascribing to this fluid such very minute and smooth parts as might remove all adhesion or friction between them, whereby all resistance would be lost, Sir Isaac proves, that fluids must resist from the inactivity of their particles, and that water and the air resist almost entirely on this account; so that in this subtle fluid, however smooth and lubricated the particles might be, yet if the whole were as dense as water, it would resist very near as much as water does: And whereas such a fluid, whose parts are absolutely close together without any intervening spaces, must be a great deal more dense than water, it must also resist more in proportion to its density, unless we suppose the matter of which this fluid is composed not to be endowed with the same degree of inactivity with other matter: But if you deprive any substance of the property so universally belonging to all other matter, without impropriety of speech it can scarce be called by this name. Sir Isaac also made an experiment to try in particular, whether the internal parts of bodies suffered any resistance; and the result did indeed appear to favour some small degree of resistance, but to very little as to leave it doubtful whether the effect did not arise from some other latent cause.

§ 2. Of the Motions of the Primary Planets.

Since the planets thus move in a space void of all resistance, they would, if once set in motion, continue to move on for ever in a straight line. We have, however, already observed, that the primary planets move about the sun in such a manner that a line extended from the sun to the planet would describe equal spaces in equal times; and this single property in the planetary motions proves, that they are continually acted upon by a power directed towards the sun as the centre. It has also been observed, that if the strength of the centripetal power were suitably accommodated everywhere to the motion of any body round a centre, the body might be carried in any bent line whatever, whose concavity should be every where turned towards the centre of that force; and likewise that the strength of the centripetal force in each place was to be collected from the nature of the line wherein the body moved. Now since each of the planets moves in an ellipse, having the sun in one of its foci, Sir Isaac Newton demonstrates, that the strength of this power is reciprocally in the duplicate proportion of the distance from the sun. This proportion may be explained in the following manner: Suppose several distances to bear to each other the proportions of the numbers 1, 2, 3, 4, 5; that is, let the second distance be double the first, the third three times, the fourth four times, and the fifth five times as great as the first: multiply each of these numbers by itself, and 1 multiplied by 1 produces still 1, 2 multiplied by 2 produces 4, 3 by 3 produces 9, 4 by 4 produces 16, and 5 by 5 produces 25; this being done, the fractions $\frac{1}{4}$, $\frac{1}{9}$, $\frac{1}{16}$, and $\frac{1}{25}$, will respectively express the proportion which the centripetal power in each of the following distances bears to the power at the first distance; for in the second distance, Motions of which is double the first, the centripetal power will be the Primacy Planets. one fourth part only of the power at the first distance; at the third distance, the power will only be one ninth part of the first power; at the fourth distance, the power will be only one sixteenth; and at the fifth distance only one twenty-fifth, of the first power. Thus is found the proportion in which the centripetal power decreases, as the distance from the sun increases within the compass of one planet's motion. How it comes to pass that the planet can be carried about the sun by this centripetal power in a continual round, sometimes rising from the sun, then descending again as low, appears from what has been already said concerning centripetal forces.

In order to know whether this centripetal power extends in the same proportion throughout the system, and consequently whether all the planets are influenced by it, Sir Isaac inquires what relation there ought to be between the periods of the different planets, provided the system they were acted upon by the same power, decreasing throughout in the proportion abovementioned; and he finds, that the period of each, in this case, would have that very proportion to the greater axis of its orbit which has been already related: which puts it beyond a doubt, that the different planets are pressed towards the sun in the same proportion to the distances as one planet is in its several distances; whence it is justly concluded, that there is such a power acting towards the sun in the foresaid proportion at all distances from it. This power, when referred to the earth, Sir Isaac calls gravity; when to the sun, attraction; and to the planets, centripetal force. By these names, however, he designates only to signify a power endowed with the properties abovementioned; but by no means would it have understood as if these names referred any way to the cause of it.

"But now (says Mr Pemberton) in these demonstrations, some very minute inequalities in the motion of the planets are neglected; which is done with a great deal of judgment: for whatever be their cause, the effects are very inconsiderable, they being so exceedingly small, that some astronomers have thought fit wholly to pass them by. However, the excellencies of this philosophy, when in the hands of so great a plane-geometer as our author (Sir Isaac Newton), is such, tary that it is able to trace the least variations of things up to their causes. The only inequalities which have been observed common to all the planets are, the motion of the aphelion and the nodes. The transverse axis of each orbit does not remain always fixed, but moves about the sun with a very slow, progressive motion; nor do the planets keep constantly in the same planes, but change them and the lines by which these planes intersect each other by insensible degrees. The Motion of first of these inequalities, which is the motion of the aphelion, may be accounted for, by supposing the gravitation of the planets towards the sun to differ a little farther from the aforementioned reciprocal duplicate proportion of the distances; but the second, which is the motion of the nodes, cannot be accounted for by any power directed towards the sun; for no such power can give it any lateral impulse to divert it from the plane of its motion into any new plane, but of necessity must be derived from some other centre. Where Motions of that power is lodged, remains to be discovered. Now it is proved, as shall be afterwards explained, that the three primary planets, Saturn, Jupiter, and the Earth, which have satellites revolving about them, are endowed with a power of causing bodies, in particular those satellites, to gravitate towards them with a force which is reciprocally in the duplicate proportion of their distances; and the planets are, in all respects in which they come under our consideration, so similar and alike, that there is no reason to question but they have all the same property, though it be sufficient for the present purpose to have it proved of Jupiter and Saturn only; for these planets contain much greater quantities of matter than the rest, and proportionally exceed the others in power. But the influence of these two planets being allowed, it is evident how the planets come to shift their places continually: for each of the planets moving in a different plane, the action of Jupiter and Saturn upon the rest will be oblique to the planes of their motion, and therefore will gradually draw them into new ones. The same action of these two planets upon the rest will likewise cause a progressive motion; and therefore will gradually draw them into new ones. The same action of these two planets upon the rest will likewise cause a progressive motion of the aphelion; so that there will be no necessity for having recourse to the other cause for this motion, which was before hinted at, viz. the gravitation of the planets toward the sun differing from the exact duplicate proportion of their distances. And in the last place, the action of Jupiter and Saturn upon each other will produce in their motions the same inequalities as their joint action produces upon the rest. All this is effected in the same manner as the sun produces the same kind of inequalities and many others in the motion of the moon and other secondary planets; and therefore will be best apprehended by what is said afterwards. Those other irregularities in the motion of the secondary planets have place likewise here, but are too minute to be observable, because they are produced and rectified alternately, for the most part in the time of a single revolution; whereas the motion of the aphelion and nodes which increase continuously, become sensible after a long series of years.

And yet some of these other inequalities are discernible in Saturn in Jupiter and Saturn; in Saturn chiefly: for when Jupiter, who moves faster than Saturn, approaches to a conjunction with him, his action upon the latter will a little retard the motion of that planet; and by the reciprocal action of Saturn, he will himself be accelerated. After conjunction, Jupiter will again accelerate Saturn, and be likewise retarded in the same degree as before the first was retarded and the latter accelerated. Whatever inequalities besides are produced in the motion of Saturn by the action of Jupiter upon that planet, will be sufficiently rectified by placing the focus of Saturn's ellipse, which should otherwise be in the sun, in the common centre of gravity of the sun and Jupiter. And all the inequalities of Jupiter's motions, caused by the action of Saturn upon him, are much less considerable than the irregularities of Saturn's motion. This one principle therefore of the planets having a power as well as the sun to cause bodies gravitate towards them, which is proved by the motion of the secondary planets to obtain in fact, explains all the irregularities relating to the planetary motions ever observed by astronomers (A).

Sir Isaac Newton after this proceeds to make an Method of improvement in astronomy, by applying this theory to correcting the farther correction of their motions. For as we have here observed the planets to possess a principle of motions, gravitation as well as the sun; so it will be explained at large hereafter, that the third law of motion, which makes action and reaction equal, is to be applied in this case, and that the sun does not only attract each planet, but is also itself attracted by them; the force wherewith the planet is acted on, bearing to the force wherewith the sun itself is acted upon at the same time, the proportion which the quantity of matter in the sun bears to the quantity of matter in the planet.

From the action of the sun and planet being thus mutual, Sir Isaac Newton proves that the sun and planet round the will describe about their common centre of gravity fi. common milar ellipses; and then, that the transverse axis of the centre of gravity of ellipses, which would be described about the sun at rest, him and the in the same time, the same proportion as the quantity planets, of solid matter in the sun and planet together bears to the first of two mean proportions between this quantity and the quantity of matter in the sun only.

It will be asked, perhaps, how this correction can be admitted, when the cause of the motions of the planets was before found, by supposing them to be the centre of the power which acted upon them? for, according to the present correction, this power appears rather to be directed to the common centre of gravity. But whereas the sun was at first concluded to be the centre to which the power acting on the planets was directed, because the spaces described in equal times round the sun were found to be equal; so Sir Isaac Newton proves, that if the sun and planet move round their common centre of gravity, yet, to an eye placed in the planet, the spaces which will appear to be described about the sun will have the same relation to the times of their description as the real spaces would if the sun were at rest. I further asserted, that, supposing the planets to move round the sun at rest, and to be attracted by a power which should everywhere act with degrees of strength reciprocally in the duplicate proportions of their distances; then the periods of the planets must observe the same relations to their distances as astronomers have found them to do. But here it must not be supposed, that the observations of astronomers absolutely agree without any the least difference: and the present correction will not cause a deviation.

(A) Professor J. Robison, however, informs us in his paper on the Georgium Sidus (Edinburgh Philosophical Transactions, Vol. I.), That all the irregularities in the planetary motions cannot be accounted for from the laws of gravitation; for which reason he was obliged to suppose the existence of planets beyond the orbit of Saturn, even before the discovery of the Georgium Sidus. M. de la Lande also has observed some unaccountable inequalities in the motion of Saturn for more than 30 years past. Motions of deviation from any one astronomer's observations so much as they differ from one another; for in Jupiter, where this correction is greatest, it hardly amounts to the 3000th part of the whole axis.

Upon this head, I think it not improper to mention a reflection made by our excellent author upon against these small inequalities in the planets motions, which contains in it a very strong philosophical argument against the eternity of the world. It is this, that these inequalities must continually increase by slow degrees, till they render at length the present frame of nature unfit for the purposes it now serves. And a more convincing proof cannot be desired against the present constitution's having existed from eternity than this, that a certain period of years will bring it to an end. I am aware, that this thought of our author has been represented even as impious, and as no less than casting a reflection upon the wisdom of the Author of nature for framing a perishable work. But I think so bold an assertion ought to have been made with singular caution: for if this remark upon the increasing irregularities in the heavenly motions be true in fact, as it really is, the imputation must return upon the author, that this does not detract from the divine wisdom. Certainly we cannot pretend to know all the omniscient Creator's purposes in making this world, and therefore cannot pretend to determine how long he designed it should last; and it is sufficient if it endure the time designed by the Author. The body of every animal shows the unlimited wisdom of the Author no less, nay, in many respects more, than the larger frame of nature; and yet we see they are all designed to last but a small space of time."

§ 3. The Motions of the Secondary Planets explained from the Principles laid down in § 1.

The excellency of the Newtonian Philosophy is discoverable even more in its solution of the motions of the secondary than in those of the primary planets; for thus not only all the irregularities formerly discovered by astronomers in these motions are solved in a satisfactory manner, but several others are discovered of such a complicated nature that they could never be distinguished into proper heads. These, however, are now not only found out from their causes, which this philosophy has brought to light; but the dependence of them upon their causes is also shown in such a perfect manner, that the degree of them may be exactly computed. Thus Sir Isaac Newton found means to compute the moon's motion so exactly, that he framed a theory from which the place of that planet may at all times be computed very nearly, or altogether, as exactly as the places of the primary planets themselves; which is much beyond what the greatest astronomers could ever effect.

The first thing demonstrated of these secondary planets is, that they are drawn towards their respective primaries in the same manner as the latter are attracted by the sun. That each secondary planet is kept in its orbit by a power directed towards its primary, &c. is proved from the phenomenon of the satellites of Jupiter and Saturn; because they move in circles, as far as we can observe, about their respective primaries with an equable course, the primary being the centre of each orbit: and by comparing the times in which the different satellites of the same primary perform their periods, they are found to observe the same relation to the distances from their primary, as the primary planets observe in respect of their mean distances from the sun. The same thing holds good also with regard to the earth and moon; for she is found to move round the earth in an ellipse after the same manner as the primary planets do about the sun, excepting only some small irregularities in her motions, the cause of which will be particularly explained in what follows; whereby it will appear that there are no objections against the earth's acting on the moon in the same manner as the sun acts on the primary planets; that is, as Jupiter and Saturn act upon their satellites.

By the number of satellites which move round Jupiter and Saturn, the power of each of these planets may be measured to a very considerable distance; for covered by the distance of the outermost satellite in each of these satellites exceeds several times the distance of the innermost. The force of the earth upon the moon, however, at different distances, is more confirmed by the following consideration than any analogical reasoning. It will appear, that if the power of the earth by which Gravity retains the moon in her orbit be supposed to act at the all distances between the earth and moon, according to the rule already mentioned, this power will be sufficient to produce upon bodies near the surface of the earth all the effects ascribed to the principle of gravity. This is discovered by the following method. Let A (in fig. 122.) represent the earth, B the moon, BCD the moon's orbit; which differs little from a circle of which A is the centre. If the moon in B were left to itself to move with the velocity it has in the point B, it would leave the orbit, and proceed straight forward in the line BE which touches the orbit in B. Suppose the moon would upon this condition move from B to E in the space of one minute of time: By the action of the earth upon the moon, whereby it is retained in its orbit, the moon will really be found at the end of this minute in the point F, from whence a straight line drawn to A shall make the space BFA in the circle equal to the triangular space BEA; so that the moon in the time wherein it would have moved from B to E, if left to itself, has been impelled towards the earth from E to F. And when the time of the moon's passing from B to F is small, as here it is only one minute, the distance between E and F scarcely differs from the space through which the moon would descend in the same time if it were to fall directly down from B toward A without any other motion. AB, the distance of the moon from the earth, is about 60 of the semidiameters of the latter; and the moon completes her revolution round the earth in about 27 days 7 hours and 43 minutes: therefore the space EF will here be found by computation to be about 16 feet. Consequently, if the power by which the moon is retained in its orbit be near the surface of the earth greater than at the distance of the moon in the duplicate proportion of that distance, the number of feet a body would descend near the surface of the earth, by Calculation of the action of this power upon it, in one minute, would be equal to the number 16 multiplied twice into the city of fall-number 60; that is, to 5800. But how fast bodies falling near the surface of the earth may be known by the pendulum; and by the exactest experiments, they are... Motions of are found to descend the space of $16\frac{1}{2}$ feet in one second; and the spaces described by falling bodies being in the duplicate proportion of the times of their fall, the number of feet a body would describe in its fall near the surface of the earth in one minute of time, will be equal to $16\frac{1}{2}$ twice multiplied by $60$; the same as would be caused by the power which acts upon the moon.

In this computation the earth is supposed to be at rest; but it would have been more exact to have supposed it to move, as well as the moon, about their common centre of gravity; as will easily be understood from what has been already said concerning the motion of the sun and primary planets about their common centre of gravity. The action of the sun upon the moon is also here neglected; and Sir Isaac Newton shows, if you take in both these considerations, the present computation will best agree to a somewhat greater distance of the moon and earth, viz. to $60\frac{1}{2}$ semidiameters of the latter, which distance is more conformable to astronomical observations; and these computations afford an additional proof that the action of the earth observes the same proportion to the distance which is here contended for. In Jupiter and Saturn this power is so far from being confined to a small extent of space, that it not only reaches to several satellites at very different distances, but also from one planet to another, nay, even through the whole planetary system; consequently, there is no appearance of reason why this power should not act at all distances, even at the very surfaces of these planets, as well as farther off. But from hence it follows, that the power which retains the moon in her orbit is the same as that which causes bodies near the surface of the earth to gravitate; for since the power by which the earth acts on the moon will cause bodies near the surface of it to descend with all the velocity they are found to do, it is certain no other power can act upon them besides; because, if it did, they must of necessity descend swifter. Now, from all this, it is at length very evident, that the power in the earth which we call gravity extends up to the moon, and decreases in the duplicate proportion of the increase of the distance from the earth.

Thus far with respect to the action of the primary planets upon their secondaries. The next thing to be shown is, that the sun likewise acts upon them. For this purpose we must observe, that if to the motion of the satellite whereby it would be carried round its primary at rest, we superadd the same motion, both in regard to the velocity and direction, as the primary itself has, it will describe about the primary the same orbit with as great regularity as if the primary had been indeed at rest. This proceeds from the law of motion, which makes a body near the surface of the earth descend perpendicularly, though the earth be in so swift a motion, that if the falling body did not partake of it, its descent would be remarkably oblique; and that a body projected describes in the most regular manner the same parabola, whether projected in the direction in which the earth moves, or in the opposite direction, if the projecting force be the same. From this we learn, that if the satellite moved about its primary with perfect regularity, besides its motion about the primary it would have the same progressive velocity with which the primary is carried about the sun, Motions of in a direction parallel to that impulse of its primary: the Secondary Planets. And, on the contrary, the want of either of these, in particular of the impulse towards the sun, will occasion great inequalities in the motion of the secondary planet. The inequalities which would arise from the absence of this impulse towards the sun are so great, that by the regularity which appears in the motion of the secondary planets, it is proved, that the sun communicates to them the same velocity by its action as it gives to their primary at the same distance. For Sir Isaac Newton informs us, that upon examination he found, that if any of the satellites of Jupiter were attracted by the sun more or less than Jupiter himself at the same distance, the orbit of that satellite, instead of being concentrical to Jupiter, would have its centre at a greater or lesser distance than the centre of Jupiter from the sun, nearly in the subduplicate proportion of the difference between the sun's action upon the satellite and upon Jupiter. Therefore, if any satellite were attracted by the sun but one hundredth part more or less than Jupiter is at the same distance, the orbit of that satellite would be distant from the centre of Jupiter no less than a fifth part of the outermost satellite from Jupiter; which is almost the whole distance of the innermost satellite. By the like argument, the satellites of Saturn gravitate towards the sun as much as Saturn itself at the same distance, and the moon as much as the earth.

Thus it is proved, that the sun acts upon the secondary planets as much as upon the primaries at the same distance; but it has also been shown, that the action of the sun upon bodies is reciprocally in the duplicate proportion of the distance; therefore the secondary planets being sometimes nearer to the sun than to the primary, and sometimes more remote, they are always acted upon in the same degree with their primary, but when nearer to the sun are attracted more, and when farther off are attracted less. Hence arise various inequalities in the motion of the secondary planets. Some of these inequalities, however, would take place, though the moon if undisturbed by the sun had moved of the earth's motion; others depend on the elliptical figure and oblique situation of the moon's orbit. One of the former is, that the moon does not describe equal spaces in equal times, but is continually accelerated as she passes from the quarter to the new or full, and is retarded again by the like degrees in returning from the new and full to the next quarter; but here we consider not so much the absolute as the apparent motions of the moon with respect to us. These two may be distinguished in the following manner. Let S in fig. 123. represent the sun, A the earth moving in its orbit BC, DEFG the moon's orbit, and H the place of the moon in her orbit. Suppose the earth to have moved from A to I. Because it has been shown that the moon partakes of all the progressive motion of the earth, and likewise that the sun attracts both the earth and moon equally when they are at the same distance from it, or that the mean action of the sun upon the moon is equal to its action upon the earth; we must therefore consider the moon as carrying about with it the moon's orbit: so that when the earth is removed from A to I, the moon's orbit shall likewise be... Motions of be removed from its former situation into that denoted by KLMN. But now the earth being in I, if the moon were found in O, so that OI should be parallel to HA, though the moon would really have moved from H to O, yet it would not have appeared to a spectator upon the earth to have moved at all, because the earth has moved as much as itself; so that the moon would still appear in the same place with respect to the fixed stars. But if the moon be observed in P, it will then appear to have moved, its apparent motion being measured by the angle under OIP. And if the angle under PIS be less than the angle under HAS, the moon will have approached nearer its conjunction with the sun. Now, to explain particularly the inequality of the moon's motion already mentioned, let S in fig. 124 represent the sun, A the earth, BCDE the moon's orbit, C the place of the moon when in the latter quarter. Here it will be nearly at the same distance from the sun as the earth is. In this case, therefore, they will be both equally attracted, the earth in the direction AS, and the moon in that of CS. Whence, as the earth, in moving round the sun, is continually descending towards it, so the moon in this situation must in any equal portion of time descend as much; and therefore the position of the line AC in respect of AS, and the change which the moon's motion produces in the angle CAS, will not be altered by the sun; but as soon as the moon is advanced from the quarter toward the new or conjunction, suppose to G, the action of the sun upon it will have a different effect. Were the sun's action upon the moon here to be applied in the direction GH parallel to AS, if its action on the moon were equal to its action on the earth, no change would be wrought by the sun on the apparent motion of the moon round the earth. But the moon receiving a greater impulse in G than the earth receives in A, were the sun to act in the direction GH, yet it would accelerate the description of the space DAG, and cause the angle under GAD to decrease faster than it otherwise would. The sun's action will have this effect upon account of the obliquity of its direction to that in which the earth attracts the moon. For the moon by this means is drawn by two forces oblique to one another; one drawing from G towards A, the other from G towards H; therefore the moon must necessarily be impelled toward D. Again, because the sun does not act in the direction GH parallel to SA, but in the direction GS oblique to it, the sun's action on the moon will, by reason of this obliquity, farther contribute to the moon's acceleration. Suppose the earth, in any short space of time, would have moved from A to I, if not attracted by the sun, the point I being in the straight line CE, which touches the earth's orbit in A. Suppose the moon in the same time would have moved in her orbit from G to K, and besides have partook of all the progressive motion of the earth. Then, if KL be drawn parallel to AI, and taken equal to it, the moon, if not attracted to the sun, would be found in L. But the earth, by the sun's action, is removed from I. Suppose it were moved down to M in the line IMN parallel to SA, and if the moon were attracted but as much, and in the same direction as the earth is here supposed to be attracted, so as to have descended during the same time in the line LO parallel also to AS, down as far as P, till LP were equal to Motions of IM, the angle under PMN would be equal to that under LIN; that is, the moon will appear advanced as much farther forward than if neither it nor the earth had been subject to the sun's action. But this is on the supposition that the actions of the sun upon the earth and moon are equal; whereas the moon being acted upon more than the earth, did the sun's action draw the moon in the line LO parallel to AS, it would draw it down so far as to make LP greater than IM, whereby the angle under PMN will be rendered greater than that under LIN. But, moreover, as the sun draws the earth in a direction oblique to IN, the earth will be found in its orbit somewhat short of the point M. However, the moon is attracted by the sun still more out of the line LO than the earth is out of the line IN; therefore this obliquity of the sun's action will yet farther diminish the angle under PMN. Thus the moon at the point G receives an impulse from the sun whereby her motion is accelerated; and the sun producing this effect in every place between the quarter and the conjunction, the moon will move from the quarter with a motion continually more and more accelerated; and therefore, by acquiring from time to time an additional degree of velocity in its orbit, the spaces which are described in equal times by the line drawn from the earth to the moon will not be every where equal, but those toward the conjunction will be greater than those toward the quarter. But in the moon's passage from the conjunction D to the next quarter, the sun's action will again retard the moon, till, at the next quarter at E, it be restored to the first velocity which it had in C. When the moon moves from E to the full, or opposition to the sun in B, it is again accelerated; the deficiency of the sun's action on the moon from what it has upon the earth producing here the same effect as before the excess of its action.

Let us now consider the moon in Q as moving from E towards B. Here, if she were attracted by the sun in a direction parallel to AS, yet being acted on less than the earth, as the latter descends towards the sun, the moon will in some measure be left behind. Therefore, QF being drawn parallel to SB, a spectator on the earth would see the moon move as if attracted from the point Q in the direction QF, with a degree of force equal to that whereby the sun's action on the moon falls short of its action on the earth. But the obliquity of the sun's action has here also an effect. In the time the earth would have moved from A to I without the influence of the sun, let the moon have moved in its orbit from Q to R. Drawing, therefore, RT parallel and equal to AI, the moon, by the motion of its orbit, if not attracted by the sun, must be found in T; and therefore, if attracted in a direction parallel to SA, would be in the line TV parallel to AS; suppose in W. But the moon in Q being farther off the sun than the earth, it will be less attracted; that is, TW will be less than IM; and if the line SM be prolonged towards X, the angle under XMW will be less than XIT. Thus, by the sun's action, the moon's passage from the quarter to the full would be accelerated, if the sun were to act on the earth and moon in a direction parallel to AS; and the obliquity of the sun's action will still increase this acceleration; Motions of the sun on the moon is oblique to the line SA the whole time of the moon's passage from Q to T, and will carry her out of the line TV towards the earth. Here we suppose the time of the moon's passage from Q to T so short, that it shall not pass beyond the line SA. The earth will also come a little short of the line IN, as was already mentioned; and from these causes the angle under XMW will be still farther lessened. The moon, in passing from the opposition B to the next quarter, will be retarded again by the same degrees as it was accelerated before its appulse to the opposition; and thus the moon, by the sun's action upon it, is twice accelerated and twice restored to its first velocity every circuit it makes round the earth; and this inequality of the moon's motion about the earth is called by astronomers its variation.

The next effect of the sun upon the moon is, that it gives the orbit of the latter in the quarters a greater degree of curvature than it would receive from the earth alone; and, on the contrary, in the conjunction and opposition the orbit is less inflected. When the moon is in the conjunction with the sun at D, the latter attracting her more forcibly than it does the earth, the moon is by that means impelled less to the earth than otherwise it would be, and thus the orbit less incurvated: for the power by which the moon is impelled towards the earth being that by which it is inflected from a rectilinear course, the less that power is, the less it will be inflected. Again, when the moon is in the opposition in B farther removed from the sun than the earth is, it follows then, that though the earth and moon are both continually descending toward the sun, that is, are drawn by the sun towards itself out of the place they would otherwise move into, yet the moon descends with less velocity than the earth; inasmuch that, in any given space of time from its passing the point of opposition, it will have less approached the earth than otherwise it would have done; that is, its orbit, in respect to the earth, will approach nearer to a straight line. Lastly, when the motion is in the quarter in F, and equally distant from the sun as the earth, it was before observed, that they would both descend with equal velocity towards the sun, so as to make no change in the angle FAS; but the length of the line FA must necessarily be shortened. Therefore the moon, in moving from F toward the conjunction with the sun, will be impelled more toward the earth by the sun's action than it would have been by the earth alone, if neither the earth nor the moon had been acted upon by the sun; so that, by this additional impulse, the orbit is rendered more curve than it otherwise should be. The same effect will also be produced in the other quarter.

A third effect of the sun's action, and which follows from that just now explained, is, that though the moon undisturbed by the sun might move in a circle, having the earth for its centre, by the sun's action, if the earth were to be in the very middle or centre of the moon's orbit, yet the moon would be nearer the earth at the new and full than in the quarters. This may at first appear somewhat difficult to be understood, that the moon should come nearest to the earth where it is least attracted by it: yet, upon a little consideration, it will evidently appear to flow from that very cause, because her orbit, in the conjunction and opposition, is rendered less curve; for the less curve the orbit is, the less will the moon have descended from the place it would move into without the action of the earth. Now, if the moon were to move from any place without further disturbance from that action, since it would proceed on the line touching the orbit in that place, it would continually recede from the earth; and therefore, if the power of the earth upon the moon be sufficient to retain it at the same distance, this diminution of that power will cause the distance to increase, though in a less degree. But, on the other hand, in the quarters, the moon being pressed in a less degree towards the earth than by the earth's single action, will be made to approach it: so that, in passing from the conjunction or opposition to the quarters, the moon ascends from the earth; and in passing from the quarters to the opposition or conjunction, it descends again, becoming nearer in these last mentioned places than in the other.

All the inequalities we have mentioned are different in degree as the sun is more or less distant from the earth; being greatest when the earth is in its perihelion, and smallest when it is in its aphelion: for in the quarters, the nearer the moon is to the sun the greater is the addition to the earth's action upon it by the power of the sun; and in the conjunction and opposition, the difference between the sun's action upon the earth and upon the moon is likewise so much the greater. This difference in the distance between the earth and the sun produces a further effect upon the moon's dilatation of motion; causing her orbit to dilate when less remote from the sun, and become greater than when at a farther distance: For it is proved by Sir Isaac Newton, that the action of the sun by which it diminishes the earth's power over the moon in the conjunction or opposition, is about twice as great as the addition to the earth's action by the sun in the quarters; so that upon the whole, the power of the earth on the moon is diminished by the sun, and therefore is most diminished when that action is strongest: but as the earth by its approach to the sun, has its influence lessened, the moon, being less attracted, will gradually recede from the earth; and as the earth, in its recession from the sun, recovers by degrees its former power, the orbit of the moon must again contract. Two consequences follow from hence, viz. that the moon will be more remote from the earth when the latter is nearest the sun, and also will take up a longer time in performing its revolution through the dilated orbit than through the more contracted.

These irregularities would be produced if the moon, without being acted upon unequally by the sun, should describe a perfect circle about the earth and in the plane of its motion; but though neither of these circumstances take place, yet the above-mentioned inequalities occur only with some little variation with regard to the degree of them; but some others are observed to take place from the moon's motion being performed in the manner already described: For, as the moon describes an ellipse, having the earth in one of its foci, this curve will be subjected to various changes, neither preserving constantly the same figure nor position; and because the plane of this ellipse is not the same with that of the earth's orbit, it thence follows, that the former will continually change; so that Motions of that neither the inclination of the two planes towards each other, nor the line in which they intersect, will remain for any length of time unaltered.

As the moon does not move in the same plane with the earth, the sun is but seldom in the plane of her orbit, viz. only when the line made by the common intersection of the two planes, if produced, will pass through the sun. Thus, let S in fig. 125 denote the sun, T the earth, ATB the plane of the earth's orbit, CDEF the moon's orbit; the part CDE being raised above, and the part CFE depressed under, the former. Here the line CE, in which the two planes intersect each other, being continued, passes through the sun in S. When this happens, the action of the sun is directed in the plane of the moon's orbit, and cannot draw her out of this plane, as will evidently appear from an inspection of the figure; but in other cases the obliquity of the sun's action to the plane of the orbit will cause this plane continually to change.

Let us now suppose, in the first place, the line in which the two planes intersect each other to be perpendicular to the line which joins the earth and sun. Let T, in fig. 126, 127, 128, 129, represent the earth; S the sun; the plane of the scheme the plane of the earth's orbit, in which both the sun and earth are placed. Let AC be perpendicular to ST, which joins the earth and sun; and let the line AC be that in which the plane of the moon's orbit intersects the orbit of the earth. On the centre T describe in the plane of the earth's motion the circle ABCD; and in the plane of the moon's orbit describe the circle AECF; one half of which, AEC, will be elevated above the plane of this scheme, and the other half AFC, as much depressed below it. Suppose then the moon to set out from the point A in fig. 127, in the direction of the plane AEC. Here she will be continually drawn out of this plane by the action of the sun; for this plane AEC, if extended, will not pass through the sun, but above it; so that the sun, by drawing the moon directly toward itself, will force it continually more and more from that plane towards the plane of the earth's motion in which itself is, causing it to describe the line AKGHI, which will be convex to the plane AEC, and concave to the plane of the earth's motion. But here this power of the sun, which is said to draw the moon toward the plane of the earth's motion, must be understood principally of as much only of the sun's action upon the moon as it exceeds the action of the same upon the earth: For suppose the last mentioned figure to be viewed by the eye placed in the plane of that scheme, and in the line CTA, on the side A, it will appear as the straight line DTB in fig. 130, and the plane AECF as another straight line FE, and the curve line AKGHI under the form of the line TKGHI. Now it is plain, that the earth and moon being both attracted by the sun, if the sun's action upon both was equally strong, the earth T, and with it the plane AECF, or the line FTE, would be carried towards the sun with as great velocity as the moon, and therefore the moon not drawn out of it by the sun's action, except only from the small obliquity of direction of this action upon the moon to that of the sun's action upon the earth, which arises from the moon being out of the plane of the earth's motion, and is not considerable; but the action of the sun upon the moon being greater than upon the earth all the time the moon is nearer to the sun than the earth is, it will be drawn from the plane AEC, or the line TE, by that excess, and made to describe the curve line AGI or TGI.

But it is the custom of astronomers, instead of considering the moon as moving in such a curve line, to refer its motion continually to the plane which touches the true line wherein it moves at the point where at any time the moon is. Thus, when the moon is in the point A, its motion is considered as being in the plane AEC, in whose direction it then attempts to move; and when in the point K, fig. 127, its motion is referred to the plane which passes through the earth and touches the line AKGHI in the point K. Thus the moon, in passing from A to I, will continually change the plane of her motion in the manner we shall now more particularly explain.

Let the plane which touches the line AKI in the point K, fig. 127, intersect the plane of the earth's orbit in the line LTM. Then, because the line AKI is concave to the plane ABC, it falls wholly between that plane and the plane which touches it in K; so that the plane MKL will cut the plane AEC before it meets the plane of the earth's motion, suppose in the line YT, and the point A will fall between K and L. With a radius equal to TY or TL describe the semicircle LYM. Now, to a spectator on the earth, the moon when in A will appear to move in the circle AECF; and when in K, will appear to be moving in the semicircle LYM. The earth's motion is performed in the plane of this scheme; and to a spectator on the earth the sun will always appear to move in that plane. We may therefore refer the apparent motion of the sun to the circle ABCD described in this plane about the earth. But the points where this circle in which the sun seems to move, intersecting the circle in which the moon is seen at any time to move, are called the nodes of the moon's orbit at that time. When the moon is seen moving in the circle AECF, the points A and C are the nodes of the orbit; when she appears in the semicircle LYM, then L and M are the nodes. It will now appear, from what has been said, that while the moon has moved from A to K, one of the nodes has been carried from A to L, and the other as much from C to M. But the motion from A to L and from C to M is backward in regard to the motion of the moon, which is the other way from A to K, and from thence toward C. Again, the angle which the plane wherein the moon at any time appears makes with the plane of the earth's motion, is called the inclination of the moon's orbit at that time: we shall now therefore proceed to show, that this inclination of the orbit, when the moon of her own is in K, is less than when she was in A; or, that bit the plane LYM, which touches the line of the moon's motion in K, makes a less angle with the plane of the earth's motion, or with the circle ABCD, than the plane AEC makes with the same. The semicircle LYM intersects the semicircle AEC in Y, and the arch AY is less than LY, and both together less than half a circle. But it is demonstrated by spheric geometry, that when a triangle is made as here, by three arches of circles AL, AY, and YL, the angle under YAB without the triangle is greater than the angle YLA within, if the two arches AY, YL, taken together, Motions of the secondary Planets.

If the two arches make a complete semicircle, the two angles will be equal; but if the two arches taken together exceed a semicircle, the inner angle YLA is greater than the other. Here then the two arches AY and LY together being less than a semicircle, the angle under ALY is less than the angle under BAE. But from the doctrine of the sphere it is also evident, that the angle under ALY is equal to that in which the plane of the circle LYKM, that is, the plane which touches the line AKGH in K is inclined to the plane of the earth's motion ABC; and the angle under BAE is equal to that in which the plane AEC is inclined to the same plane. Therefore the inclination of the former plane is less than that of the latter. Suppose, now, the moon to be advanced to the point G in fig. 128, and in this point to be distant from its node a quarter part of the whole circle; or, in other words, to be in the mid-way between its two nodes. In this case the nodes will have receded yet more, and the inclination of the orbit be still more diminished: for suppose the line AKGH to be touched in the point G by a plane passing through the earth T, let the intersection of this plane with the plane of the earth's motion be the line WTO, and the line TP its intersection with the plane LKM. In this plane let the circle NGO be described with the semidiameter TP or NT cutting the other circle LKM in P. Now, the line AKGH is convex to the plane LKM which touches it in K; and therefore the plane NGO, which touches it in G, will intersect the other touching plane between G and K; that is, the point P will fall between these two points, and the plane continued to the plane of the earth's motion will pass beyond L; so that the points N and O, or the places of the nodes when the moon is in G, will be farther from A and C than L and M; that is, will have moved farther backward. Besides, the inclination of the plane NGO to the plane of the earth's motion ABC is less than the inclination of the plane LKM to the same; for here also the two arches LP and NP, taken together, are less than a semicircle, each of them being less than a quadrant, as appears, because GN, the distance of the moon in G from its node N, is here supposed to be a quarter part of a circle. After the moon is passed beyond G, the case is altered: for then these arches will be greater than quarters of a circle; by which means the inclination will be again increased, though the nodes still go on to move the same way. Suppose the moon in H (fig. 129.), and that the plane which touches the line AKGH in H intersects the plane of the earth's motion in the line QTR, and the plane NGO in the line TV, and besides, that the circle QHR be described in that plane: then, for the same reason as before, the point V will fall between H and G, and the plane RVQ will pass beyond the last plane OVN, causing the points Q and K to fall farther from A and C than N and O. But the arches NV, VQ are each greater than the quarter of a circle; consequently the angle under BQV will be greater than that under BNV. Lastly, when the moon is by this attraction of the sun drawn at length into the plane of the earth's orbit, the node will have receded yet more, and the inclination be so much increased, as to become somewhat more than at first; for the line AKGH being convex to all the planes which touch it, the part HI will wholly fall between the plane QVR and the second plane ABC; so that the point I will fall between B and R; and, drawing ITW, the point W will be farther removed from A than Q. But it is evident, that the plane which passes through the earth T and touches the line AGI in the point I, will cut the plane of the earth's motion ABCD in the line ITW, and be inclined to the same in the angle under HIB; so that the node which was first in A, after having passed into L, N, and Q, comes at last in the point W, as the node which was at first in C has passed from thence successively through the points M, O, and R, to I. But the angle HIB, which is now the inclination of the orbit to the plane of the ecliptic, is manifestly not less than the angle under ECB or EAB, but rather something greater. Thus the moon, while it passes from the plane of the earth's motion in the quarter, till it comes again into the same plane, has the nodes of its orbit continually moved backward, and the inclination of it at first diminished till it comes to G in fig. 128, which is near to its conjunction with the sun, but afterwards is increased again almost by the same degrees, till upon the moon's arrival again to the plane of the earth's motion the inclination of the orbit is restored to something more than its first magnitude, though the difference is not very great, because the points I and C are not far distant from each other.

In like manner, if the moon had departed from the quarter at C, it should have described the curve line CXW in fig. 126. between the planes AFC and ADC, which would be convex to the former planes and concave to the latter; so that here also the nodes would continually recede, and the inclination of the orbit gradually diminish more and more, till the moon arrived near its opposition to the sun in X; but from that time the inclination should again increase till it become a little greater than at first. This will easily appear by considering, that as the action of the sun upon the moon, by exceeding its action upon the earth, drew it out of the plane AEC towards the sun, while the moon passed from A to I; so during its passage from C to W, the moon being all that time farther from the sun than the earth, it will be attracted less; and the earth, together with the plane AECF, will as it were be drawn from the moon, in such a manner, that the path the moon describes shall appear from the earth as it did in the former case by the moon being drawn away.

Such are the changes which the nodes and inclination of the moon's orbit undergo when the nodes are in the quarters; but when the nodes by their motion, and the motion of the sun together, come to be situated between the quarter and conjunction or opposition, their motion and the change made in the inclination of the orbit are somewhat different.—Let AGH, in fig. 131. be a circle described in the plane of the earth's motion, having the earth in T for its centre, A the point opposite to the sun, and G a fourth part of the circle distant from A. Let the nodes of the moon's orbit be situated in the line BT'D, and B the node falling between A, the place where the moon would be in the full, and G the place where she would be in the quarter. Suppose BEDF to be the plane in which the moon attempts to move when it proceeds from the point B; then, because the moon in B is more distant... Motions of distant from the sun than the earth, it will be less attracted by the sun, and will not descend towards the sun so fast as the earth, consequently it will quit the plane BEDF, which is supposed to accompany the earth, and describe the line BIK convex to it, till such time as it comes to the point K, where it will be in the quarter; but from thenceforth being more attracted than the earth, the moon will change its course, and the following part of the path it describes will be concave towards the plane BED or BG-D, and continue concave to the plane BGD till it crosses that plane in L just as in the preceding case. Now, to show that the nodes, while the moon is passing from B to K, will proceed forward, or move the same way with the moon, and at the same time the inclination of the orbit will increase when the moon is in the point I, let the line MIN pass through the earth T, and touch the path of the moon in I, cutting the plane of the earth's motion in the line MTN, and the line BED in TO. Because the line BIK is convex to the plane BED, which touches it in B, the plane NIM must cross the plane DEB before it meets the plane CGB; and therefore the point M will fall from G towards B; and the node of the moon's orbit being translated from B towards M is moved forward.

Again, the angle under OMG, which the plane MON makes with the plane BGC, is greater than the angle OBG, which the plane BOD makes with the same. This appears from what has been already demonstrated, because the arches BO and OM are each of them less than the quarter of a circle; and therefore, taken both together are less than a semicircle. But further, when the moon is come to the point K in its quarter, the nodes will be advanced yet farther forward, and the inclination of the orbit also more augmented. Hitherto we have referred the moon's motion to that plane which, passing through the earth, touches the path of the moon in the point where the moon is, as we have already said that the custom of astronomers is. But in the point K no such plane can be found: on the contrary, seeing the line of the moon's motion on one side the point K is convex to the plane BED, and on the other side concave to the same, so that no plane can pass through the points T and K, but will cut the line BKL in that point; therefore, instead of such a touching plane, we must make use of PKQ, which is equivalent, and with which the line BKL shall make a less angle than with any other plane; for this does as it were touch the line BK in the point K, since it cuts it in such a manner that no other plane can be drawn so as to pass between the line BK and the plane PKQ. But now it is evident, that the point P, or the node, is removed from M towards G, that is, has moved yet farther forward; and it is likewise as manifest, that the angle under KPG, or the inclination of the moon's orbit in the point K, is greater than the angle under IMG for the reason already given.

After the moon has passed the quarter, her plane being concave to the plane AGCH, the nodes will recede as before till she arrives at the point L; which shows, that considering the whole time of the moon's passing from B to L, at the end of that time the nodes shall be found to have receded, or to be placed more backward, when the moon is in L than when it was in B; for the moon takes a longer time in passing from K to L than in passing from B to K; and therefore the nodes continue to recede a longer time than they moved forwards; so that their recess must surmount their advance. In the same manner, while the moon is in its passage from K to L, the inclination of the orbit shall diminish till the moon come to the point in which it is one quarter part of a circle distant from its node, suppose in the point R; and from that time the inclination will again increase. Since, therefore, the inclination of the orbit increases while the moon is passing from B to K, and diminishes itself again only while the moon is passing from K to R, then augments again while the moon passes from B to L; it thence comes to be much more increased than diminished, and thus will be distinguishably greater when the moon comes to L than when it sets out from B. In like manner, when the moon is passing from L on the other side the plane AGCH, the node will advance forward as long as the moon is between the point L and the next quarter; but afterwards it will recede till the moon come to pass the plane AGCH again, in the point V between B and A; and because the time between the moon's passing from L to the next quarter is less than the time between that quarter and the moon's coming to the point V, the node will have receded more than it has advanced; so that the point V will be nearer to A than L is to C. So also the inclination of the orbit, when the moon is in V, will be greater than when she was in L; for this inclination increases all the time the moon is betwixt L and the next quarter, decreasing only when she is passing from this quarter to the mid-way between the two nodes, and from thence increases again during the whole passage through the other half of the way to the next node.

In this manner we see, that at every period of the moon the nodes will have receded, and thereby have approached towards a conjunction with the sun; but this will be much forwarded by the motion of the earth, or the apparent motion of the sun himself. In the last scheme the sun will appear to have moved from S towards W. Let us suppose it had appeared to have moved from S to W while the moon's node has receded from B to V; then drawing the line WTX, the arch VX will represent the distance of the line drawn between the nodes from the sun when the moon is in V; whereas the arch BA represented that distance when the moon was in B. This visible motion of the sun is much greater than that of the node; for the sun appears to revolve quite round in one year, while the node is near nineteen in making its revolution. We have also seen, that when the moon was in the quadrature, the inclination of her orbit decreased till she came to the conjunction or opposition, according to the node it set out from; but that afterwards it again increased till it became at the next node rather greater than at the former. When the node is once removed from the quarter nearer to a conjunction with the sun, the inclination of the moon's orbit when she comes into the node is more sensibly greater than it was in the node preceding; the inclination of the orbit by this means more and more increasing till the node comes into conjunction with the sun: at which time it has been shown that the latter has no power to change the plane of her orbit. As soon, however, as the nodes are got out of conjunction towards the other quarters, they begin to recede as before; but the in- clination of the orbit in the appulse of the moon to each succeeding node is less than at the preceding, till the nodes come again into the quarters. This will appear as follows: Let A, in fig. 132., represent one of the moon's nodes placed between the point of oppo- sition B and the quarter C. Let the plane ADE pass through the earth T, and touch the path of the moon in A. Let the line AFGH be the path of the moon in her passage from A to H, where she crosses again the plane of the earth's motion. This line will be convex towards the plane ADE, till the moon comes to G, where she is in the quarter; and after this, between G and H, the same line will be concave towards this plane. All the time this line is convex towards the plane ADE, the nodes will recede; and, on the contrary, move forward when the line is con- cave towards that plane. But the moon is longer in passing from A to G, and therefore the nodes go back- ward farther than they proceed; and therefore, on the whole, when the moon has arrived at H, the nodes will have receded, that is, the point H will fall between B and E. The inclination of the orbit will decrease till the moon is arrived at the point F in the middle between A and H. Through the passage between F and G the inclination will increase, but decrease again in the remaining part of the passage from G to H, and con- sequently at H must be less than at A. Similar ef- fects, both with respect to the nodes and inclination of the orbit, will take place in the following passage of the moon on the other side of the plane ABEC from H, till it comes over that plane again in I.

Thus the inclination of the orbit is greatest when the line drawn between the moon's nodes will pass through the sun, and least when this line lies in the quarters; especially if the moon at the same time be in conjunction with the sun, or in the opposition. In the first of these cases the nodes have no motion; in all others, the nodes will each month have receded; and this retrograde motion will be greatest when the nodes are in the quarters, for in that case they will have no progressive motion during the whole month; but in all other cases they at some times go forward, viz. whenever the moon is between either of the quarters and the node which is least distant from that quarter than the fourth part of a circle.

We have now only to explain those irregularities of the lunar motion which arise from her motion in an ellipsis. From what has been already said it appears, that the earth acts on the moon in the reciprocal du- plicate proportion of the distance; therefore the moon, if undisturbed by the sun, would move round the earth in a true ellipsis, and a line drawn from the earth to the sun would pass over equal spaces in equal times. We have, however, already shown, that this equality is disturbed by the sun, and likewise how the figure of the orbit is changed each month; that the moon is nearer the earth at the new and full, and more re- mote in the quarters, than it would be without the sun. We must, however, pass by those monthly changes, and consider the effect which the sun will have in the dif- ferent situations of the axis of the orbit in respect of that luminary. This action varies the force wherewith the moon is drawn towards the earth. In the quarters the force of the earth is directly increased by the sun, the Sec- ondary Planets. but diminished at the new and full; and in the inter- mediate places the influence of the earth is sometimes lessered, sometimes afflited, by the action of that lumi- nary. In these intermediate places, however, between the quarters and the conjunction or opposition, the sun's action is so oblique to that of the earth on the moon, as to produce that alternate acceleration and re- tardation of her motion so often mentioned. But be- sides this effect, the power by which the moon attracts the earth towards itself, will not be at full liberty to act with the same force as if the sun acted not at all on the moon; and this effect of the sun's action, whereby it corroborates or weakens the action of the earth, is here only to be considered; and by means of this influence it comes to pass, that the power by which the moon is impelled towards the earth is not perfectly in the reciprocal duplicate proportion of the distance, and of consequence the moon will not describe a per- fect ellipsis. One particular in which the lunar orbit will differ from a perfect elliptical figure, consists in the places where the motion of the moon is perpendicular to the line drawn from itself to the earth. In an ellip- sis, after the moon should have set out in the direction perpendicular to this line, drawn from itself to the earth, and at its greatest distance from the earth, its motion would again become perpendicular to this line drawn between itself and the earth, and the moon be at its nearest distance from the earth, when it should have performed half its period: after having performed the other half period of its motion, it would again be- come perpendicular to the forementioned line, and the moon return to the place whence it set out; and have recovered again its greatest distance. But the moon in its real motion, after setting out as before, some- times makes more than half a revolution before its mo- tion comes again to be perpendicular to the line drawn from itself to the earth, and the moon is at its nearest distance, and then performs more than another half of an entire revolution before its motion can a second time recover its perpendicular direction to the line drawn from the moon to the earth, and the former arrive again at its greatest distance from the earth. At other times the moon will descend to her nearest distance be- fore she has made half a revolution, and recover again its greatest distance before it has made an entire revolu- tion. The place where the moon is at its greatest di- stance is called the moon's apogee, and the place of her and peri- nearest distance her perigee; and this change of place, geo of the where the moon comes successively to its greatest di- stance from the earth, is called the motion of the apogee. The manner in which this motion of the apogee is caused by the sun, comes now to be explained.

Sir Isaac Newton has shown, that if the moon were attracted toward the earth by a composition of two powers, one of which were reciprocally in the dupli- cate proportion of the distance from the earth, and the other reciprocally in the triplicate proportion of the Tame distance; then, though the line described by the moon would not be in reality an ellipsis, yet the moon's motion might be perfectly explained by an ellipsis whose axis should be made to move round the earth; this motion being in consequence, as astronomers express themselves, that is, the same way as the moon itself moves, Motions of moves; if the moon be attracted by the sum of the two powers; but the axis must move in antecedence, or the contrary way, if the moon be acted upon by the difference of these forces. We have already explained what is meant by duplicate proportion, namely, that if three magnitudes, as A, B, and C, are so related that the second B bears the same proportion to the third C as the first A bears to the second B; then the proportion of the first A to the third C is the duplicate of the proportion of the first A to the second B. Now if a fourth magnitude as D be assumed, to which C shall bear the same proportion as A bears to B, and B to C; then the proportion of A to D is the triplicate of the proportion of A to B.

Let now T (fig. 133, 134.) denote the earth, and suppose the moon in the point A its apogee or greatest distance from the earth, moving in the direction AF perpendicular to AB, and acted upon from the earth by two such forces as already mentioned. By that power alone, which is reciprocally in the duplicate proportion of the distance, if the moon set out with a proper degree of velocity, the ellipsis AMB may be described; but if the moon be acted upon by the sum of the aforementioned powers, and her velocity in the point A be augmented in a certain proportion; or if that velocity be diminished in a certain proportion, and the moon be acted upon by the difference of those powers; in both these cases the line AE, which shall be described by the moon, shall thus be determined.

Let the point M be that into which the moon would have arrived in any given point of time, had it moved in the ellipsis AMB; draw MT and likewise CTD in such a manner that the angle ATM shall bear the same proportion to the angle under ATC as the velocity with which the ellipsis must have been described bears to the difference between this velocity and that with which the moon must set out from the point A, in order to describe the path AE. Let the angle ATC be taken toward the moon, as in fig. 133, if the moon be attracted by the sum of the powers; but the contrary way (as in fig. 134.) if by their difference. Then let the line AB be moved into the position CD, and the ellipsis AMB into the situation CND, so that the point M be translated to L; then the point L shall fall upon the path of the moon AE.

Now the angular motion of the line AT, whereby it is removed into the situation CT, represents the motion of the apogee; by the means of which the motion of the moon might be fully explained by the ellipsis AMB, if the action of the sun upon it was directed to the centre of the earth, and reciprocally in the triplicate proportion of the moon's distance from it; but that not being so, the motion of the apogee will not proceed in the regular manner now described. It is, however, to be observed here, that in the first of the two preceding cases, where the apogee moves forward, the whole centripetal power increases faster, with the decrease of distance, than if the entire power were reciprocally in the duplicate proportion of the distance; because one part only is already in that proportion, and the other part, which is added to this to make up the whole power, increases faster with the decrease of distance. On the other hand, when the centripetal power is the difference between these two bodies, it increases less with the decrease of the distance, than if it were simply in the reciprocal duplicate proportion of the distance. Therefore, if we choose the secondary Pla-

Sir Isaac shows, in the next place, that when the inequality line AB coincides with the line that joins the sun and the moon, the progressive motion of the apogee, when the moon is in conjunction or opposition, exceeds the retrograde, in the quadratures, more than in any other situation of the line AB. On the contrary, when the line AB makes right angles with that which joins the earth and sun, the retrograde motion will be more considerable, nay, is found to great as to exceed the progressive; so that in this case the apogee, in the compass of an entire revolution of the moon, is carried in antecedence. Yet from the considerations already mentioned, the progressive motion exceeds the other; so that on the whole, the motion of the apogee is in consequence. The line AB also changes its situation with that which joins the earth and sun by such slow degrees, that the inequalities of the motion of the apogee, arising from this last consideration, are much greater than what arise from the other.

This unsteady motion of the apogee gives rise to occasions another inequality in the motion of the moon herself, another in- so that it cannot at all times be explained by the same equality in the eccentricity of the orbit. For whenever the apogee moves in consequence, the motion of the luminous must be referred to an orbit more eccentric than what the moon would orbit, describe, if the whole power by which the moon was acted upon in its passing from the apogee changed according to the reciprocal duplicate proportion of its distance from the earth, and by that means the moon did describe an immoveable ellipsis; and when the apogee moves in antecedence, the moon's motion must be referred to an orbit less eccentric. In the former of the two figures last referred to, the true place of the moon L falls without the orbit AMB, to which its motion is referred; whence the orbit ALE truly described by the moon, is less incurvated in the point A than is the orbit AMB; therefore this orbit is more oblong, and differs farther from a circle than the ellipsis would, whose curvature in A were equal to that of the line ALB: that is, the proportion of the distance of the earth T from the centre of the ellipsis to its axis, will be greater in AMB than in the other; but that Motions of that other is the ellipse which the moon would describe, if the power acting upon it in the point A were altered in the reciprocal duplicate proportion of the distance; and consequently the moon being drawn more forcibly toward the earth, it will descend nearer to it. On the other hand, when the apogee recedes, the power acting on the moon increases with the decrease of distance, in less than the duplicate proportion of the distance; and therefore the moon is less impelled towards the earth, and will not descend so low. Now, suppose, in the former of these figures, that the apogee A is in the situation where it is approaching towards the conjunction or opposition of the sun; in this case its progressive motion will be more and more accelerated. Here suppose the moon, after having descended from A through the orbit AE as far as F, where it is come to its nearest distance from the earth, ascends again up the line FG. As the motion of the apogee is here more and more accelerated, it is plain that the cause of its motion must also be on the increase; that is, the power by which the moon is drawn to the earth, will decrease with the increase of the moon's distance in her ascent from F, in a greater proportion than that wherewith it is increased with the decrease of distance in the moon's descent to it. Consequently the moon will ascend to a greater distance than AT from whence it is descended; therefore the proportion of the greatest distance of the moon to the least is increased. But further, when the moon again descends, the power will increase yet farther with the decrease of distance than in the last ascent it increased with the augmentation of distance. The moon therefore must descend nearer to the earth than it did before, and the proportion of the greatest distance to the least be yet more increased. Thus, as long as the apogee is advancing to the conjunction or opposition, the proportion of the greatest distance of the moon from the earth to the least will continually increase; and the elliptical orbit to which the moon's motion is referred, will become more and more eccentric. As soon, however, as the apogee is past the conjunction or opposition with the sun, its progressive motion abates, and with it the proportion of the greatest distance of the moon from the earth to the least will also diminish; and when the apogee becomes retrograde, the diminution of this proportion will be still farther continued, until the apogee comes into the quarter; from thence this proportion, and the eccentricity of the orbit, will increase again. Thus the orbit of the moon is most eccentric when the apogee is in conjunction with the sun, or in opposition to it, and least of all when the apogee is in the quarters. These changes in the nodes, the inclination of the orbit to the plane of the earth's motion, in the apogee and in the eccentricity, are varied like the other inequalities in the motion of the moon, by the different distance of the earth from the sun being greatest when their cause is greatest; that is, when the earth is nearest the sun. Sir Isaac Newton has computed the very quantity of many of the moon's inequalities. That acceleration of the moon's motion which is called the variation, when greatest, removes the luminary out of the place in which it would otherwise be found, somewhat more than half a degree. If the moon, without disturbance from the sun, would have described a circle concentric to the earth, his action will cause her approach Nature and nearer in the conjunction and opposition than in the Motions of the Co-quarters, nearly in the proportion of 69 to 70. It has already been mentioned, that the nodes perform their period in almost 19 years. This has been found by observation; and the computations of Sir Isaac assign to them the same period. The inclination of the moon's orbit, when least, is an angle about one-eighth part of that which constitutes a right angle; and the difference between the greatest and least inclination is about one eighteenth of the least inclination, according to our author's computations; which is also agreeable to the general observations of astronomers. The motion of the apogee and the changes in the eccentricity have not been computed by Sir Isaac.

The same incomparable geometer shows how, by comparing the periods of the motions of the satellites which revolve round Jupiter and Saturn with the period of our moon round the earth, and the periods of those planets round the sun with our earth's motion, Jupiter's inequalities in the motion of those satellites may be telltles, computed from those of our moon, excepting only the motion of the apogee; for the orbits of those satellites, as far as can be discerned by us at this distance, appearing little or nothing eccentric, this motion, as deduced from the moon, must be diminished.

§ 4. Of the Nature and Motions of the Comets.

That these bodies are not meteors in our air is manifest, because they rise and set in the same manner as the moon and stars. The astronomers had gone so far in their inquiries concerning them, as to prove by their observations that they moved in the celestial spaces beyond the moon; but they had no notion of the path which they described. Before the time of our author, it was supposed that they moved in straight lines; and Descartes, finding that such a motion would interfere with his vortices, removed them entirely out of the solar system. Sir Isaac Newton, however, distinctly proves comets generally through the planetary regions, and are generally invisible until they come nearer than that of Jupiter. Hence, finding that they were evidently within the sphere of Jupiter, the sun's action, he concludes, that they must necessarily move about the sun as the planets do; and he proves, that the power of the sun being reciprocally in the duplicate proportion of the distance, every body acted upon by him must either fall directly down, or move about him in one of the conic sections; viz. either the ellipse, parabola, or hyperbola. If a body which descends towards the sun as low as the orbit of any planet, move with a swifter motion than the planet, it will describe an orbit of a more oblong figure than that of the planet, and have at least a longer axis. The velocity of the body may be so great, that it shall move in a parabola; so that having once passed the sun, it shall ascend forever without returning, though the sun will still continue in the focus of that parabola; and with a velocity still greater, they will move in an hyperbola. It is, however, most probable, that the comets move in very eccentric ellipses, such as is represented in fig. 135., where S represents the sun, C the comet, and ABDE its orbit; wherein the distance of S and D far exceeds that of S and A. Hence those bodies are sometimes found at a moderate distance from the sun, and appear within Nature and within the planetary regions; at other times they approach the comets.

That the comets do move in this manner is proved by our author from computations built upon the observations made by many astronomers. These computations were made by Sir Isaac Newton himself upon the comet which appeared toward the latter end of the year 1680 and beginning of 1681, and the same were prosecuted more at large by Dr Halley upon this and other comets. They depend on this principle, that the eccentricity of the orbits of the comets is so great, that if they are really elliptical, yet that part of them which comes under our view approaches so near to a parabola that they may be taken for such without any sensible error, as in the foregoing figure the parabola FAG, in the lower part of it about A, differs very little from the ellipse DEAB; on which foundation Sir Isaac teaches a method of finding the parabola in which any comet moves, by three observations made upon it in that part of its orbit where it agrees nearest with a parabola; and this theory is confirmed by astronomical observations; for the places of the comets may thus be computed as exactly as those of the primary planets. Our author afterwards shows how to make use of any small deviation from the parabola which may be observed, to determine whether the orbits of the comets be elliptical or not; and thus to know whether or not the same comet returns at different seasons. On examining by this rule the comet of 1680, he found its orbit to agree more exactly with an ellipse than a parabola, though the ellipse be so very eccentric, that it cannot perform its revolution in 500 years. On this Dr Halley observed, that mention is made in history of a comet with a similar large tail, which appeared three several times before. The first was before the death of Julius Caesar; and each appearance happened at the interval of 575 years, the last coinciding with the year 1680. He therefore calculated the motion of this comet to be in such an eccentric orbit, that it could not return in less than 575 years; which computations agree yet more perfectly with the observations made on this comet than any parabolic orbit will do. To compare together different appearances of the same comet, is indeed the only method of discovering with certainty the form of its orbit; for it is impossible to discover the form of one so exceedingly eccentric from observations taken in a small part of it. Sir Isaac Newton therefore proposes to compare the orbits, on the supposition that they are parabolical, of such comets as appear at different times; for if we find the same orbit described by a comet at different times, in all probability it will be the same comet that describes it. Here he remarks from Dr Halley, that the same orbit very nearly agrees to two appearances of a comet about the space of 75 years distance; so that if these two appearances were really of the same comet, the transverse axis of its orbit would be 18 times that of the axis of the earth's orbit; and therefore, when at its greatest distance from the sun, this comet would be removed not less than 35 times the mean distance of the earth from the same luminary.

Even this is the least distance assigned by our author to any comet in its greatest elongation from the sun; and on the foundation of Dr Halley's computations it was expected in the year 1758 or 1759.

The Astronomer Royal advertises us of the expected return of the comet of 1532 and 1661, in the latter end of the year 1788, or beginning of 1789, in the following particulars.

"The elements of the orbits of the comets observed by Appian in 1532, and by Hevelius in 1661, are so much alike as to have induced Mr Halley to suppose them to be one and the same comet; and astronomers since have joined in the same opinion. Hence it should return to its perihelion the 27th of April 1789. But from the disturbances of the planets, it will probably come a few months sooner. It will first be seen in the southern parts of the heavens, if any astronomers should watch for it in situations near the line, or in southern climates, in the course of the year 1788, and probably not before the month of September. Astronomers who may happen to be in those parts will be enabled to direct their telescopes for discovering it as early as possible, by being furnished with the following elements of its orbit:

- Perihelion distance: 0.4485 - Place of ascending node: 2° 24' 18" - Inclination of the orbit to the ecliptic: 32° 36' - Perihelion forwarder in the orbit than the ascending node: 33° 28'

Time of the perihelion in the latter end of the year 1788, or beginning of 1789. Its motion is direct. If it should come to its perihelion on January 1, 1789, it might be seen in the southern parts of the world with a good achromatic telescope about the middle of September, towards the middle of Pisces, with 55° south latitude, and 53° south declination."

Sir Isaac Newton observes, that as the great eccentricity of the orbits of comets renders them very liable to be disturbed by the attraction of the planets and other comets, it is probably to prevent too great disturbances from these, that while all the planets revolve nearly in the same plane, the comets are disposed in very different ones, and disposed all over the heavens; that when in their greatest distance from the sun, and moving slowest, they might be removed as far as possible out of the reach of each other's action. The same end is likewise answered in these comets, which by moving slowest in the aphelion or remotest distance from the sun, descend nearest to it by placing their aphelion at the greatest height from the sun. See more on the subject of comets by Sir Isaac, Sect. III, no 169.

§ 5. Of the Bodies of the Sun and Planets, with the Method of computing the Quantity of Matter they contain.

Our author having proved, as has been related, that the primary planets and comets are retained in their orbits by a power directed towards the sun, and that the secondaries are also retained by a power of the like kind directed to the centre of their primaries, proceeds next to demonstrate, that the same power is attractive diffused through their whole substance, and inherent in every particle. For this purpose he shows first, that throughout each of the heavenly bodies attracts the rest and other sub-bodies with such different degrees of force, as that the stance of all force of the same attracting body is exerted on others matter, exactly in proportion to the quantity of matter contained in the body attracted. The first proof of this he brings from experiments made on bodies here on earth. The power by which the moon is influenced has been already shown to be the same with that which we call gravity. Now, one of the effects of the principle of gravity is, that all bodies descend by this force from equal heights in equal times. This was taken notice of long ago; and particular methods have been invented to show, that the only cause why some bodies were observed to fall in a shorter time than others was the resistance of the air. As these methods, however, have been found liable to some uncertainty, Sir Isaac Newton had recourse to experiments made on pendulums. These vibrate by the same power which makes heavy bodies fall to the ground; but if the ball of any pendulum of the same length with another were more or less attracted in proportion to the quantity of solid matter it contains, that pendulum must then vibrate faster or slower than the other. Now the vibrations of pendulums continue for a long time, and the number of vibrations they make may be easily determined without any suspicion of error; so that this experiment may be extended to what exactness we please: and Sir Isaac assures us, that he examined in this way several substances, as gold, silver, lead, glass, sand, common salt, wood, water, and wheat; in all which he found not the least deviation from the theory, tho' he made the experiment in such a manner, that in bodies of the same weight, a difference in the quantity of their matter less than the thousandth part of the whole would have discovered itself. It appears, therefore, that all bodies are made to descend here by the power of gravity with the same degree of swiftness. This descent has already been determined at 16 feet in a second from the beginning of their fall. It has also been observed, that if any terrestrial body could be conveyed as high up as the moon, it would descend with the very same degree of velocity with which the moon is attracted toward the earth; and therefore that the power of the earth upon the moon bears the same proportion it would have upon those bodies at the same distance as the quantity of matter in the moon bears attraction to the quantity in those bodies. Thus the assertion is proportioned in the earth, that its power on every body it attracts is, at the same distance from the earth, proportional to the quantity of solid matter in the body acted upon. As to the sun, it has been shown, that the power of his action upon the same primary planet is reciprocally in the duplicate proportion of its distance; and that the power of the sun decreases throughout in the same proportion, is testified by the motion of the planets traversing the whole planetary region. This proves, that if any planet were removed from the sun to any distance whatever, the degree of its acceleration towards the sun would yet be reciprocally in the duplicate proportion of their distance. But it has already been proved, that the degree of acceleration given to the planets by the sun is reciprocally in the duplicate proportion of their respective distances; all which, compared together, puts it out of doubt, that the power of the sun upon any planet removed into the place of any other, would give it the same velocity of descent as it gives that other; and consequently that the sun's action upon different planets at the same distance would be proportionable to the quantity of matter in each. It has likewise been shown, that the sun attracts the primary planets and their respective secondaries, when at the same distance, in such a manner as to communicate to both the same degree of velocity; and therefore the force wherewith the sun acts on the secondary planet bears the same proportion to the force wherewith it attracts the primary, as the quantity of matter in the secondary planet bears to the quantity of matter in the primary. This property therefore is found in the sun with regard to both kinds of planets; so that he possesses the same quality found in the earth, viz., that of acting on bodies with a degree of force proportional to the quantity of matter they contain.

This point being granted, it is hardly to be supposed that the power of attraction with which the other planets are endowed, should be different from that of the earth, if we consider the similitude of these bodies; and that it does not in this respect, is farther evident from the satellites of Saturn and Jupiter, which are attracted according to this law; that is, in the same proportion to their distances that their primaries are attracted by the sun. So that what has been concluded of the sun in relation to the primary planets, may be justly concluded of those primaries in respect to their secondaries; and in consequence of that likewise in regard to all other bodies, viz., that they will attract every other body in proportion to the quantity of solid matter it contains. Hence it follows, that this attraction extends itself to every particle of matter in the attracted body, and that no portion of matter is exempted from the influence of these bodies to which this attractive power has been proved to belong.

Here we may remark, that the attractive power of both the sun and planets appears to be the same in acts equally all; for it acts in each in the same proportion to the distance, and in the same manner acts alike upon every in the unit particle of matter. This power, therefore, in the sun and planets, is not of a different nature from the power of gravity in the earth; and this enables us to prove, that the attracting power lodged in the sun and planets belongs likewise to every part of them; and that their respective powers upon the same body are proportional to the quantity of matter of which they are composed; for instance, that the force with which the earth attracts the moon, is to the force with which the sun would attract it at the same distance, as the quantity of solid matter in the earth is to that in the sun.

Before we proceed to a full proof of these assertions, it will be necessary to show that the third law of motion, viz. That action is equal to reaction, holds good in attractive powers as well as in any other. The most remarkable force of this kind with which we are acquainted, next to that of gravity, is the attraction the loadstone has for iron. Now if a loadstone and piece of iron were both made to swim on water, both of them would move towards each other, and thus the attraction would be shown to be mutual; and when they meet, they will mutually stop each other: which shows that their velocities are reciprocally proportioned to the quantities of solid matter in each; and that by the stone's attracting the iron, it receives as much motion itself, in the strict philosophical sense of the word, as it communicates to the iron; for it is proved from experiments on the percussion of bodies, that if two meet with velocities reciprocally proportional to the respective bodies, they will be stopped by the concurrence, unless they meet with some other velocity, or their elasticity. Of the Bodies which will attract the particles of any other body in the reciprocal proportion of their distances, the whole globe will attract the same in the reciprocal duplicate proportion of their distances from the centre of the globe, provided it be of equal density throughout. Hence also he deduces the reverse; that if a globe acts upon distant bodies by the law just now specified, and the power of the globe be derived from its being composed of attracting particles, each of these will attract after the same proportion. The manner of deducing this is as follows: The globe is supposed to act upon the particles of a body without it constantly in the reciprocal duplicate proportion of their distances from the centre; and therefore, at the same distance from the globe, on which side forever the body be placed, the globe will act equally upon it.

Now because if the particles of which the globe is composed acted upon those without in the reciprocal duplicate proportion of their distances, the whole globe would act upon them in the same manner as it does; therefore, if the particles of the globe have not all of them that property, some must act stronger than in that proportion, whilst others act weaker; and if this be the condition of the globe, it is plain, that when the body attracted is in such a situation in respect of the globe, that the greater number of the strongest particles are nearest to it, the body will be more forcibly attracted than when, by turning the globe about, the greater quantity of weak particles should be nearest, though the distance of the body should remain the same from the centre of the globe; which is contrary to what was at first remarked, that the globe acts equally on all sides.

It is further deduced from these propositions, that if all the particles of one globe attract all the particles of another in the proportion already mentioned, the attracting globe will act upon the other in the same proportion to the distance between the centre of the globe which attracts, and the centre of that which is attracted; and further, that the proportion holds true, though either or both of the globes be composed of dissimilar parts, some rarer, and some more dense; provided only, that all the parts in the same globe, equally distant from the centre, be homogeneous, and likewise if both globes attract each other.

Thus has our author shown that this power in the great bodies of the universe is derived from the same and universal being lodged in every particle of the matter which composes them; and consequently that it is no less than universal in matter, though the power be too minute to produce any visible effects on the small bodies with which we are conversant, by their action on one another. In the fixed stars indeed we have no particular proof that they have this power, as we find no appearance to demonstrate that they either act or are acted upon by it. But since this power is found to belong to all bodies whereon we can make observation, and we find that it is not to be altered by any change in the shape of bodies, but accompanies them in every form, without diminution, remaining ever proportional to the quantity of solid matter in each; such a power must without doubt universally belong to matter. All this naturally follows from a consideration of the phenomena of those planets which have secondaries revolving about them. By the times in which these satellites perform their revolutions, compared with their distances from their respective primaries, the proportion between the power with which one primary attracts his satellites and the force with which any other attracts his, will be known; and the proportion of the power with which any planet attracts its secondary to the power with which it attracts a body at its surface, is found by comparing the distance of the secondary planet from the centre of the primary to the distance of the primary planet's surface from the same; and from hence is deduced the proportion between the power of gravity upon the surface of one planet to the gravity upon the surface of another. By the like method of comparing the periodical time of a primary planet about the sun with the revolution of a satellite about its primary, may be found the proportion of gravity or of the weight of any body on the surface of the sun, to the gravity or to the weight of the same body upon the surface of the planet which carries about the satellite. By computations of this kind it is found, that the weight of any body on the surface of the sun will be about 23 times as great as on the surface of the earth; about 10 times as great as on the surface of Jupiter; and near 19 times as great as on Saturn.

The quantity of matter contained in each of these bodies is proportional to the power it has upon a body at a given distance. Thus it is found, that the sun contains 1067 times as much matter as Jupiter; Jupiter 158½ times as much as the earth, and 2¾ times as much as Saturn. The diameter of the sun, according to the data with which Sir Isaac Newton was furnished, was calculated at 92 times, that of Jupiter about 9 times, and that of Saturn about 7 times as large as the diameter of the earth.

By comparing the quantities of matter in each of the heavenly bodies with their respective magnitudes, their densities are likewise easily discovered; the density of every body being measured by the quantity of matter contained under the same bulk. Thus the earth is found 4½ times more dense than Jupiter, while Saturn has only between two-thirds and three-fourths of the density of the latter, and the sun has only one-fourth part of the density of the earth. From all this our author draws the following conclusions, viz. That the sun is rarefied by its great heat; and of the three planets above-mentioned, the most dense is that nearest the sun. This it was highly reasonable to expect, the densest bodies requiring the greatest heat to agitate and keep their parts in motion; as on the contrary, the planets which are more rare would be rendered unfit for their office by the intense heat to which the denser are exposed. Thus the waters of our seas, if removed to the distance of Saturn, would remain perpetually frozen, and at Mercury would constantly boil. The densities of the planets Mars, Venus, Mercury, and the Georgium Sidus, as they are not attended with planets on which many observations have been made, cannot be ascertained. From analogy, however, we ought to conclude, that the inferior planets, Venus and Mercury, are more dense than the earth, Mars more rare, and the Georgium Sidus much more rare, than any of the rest.

Sect. V. The Newtonian Doctrine applied more particularly to the Explication of the Celestial Phenomena.

From the general account of those laws by which the universe is upheld, we now proceed to give an explanation of the particular parts of which it is composed. Those which are most exposed to our researches, besides the earth we inhabit, are the Sun, Moon, Mercury, Venus, Mars, Jupiter, Saturn, and the Georgium Sidus (see fig. 119). The sun, an immense globe of fire, is situated near the centre of the system, round which he turns by a small and irregular motion, according as the common centre of gravity betwixt him and the planets, which is the true centre of the system, varies by their different positions on this or that side of him. All the planets move round this common centre of gravity together with the sun; but the latter, by reason of his vast bulk, is so near the true centre, that the motions of the celestial bodies are by astronomers always referred to the centre of the sun as the point round which they are directed. The motions of all of them are performed the same way, viz. from west to east; and some comets have been observed to move also in this way, though the motion of others has been directly contrary. This motion, from west to east, is said to be in the order of the signs, or in consequence, as has been already mentioned, with regard to the moon; while that from east to west is in antecedence, or contrary to the order of the signs. Though all of them, however, revolve round the sun, their motions, as we have already observed, are not in the same plane, but inclined to one another by small angles; and the way in which we compute this inclination is by considering the orbit of the earth as a standard, and calculating the angle which each of their orbits makes with it.

To a spectator placed in the sun, all the planets would appear to describe circles annually in the heliocentric circles; for though their motions are really elliptical, of the planes the eccentricity is so small, that the difference between them and true circles is not easily perceived even on earth; and at the sun, whether great or small, it would entirely vanish. These circles, which in such a situation would appear to be annually described among the fixed stars, are called the heliocentric circles of the planets; and if we suppose the orbits of the planets to be extended to the extreme bounds of the creation, they would describe among the fixed stars those circles just mentioned. To a spectator in the sun, the comets, though moving in the most eccentric orbits, would also appear to describe circles in the heavens; for though their orbits are in reality very long ellipses, the planes of them extended to the heavens would mark a great circle thereon, whereof the eye would be the centre; only, as the real motion is in an ellipse, the body would appear to move much more slowly in some part of the circle than another, and to differ excessively in magnitude. To an inhabitant of any planet, however, the sun appears to go round in its own heliocentric circle, or to describe in the heavens that same curve which the planet would appear to do if seen from the sun. Thus (fig. 137), when the earth is at a, if we draw a line from \(a\) through the sun as \(S\), the point \(G\), in the sphere of the heavens where the line terminates, is the place where the sun then appears to an inhabitant of the earth. In a month's time the earth will be got from \(a\) to \(b\); draw a line then through the sun, and its extremity at \(H\) will point out his apparent place at that time. In like manner, if we draw lines from the earth in the twelve several situations in which it is represented for the twelve months of the year, the sun's apparent place will be found as above, and so it would be found by a spectator placed in Venus or any other planet.

The heliocentric circle of the earth is called the ecliptic; because eclipses of the sun or moon can only happen when the latter is in or near it, as will afterwards be more particularly explained. By some ancient writers, it has been called the circle of the sun, or the oblique circle, because it cuts the equator at oblique angles. It is also called by Ptolemy the circle which passes through the midst of the animals; because the twelve constellations through which it passes were anciently all represented by animals, or parts of them, though now the balance is introduced in place of the claws of the scorpion. For this reason, a belt or hoop taken in the concave sphere of the heavens about 10 degrees on each side of the ecliptic, is called the zodiac, from a Greek word which signifies an animal; and the constellations through which the ecliptic is drawn, are called the constellations of the zodiac.

Though the sun, as we have said, apparently goes round the earth annually in the circle just mentioned, we cannot determine his place by mere inspection as we can do that of any of the other heavenly bodies; for the fixed stars are the only marks by which we can determine the place of any of the celestial bodies, and the superior brightness of the sun renders them totally invisible, except in the time of a great eclipse, when his light is for a time totally obscured. But though we cannot know the place of the sun directly, it is easily found from a knowledge of those fixed stars which are opposite to him. Thus, in fig. 137, suppose it the time of the year in which the earth is at \(g\); if we know that the point \(G\) is then diametrically opposite to the sun, we know that \(A\), its opposite, is the sun's place, and consequently, by finding the places throughout the year diametrically opposite to the sun, as \(GHIKLMABCDEF\), we may be assured that in these times the sun's place was in the points \(ABCDEFGHKLM\). The point in the heavens diametrically opposite to the sun may be known every night at twelve o'clock when the stars are visible; for the star which has an elevation above the horizon at that time equal to the sun's depression below it, is directly opposite to him.

The ecliptic being thus found, the latitude of the moon or any star is counted by its distance from the ecliptic, as the latitude of places on earth is counted by their distance from the equator; and is marked upon circles drawn through the pole of the ecliptic, and perpendicular to its plane, as the latitude of places is marked on one of the meridians of a terrestrial globe. These are called circles of latitude, and each of them is supposed to divide the celestial concave into two equal hemispheres; and the declination of any celestial body is its deviation from the ecliptic towards the celestial equator perpendicular to that of the particular earth.

The latitude of any planet is either heliocentric or geocentric. The heliocentric latitude is its distance from the ecliptic as seen from the sun, and its geocentric as seen from the earth, and is considerably different from the former. With the fixed stars indeed it is otherwise; for their distance is so vast, that the whole diameter of the earth's orbit is but a point in comparison with it. For this reason, whatever part of its orbit the earth may be in, the fixed stars always appear to keep the same place; but with respect to the planets, the orbit of the earth, or magnus orbis, as it has sometimes been called, bears a very considerable proportion, excepting only to the Georgium Sidus, of whose distance the diameter of the earth's orbit forms little more than a tenth part; and therefore all calculations with regard to that star are much more difficult than the rest. The apparent places of the planets therefore are considerably altered by the earth's change of place as well as by their own motions; so that though a planet should stand still for a whole year, it would nevertheless appear to us to describe a circle round the heavens, as in that space of time we would have been carried by the earth round the sun, and have continually taken a view of it from different stations. As the orbits of the planets are inclined in different angles to the ecliptic, it thence happens, that the heliocentric latitude of any planet is almost always different from its geocentric latitude. Thus, let \(AB\), fig. 138, be the orbit of the earth, \(CD\) the orbit of Venus, viewed with the eye in their common section, wherein they appear straight lines; let \(E\) and \(F\) be two opposite points of the ecliptic; and suppose Venus to be in the point \(C\) in her utmost north limit. If she were at that time viewed from the sun \(S\), she would appear in the point of the heavens marked \(H\), and her heliocentric latitude is then \(FH\); but if viewed from the earth in \(B\), she will appear at \(g\); at which time her heliocentric latitude is \(FH\), and her geocentric only \(FG\). When at \(I\), her apparent place is at \(K\), her heliocentric latitude \(FH\), and her geocentric \(FK\); but when the earth is at \(A\), her apparent place will then be at \(G\), and her geocentric latitude \(EG\), while her heliocentric is only \(FH\) as before.

The two planets, Mercury and Venus, whose orbits are included in that of the earth, are called inferior; and Mars, Jupiter, Saturn, and the Georgium Sidus, whose orbits include that of the earth, are called superior; and from the circumstance just mentioned, they must present very different appearances in the heavens, as will afterwards be particularly explained. The geocentric latitude of a superior planet may be understood from fig. 139. Let \(AB\) be the orbit of the earth, \(CD\) that of Mars, both viewed with the eye in their common section continued, by which they appear in straight lines. Let \(E\) and \(F\) be opposite points of the ecliptic, and suppose Mars to be in his south limit at \(C\). If he were at that time viewed from \(S\), the centre of the sun, he would appear in the sphere of the heavens at the point \(H\); in which case his heliocentric latitude would be \(FH\): But when viewed in \(C\) from the earth, or from its centre, which in this case is supposed to be the station of the spectator, he will appear to be in different places of the heavens according to the position of Particular the earth. When the earth, for instance, is at B, the Explication place of Mars will appear to be at g, and his geocentric latitude will be F g. When the earth is at A, his apparent place will be in G, and his geocentric latitude FG; and in like manner, supposing the earth to be in any other part of its orbit, as in I or K, it is easy to see, that his apparent places, as well as geocentric latitudes at those times, will be different.

The two points where the heliocentric circle of any planet cuts the ecliptic, are called its nodes; and that which the planet passes through as it goes into north latitude, is called the ascending node, and is marked thus $O$; and the opposite to this is called the descending node, and is marked $Q$. A line drawn from one node to the other is called the line of the nodes of the planet, which is the common section of the plane of the ecliptic, and that of the planet produced on each side to the fixed stars.

The zodiac, of which we have already given some account, is either astral or local. The astral is divided into 12 unequal parts, because it contains 12 celestial constellations, some of which are larger than others. This continues always invariably the same; because the same stars now go to the making up of the different constellations as formerly, excepting some small variations to be afterwards explained. The local zodiac is divided into twelve equal parts, each containing 30 degrees, called signs. These are counted from the point where the equator and ecliptic intersect each other at the time of the vernal equinox; and are denoted by particular marks, according to the apparent annual motion of the sun. See fig. 158. A motion in the heavens in the order of these signs, as from Aries to Taurus, is said to be a motion in consequence; and such are the true motions of all the planets; tho' their apparent motions are sometimes contrary, and then they are said to move in antecedence. The local zodiac is not always invariably the same as to the places of the several signs, though the whole always takes up the same place in the heaven, viz. 10 degrees on each side the ecliptic. The points where the celestial equator cuts into the ecliptic, are found to have a motion in antecedence of about 50 seconds in a year. This change of place of the first point of the ecliptic, from whence the signs are counted, occasions a like change in the signs themselves; which though scarce sensible for a few years, has now become very considerable. Thus, since astronomy was first cultivated among the Greeks, which is about 2000 years ago, the first point of the ecliptic is removed backward above a whole sign; and though it was then about the middle of the constellation Aries, is now about the middle of Pisces. Notwithstanding this alteration, however, the signs still retain their ancient names and marks. When the zodiac is mentioned by astronomers, the local zodiac is generally meant.

The longitude of a phenomenon in the heavens is the number of degrees counted from the first point of Aries on the ecliptic to the place where a circle of latitude drawn through the phenomenon would cut the ecliptic at right angles. Every phenomenon in the heavens, whether in the zodiac or not, is thus referred to the ecliptic by the circles of latitude, as the longitudes of terrestrial places are referred to the equator by the meridians; and whatever sign the circle of latitude passes through, the phenomenon is said to have its place in that sign, though ever so far distant from it.

Some astronomical writers have made the local zodiac invariable; for which purpose they imagine a circle of latitude drawn through the first star of the constellation Aries, marked in Bayer's catalogue by the Greek letter $\gamma$; and reckon their longitude from the point where that circle cuts the ecliptic. This star, from its use, is called the first star of the Ram; and when this method is made use of, the longitude of any phenomenon is said to be so many signs, degrees, minutes, &c. from the first star of the Ram. Thus, in Street's Caroline tables, the longitude of Jupiter's ascending node is two signs eight degrees from the first star of Aries, which is thus marked: Long. $2^{\circ} 8'$.

The longitude of a phenomenon is to take $\gamma$ for the first point of the ecliptic, and not to number the degrees quite round that circle as a continued series, but to make a new beginning at the first point of every sign, and to reckon from thence only the length of $360^{\circ}$. When this method is made use of, the longitude of any phenomenon is expressed by saying it is in such a degree and such a minute of a sign: and thus we may express the longitude of the ascending node of Mercury, $\gamma \Omega \gamma 14^{\circ} 40'$; and so of any other. The place of a phenomenon in the heaven is expressed by setting down its longitude and latitude, as is done with places situated anywhere on the surface of the earth.

Having thus explained the astronomical terms commonly made use of with respect to the planets, and likewise shown how, from their motions and that of the earth, there must be a considerable variation in their apparent places, as seen from the sun and from the earth; we shall now proceed to a more particular consideration of their phenomena, as derived from a composition of the two motions just mentioned, viz. that of the planets in their respective orbits, and that of the earth in the ecliptic. Every planet, like the moon, is sometimes in conjunction, and sometimes in opposition with the sun. Its conjunction is when the geocentric place of the planet is the same with that of the sun; though an exact or central conjunction can only take place when the line of its nodes passes through the earth, and the planet itself is in one of its nodes at the time. It is however, in general, called a conjunction or opposition, when the same circle of latitude passes through the sun and planet at the same time. When the geocentric place of a planet is $90^{\circ}$, or a quarter of a circle from the sun's place, the planet is said to be in quadrature or in a quartile aspect with the sun; and these terms are used in a like sense when applied to any two of the heavenly bodies. Thus the sun and moon, or the moon and any planet, or any two planets, may be in conjunction, opposition, or quadrature. Besides these, the ancients reckoned other two aspects, the trine and the sextile; the former when the bodies were distant $120^{\circ}$, and the latter when only half that distance. These aspects they marked thus:

Conjunction. Opposition. Quadrature. Trine. Sextile.

$\sigma \quad \& \quad \Box \quad \triangle \quad *$

The aspects were supposed to influence the affairs of mankind; and many conclusions drawn from them too absurd to be mentioned here, and now indeed almost entirely buried in oblivion. The inferior planets have two kinds of conjunction with the sun; one in the inferior part of their semi-circles, where they are nearer to the earth than the sun; the other in the superior part, where they are farther off. In the former, the planet is between the earth and the sun; and in the latter, the sun is between the earth and planet. The inferior planets can never be in opposition to the sun, nor even appear at a great distance from him. The length they go is called their elongation. Thus, in fig. 140, let OPQRT be part of the ecliptic; S the sun; and the three circles round him, the orbits of Mercury, Venus, and the Earth.

Suppose the earth to be at A, the sun's geocentric place will be at Q. If Mercury be then at I, his geocentric place is likewise at Q; so that he is in conjunction with the sun in his inferior semicircle; if at M, his geocentric place is likewise at Q; so that he is in conjunction with the sun in his superior semicircle. In like manner, Venus at E is in conjunction in her inferior semicircle, at G in her superior; but if we suppose the earth to be at A, and Venus at H, her geocentric place is T, and her elongation QT, which in this figure is the greatest possible; for this always takes place when a straight line from the earth touches the orbit of the planet, as is evident from the figure; that is, provided the planet be in its aphelion at the time. Thus the greatest possible elongation of Mercury is QP when he is in his aphelion at L; and the quantity of this is found by astronomical observations to be about 28 degrees, that of Venus about 48. The inferior planets in their elongations are sometimes eastward and sometimes westward of the sun; in the former case they appear in the evening, and in the latter in the morning. The smallness of Mercury and his nearness to the sun prevent him from being often taken notice of; but the largeness and beauty of Venus have made her, in all ages, celebrated as the evening and morning star.

The irregular apparent motion of the planets has been already taken notice of; sometimes going forward, sometimes backward, and sometimes appearing to stand still for a little. These different conditions are by astronomers called direct, retrograde, and stationary. Were they to be viewed from the sun, they would always appear direct, as has been already shown; but when viewed from the earth, the inferior planets appear direct while moving in their upper semicircles, and retrograde when in their lower ones. Thus, in fig. 140, suppose the earth at rest at A, while Mercury is going on in his orbit from N to L; from L to I his motion appears to an observer at A to be retrograde, or contrary to the order of the signs, namely, from R to Q and from Q to P; but when in that part of his orbit which lies between L and N, his motion appears direct, or from P to Q and from Q to R.

When the earth is in the line of nodes of an inferior planet, the apparent motion of the former is then in a straight line, because the plane of it passes through the eye; if in a conjunction in his upper semicircle, he passes behind the sun; if in his lower semicircle, he passes before it, and will then be seen by an observer on earth to pass over the sun's disk like a round and very black spot. Were the plane of his orbit coincident with the ecliptic, this appearance would be seen every year; but by reason of the obliquity of the two planes to each other, it is much more rare. However, he was seen in this manner November 12th 1782, at 3 h. 44' in the afternoon; May 4th 1786, at 6 h. 57' in the morning; and will be seen again December 6th 1789, at 3 h. 55' in the afternoon: but from that time not, in this island at least, until the year 1799, May 7th, at 2 h. 34' in the afternoon. In like manner, Venus sometimes appears as a black spot on the sun, but much more seldom than Mercury. She was first seen by Mr Horrocks, as we have already related, in the 1639; afterwards in the years 1761 and 1769; but will not afterwards be visible in this manner till the year 1874.

When the earth is out of the line of the nodes of an inferior planet, its orbit appears an ellipse, more or less eccentric according to the situation of the eye of the spectator. Thus, suppose the earth to be as far as possible (that is, 90°) out of the line of the nodes of Mercury, the projection of his orbit will be in such an elliptic curve as is represented fig. 141, wherein he will appear to move in the order of the letters; direct when in his upper semicircle from a to b, from b to c, being above the sun at b in his superior conjunction; but in his inferior semicircle his motion will appear retrograde from c to d, and from d to a; in conjunction he will be at d below the sun. In these cases, the motion of Mercury is unequal; faster near the inferior conjunction, but most unequal in the inferior semicircle, going through the unequal spaces into which the ellipse is divided. The motions of the inferior planets, both direct and retrograde, are very unequal; and this inequality proceeds not from the eccentricity of their orbits, but from the projection of their orbits into long ellipses; and is therefore a mere optical deception.

These planets appear stationary while changing their motion from direct to retrograde, or from retrograde to direct. If the earth stood still, the times of their appearing stationary would be at their greatest elongation; for though it be a property of the circle that a straight line can only touch it in one point, yet when the circle is very large, the recess from the tangent is not perceptible for a considerable time. Thus, in fig. 140, suppose the earth to be at rest in A, Venus would appear stationary, her geocentric place continuing at T all the while she is going in her orbit from a to b; because her deviation from the visual line AT would scarcely be perceptible so near the point of contact H.

To an inhabitant of the earth, therefore, the inferior planets appear always near the sun; alternately going from and returning to him, sometimes in straight lines, at others in elliptical curves, first on one side and then on the other; sometimes so near as to be rendered invisible by his stronger light. Sometimes, when in or near their nodes, they pass behind the sun in their superior semicircles, or pass between him and us; in which case they appear like black spots on his disk, as has been just now mentioned. For the better comprehending of these motions, however, we have hitherto supposed the earth to stand still in some part of its orbit, while they go round the sun in theirs: but as this is not the case, it now remains to consider the changes which take place in consequence of the earth's appearance motion. Were the earth to stand still in any part of its orbit, Particular its orbit as at A, the places of conjunction both in the superior and inferior semicircle, as also of the greatest elongation; and consequently the places of direct and retrograde motion, and of the stations of an inferior planet, would always be in the same part of the heavens. Thus, in fig. 140, upon this supposition, the places of Mercury's stations would always be the points P and R, the arc of his motion PR, and of his retrograde motion RP; whereas, on account of the earth's motion, the places where these appearances happen are continually advancing forward in the ecliptic according to the order of the signs. In fig. 142, let ABCD be the orbit of the earth; e f g h that of Mercury; O the sun; GKI an arc of the ecliptic extended to the fixed stars. When the earth is at A, the sun's geocentric place is at F; and Mercury, in order to a conjunction, must be in the line AF; that is, in his orbit he must be at f or b. Suppose him to be at f in his inferior semicircle: If the earth stood still at A, his next conjunction would be when he is in his superior semicircle at h; the places of his greatest elongation also would be at e and g, and in the ecliptic at E and G: but supposing the earth to go on in its orbit from A to B; the sun's geocentric place is now at K; and Mercury, in order to be in conjunction, ought to be in the line BK at m. As by the motion of the earth the places of Mercury's conjunctions with the sun are thus continually carried round in the ecliptic in consequence, so the places of his utmost elongations must be carried in consequence also. Thus, when the earth is at A, the places of his greatest elongation from the sun are in the ecliptic E and G; the motion of the earth from A to B advances them forward from G to L and from E to I. But the geocentric motion of Mercury will best be seen in fig. 146. Here we have part of the extended ecliptic marked Ψ, ψ, Π, &c. in the centre of which S represents the sun, and round him are the orbits of Mercury and the earth. The orbit of Mercury is divided in 11 equal parts, such as he goes through once in eight days: and the divisions are marked by numeral figures 1, 2, 3, &c. Part of the orbit of the earth is likewise divided into 22 equal arcs, each arc being as much as the earth goes through in eight days. The points of division are marked with the letters a, b, c, d, e, f, &c. and show as many several stations from whence Mercury may be viewed from the earth. Suppose then the planet to be at 1 and the earth at a; draw a line from a to 1, and it shows Mercury's geocentric place at A. In eight days he will be got to 2, and the earth to b; draw a line 2 to b, and it shows his geocentric place at B. In other eight days he will have proceeded to 3, and the earth to c; a line drawn from 3 to c will show his geocentric place at C. In this manner, going through the figure, and drawing lines from the earth at d, e, f, g, &c. through 4, 5, 6, 7, &c. we shall find his geocentric places successively at the points D, E, F, G, &c. where we may observe, that from A to B, and from B to C, the motion is direct; from C to D, and from D to E, retrograde. In this figure 22 stations are marked in the earth's orbit, from whence the planet may be viewed; corresponding to which there ought to be as many in the orbit of Mercury: and for this purpose the place of that planet is marked at the end of every eight days for two of his particular periodical revolutions; and to denote this, two numerical figures are placed at each division.

The geocentric motion of Venus may be explained in a similar manner; only as the motion of Venus is much slower than that of Mercury, his conjunctions, oppositions, elongations, and stations, all return much more frequently than those of Venus.

To explain the stationary appearances of the planets, it must be remembered, that the diameter of the earth's orbit, and even of that of Saturn, are but mere points in comparison of the distance of the fixed stars; and therefore, any two lines absolutely parallel, though drawn at the distance of the diameter of Saturn's orbit from each other, would, if continued to the fixed stars, appear to us to terminate in the same point. Let, then, the two circles fig. 143. represent the orbits of Venus and of the Earth; let the lines AE, BF, CG, DH, be parallel to SP, we may nevertheless affirm, that if continued to the distance of the fixed stars, they would all terminate in the same point with the line SP. Suppose, then, Venus at E while the earth is at A, the visual ray by which she is seen is the line AE. Suppose again, that while Venus goes from E to F, the Earth goes from A to B, the visual ray by which Venus is now seen is BF parallel to AE; and therefore Venus will be all that time stationary, appearing in that point of the heaven where SP extended would terminate: this station is at her changing from direct to retrograde. Again, suppose, when the Earth is at C, Venus is at G, and the visual line CG; if, while the Earth goes from C to D, Venus goes from G to H, so that she is seen in the line GH parallel to CG, she will be all that time stationary, appearing in the point where a line drawn from S through P would terminate. This station is at her changing from retrograde to direct; and both are in her inferior semicircle.

An inferior planet, when in conjunction with the sun perigee in its inferior semicircle, is said to be in perigee, and apogee in the other in apogee, on account of its different distances from the earth. Their real distances from the earth when in perigee are variable, partly owing to the eccentricities of their orbits, as well as that of the earth; and partly owing to the motions of the different bodies, by which it happens that they are in perigee in different parts of their orbits. The least possible distance is when the perigee happens when the earth is in its perihelion, and the planet in its aphelion.

The difference of distance between the earth and inferior planets at different times, makes a considerable variation in their apparent diameters, which indeed is very observable in all the planets; and thus they sometimes look very considerably larger than others. This difference in magnitude in Mercury is nearly at 5½ to 1; and in Venus, no less than 32 to 1. A common spectator, unassisted by any instrument, may observe an inferior planet alternately approach nearer and nearer the sun, until at last it comes into conjunction with him, and then to recede farther and farther till it is at its greatest elongation, which will be first on one side and then on the other: but if we observe the apparent change of place of an inferior planet in the sphere of the heavens, its direct motions, stations, and retrogradations, measuring its diameter frequently with the micrometer, Particular micrometer, we shall find by its decrease at some times and increase at others, that its distance from us is very considerably varied; so that, taking the whole of its course into consideration, it appears to move in a very complicated curve. See fig. i. at C.

As the superior planets move in a larger orbit than the earth, they can only be in conjunction with the sun when they are on that side opposite to the earth; as, on the other hand, they are in opposition to him when the earth is between the sun and them. They are in quadrature with him when their geocentric places are 90° distant from that of the sun. In order to understand their apparent motions, we shall suppose them to stand still in some part of their orbit while the earth makes a complete revolution in hers; in which case, any superior planet would then have the following appearances:

1. While the earth is in her most distant semicircle, the motion of the planet will be direct. 2. While the earth is in her nearest semicircle, the planet will be retrograde. 3. While the earth is near those places of its orbit where a line drawn from the planet would be a tangent, it would appear to be stationary.

Thus, in fig. 147, let \(a b c d\) represent the orbit of the earth; \(S\) the Sun; \(EFG\) an arc of the orbit of Jupiter; \(ABC\) an arc of the ecliptic projected on the sphere of the fixed stars. Suppose Jupiter to continue at \(F\), while the earth goes round in her orbit according to the order of the letters \(a b c d\). While the earth is in the semicircle most distant from Jupiter, going from \(a\) to \(b\) and from \(b\) to \(c\), his motion in the heaven would appear direct, or from \(A\) to \(B\) and from \(B\) to \(C\); but while the earth is in its nearest semicircle \(c d e\), the motion of Jupiter would appear retrograde from \(C\) to \(B\) and from \(B\) to \(A\); for \(a\), \(b\), \(c\), \(d\), may be considered as so many different stations from whence an inhabitant of the earth would view Jupiter at different seasons of the year, and a straight line drawn from each of these stations, through \(P\) the place of Jupiter, and continued to the ecliptic, would show his apparent place there to be successively at \(A\), \(B\), \(C\), \(B\), \(A\). While the earth is near the points of contact \(a\) and \(c\), Jupiter would appear stationary, because the visual ray drawn through both planets does not sensibly differ from the tangent \(Pa\) or \(Fc\). When the earth is at \(b\), a line drawn from \(b\) through \(S\) and \(F\) to the ecliptic, shows Jupiter to be in conjunction with the sun at \(B\). When the earth is at \(d\), a line drawn from \(d\) through \(S\), continued to the ecliptic, would terminate in a point opposite to \(B\); which shows Jupiter then to be in opposition to the sun; and thus it appears that his motion is direct in the conjunction, but retrograde when in opposition, with the sun.

The direct motion of a superior planet is swifter the nearer it is to a conjunction, and slower as it approaches to a quadrature with the sun. Thus, in fig. 144, let \(O\) be the sun; the little circle round it, the orbit of the earth, whereof \(a b c d e f g\) is the most distant semicircle; \(OPQ\), an arc of the orbit of Jupiter; and \(ABCDEFG\), an arc of the ecliptic in the sphere of the fixed stars. If we suppose Jupiter, to stand still at \(P\), by the earth's motion from \(a\) to \(g\), he would appear to move direct from \(A\) to \(G\), describing the unequal arcs \(AB\), \(BC\), \(CD\), \(DE\), \(EF\), \(FG\), in equal times. When the earth is at \(d\), Jupiter is in conjunction with the sun at \(D\), and there his direct motion is swiftest. When the earth is in that part of her orbit where a line drawn from Jupiter would touch it, as in the points \(e\) or \(g\), Jupiter is nearly in quadrature with the sun; and the nearer the earth is to any of those points, the slower is the geocentric motion of Jupiter; for the arcs \(CD\) and \(DE\) are greater than \(BC\) or \(EF\), and the arcs \(BC\) and \(EF\) are greater than \(AB\) or \(FG\).

The retrograde motion of a superior planet is swifter the nearer it is to an opposition, and slower as it approaches to a quadrature with the sun. Thus, let \(O\), fig. 145, be the sun; the little circle round it the orbit of the earth, whereof \(g h i k l m n\) is the nearest semicircle; \(OPQ\), an arc of the orbit of Jupiter; \(NKG\) an arc of the ecliptic: If we suppose Jupiter to stand still at \(P\), by the earth's motion from \(g\) to \(n\), he would appear to move retrograde from \(G\) to \(N\), describing the unequal arcs \(GH\), \(HI\), \(IK\), \(KL\), \(LM\), \(MN\), in equal times. When the earth is at \(k\), Jupiter appears at \(K\), in opposition to the sun, and there his retrograde motion is swiftest. When the earth is either at \(g\) or \(n\), the points of contact of the tangents \(Pg\) and \(Pn\), Jupiter is nearly in quadrature with the sun; and the nearer he is to either of these points, the slower is his retrogradation; for the arcs \(IK\) and \(KL\) are greater than \(HI\) or \(LM\); and the arcs \(HI\) and \(LM\) are greater than \(GH\) or \(MN\). Since the direct motion is swiftest when the earth is at \(d\), and continues diminishing till it changes to retrograde, it must be infensible near the time of change; and, in like manner, the retrograde motion being swiftest when the earth is in \(k\), and diminishing gradually till it changes to direct, must also at the time of that change be infensible; for any motion gradually decreasing till it changes into a contrary one gradually increasing, must at the time of the change be altogether infensible.

The same changes in the apparent motions of this planet will also take place if we suppose him to go on slowly in his orbit; only they will happen every year when the earth is in different parts of her orbit, and consequently at different times of the year. Thus, (fig. 147.) let us suppose, that while the earth goes round her orbit Jupiter goes from \(F\) to \(G\), the points of the earth's orbit from which Jupiter will now appear to be stationary will be \(a\) and \(y\); and consequently his stations must be at a time of the year different from the former. Moreover, the conjunction of Jupiter with the sun will now be when the earth is at \(f\); and his opposition when it is at \(e\); for which reason these also will happen at times of the year different from those of the preceding opposition and conjunction. The motion of Saturn is so slow, that it makes but little alteration either in the times or places of his conjunction or opposition; and no doubt the same will take place in a more eminent degree in the Georgium Sidus; but the motion of Mars is so much swifter than even that of Jupiter, that both the times and places of his conjunctions and oppositions are thereby very much altered.

Fig. 148. exemplifies the geocentric motion of Jupiter in a very intelligible manner; where \(O\) represents the sun; the circle \(1, 2, 3, 4\), the orbit of the earth, divided into twelve equal arcs for the twelve months of the year; \(PQ\) an arc of the orbit of Jupiter, containing as much as he goes through in a year, and divided in like manner into twelve equal parts, each as much Particular as he goes through in a month. Now, suppose the Explication earth to be at \( t \) when Jupiter is at \( a \); a line drawn of the Celestial Phe- through \( i \) and \( a \) shows Jupiter's place in the celestial phenomena. In a month's time the earth will have moved from \( i \) to \( z \), Jupiter from \( a \) to \( b \); and a line drawn from \( z \) to \( b \) will show his geocentric place to be in \( B \). In another month, the earth will be in \( g \), and Jupiter at \( C \), and consequently his geocentric place will be at \( C \); and in like manner his place may be found for the other months at \( D, E, F, \) &c. It is likewise easy to observe, that his geocentric motion is direct in the arcs \( A B, B C, E D, D E; \) retrograde in \( E F, F G, G H, H I; \) and direct again in \( I K, K L, L M, M N \). The inequality of his geocentric motion is likewise apparent from the figure.

A superior planet is in apogee when in conjunction with the sun, and in perigee when in opposition; and every one of the superior planets is at its least possible distance from the earth where it is in perigee and perihelion at the same time. Their apparent diameters are variable, according to their distances, like those of the inferior planets; and this, as might naturally be expected, is most remarkable in the planet Mars, who is nearest us. In his nearest approach, this planet is 25 times larger than when farthest off, Jupiter twice and a half, and Saturn once and a half.

The honour of discovering the new attendant of the sun, called the Georgium Sidus, is undoubtedly due to Mr Herschel; though Mr Robison, professor of Natural Philosophy in Edinburgh, has given strong reasons for supposing that it had been marked by several astronomers as a fixed star. It was first observed by Mr Herschel on the 13th of March 1781, near the foot of Caistor, and his attention was drawn by its steady light. On applying an higher magnifying power to his telescope, it appeared manifestly to increase in diameter; and two days after, he observed that its place was changed. From these circumstances he concluded, that it was a comet; and sent an account of it as such to the astronomer-royal, which very soon spread all over Europe. It was not long, however, before it was known, by the English astronomers especially, to be a planet. The circumstances which led to this discovery were, its vicinity to the ecliptic, the direction of its motion, and its being nearly stationary at the time, in such a manner as corresponds with the like appearances of the other planets. The French astronomers, however, still imagined it to be a comet, although it had not that faint train of light which usually accompanies these bodies, nor would its successive appearances correspond with such an hypothesis; so that they were at last obliged to own that it went round the sun in an orbit nearly circular. Its motion was first computed on this principle by Mr Lexell professor of astronomy at St Petersburgh; who showed, that a circular orbit, whose radius is about 19 times the distance of the earth from the sun, would agree very well with all the observations which had been made during the year 1781. On the 1st of December that year it was in opposition with the sun; whence one of its stations was certainly determined. In the mean time, however, as astronomers were every where engaged in making observations on the same star, it occurred to some, that it might possibly have been observed before, though not known to be a planet.

Mr Bade of Berlin, who had just published a work containing all the catalogues of zodiacal stars which had appeared, was induced, by the observations which had been already made on the new planet, to consult these catalogues, in order to discover whether any star, marked by one astronomer and omitted by another, might not be the new planet in question. In the course of this inquiry, he found, that the star, No 64 of Mayer's catalogue, had been unobserved by others, and only once by Mr Mayer himself, so that no motion could have been perceived by him. On this Mr Bade immediately directed his telescope to that part of the heavens where he might expect to find the star marked in Mayer's catalogue, but without success. At the same time, by the calculations already made concerning the new planet, he discovered, that its apparent place in the year 1756 ought to have been that of Mayer's star, and this was one of the years in which he was busied in his observations; and on further inquiry it was found, that the star 964 had been discovered by Mr Mayer on the 15th of September 1756: So that it is now generally believed, that the star No 964 of Mayer's catalogue was the new planet of Herschel.

Before the end of the year 1782, it was found, that the angular motion of the planet was increasing; which showed, that it was not moving in a circle, but in an eccentric orbit, and was approaching towards the sun. Astronomers, therefore, began to investigate the inequality of this angular heliocentric motion, in order to discover the form and position of the ellipse described. This was a very difficult task, as the small inequality of motion showed that the orbit was nearly circular, and the arch already described was no more than one-fiftieth part of the whole circumference. It was, however, by no means easy, from the variation of curvature discoverable in this small arch, to determine to what part of the circumference it belongs; though the Professor is of opinion, that the supposition of its being the star 964 of Mayer's catalogue renders the calculation easy. On this supposition, its motion has been calculated by several astronomers, as well as by Mr Robison himself. He observes, however, that if we do not admit the identity of these stars, near half a century must elapse before we can determine the elements of this planet's motion with a precision equal to that of the others.

Some astronomers are of opinion, that the new planet is the same with the star No 34 Tauri of the Britannic catalogue. "In this case (says Mr Robison), the elements will agree very well with Flamstead's observation of that star on December 13th 1690, being only 40", or perhaps only 12", to the westward of it; but the latitude differs more than two minutes from Flamstead's latitude, which is properly deduced from the zenith distance. This is too great an error for him to commit in the observation; and we should therefore reject the supposition on this account alone: But there are stronger reasons for rejecting it, arising from the disagreement of those elements with the observations made on the stations of the planet in October 1781 and in March and October 1782, which gave a very near approximation of its distance from the sun. When compared with observations of the planet near its stationary tionary points in the spring, they give the geocentric latitude considerably too great, while they give it too small for the similar observations in autumn."

As the times of conjunction, utmost elongation, direct or retrograde motions of the inferior planets, depend on the combinations of their motions in their orbits with the motion of the Earth in its orbit; any of these appearances will be more frequent in Mercury than in Venus, because the former moves with a swifter motion in his orbit, and consequently must more frequently pass through those places where he is in conjunction, &c. The time in which any of the inferior planets will return into a given situation, may be known by the following examples. Let fig. 149. represent the orbits of Venus and the Earth. Let the Earth be at E, Venus at V, when she is in the inferior conjunction with the sun in Φ. From S, Venus and the Earth would appear in conjunction in Ω: let Venus go round her orbit, and return to V; the earth taking longer time to go round than Venus will, in the mean time, go from E, only through a part of her orbit, and Venus must overtake the Earth before she can have another inferior conjunction; that is, she must, besides an entire revolution, which is equal to four right angles, go through as much more angular motion round the sun as the earth has done in the mean time, so as to be in a right line between the sun and the earth. Suppose this is to happen when the earth is got to F and Venus to T, the angular motions of the Earth and Venus performed in the same times are reciprocally as their periodical times: and therefore as the periodical time of the Earth is to the periodical time of Venus; so is the angular motion of Venus, which is equal to four right angles, added to the angular motion of the earth, in the time between two like conjunctions of Venus, to the angular motion of the earth in the same time: and therefore, by division of proportion, as the difference between the periodical times of Venus and the Earth is to the periodical time of Venus; so are four right angles, or 360°, to a fourth quantity; namely, to the angular motion or number of degrees which the Earth goes in her orbit from the time of one conjunction of Venus to the next conjunction of the same kind. Now the periodical time of the earth is 365 days 6 hours or 8766 hours; the period of Venus 224 days 16 hours or 5392 hours; the difference is 3374 hours. Say then, As 3374 is to 5392, so are four right angles, or 360°, to a fourth number, which is 575°; which the earth goes through in a year and 218 days. Were Venus therefore this day in an inferior conjunction with the sun, it would be a year and 218 days before she come into another conjunction of the same kind; and this alteration in time occasions a proportional change in place; so that if one conjunction be in Φ, the next similar conjunction will be in Ω. The time between any situation of Mercury, with regard to the sun and the earth, and another like situation, may be found by the same method. The periodical time of the earth is 8766 hours; the period of Mercury 87 days 23 hours, or 2111 hours; the difference 6655 hours. Say then, As 6655 is to 2111, so are four right angles or 360° to 114°, through which the earth passes in 116 days. If therefore Mercury were to be this day in his inferior conjunction, it would be 116 days before he were in a similar situation.

This problem is commonly resolved in another manner. Astronomers compute the diurnal heliocentric motions of Venus and of the Earth: the difference of these motions is the diurnal motion of Venus from the Earth, or the quantity by which Venus would be seen to recede from the Earth every day by a spectator placed in the sun: thus the mean motion of Venus is every day about 59 minutes and 8 seconds; the difference is 37 minutes. Say, therefore, As 37 minutes is to 36°, or to 21,600 minutes, so is one day to the time wherein Venus, having left the earth, recedes from her 360 degrees; that is, to the time wherein she returns to the earth again, or the time between two conjunctions of the same kind.

The times are here computed according to the mean or equable motions of the planets; and this is true times therefore called a mean conjunction: but because Venus and the Earth are really carried in elliptic orbits, conjunctions, in which their motions are sometimes swifter and sometimes slower, the true conjunctions may happen some days either sooner or later than what these rules will give. The time of the true conjunction is to be computed from that of the mean conjunction in the following manner. Find by astronomical tables the places of Venus and the earth in the ecliptic, from which we shall have the distance of the two as seen from the sun; compute also for the same time the angular motions of these two planets for any given time, suppose six hours; the difference of these two motions will give the accels of Venus to the earth, or her receds from it in six hours: then say, as this difference is to the arc between the places of Venus and the earth at the time of a mean conjunction, so is six hours to the time between the mean conjunction and the true. This time added to or subtracted from the time of the mean conjunction, according as Venus is in antecedence or consequence from the earth, shows the time of their true conjunction.

With regard to the conjunctions, oppositions, direct and retrograde motions, &c. of the superior planets, as they depend on the combinations of their motions with that of the earth, they will be more frequent in Saturn than in Jupiter, in Jupiter than in Mars, but most frequent of all in the Georgium Sidus; because the slower the motion of the planet is, the sooner the earth will overtake it, so as to have it again in any given situation. Thus, suppose Saturn to be in conjunction with the sun in Φ, if he were to stand still for one year, then he would again be in conjunction in Φ; but as he goes on slowly, according to the order of the signs, about 12° annually, the earth must go through almost 13° more than an entire revolution; so that there will be almost a year and 13 days between any conjunction between the Sun and Saturn and the conjunction immediately following. As Jupiter moves in his orbit with greater velocity than Saturn, the Earth must have a proportionably larger space added to the year; and as Mars moves swifter still, the time betwixt any two of his conjunctions must be still longer.

The time when a superior planet will return into any given situation, may be found by the methods already ready laid down for the inferior planets. Thus, the mean diurnal motion of the earth is about $59'8''$; the mean motion of Saturn in a day is only two minutes; the difference $57'8''$. Say therefore, As $57'8''$ are to $360^\circ$, or $21,600$ minutes, so is one day to the space of time wherein the earth having left Saturn, recedes from him $360^\circ$; that is, to the time of her return to Saturn again, or the time between two conjunctions, oppositions, or other like aspects. This time will be found $378$ days, or one year and $13$ days.

The mean motion of Jupiter in a day is $4'59''$; the difference between this and the earth's diurnal motion is $54'9''$. Say then, As $54'9''$ are to $360^\circ$ or $21,600$, so is one day to the space of time when the earth, having left Jupiter, will overtake him again; which will be found to be $398$ days, or one year and $33$ days.

The mean motion of Mars is $31'27''$; the difference between which and the earth's diurnal motion is $27'4''$. Say then, As $27'4''$ are to $360^\circ$ or $21,600$, so is one day to the space of time wherein the earth, having left Mars, recedes from him $360^\circ$; which will be found two years and $50$ days. The true conjunctions, &c., may be found in the superior planets as in the inferior.

The Earth is the next planet above Venus in the system. It is $95,173,000$ miles from the sun; and goes round him in $365$ days $5$ hours $49$ minutes, from any equinox or solstice to the same again; but from any fixed star to the same again, as seen from the sun, in $365$ days $6$ hours $9$ minutes; the former being the length of the tropical year, and the latter the length of the sidereal. It travels at the rate of $68,000$ miles every hour; which motion, though upwards of $140$ times swifter than that of a cannon ball, is little more than half as swift as Mercury's motion in his orbit. The earth's diameter is $7970$ miles; and by turning round its axis every $24$ hours from west to east, it causes an apparent diurnal motion of all the heavenly bodies from east to west. By this rapid motion of the earth on its axis, the inhabitants about the equator are carried $1042$ miles every hour, whilst those on the parallel of London are carried only about $580$, besides the $68,000$ miles by the annual motion abovementioned, which is common to all places whatever.

That the earth is of a globular figure may be proved from several different and evident circumstances.

1. When we are at sea on board a ship, we may be out of sight of land when the land is near enough to be visible if it were not hid from our eye by the convexity of the water. Thus, let $ABCD$ (fig. 154.) represent a portion of the globe of our earth. Let $M$ be the top of a mountain; this cannot be seen by a person on board the ship at $B$, because a line drawn from $M$ to his eye at $E$ is intercepted by the convexity of the water; but let the ship come to $C$, then the mountain will be visible, because a line may be drawn from $M$ to his eye at $E$.

2. The higher the eye, the farther will the view be extended. It is very common for sailors from the top of the mast of a ship to discover land or ships at a much greater distance than they can do when they stand upon deck.

3. When we stand on shore, the highest part of a ship is visible at the greatest distance. If a ship is going from us out to sea, we shall continue to see the mast after the hull or body of the ship disappears, and the top of the mast will continue to be seen the longest. If a ship is coming towards us, the top of the mast comes first in view, and we see more and more till at last the hull appears. If the surface of the sea were a flat plain (fig. 155.), a line might be drawn from any object situated upon it, as the ship $D$, to the eye, whether placed high or low, at $A$ or $B$. In this case, any object upon the earth or sea, would be visible at any distance which was not so great as to make the appearance of it too faint, or the angle under which it appears too small, to be seen by us. An object would be visible at the same distance, whether the eye were high or low. Not the highest, but the largest, objects would be visible to the greatest distance, so that we should be able to see the hulk of a ship farther off than the mast: All of which is contrary to experience.

4. Several navigators, such as Ferdinand Magellan, Sir Francis Drake, Captain Cook, have sailed round the globe; not in an exact circle, the land preventing them, but by going in and out as the shores happened to lie.

5. All the appearances in the heavens are the same, whether at land or sea. 6. Eclipses of the moon arise from the shadow of the earth, and this shadow is always circular. Although the earth presents, during several hours, different portions of its surface to the moon, yet still the shadow is round. The small inequalities upon the surface of the earth bear no kind of proportion to its magnitude sufficient to alter the appearance of its shadow.

The earth's axis makes an angle of $23\frac{1}{2}$ degrees with the axis of its orbit, and keeps always the same oblique direction, inclining nearly to the same fixed stars ($A$) throughout its annual course, which causes the returns of spring, summer, autumn and winter. That the sun, and not the earth, is the centre of our solar system, may be demonstrated beyond a possibility of doubt, from considering the forces of gravitation and projection, by which all the celestial bodies are retained in their orbits. For, if the sun moves about the earth, the earth's attractive power must draw the sun towards it from the line of projection so as to bend its motion into a curve: But the sun being at least $227,000$ times as heavy as the earth, by being so much weightier as its quantity of matter is greater, it must move $227,000$ times as slowly towards the earth as the earth does towards the sun; and consequently the earth would fall to the sun in a short time, if it had not a very strong projectile motion to carry it off. The earth, therefore, as well as every other planet in the system, must have a rectilineal impulse, to prevent its falling into the sun. To say, that gravitation retains

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(a) This is not strictly true, as will appear when we come to treat of the recession of the equinoctial points in the heavens, which recession is equal to the deviation of the earth's axis from its parallelism: but this is rather too small to be sensible in an age, except to those who make very nice observations. Particular tains all the other planets in their orbits without affecting the earth, which is placed between the orbits of Mars and Venus, is as absurd as to suppose that six cannon-bullets might be projected upwards to different heights in the air, and that five of them should fall down to the ground; but the sixth, which is neither the highest nor the lowest, should remain suspended in the air without falling, and the earth move round about it.

There is no such thing in nature as a heavy body moving round a light one as its centre of motion. A pebble fastened to a mill-stone by a string, may by an easy impulse be made to circulate round the mill-stone: but no impulse can make a mill-stone circulate round a loose pebble; for the mill-stone would go off, and carry the pebble along with it.

The sun is so immensely bigger and heavier than the earth, that, if he was moved out of his place, not only the earth, but all the other planets, if they were united into one mass, would be carried along with the sun as the pebble would be with the mill-stone.

By considering the law of gravitation, which takes place throughout the solar system, in another light, it will be evident that the earth moves round the sun in a year, and not the sun round the earth. It has been observed, that the power of gravity decreases as the square of the distance increases; and from this it follows with mathematical certainty, that when two or more bodies move round another as their centre of motion, the squares of their periodic times will be to one another in the same proportion as the cubes of their distances from the central body. This holds precisely with regard to the planets round the sun, and the satellites round the planets; the relative distances of all which are well known. But, if we suppose the sun to move round the earth, and compare its period with the moon's by the above rule, it will be found that the sun would take no less than 173510 days to move round the earth; in which case our year would be 475 times as long as it now is. To this we may add, that the aspects of increase and decrease of the planets, the times of their seeming to stand still, and to move direct and retrograde, answer precisely to the earth's motion; but not at all to the sun's, without introducing the most absurd and monstrous suppositions, which would destroy all harmony, order, and simplicity, in the system. Moreover, if the earth be supposed to stand still, and the stars to revolve in free spaces about the earth in 24 hours, it is certain that the forces by which the stars revolve in their orbits are not directed to the earth, but to the centres of the several orbits; that is, of the several parallel circles which the stars on different sides of the equator describe every day: and the like inferences may be drawn from the supposed diurnal motion of the planets, since they are never in the equinoctial but twice in their courses with regard to the starry heavens. But, that forces should be directed to no central body, on which they physically depend, but to innumerable imaginary points in the axis of the earth produced to the poles of the heavens, is an hypothesis too absurd to be allowed of by any rational creature. And it is still more absurd to imagine that these forces should increase exactly in proportion to the distances from this axis; for this is an indication of an increase to infinity; whereas the force of attraction is found to decrease in receding from the fountain from whence it flows. But the farther any star is from the quiescent pole, the greater must be the orbit which it describes; and yet it appears to go round in the same time as the nearest star to the pole does. And if we take into consideration the twofold motion observed in the stars, one diurnal round the axis of the earth in 24 hours, and the other round the axis of the ecliptic in 25920 years, it would require an explication of such a perplexed composition of forces, as could by no means be reconciled with any physical theory.

The strongest objection that can be made against the earth's motion round the sun is, that in opposite directions against the points of the earth's orbit, its axis, which always keeps a parallel direction, would point to different fixed stars; which is not found to be fact. But this objection is easily removed, by considering the immense distance of the stars in respect of the diameter of the earth's orbit; the latter being no more than a point when compared to the former. If we lay a ruler on the side of a table, and along the edge of the ruler view the top of a spire at ten miles distance; then lay the ruler on the opposite of the table in a parallel situation to what it had before, and the spire will still appear along the edge of the ruler; because our eyes, even when assisted by the best instruments, are incapable of distinguishing so small a change at so great a distance.

Dr Bradley, our late astronomer-royal, found by a long series of the most accurate observations, that there is a small apparent motion of the fixed stars, occasioned by the aberration of their light; and so exactly answering to an annual motion of the earth, as evinces of light, the same, even to a mathematical demonstration. He considered this matter in the following manner: he imagined CA, fig. 33., to be a ray of light falling perpendicularly upon the line BD; that, if the eye is at rest at A, the object must appear in the direction AC, whether light be propagated in time or in an instant. But if the eye is moving from B towards A, and light is propagated in time, with a velocity that is to the velocity of the eye, as CA to BA; then light moving from C to A, whilst the eye moves from B to A, that particle of it by which the object will be discerned when the eye comes to A, is at C when the eye is at B. Joining the points BC, he supposed the line CB to be a tube, inclined to the line BD in the angle DBC, of such diameter as to admit but one particle of light. Then it was easy to conceive, that the particle of light at C, by which the object must be seen, when the eye, as it moves along, arrives at A, would pass through the tube BC, if it is inclined to BD, in the angle DBC, and accompanies the eye in its motion from B to A; and that it could not come to the eye placed behind such a tube, if it had any other inclination to the line BD. If, instead of supposing CB so small a tube, we imagine it to be the axis of a larger; then, for the same reason, the particle of light at C would not pass through the axis, unless it is inclined to BD in the angle CBD. In like manner, if the eye moved the contrary way, from D towards A, with the same velocity, then the tube must be inclined in the angle BCD. Although, therefore, the true or real place of an object is perpendicular to the line in which the eye is moving, yet the visible place will not be so; since that, no doubt, must be in the direction of the tube; but the difference between the true and apparent place will be *ceteris paribus* greater or less, according to the different proportion between the velocity of light and that of the eye. So that, if we could suppose that light was propagated in an instant, then there would be no difference between the real and visible place of an object, although the eye was in motion; for in that case, AC being infinite with respect to AB, the angle ACB, the difference between the true and visible place, vanishes. But if light be propagated in time, it is evident, from the foregoing considerations, that there will be always a difference between the real and visible place of an object, unless the eye is moving either directly towards or from the object. And in all cases the fine of the difference between the real and visible place of the object will be to the fine of the visible inclination of the object to the line in which the eye is moving, as the velocity of eye is to the velocity of light.

He then shows, that if the earth revolve round the sun annually, and the velocity of light be to the velocity of the earth's motion in its orbit, as 1000 to 1, that a star really placed in the very pole of the ecliptic would, to an eye carried along with the earth, seem to change its place continually; and neglecting the small difference on the account of the earth's diurnal revolution on its axis, would seem to describe a circle round that pole every way distant from it $3\frac{1}{2}$; so that its longitude would be varied through all the points of the ecliptic every year, but its latitude would always remain the same. Its right ascension would also change, and its declination, according to the different situation of the sun with respect to the equinoctial points, and its apparent distance from the north pole of the equator, would be $\gamma$ less at the autumnal than at the vernal equinox.

By calculating exactly the quantity of aberration of the fixed stars from their place, he found that light came from the sun to us in $8'13''$; so that its velocity is to the velocity of the earth in its orbit as 10,201 to 1.

It must here be taken notice of, however, that Mr Nevil Maskelyne, in attempting to find the parallax of Sirius with a ten feet sector, observed, that by the friction of the plummet line on the pin which suspended it, an error of $10''$, $20''$, and sometimes $30''$, was committed. The pin was $\frac{1}{2}$ of an inch diameter; and though he reduced it to $\frac{1}{50}$ of an inch, the error still amounted to $3''$. All observations, therefore, that have hitherto been made in order to discover the parallax of the fixed stars, are to be disregarded.

It is also objected, that the sun seems to change his place daily, so as to make a tour round the starry heavens in a year. But whether the sun or earth moves, this appearance will be the same; for when the earth is in any part of the heavens, the sun will appear in the opposite. And therefore, this appearance can be no objection against the motion of the earth.

It is well known to every person who has sailed on smooth water, or been carried by a stream in a calm, that, however fast the vessel goes, he does not feel its progressive motion. The motion of the earth is incomparably more smooth and uniform than that of a ship, or any machine made and moved by human art; and therefore it is not to be imagined that we can feel its motion.

We find that the sun, and those planets on which there are visible spots, turn round their axes: for the spots move regularly over their disks (b). From hence we may reasonably conclude, that the other planets on which we see no spots, and the earth, which is likewise a planet, have such rotations. But being incapable of leaving the earth, and viewing it at a distance, and its rotation being smooth and uniform, we can neither see it move on its axis as we do the planets, nor feel ourselves affected by its motion. Yet there is one effect of such a motion, which will enable us to judge with certainty whether the earth revolves on its axis or not.

All globes which do not turn round their axes will be perfect spheres, on account of the equality of the weight of bodies on their surfaces; especially of the fluid parts. But all globes which turn on their axes will be oblate spheroids; that is, their surfaces will be higher or farther from the centre in the equatorial than in the polar regions: for as the equatorial parts move quickest, they will recede farthest from the axis of motion, and enlarge the equatorial diameter. That our earth is really of this figure, is demonstrable from the unequal vibrations of a pendulum, and the unequal lengths of degrees in different latitudes. Since then the earth is higher at the equator than at the poles, the sea, which naturally runs downward, or toward the places which are nearest the centre, would run towards the polar regions, and leave the equatorial parts dry, if the centrifugal force of these parts, by which the waters were carried thither, did not keep them from returning. The earth's equatorial diameter is 36 miles longer than its axis.

Bodies near the poles are heavier than those towards the equator, because they are nearer the earth's centre, where the whole force of the earth's attraction is accumulated. They are also heavier, because their centripetal force is less, on account of their diurnal motion being slower. For both these reasons, bodies carried from the poles toward the equator gradually lose their weight. Experiments prove, that a pendulum which vibrates seconds near the poles, vibrates slower near the equator, which shews that it is lighter or less attracted there. To make it oscillate in the same time, it is found necessary to diminish its length. By comparing the different lengths of pendulums swinging seconds at the equator and at London, it is found that a pendulum must be $2\frac{1}{60}$ lines shorter at the equator than at the poles. A line is a twelfth part of an inch.

If the earth turned round its axis in 34 minutes 43 seconds, the centrifugal force would be equal to the power of gravity at the equator; and all bodies there would entirely lose their weight. If the earth revolved quicker, they would all fly off and leave it.

A person on the earth can no more be sensible of its undisturbed motion on its axis, than one in the cabin of

(b) This, however, must be understood with some degree of limitation, as will evidently appear from what has been already said concerning the variable motion both of the spots of the sun and planets. Particular Explication of the Celestial Phenomena.

A ship on smooth water can be sensible of the ship's motion when it turns gently and uniformly round. It is therefore no argument against the earth's diurnal motion, that we do not feel it; nor is the apparent revolutions of the celestial bodies every day a proof of the reality of these motions; for whether we or they revolve, the appearance is the very same. A person looking through the cabin-windows of a ship, as strongly fancies the objects on land to go round when the ship turns as if they were actually in motion.

If we could translate ourselves from planet to planet, we should still find that the stars would appear of the same magnitudes, and at the same distances from each other, as they do to us here; because the width of the remote planet's orbit bears no sensible proportion to the distance of the stars. But then the heavens would seem to revolve about very different axes; and consequently, those quiescent points, which are our poles in the heavens, would seem to revolve about other points, which, though apparently in motion as seen from the earth, would be at rest as seen from any other planet. Thus the axis of Venus, which lies at right angles to the axis of the earth, would have its motionless poles in two opposite points of the heavens lying almost in our equinoctial, where the motion appears quickest, because it is seemingly performed in the greatest circle; and the very poles, which are at rest to us, have the quickest motion of all as seen from Venus. To Mars and Jupiter the heavens appear to turn round with very different velocities on the same axis, whose poles are about 23½ degrees from ours. Were we on Jupiter, we should be at first amazed at the rapid motion of the heavens; the sun and stars going round in 9 hours 56 minutes. Could we go from thence to Venus, we should be as much surprised at the slowness of the heavenly motions; the sun going but once round in 584 hours, and the stars in 540. And could we go from Venus to the moon, we should see the heavens turn round with a yet slower motion; the sun in 708 hours, the stars in 655. As it is impossible these various circumvolutions in such different times, and on such different axes, can be real, so it is unreasonable to suppose the heavens to revolve about our earth more than it does about any other planet. When we reflect on the vast distance of the fixed stars, to which 190,000,000 of miles, the diameter of the earth's orbit, is but a point, we are filled with amazement at the immensity of their distance. But if we try to frame an idea of the extreme rapidity with which the stars must move, if they move round the earth in 24 hours, the thought becomes so much too big for our imagination, that we can no more conceive it than we do infinity or eternity. If the sun was to go round the earth in 24 hours, he must travel upwards of 300,000 miles in a minute; but the stars being at least 400,000 times as far from the sun as the sun is from us, those about the equator must move 400,000 times as quick. And all this to serve no other purpose than what can be as fully and much more simply obtained by the earth's turning round eastward, as on an axis, every 24 hours, causing thereby an apparent diurnal motion of the sun westward, and bringing about the alternate returns of day and night.

As to the common objections against the earth's motion on its axis, they are all easily answered and set aside. That it may turn without being seen or felt by us to do so, has been already shown. But some particular are apt to imagine, that if the earth turns eastward (as it certainly does if it turns at all), a ball fired perpendicularly upward in the air must fall considerably westward of the place it was projected from. The objection, which at first seems to have some weight, will be found to have none at all, when we consider that the gun and ball partake of the earth's motion; and therefore the ball being carried forward with the air as quick as the earth and air turn, must fall down on the same place. A stone let fall from the top of a mainmast, if it meets with no obstacle, falls on the deck as near the foot of the mast when the ship sails as when it does not. If an inverted bottle full of liquor be hung up to the ceiling of the cabin, and a small hole be made in the cork to let the liquor drop through on the floor, the drops will fall just as far forward on the floor when the ship sails as when it is at rest. And gnats or flies can as easily dance among one another in a moving cabin as in a fixed chamber. As for those scripture expressions which seem to contradict the earth's motion, this general answer may be made to them all, viz. It is plain from many instances, that the scriptures were never intended to instruct us in philosophy or astronomy; and therefore on those subjects expressions are not always to be taken in the literal sense, but for the most part as accommodated to the common apprehensions of mankind. Meh of sense in all ages, when not treating of the sciences purposely, have followed this method; and it would be in vain to follow any other in addressing ourselves to the vulgar, or bulk of any community.

The following experiment will give a plain idea of Diurnal the diurnal or annual motions of the earth, together with the different lengths of days and nights, and all the beautiful variety of seasons, depending on those motions.

Take about seven feet of strong wire, and bend it into a circular form, as a b c d, which being viewed obliquely, appears elliptical, as in the figure. Place a lighted candle on a table; and having fixed one end of a silk thread K to the north pole of a small terrestrial globe H, about three inches diameter, cause another person to hold the wire circle, so that it may be parallel to the table, and as high as the flame of the candle I, which should be in or near the centre. Then having twisted the thread as towards the left hand, that by untwisting it may turn the globe round eastward, or contrary to the way that the hands of a watch move, hang the globe by the thread within this circle, almost contiguous to it; and as the thread untwists, the globe (which is enlightened half round by the candle as the earth is by the sun) will turn round its axis, and the different places upon it will be carried through the light and dark hemispheres, and have the appearance of a regular succession of days and nights, as our earth has in reality by such a motion. As the globe turns, move your hand slowly, so as to carry the globe round the candle according to the order of the letters a b c d, keeping its centre even with the wire circle; and you will perceive, that the candle, being still perpendicular to the equator, will enlighten the globe from pole to pole in its whole motion round the circle; and that every place on the globe goes equally through the light and the dark, as it turns round by the untwisting. Particular ing of the thread, and therefore has a perpetual equi- nox. The globe thus turning round represents the earth turning round its axis; and the motion of the globe round the candle represents the earth's annual motion round the sun; and shows, that if the earth's orbit had no inclination to its axis, all the days and nights of the year would be equally long, and there would be no different seasons. Hence also it appears why the planets Mars and Jupiter have a perpetual e- quinox, namely, because their axis is perpendicular to the plane of their orbit, as the thread round which the globe turns in this experiment is perpendicular to the plane of the area inclosed by the wire.—But now de- fire the person who holds the wire to hold it obliquely in the position ABCD, raising the side FG just as much as he depresses the side FG, that the flame may be still in the plane of the circle; and twisting the thread as before, that the globe may turn round its axis the same way as you carry it round the candle, that is, from west to east; let the globe down into the lowermost part of the wire circle at FG; and if the circle be pro- perly inclined, the candle will shine perpendicularly on the tropic of Cancer; and the frigid zone, lying within the arctic or north polar circle, will be all in the light, as in the figure; and will keep in the light, let the globe turn round its axis ever so often. From the equator to the north polar circle, all the places have longer days and shorter nights; but from the equator to the south polar circle, just the reverse. The sun does not set to any part of the north frigid zone, as shown by the candle's shining on it, so that the motion of the globe can carry no place of that zone into the dark; and at the same time the south frigid zone is involved in darkness, and the turning of the globe brings none of its places into the light. If the earth were to continue in the like part of its orbit, the sun would never set to the inhabitants of the north frigid zone, nor rise to those of the south. At the equator it would be always equal day and night; and as places are gradually more and more distant from the equator towards the arctic circle, they would have longer days and shorter nights; whilst those on the south side of the equator would have their nights longer than their days. In this case, there would be continual summer on the north side of the equator, and continual winter on the south side of it.

But as the globe turns round its axis, move your hand slowly forward, so as to carry the globe from H towards E, and the boundary of light and darkness will approach towards the north pole, and recede to- wards the south pole; the northern places will go through less and less of the light, and the southern places through more and more of it; showing how the northern days decrease in length and the southern days increase, whilst the globe proceeds from H to E. When the globe is at E, it is at a mean state between the lowest and highest parts of its orbit; the candle is directly over the equator, the boundary of light and darkness just reaches to both the poles, and all places on the globe go equally through the light and dark hemispheres, showing that the days and nights are then equal at all places of the earth, the poles only except- ed; for the sun is then setting to the north pole and rising to the south pole.

Continue moving the globe forward, and as it goes through the quarter A, the north pole recedes still far- ther into the dark hemisphere, and the south pole ad- vances more into the light, as the globe comes nearer to FG; and when it comes there at F, the candle is directly over the tropic of Capricorn; the days are at the shortest and nights at the longest, in the northern hemisphere, all the way from the equator to the arctic circle; and the reverse in the southern hemisphere from the equator to the antarctic circle; within which circles it is dark to the north frigid zone, and light to the south.

Continue both motions; and as the globe moves through the quarter B, the north pole advances to- wards the light, and the south pole recedes towards the dark; the days lengthen in the northern hemisphere and shorten in the southern; and when the globe comes to G, the candle will be again over the equator (as when the globe was at E), and the days and nights will again be equal as formerly; and the north pole will be just coming into the light, the south pole going out of it.

Thus we see the reason why the days lengthen and shorten from the equator to the polar circles every year; why there is sometimes no day or night for many turnings of the earth, within the polar circles; why there is but one day and one night in the whole year at the poles; and why the days and nights are equally long all the year round at the equator, which is always equally cut by the circle bounding light and darkness.

The inclination of an axis or orbit is merely relative, because we compare it with some other axis or orbit which we consider as not inclined at all. Thus, our horizon being level to us, whatever place of the earth we are upon, we consider it as having no inclination; and yet, if we travel 90 degrees from that place, we shall then have an horizon perpendicular to the former, but it will still be level to us.

Let us now take a view of the earth in its annual Differ- course round the sun, considering its orbit as having no deflection inclination; and its axis as inclining 23° degrees from circularly a line perpendicular to the plane of its orbit, and keep- ing the same oblique direction in all parts of its annual course; or, as commonly termed, keeping always pa- rallel to itself.

Let \(a, b, c, d, e, f, g, h\) be the earth in eight diffe- rent parts of its orbit, equidistant from one another; \(N\) its axis, \(N'\) its north pole, \(S\) its south pole, and \(S'\) the sun nearly in the centre of the earth's orbit. As the earth goes round the sun according to the order of the letters abcd, &c., its axis \(N\) keeps the same ob- liquity, and is still parallel to the line \(MN\). When the earth is at \(a\), its north pole inclines towards the sun \(S\), and brings all the northern places more into the light than at any other time of the year. But when the earth is at \(e\) in the opposite time of the year, the north pole declines from the sun, which occasions the northern places to be more in the dark than in the light, and the reverse at the southern places; as is evi- dent by the figure which is taken from Dr Long's as- tronomy. When the earth is either at \(c\) or \(g\), its axis inclines not either to or from the sun, but lies side- wise to him, and then the poles are in the boundary of light and darkness; and the sun, being directly over the equator, makes equal day and night at all places. When the earth is at b, it is half-way between the summer solstice and harvest equinox; when it is at d, it is half-way from the harvest equinox to the winter solstice; at f, half-way from the winter solstice to the spring equinox; and at h, half-way from the spring equinox to the summer solstice.

From this oblique view of the earth's orbit, let us suppose ourselves to be raised far above it, and placed just over its centre S, looking down upon it from its north pole; and as the earth's orbit differs but very little from a circle, we shall have its figure in such a view represented by the circle ABCDEFG. Let us suppose this circle to be divided into 12 equal parts, called signs, having their names affixed to them; and each sign into 30 equal parts, called degrees, numbered 10, 20, 30, as in the outermost circle of the figure, which represents the great ecliptic in the heavens. The earth is shown in eight different positions in this circle; and in each position A is the equator, T the tropic of Cancer, the dotted circle the parallel of London, U the arctic or north polar circle, and P the north pole, where all the meridians or hour-circles meet. As the earth goes round the sun, the north pole keeps constantly towards one part of the heavens, as it keeps in the figure towards the right-hand side of the plate.

When the earth is at the beginning of Libra, namely on the 20th of March, in this figure the sun S as seen from the earth, appears at the beginning of Aries in the opposite part of the heavens, the north pole is just coming into the light, and the sun is vertical to the equator; which, together with the tropic of Cancer, parallel of London, and arctic circle, are all equally cut by the circle bounding light and darkness, coinciding with the six-o'clock hour-circle, and therefore the days and nights are equally long at all places; for every part of the meridian A'TLa comes into the light at six in the morning, and, revolving with the earth according to the order of the hour-letters, goes into the dark at six in the evening. There are 24 meridians or hour-circles drawn on the earth in this figure, to show the time of rising and setting at different seasons of the year.

As the earth moves in the ecliptic according to the order of the letters ABCD, &c. through the signs Libra, Scorpio, and Sagittarius, the north pole P comes more and more into the light; the days increase as the nights decrease in length, at all places north of the equator A'; which is plain by viewing the earth at b on the 5th of May, when it is in the 15th degree of Scorpio, and the sun as seen from the earth appears in the 15th degree of Taurus. For then the tropic of Cancer T is in the light from a little after five in the morning till almost seven in the evening; the parallel of London, from half an hour past four till half an hour past seven; the polar circle U, from three till nine; and a large track round the north pole P has day all the 24 hours, for many rotations of the earth on its axis.

When the earth comes to c (fig. 174.) at the beginning of Capricorn, and the sun as seen from the earth appears at the beginning of Cancer, on the 21st of June, as in this figure, it is in the position C in fig. 177; and its north pole inclines towards the sun, so as to bring all the north frigid zone into the light, and the northern parallels of latitude more into the light than the dark from the equator to the polar circle; and the more so as they are farther from the equator. The tropic of Cancer is in the light from five in the morning till seven at night, the parallel of London from a quarter before four till a quarter after eight; and the polar circle just touches the dark, so that the sun has only the lower half of his disk hid from the inhabitants on that circle for a few minutes about midnight, supposing no inequalities in the horizon, and no refractions.

A bare view of the figure is enough to show, that as the earth advances from Capricorn towards Aries, and the sun appears to move from Cancer towards Libra, the north pole recedes from the light, which causes the days to decrease, and the nights to increase in length, till the earth comes to the beginning of Aries, and then they are equal as before; for the boundary of light and darkness cuts the equator and all its parallels equally or in halves. The north pole then goes into the dark, and continues therein until the earth goes halfway round its orbit; or, from the 23rd of September till the 20th of March. In the middle between these times, viz. on the 22nd of December, the north pole is as far as it can be in the dark, which is $2\frac{1}{2}$ degrees, equal to the inclination of the earth's axis from a perpendicular to its orbit; and then the northern parallels are as much in the dark as they were in the light on the 21st of June; the winter nights being as long as the summer days, and the winter days as short as the summer nights. Here it must be noted, that of all that has been said of the northern hemisphere, the contrary must be understood of the southern; for on different sides of the equator the reasons are contrary, because, when the northern hemisphere inclines towards the sun, the southern declines from him.

The earth's orbit being elliptical, and the sun constantly keeping in its lower focus, which is 1,617,941 miles from the middle point of the longer axis, the earth bigger in comes twice so much, or 3,235,882 miles nearer the sun than at one time of the year than another; for the sun appearing under a larger angle in our winter than summer, proves that the earth is nearer the sun in winter. But here this natural question will arise, Why have we not the hottest weather when the earth is nearest the sun? In answer it must be observed, that the eccentricity of the earth's orbit, or 1,617,941 miles, bears no greater proportion to the earth's mean distance from the sun than 17 does to 1000; and therefore this small difference of distance cannot occasion any great difference of heat or cold. But the principal cause of this difference is, that in winter the sun's rays fall so obliquely upon us, that any given number of them is spread over a much greater portion of the earth's surface where we live, and therefore each point must then have fewer rays than in summer. Moreover, there comes a greater degree of cold in the long winter-nights than there can return of heat in so short days; and on both these accounts the cold must increase. But in summer the sun's rays fall more perpendicularly upon us; and therefore come with greater force, and in greater numbers, on the same place; and by their long continuance, a much greater degree of heat is imparted by day than can fly off by night. Besides, those parts which are once heated, retain the heat for some time; which, with the additional heat daily imparted, makes Particular makes it continue to increase, though the sun declines towards the south; and this is the reason why July is hotter than June, although the sun has withdrawn from the summer tropic; as we find it is generally hotter at three in the afternoon, when the sun has gone towards the west, than at noon when he is on the meridian. Likewise those places which are well cooled require time to be heated again; for the sun's rays do not heat even the surface of any body till they have been some time upon it. And therefore we find January for the most part colder than December, although the sun has withdrawn from the winter tropic, and begins to dart his beams more perpendicularly upon us. An iron bar is not heated immediately upon being in the fire, nor grows cold till some time after it has been taken out.

It has been already observed, that by the earth's motion on its axis, there is more matter accumulated all around the equatorial parts than anywhere else on the earth.

The sun and moon, by attracting this redundancy of matter, bring the equator sooner under them in every return towards it, than if there was no such accumulation. Therefore, if the sun sets out, as from any star, or other fixed point in the heavens, the moment when he is departing from the equinoctial or from either tropic, he will come to the same equinox or tropic again 20 minutes 17 seconds of time, or 50 seconds of a degree, before he completes his course, so as to arrive at the same fixed star or point from whence he set out. For the equinoctial points recede 50 seconds of a degree westward every year, contrary to the sun's annual progressive motion.

When the sun arrives at the same equinoctial or solstitial point, he finishes what we call the Tropical Year; which, by observation, is found to contain 365 days 5 hours 48 minutes 57 seconds; and when he arrives at the same fixed star again, as seen from the earth, he completes the sidereal year, which contains 365 days 6 hours 9 minutes 14 seconds. The sidereal year is therefore 20 minutes 17 seconds longer than the solar or tropical year, and 9 minutes 14 seconds longer than the Julian or the civil year, which we state at 365 days 6 hours, so that the civil year is almost a mean between the sidereal and tropical.

As the sun describes the whole ecliptic, or 360 degrees, in a tropical year, he moves 5° 8' of a degree every day at a mean rate; and consequently 5° of a degree in 20 minutes 17 seconds of time; therefore he will arrive at the same equinox or solstice when he is 5° of a degree short of the same star or fixed point in the heavens from which he set out the year before. So that, with respect to the fixed stars, the sun and equinoctial points fall back (as it were) 30 degrees in 2160 years, which will make the stars appear to have gone 30 deg. forward with respect to the signs of the ecliptic in that time; for the same signs always keep in the same points of the ecliptic, without regard to the constellations.

To explain this by a figure, let the sun be in conjunction with a fixed star at S, suppose in the 30th degree of γ, at any given time. Then, making 2160 revolutions through the ecliptic VWX, at the end of so many sidereal years, he will be found again at S; but at the end of so many Julian years, he will be found at M, short of S; and at the end of so many tropical years he will be found short of M, in the 30th degree of Taurus at T, which has receded back from S to T in that time, by the precession of the equinoctial points Ψ Aries and Ω Libra. The arc ST will be equal to the amount of the precession of the equinox in 2160 years, at the rate of 5° of a degree, or 20 minutes 17 seconds of time annually; this, in so many years, makes 30 days 10 hours, which is the difference between 2160 sidereal and tropical years; and the arc MT will be equal to the space moved through by the sun in 2160 times 1 min. 8 sec. or 16 days 13 hours 48 minutes, which is the difference between 2160 Julian and tropical years.

The anticipation of the equinoxes, and consequently of the seasons, is by no means owing to the precession of the equinoctial and solstitial points in the heavens (which can only affect the apparent motions, places, and declinations, of the fixed stars), but to the difference between the civil and solar year, which is 11 minutes 3 seconds; the civil year containing 365 days 6 hours, and the solar year 365 days 5 hours 48 minutes 57 seconds.

The above 11 minutes 3 seconds, by which the civil or Julian year exceeds the solar, amounts to 11 days in 1433 years; and so much our seasons have fallen back with respect to the days of the months, since the time of the Nicene council in A.D. 325; and therefore, in order to bring back all the saints and festivals to the days then settled, it was requisite to supply 11 nominal days; and, that the same seasons might be kept to the same times of the year for the future, to leave out the bissextile-day in February at the end of every century of years not divisible by 4; reckoning them only common years, as the 17th, 18th, and 19th centuries, viz. the years 1700, 1800, 1900, &c., because a day intercalated every fourth year was too much; and retaining the bissextile-day at the end of those centuries of years which are divisible by 4, as the 16th, 20th, and 24th centuries, viz. the years 1600, 2000, 2400, &c.; otherwise, in length of time, the seasons would be quite reversed with regard to the months of the year; though it would have required near 23,783 years to have brought about such a total change. If the earth had made exactly 365 1/4 diurnal rotations on its axis, whilst it revolved from any equinoctial or solstitial point to the same again, the civil and solar years would always have kept pace together, and the style would never have needed any alteration.

Having thus mentioned the cause of the precession of the equinoctial points in the heavens, which occasions a slow deviation of the earth's axis from its parallelism, and thereby a change of the declination of the stars from the equator, together with a slow apparent motion of the stars forward with respect to the signs of the ecliptic, we shall now explain the phenomena by a diagram.

Let NZSVL be the earth, SONA its axis produced to the starry heavens, and terminating in A, the present north pole of the heavens, which is vertical to N the north pole of the earth. Let EOQ be the equator, TΣZ the tropic of Cancer, and VTΨ the tropic of Capricorn; VOZ the ecliptic, and BO its axis, both which are immovable among the stars. But as the equinoctial points recede in the ecliptic, the earth's axis... axis SON is in motion upon the earth's centre O, in such a manner as to describe the double cone NO and SO, round the axis of the ecliptic BO, in the time that the equinoctial points move quite round the ecliptic, which is 25,920 years; and in that length of time, the north pole of the earth's axis produced, describes the circle ABCDA in the starry heavens, round the pole of the ecliptic, which keeps immovable in the centre of that circle. The earth's axis being 23° 27' degrees inclined to the axis of the ecliptic, the circle ABCDA described by the north pole of the earth's axis produced to A, is 47 degrees in diameter, or double the inclination of the earth's axis. In consequence of this, the point A, which at present is the north pole of the heavens, and near to a star of the second magnitude in the tail of the constellation called the Little Bear, must be deflected by the earth's axis; which moving backwards a degree every 72 years, will be directed towards the star or point B in 6480 years hence; and in double of that time, or in 12,960 years, it will be directed towards the star or point C, which will then be the north pole of the heavens, although it is at present 8½ degrees south of the zenith of London L. The present position of the equator EOQ will then be changed into EQ, the tropic of Cancer TΣZ into VΣB, and the tropic of Capricorn VTΣ into TΣZ; as is evident by the figure. And the sun, in the same part of the heavens where he is now over the earthly tropic of Capricorn, and makes the shortest days and longest nights in the northern hemisphere, will then be over the earthly tropic of Cancer, and make the days longest and nights shortest. So that it will require 12,960 years yet more, or 25,920 from the then present time, to bring the north pole N quite round, so as to be directed toward that point of the heavens which is vertical to it at present. And then, and not till then, the same stars which at present describe the equator, tropics, and polar circles, &c. by the earth's diurnal motion, will describe them over again.

From the shifting of the equinoctial points, and with them all the signs of the ecliptic, it follows that those stars which in the infancy of astronomy were in Aries are now got into Taurus; those of Taurus into Gemini, &c. Hence likewise it is that the stars which rose or set at any particular season of the year, in the times of Hesiod, Eudoxus, Virgil, Pliny, &c. by no means answer at this time to their descriptions.

The moon is not a planet, but only a satellite, or attendant of the earth, going round the earth from change to change in 29 days 12 hours and 44 minutes, and round the sun with it every year. The moon's diameter is 2180 miles; and her distance from the earth's centre is 240,000. She goes round her orbit in 27 days 7 hours 43 minutes, moving about 2290 miles every hour; and turns round her axis exactly in the time that she goes round the earth, which is the reason of her keeping always the same side towards us, and that her day and night taken together is as long as our lunar month.

The moon is an opaque globe like the earth, and shines only by reflecting the light of the sun; therefore, whilst that half of her which is towards the sun is enlightened, the other half must be dark and invisible. Hence it disappears when the comes between us and the sun; because her dark side is then towards us. When she is gone a little way forward, we see a little of her enlightened side; which still increases to our view as she advances forward, until she comes to be opposite to the earth, and she appears with a round illumined orb, which we call the full moon; her dark side being then turned away from the earth. From the full she seems to decrease gradually as she goes through the other half of her course; showing us less and less of her enlightened side every day, till her next change or conjunction with the sun, and then she disappears as before.

The moon has scarce any difference of seasons; her axis being almost perpendicular to the ecliptic. What is very singular, one half of her has no darkness at all; the earth constantly affording it a strong light in the sun's absence; while the other half has a fortnight's darkness and a fortnight's light by turns.

Our earth is thought to be a moon to the moon; waxing and waning regularly, but appearing 13 times pears as big, and affording her 13 times as much light as the moon does us. When she changes to us the earth appears our moon full to her; and when she is in her first quarter to us, the earth is in its third quarter to her; and vice versa.

But from one half of the moon the earth is never seen at all; from the middle of the other half, it is always seen over head; turning round almost 30 times as quick as the moon does. From the circle which limits our view of the moon, only one half of the earth's side next her is seen; the other half being hid below the horizon of all places on that circle. To her the earth seems to be the biggest body in the universe; for it appears 13 times as big as she does to us.

As the earth turns round its axis, the several continents, seas, and islands, appear to the moon's inhabitants like to many spots of different forms and brightness, moving over its surface; but much fainter at some times than others, as our clouds cover them or leave them. By these spots the Lunarians can determine the time of the earth's diurnal motion, just as we do the motion of the sun; and perhaps they measure their time by the motion of the earth's spots; for they cannot have a truer dial.

The moon's axis is so nearly perpendicular to the ecliptic, that the sun never removes sensibly from her equator; and the obliquity of her orbit, which is next to nothing as seen from the sun, cannot cause the sun to decline sensibly from her equator. Yet her inhabitants are not destitute of means for ascertaining the length of their year, though their method and ours must differ. For we can know the length of our year by the return of our equinoxes; but the Lunarians, having always equal day and night, must have recourse to another method; and we may suppose, they measure their year by observing when either of the poles of our earth begins to be enlightened and the other to disappear, which is always at our equinoxes; they being conveniently situated for observing great tracts of land about our earth's poles, which are entirely unknown to us. Hence we may conclude, that the year is of the same absolute length both to the earth and moon, though very different as to the number of days; we having 365 natural days, and the Lunarians only 12½, every day and night in the moon being as long as 29½ on the earth.

The moon's inhabitants on the side next the earth Particular may as easily find the longitude of their places as we can find the latitude of ours. For the earth keeping constantly, or very nearly so, over one meridian of the moon, the east or west distances of places from that meridian are as easily found as we can find our distance from the equator by the altitude of our celestial poles.

As the sun can only enlighten that half of the earth which is at any moment turned towards him, and, being withdrawn from the opposite half, leaves it in darkness; so he likewise doth to the moon; only with this difference, that as the earth is surrounded by an atmosphere, we have twilight after the sun sets; but if the moon has none of her own, nor is included in that of the earth, the lunar inhabitants have an immediate transition from the brightest sun-thine to the blackest darkness. For, let \( t r k s w \) be the earth, and \( A, B, C, D, E, F, G, H \), the moon in eight different parts of her orbit. As the earth turns round its axis from west to east, when any place comes to \( t \), the twilight begins there, and when it revolves from thence to \( r \) the sun rises; when the place comes to \( s \) the sun sets, and when it comes to \( w \) the twilight ends. But as the moon turns round her axis, which is only once a month, the moment that any part of her surface comes to \( r \) (see the moon at \( G \)), the sun rises there without any previous warning by twilight; and when the same point comes to \( s \) the sun sets, and that point goes into darkness as black as at midnight.

The moon being an opaque spherical body (for her hills take off no more from her roundness than the inequalities on the surface of an orange takes off from its roundness), we can only see that part of the enlightened half of her which is towards the earth. And therefore, when the moon is at \( A \), in conjunction with the sun \( S \), her dark half is towards the earth, and the difference, as at \( a \), there being no light on that half to render it visible. When she comes to her first octant at \( B \), or has gone an eighth part of her orbit from her conjunction, a quarter of her enlightened side is towards the earth, and she appears horned, as at \( b \). When she has gone a quarter of her orbit from between the earth and sun to \( C \), she shows us one half of her enlightened side, as at \( c \), and we say she is a quarter old. At \( D \), she is in her second octant; and by showing us more of her enlightened side she appears gibbous, as at \( d \). At \( E \), her whole enlightened side is towards the earth; and therefore she appears round, as at \( e \); when we say it is full moon. In her third octant at \( F \), part of her dark side being towards the earth, she again appears gibbous, and is on the decrease, as at \( f \). At \( G \), we see just one half of her enlightened side; and she appears half decreased, or in her third quarter, as at \( g \). At \( H \), we only see a quarter of her enlightened side, being in her fourth octant; where she appears horned, as at \( h \). And at \( A \), having completed her course from the sun to the sun again, she disappears; and we say it is new moon.

Thus, in going from \( A \) to \( E \), the moon seems continually to increase; and in going from \( E \) to \( A \), to decrease in the same proportion; having like phases at equal distances from \( A \) to \( E \), but as seen from the sun \( S \) she is always full.

The moon appears not perfectly round when she is full in the highest or lowest part of her orbit, because we have not a full view of her enlightened side at that particular time. When full in the highest part of her orbit, a small deficiency appears on her lower edge; and the contrary when full in the lowest part of her orbit.

It is plain by the figure, that when the moon changes to the earth, the earth appears full to the moon; and vice versa. For when the moon is at \( A \), new to the earth, the whole enlightened side of the earth is towards the moon; and when the moon is at \( E \), full to the earth, its dark side is towards her. Hence a new moon answers to a full earth, and a full moon to a new earth. The quarters are also reversed to each other.

Between the third quarter and change, the moon is frequently visible in the forenoon, even when the sun prefers shines; and then she affords us an opportunity of seeing a very agreeable appearance, wherever we find a globular stone above the level of the eye, as suppose on the top of a gate. For, if the sun shines on the stone, and we place ourselves so as the upper part of the stone may just seem to touch the point of the moon's lowermost horn, we shall then see the enlightened part of the stone exactly of the same shape with the moon; horned as she is, and inclined the same way to the horizon. The reason is plain; for the sun enlightens the stone the same way as he does the moon: and both being globes, when we put ourselves into the above situation, the moon and stone have the same position to our eyes; and therefore we must see as much of the illuminated part of the one as of the other.

The position of the moon's cusps, or a right line touching the points of her horns, is very differently inclined to the horizon at different hours of the same days of her age. Sometimes she stands, as it were, upright on her lower horn, and then such a line is perpendicular to the horizon: when this happens, she is in what the astronomers call the nonageinal degree; Nonage, which is the highest point of the ecliptic above the horizon at that time, and is 90 degrees from both sides of the horizon where it is then cut by the ecliptic. But this never happens when the moon is on the meridian, except when she is at the very beginning of Cancer or Capricorn.

That the moon turns round her axis in the time that she goes round her orbit, is quite demonstrable; for, a spectator at rest, without the periphery of the moon's orbit, would see all her sides turned regularly towards him in that time. She turns round her axis from any star to the same star again in 27 days 8 hours; from the sun to the sun again in 29\(\frac{1}{2}\) days: the former is the length of her sidereal day, and the latter the length of her solar day. A body moving round the sun would have a solar day in every revolution, without turning on its axis; the same as if it had kept all the while at rest, and the sun moved round it: but without turning round its axis it could never have one sidereal day, because it would always keep the same side towards any given star.

If the earth had no annual motion, the moon would go round it so as to complete a lunation, a sidereal, and a solar day, all in the same time. But, because the earth goes forward in its orbit while the moon goes round the earth in her orbit, the moon must go as much more than round her orbit from change to change in particular in completing a solar day, as the earth has gone forward in its orbit during that time, i.e., almost a twelfth part of a circle.

If the earth had no annual motion, the moon's motion round the earth, and her track in open space, would be always the same (c). But as the earth and moon move round the sun, the moon's real path in the heavens is very different from her visible path round the earth; the latter being in a progressive circle, and the former in a curve of different degrees of concavity, which would always be the same in the same parts of the heavens, if the moon performed a complete number of lunations in a year without any friction.

Let a nail in the end of the axle of a chariot-wheel represent the earth, and a pin in the nave the moon; if the body of the chariot be propped up so as to keep that wheel from touching the ground, and the wheel be then turned round by hand, the pin will describe a circle both round the nail and in the space it moves through. But if the props be taken away, the horses put to, and the chariot driven over a piece of ground which is circularly convex; the nail in the axle will describe a circular curve, and the pin in the nave will still describe a circle round the progressive nail in the axle, but not in the space through which it moves. In this case, the curve described by the nail will resemble in miniature as much of the earth's annual path round the sun, as it describes whilst the moon goes as often round the earth as the pin does round the nail; and the curve described by the pin will have some resemblance of the moon's path during so many lunations.

Let us now suppose that the radius of the circular curve described by the nail in the axle is to the radius of the circle which the pin in the nave describes round the axle, as \(337\frac{1}{2}\) to 1; (n) which is the proportion of the radius or semidiameter of the earth's orbit to that of the moon's, or of the circular curve A 1 2 3 4 5 6 7 B, &c. to the little circle a; and then, whilst the progressive nail describes the said curve from A to E, the pin will go once round the nail with regard to the centre of its path, and in so doing will describe the curve a b c d e. The former will be a true representation of the earth's path for one lunation, and the latter of the moon's for that time. Here we may set aside the inequalities of the moon's motion, and also the earth's moving round its common centre of gravity and the moon's; all which, if they were truly copied in this experiment, would notensibly alter the figure of the paths described by the nail and pin, even though they should rub against a plain upright surface all the way, and leave their tracks visible upon it. And if the chariot were driven forward on such a convex piece of ground, so as to turn the wheel several times round, the track of the pin in the nave would still be concave towards the centre of the circular curve described by the nail in the axle; as the moon's path is always concave to the sun in the centre of the earth's particular annual orbit.

In this diagram, the thickest curve line ABCDE, with the numeral figures set to it, represents as much phenomena of the earth's annual orbit as it describes in 32 days from west to east; the little circles at A, B, C, D, E, show the moon's orbit in due proportion to the earth's; and the smallest curve a C f represents the line of the moon's path in the heavens for 32 days, accounted from any particular new moon at a. The sun is supposed to be in the centre of the curve A 1 2 3 4 5 6 7 B, &c. and the small dotted circles upon it represent the moon's orbit, of which the radius is in the same proportion to the earth's path in this scheme, that the radius of the moon's orbit in the heavens was supposed to bear to the radius of the earth's annual path round the sun; that is, as 240,000 to 81,000,000, or as 1 to 337\(\frac{1}{2}\).

When the earth is at A, the new moon is at a; and in the seven days that the earth describes the curve 1 2 3 4 5 6 7, the moon in accompanying the earth describes the curve ab; and is in her first quarter at b when the earth is at B. As the earth describes the curve B 8 9 10 11 12 13 14, the moon describes the curve b c; and is at c, opposite to the sun, when the earth is at C. Whilst the earth describes the curve C 15 16 17 18 19 20 21 22, the moon describes the curve c d; and is in her third quarter at d when the earth is at D. And lastly, whilst the earth describes the curve D 23 24 25 26 27 28 29, the moon describes the curve d e; and is again in conjunction at e with the sun when the earth is at E, between the 29th and 30th days of the moon's age, accounted by the numeral figures from the new moon at A. In describing the curve a C e, the moon goes round the progressive earth as really as if she had kept in the dotted circle A, and the earth continued immovable in the centre of that circle.

And thus we see, that although the moon goes Her path round the earth in a circle, with respect to the earth's always centre, her real path in the heavens is not very different in appearance from the earth's path. To show that the moon's path is concave to the sun, even at the time of change, it is carried on a little farther into a second lunation as to f.

The curves which Jupiter's satellites describe, are all of different sorts from the path described by our moon, although these satellites go round Jupiter as the moon goes round the earth. Let ABCDE, &c. Fig. 185, be as much of Jupiter's orbit as he describes in 18 days from A to T; and the curves a, b, c, d will be the paths of his four moons going round him in his progressive motion. Now let us suppose all these moons to let out from a conjunction with the sun, as seen from Jupiter at A; then, his first or nearest moon will be at a, his second at b, his third at c, and his fourth at d. At the end of 24 terrestrial hours after this.

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(c) In this place, we may consider the orbits of all the satellites as circular, with respect to their primary planets; because the eccentricities of their orbits are too small to affect the phenomena here described.

(n) The figure by which this is illustrated is borrowed from Mr Ferguson. Later observations have determined the proportions to be different: but we cannot find that any delineation of this kind hath been given by astronomers, according to the new proportions. Particular conjunction, Jupiter has moved to B, his first moon or satellite has described the curve a 1, his second the curve b 1, his third c 1, and his fourth d 1. The next day, when Jupiter is at C, his first satellite has described the curve a 2 from its conjunction, his second the curve b 2, his third the curve c 2, and his fourth the curve d 2, and so on. The numeral figures under the capital letters show Jupiter's place in his path every day for 18 days, accounted from A to T; and the like figures set to the paths of his satellites, show where they are at the like times. The first satellite, almost under C, is stationary at + as seen from the sun, and retrograde from + to 2: at 2 it appears stationary again, and thence it moves forward until it has passed 3, and is twice stationary and once retrograde between 3 and 4. The path of this satellite intersects itself every 42½ hours, making such loops as in the diagram at 2, 3, 5, 9, 10, 12, 14, 16, 18, a little after every conjunction. The second satellite b, moving slower, barely crosses its path every 3 days 13 hours; as at 4, 7, 11, 14, 18, making only five loops and as many conjunctions in the time that the first makes ten. The third satellite c moving still slower, and having described the curve c 1, 2, 3, 4, 5, 6, 7, comes to an angle at 7 in conjunction with the sun at the end of 7 days 4 hours; and so goes on to describe such another curve 7, 8, 9, 10, 11, 12, 13, 14, and is at 14 in its next conjunction. The fourth satellite d is always progressive, making neither loops nor angles in the heavens; but comes to its next conjunction at e between the numeral figures 16 and 17, or in 16 days 18 hours.

The method used by Mr Ferguson to delineate the paths of these satellites was the following. Having drawn their orbits on a card, in proportion to their relative distances from Jupiter, he measured the radius of the orbit of the fourth satellite, which was an inch and ¼ parts of an inch; then multiplied this by 424 for the radius of Jupiter's orbit, because Jupiter is 424 times as far from the sun's centre as his fourth satellite is from his centre; and the product thence arising was 483½ inches. Then taking a small cord of this length, and fixing one end of it to the floor of a long room by a nail, with a black-lead pencil at the other end, he drew the curve ABCD, &c., and set off a degree and half thereon from A to T; because Jupiter moves only so much, whilst his outermost satellite goes once round him, and somewhat more; so that this small portion of so large a circle differs but very little from a straight line. This done, he divided the space AT into 18 equal parts, as AB, BC, &c., for the daily progress of Jupiter; and each part into 24 for his hourly progress. The orbit of each satellite was also divided into as many equal parts as the satellite is hours in finishing its synodical period round Jupiter. Then drawing a right line through the centre of the card, as a diameter to all the four orbits upon it, he put the card upon the line of Jupiter's motion, and transferred it to every horary division thereon, keeping always the said diameter-line on the line of Jupiter's path; and running a pin through each horary division in the orbit of each satellite as the card was gradually transferred along the line ABCD, &c., of Jupiter's motion, he marked points for every hour through the card for the curves described by the satellites, as the primary planet in the centre of the card was carried forward on the line; and so finished the figure, by drawing the lines of each satellite's motion through those (almost innumerable) points: by which means, this is perhaps as true a figure of the paths of the satellites as can be desired. And in the same manner might those of Saturn's satellites be delineated.

It appears by the scheme, that the three first satellites come almost into the same line or position every seventh day; the first being only a little behind with the second, and the second behind with the third. But the period of the fourth satellite is so incommensurate to the periods of the other three, that it cannot be guessed at by the diagram when it would fall again into a line of conjunction with them, between Jupiter and the sun. And no wonder; for supposing them all to have been once in conjunction, it will require 3,087,043,493,260 years to bring them in conjunction again.

The moon's absolute motion from her change to her first quarter, or from a to b, is to much slower than the earth's that she falls 240,000 miles (equal to the femidiameter of her orbit) behind the earth at her first quarter in b, when the earth is in B; that is, she falls back a space equal to her distance from the earth. From that time her motion is gradually accelerated to her opposition or full at c; and then she is come up as far as the earth, having regained what she lost in her first quarter from a to b. From the full to the last quarter at d, her motion continues accelerated so as to be just as far before the earth at d as she was behind it at her first quarter in b. But from d to e her motion is retarded so, that she loses as much with respect to the earth as is equal to her distance from it, or to the femidiameter of her orbit; and by that means she comes to e, and is then in conjunction with the sun as seen from the earth at E. Hence we find, that the moon's absolute motion is slower than the earth's from her third quarter to her first, and swifter than the earth's from her first quarter to her third; her path being less curved than the earth's in the former case and more in the latter. Yet it is still bent the same way towards the sun; for if we imagine the concavity of the earth's orbit to be measured by the length of a perpendicular line C g, let down from the earth's place upon the straight line b g d at the full of the moon, and connecting the places of the earth at the end of the moon's first and third quarters, that length will be about 640,000 miles; and the moon when new only approaching nearer to the sun by 240,000 miles than the earth is, the length of the perpendicular let down from her place at that time upon the same straight line, and which shows the concavity of that part of her path, will be about 400,000 miles.

The moon's path being concave to the sun throughout, demonstrates that her gravity towards the sun, at her conjunction, exceeds her gravity towards the earth; and if we consider that the quantity of matter in the sun is vastly greater than the quantity of matter in the earth, and that the attraction of each body diminishes as the square of the distance from it increases, we shall soon find, that the point of equal attraction between the earth and the sun, is much nearer the earth than the moon is at her change. It may then appear surprising that the moon does not abandon the earth when when she is between it and the sun, because she is considerably more attracted by the sun than by the earth at that time. But this difficulty vanishes when we consider, that a common impulse on any system of bodies affects not their relative motions; but that they will continue to attract, impel, or circulate round one another, in the same manner as if there was no such impulse. The moon is so near the earth, and both of them so far from the sun, that the attractive power of the sun may be considered as equal on both; and therefore the moon will continue to circulate round the earth in the same manner as if the sun did not attract them at all: like bodies in the cabin of a ship, which may move round or impel one another in the same manner when the ship is under sail as when it is at rest; because they are all equally affected by the common motion of the ship. If by any other cause, such as the near approach of a comet, the moon's distance from the earth should happen to be so much increased, that the difference of their gravitating forces towards the sun should exceed that of the moon towards the earth; in that case, the moon, when in conjunction, would abandon the earth, and be either drawn into the sun, or comet, or circulate round about it.

The ruggedness of the moon's surface mentioned n° 146, 147, is of great use to us, by reflecting the sun's light to all sides: for if the moon were smooth and polished like a looking-glass, or covered with water, she could never distribute the sun's light all round; only in some positions she would show us his image, no bigger than a point, but with such a lustre as would be hurtful to our eyes.

The moon's surface being so uneven, many have wondered why her edge appears not jagged, as well as the curve bounding the light and dark places. But if we consider, that what we call the edge of the moon's disk is not a single line set round with mountains, in which case it would appear irregularly indented, but a large zone having many mountains lying behind one another from the observer's eye, we shall find that the mountains in some rows will be opposite to the vales in others; and so fill up the inequalities as to make her appear quite round: just as when one looks at an orange, although its roughness be very discernible on the side next the eye, especially if the sun or a candle shines obliquely upon that side, yet the line terminating the visible part still appears smooth and even.

Having said thus much of the moon's Period, Phases, Path, &c. it may not be amiss to describe, in a summary manner, the irregularities of her motion; and though these have been already treated of on the principles of the Newtonian system, yet as the subject has much embarrassed the astronomical world, it is hoped, that the following explanation of the planetary irregularities upon common mechanical principles, from Mr Nicholson's Natural Philosophy, may not appear superfluous to uninformed readers.

"If the sun were at rest, and the planets did not mutually gravitate towards each other, they would describe ellipses, having the sun in the common focus: But since they mutually act on the sun and on each other, it must follow that the sun is perpetually moved about the centre of gravity of all the planets; which centre is the common focus of all their orbits. This centre, by reason of the sun's very great bulk, can in no situation exceed the distance of its semidiameter from its surface. Particular Some small irregularities arise from these mutual actions, but much less would ensue if the sun were at rest, or not subject to the reaction of the other planets. Phenomena The irregularities in the motions of the primary planets are scarcely considerable enough to come under observation in the course of many revolutions; but those of the moon are very perceptible on account of its nearness to us, and from other causes. It will therefore be sufficient to explain the latter, and apply the explanation to the former, being effects of the same kind.

"If the actions of the sun upon the earth and moon were equal upon each, according to their masses, and tended to produce motions in parallel directions, their relative motions would be the same as if no such force acted upon them. But these forces vary both in quantity and direction according to the various relative situations of the earth and moon.

"Let the point S (fig. 162.) represent the sun, and ADBC the orbit of the moon. Then if the moon be at the quadrature A, the distances ES and AS of the earth and moon from the sun being equal, their gravities towards S will also be equal, and may be represented by those lines ES and AS. Draw the line A parallel and equal to ES, and join LS, which will be parallel to AE. The force AS may be resolved (from principles of compound motion) into the two forces AL and AE, of which AL, by reason of its parallelism and equality to ES, will not disturb their relative motions or situation: but the force AE conspiring with that of gravity, will cause the moon to fall farther below the tangent of its orbit than it would have done if no such disturbing force had existed. Therefore, at or near the quadratures, the moon's gravity towards the earth is increased more than according to the regular course, and its orbit is rendered more curve.

"When the moon is at the conjunction C, the distances ES and CS not being equal, the moon's gravitation towards the sun exceeds that of the earth in the same proportion as the square of ES exceeds the square of CS. And because the excess acts contrary to the direction of the moon's gravity towards the earth, it diminishes the effect thereof, and causes the moon to fall less below the tangent of its orbit than it would if no such disturbing force existed. A like and very nearly equal effect follows, when the moon is at the opposition D, by the earth's gravitation towards the sun being greater than that of the moon; whence their mutual gravitation is diminished as in the former case. Therefore, at or near the conjunction or opposition, the moon's gravity is diminished, and its orbit rendered less curve.

"It is found that the force added to the moon's gravity at the quadratures, is to the gravity with which it would revolve about the earth in a circle, as its present mean distance, if the sun had no effect on its motion, as 1 to 190; and that the force subducted from its gravity at the conjunction or opposition is about double this quantity. The influence of the sun, then, on the whole, increases the moon's distance from the earth, and augments its periodical time: and since this influence is most considerable when the earth is nearest the sun, or in its perihelion, its periodical time must then be the greatest;" and it is so found by observation. To show the effect of the sun in disturbing the moon's motion at any situation between the conjunction and one of the quadratures, suppose at M, let ES represent the earth's gravity towards the sun; draw the line MS, which continue towards G; from M set off MG, so that MG may be to ES as the square of the earth's distance ES is to the square of the moon's distance MS; and MG will represent the moon's gravity towards the sun. From M draw MF parallel and equal to ES; join FG, and draw MH parallel and equal to FG. The force MG may be resolved into MF and MH; of which MF, by reason of its parallelism and equality to ES, will not disturb the relative motions or situations of the moon and earth: MH then is the disturbing force. Draw the tangent MK to the moon's orbit, and continue the radius EM towards I; draw HI parallel to RM, and intersecting MI in I, and complete the parallelogram by drawing HK parallel to IM, and intersecting MK in K. The force MH may be resolved into MI and MK; of which MI affects the gravity, and MK the velocity, of the moon. When the force MH coincides with the tangent, that is, when the moon is $35^\circ 16'$ distant from the quadrature, the force MI, which affects the gravity, vanishes; and when the force MH coincides with the radius, that is, when the moon is either in the conjunction or quadrature, the force MK vanishes.

Between the quadrature and the distance of $35^\circ 16'$ from it, the line or force MH falls within the tangent, and consequently the force MI is directed towards E, and the moon's gravity is increased: but, at any greater distance from the quadrature, the line MH falls without the tangent, and the force MI is directed from E, the moon's gravity being diminished. It is evident that the force MK is always directed to some point in the line which passes through the sun and earth; therefore it will accelerate the moon's motion while it is approaching towards that line, or the conjunction, and similarly retard it as it recedes from it, or approaches towards the quadrature, by conspiring with the motion in one case, and subduing from it in the other.

As the moon's gravity towards the sun, at the conjunction, is diminished by a quantity which is as the difference of the squares of their distances; and as this difference, on account of the very great distance of the sun, is nearly the same when the moon is at the opposition, the mutual tendency to separate, or diminution of gravity, will be very nearly the same. Whence it easily follows, that all the irregularities which have been explained as happening between the quadratures and conjunction, must, in like circumstances, take place between the quadratures and opposition.

If the moon revolved about the earth in a circular orbit, the sun's disturbing influence being supposed not to act, then this influence being supposed to act would convert the orbit into an ellipse. For the increase of gravity renders it more curve at the quadratures, by causing the moon to fall further below the tangent; and the diminution of gravity, as well as the increased velocity, render the orbit less curve at the conjunction and opposition, by causing the moon to fall less below the tangent in a given time. Therefore, an ellipse would be described whose less or more convex parts would be at the quadratures, and whose longest diameter would pass through them; consequently the moon would be farthest from the earth at the quadratures, and nearest at the conjunction and opposition.

Neither is it strange that the moon should approach or come nearer to the earth at the time when its gravity is the least, since that approach is not the immediate consequence of the decrease of gravity, but of the curvature of its orbit near the quadratures; and in like manner, its recess from the earth does not arise immediately from its diminished gravity, from the velocity and direction acquired at the conjunction or opposition. But as the moon's orbit is, independent of the sun's action, an ellipse, then effects take place only as far as circumstances permit. The moon's gravity towards the earth being thus subject to a continual change in its ratio, its orbit is of no constant form. The law of its gravity being nearly in the inverse proportion of the squares of the distances, its orbit is nearly a quiescent ellipse; but the deviation from this law occasions its apsides to move direct or retrograde, according as those deviations are in defect or in excess. Astronomers, to reduce the motion of the apsides to computation, suppose the revolving body to move in an ellipse, whose transverse diameter, or line of the apsides, revolves at the same time about the focus of the orbit. When the moon is in the conjunction or opposition, the sun subducts from its gravity, and that the more the greater its distance is from the earth; so that its gravity follows a greater proportion than the inverted ratio of the square of the distance, and consequently the apsides of its orbit must then move in consequentia or direct. In the quadratures the sun adds to the moon's gravity; and that the more the greater its distance from the earth; so that its gravity follows a less proportion than the inverted ratio of the square of her distance, and consequently the apsides of its orbit must then move in antecedentia or retrograde. But because the action of the sun subducts more from the moon's gravity in the conjunction and opposition than it adds to it in the quadratures, the direct motion exceeds the retrograde, and at the end of each revolution the apsides are found to be advanced according to the order of the signs.

If the plane of the moon's orbit coincided with that of the ecliptic, there would be the only irregularities arising from the sun's action; but because it is inclined to the plane of the ecliptic in an angle of about 5 degrees, the whole disturbing force does not act upon the moon's motion in its orbit; a small part of the force being employed to draw it out of the plane of the orbit into that of the ecliptic.

Of the forces MK and MI (fig. 162.) which disturb the moon's motion, MI, being always in the direction of the radius, can have no effect in drawing it out of the plane of its orbit: And if the force MK really coincided with the tangent, as we, neglecting the small deviation arising from the obliquity of the moon's orbit, have hitherto supposed, it is evident that its only effect would be that of accelerating or retarding the moon's motion, without affecting the plane of its orbit: But because that force is always directed to some point in the line which passes through the centres of the sun and earth, it is evident that it can coincide with the tangent only when that line is in the plane of the moon's orbit; that is to say, when the nodes are in the conjunction and opposition. At all other times times the force MK must decline to the northward or southward of the tangent, and, compounding itself with the moon's motion, will not only accelerate or retard it, according to the circumstances before explained, but will likewise alter its direction, deflecting it towards that side of the orbit on which the point, the force MK, tends to be situated. This deflection causes the moon to arrive at the ecliptic either sooner or later than it would otherwise have done; or, in other words, it occasions the intersection of its orbit with the ecliptic to happen in a point of the ecliptic, either nearer to or farther from the moon, than that in which it would have happened if such deflection had not taken place.

"To illustrate this, let the elliptical projection COON (fig. 163.) represent a circle in the plane of the ecliptic, MOPN the moon's orbit intersecting the ecliptic in the nodes N and O. Suppose the moon to be in the northern part of its orbit at M, and moving towards the node O; the disturbing force MK, which tends towards a point in the line SE to the southward of the tangent MT, will be compounded with the tangential force, and will cause the moon to describe the arc MM', to which MR is tangent, instead of the arc MO; whence the node O is said to be moved to m. In this manner the motion of the nodes may be explained for any other situation.

"This motion evidently depends on a twofold circumstance; namely, the quantity and direction of the force MK. If the force MK be increased, its direction remaining the same, it will deflect the curve of the moon's path from its orbit in a greater degree; and, on the other hand, if its direction be altered so as to approach nearer to a right angle with the tangent, it will cause a greater deflection, though its quantity remain the same. When the moon is in the quadratures, the force MK vanishes, consequently the nodes are then stationary. When the moon is at the octant, or 45 degrees from the quadrature, the force MK is the greatest of all; and therefore the motion of the nodes is then most considerable, as far as it depends on the quantity of MK. But the direction of this force in like circumstances depends on the situation of the line of the nodes. If the line of the nodes coincides with the line passing through the centres of the sun and earth, the force MK coincides with the tangent of the moon's orbit, and the nodes are stationary; and the farther the node is removed from that line, the farther is that line removed from the plane of the moon's orbit, till the line of the nodes is in the quadratures; at which time the line passing through the centres of the sun and earth, makes an angle with the plane of the moon's orbit equal to its whole inclination, or 5 degrees; consequently, the angle formed between MK and the tangent, in like circumstances, is then greatest, MK being directed to a point in a line which is farther from the plane of the moon's orbit than at any other time, and of course the motion of the nodes is then most considerable.

"To determine the quantity and direction of the motion of the nodes, suppose the moon in the quarter preceding the conjunction, and the node towards which it is moving to be between it and the conjunction; in this case its motion is directed to a point in the ecliptic which is less distant than the point towards which the force MK is directed: the force MK then compounding with the moon's motion, causes it to be directed to a point more distant than it would otherwise have been; that is to say, the node towards which the moon moves is moved towards the conjunction. When the moon has passed the node, its course is directed to the other node, which is a point in the ecliptic more distant than the point to which MK is directed, and therefore MK compounding with its motion causes it to be directed to a point less distant than it would otherwise have been; so that, in this case likewise, the ensuing node is moved towards the conjunction. After the moon has passed the conjunction, the force MK still continues to deflect its course towards the ecliptic; and consequently the motion of the node is the same way till its arrival at the quadrature. Suppose, again, the moon to be at the conjunction, and the node towards which it is moving to be between it and the quadrature; in this case, the force MK compounding with the moon's motion, causes it to move towards a point in the ecliptic less distant than it would otherwise have done; so that the ensuing node is brought towards the conjunction.

"When the moon has passed the node, the force MK still continuing to deflect its course towards the same side of its orbit, produces a contrary effect; namely, as it before occasioned it to converge to the ecliptic, so it now causes it to diverge from it; and its motion, in consequence, tends continually to a point in the ecliptic more distant than it would otherwise have done; the ensuing node in this instance being also brought towards the conjunction.

"As the disturbing forces are very nearly the same in the half of the moon's orbit which is farthest from the sun, this last paragraph is true when it moves in that part of its orbit, if the word opposition be everywhere inserted instead of the word conjunction.

"Whence it is easy to deduce this general rule: That when the moon is in the part of its orbit nearest the sun, the node towards which it is moving is made to move towards the conjunction; and when it is in the part of its orbit farthest from the sun, the node towards which it is moving is made to move towards the opposition.

"Suppose the moon at Q (fig. 176.), or the quadrature preceding the conjunction, then the ensuing node, if at 90° distance, or at the conjunction C, will be stationary (as before observed); but if it be at a greater or less distance, it will be brought towards C. Thus, if the nodes be in the position MN, the ensuing node M, being at a less distance from Q than 90°, will move towards C, or direct, while the moon moves through the arc QM; after which N becomes the ensuing node, and likewise moves towards the conjunction C, or retrograde during the moon's motion through the arc MR; and because the arc MR exceeds QM, the retrograde motion exceeds the direct. Again, if the nodes be in the position nm, the ensuing node n being at a greater distance from Q than 90°, will move towards C, or retrograde during the moon's motion through the arc QZ; after which the node m becomes the ensuing node, and likewise moves towards the conjunction C, or direct, during the moon's motion through the arc nR; and because the arc Qn exceeds NR, the retrograde motion here also exceeds Particular the direct. If the nodes be in the quadratures Q, R, the ensuing node R removes towards C, or retrograde, during the moon's motion through the arc Q, R, or almost the whole semiorbit. The same may be shown in the other half of the orbit ROQ with respect to the opposition O; and therefore, in every revolution of the moon, the retrograde motion of the nodes exceeds the direct; and, on the whole, the nodes are carried round contrary to the order of the signs.

"The line of the conjunction is by the earth's annual motion brought into every possible situation with respect to the nodes in the course of a year, independent of their own proper motion; which last occasions the change of situation to be performed in about nineteen days less.

"The inclination of the moon's orbit being the angle which its course makes with the plane of the ecliptic, it is evident from what has been said, that this angle is almost continually changing. Suppose the line of the nodes, by its retrograde motion, to leave the conjunction C (fig. 175.) and become in the second and fourth quarters as in the position MN, and the moon to move from the node M to the node N; then, because the ensuing node N moves towards the conjunction C, while the moon is in the nearer half of its orbit, the moon's course must be continually more and more inflected towards the ecliptic till its arrival at R. This inflection in the first 90°, or MA from M, prevents its diverging so much from the ecliptic as it would otherwise have done; that is to say, it diminishes the angle of the moon's inclination. From A to R its course begins to converge towards the ecliptic; and this convergence is increased by the inflection which, in the preceding 90°, prevented its divergence; in the arc AR, then, their inclination is increased. During the moon's motion from R to N, the node is moved towards the opposition O, and consequently the angle of its course to N is rendered less than it would have been if the node had not moved; or, in other words, the inclination is diminished. And because the arc MA added to the arc RN is greater than the arc AR, the inclination at the subsequent node is less than at the precedent node; and the same may be shown in the other half revolution NQM.

"Therefore, while the nodes are moving from the conjunction and opposition to the quadratures, the inclination of the moon's orbit on the whole diminishes in every revolution till they arrive in the quadratures, at which time it is least of all. When the line of the nodes has passed the quadratures, and is in the first and third quarters, as in the position MR, it is easily shown by the same kind of argument, that the inclination is increased while the moon passes from M to Q, then diminishes for the remainder of the first 90°, or QO, and is afterwards increased for the other 90°, or OR; and the same may be proved for the other half revolution RMR. Consequently, while the nodes are moving from the quadratures to the conjunction and opposition, the inclination is increased by the same degrees as it before was diminished, till they arrive at the conjunction and opposition; at which time it returns to its first quantity, being then greatest of all.

"The line of the nodes in the course of one entire revolution with respect to the sun, is twice in the quadratures, and twice in the conjunction and opposition.

N° 84.

Therefore, the inclination of the moon's orbit to the particular ecliptic is diminished and increased by turns, twice in every revolution of the nodes.

"All the irregularities of the moon's motion are somewhat diminished when in the half of its orbit nearest the sun, than when it is in the other half; the chief reason of which is, that the difference between the squares of the moon's and earth's distances from the sun is greater in proportion to the squares themselves, in the former than in the latter case at equal elongations from the quadrature; and consequently the disturbing forces must be more considerable.

"Although the moon in reality revolves about the common centre of gravity between her and the earth, and not about the earth itself, and consequently their motions and irregularities are similar, and not confined to the moon alone; yet it may be easily conceived, that the conclusions are not affected in any degree that may be here regarded, when, for the sake of conciseness, we suppose one of the two bodies to be quiescent, and the other to revolve about it.

"Irregularities of the same kind take place among the primary planets, by their mutual actions on each other; but the quantities are not considerable. Hence the apsides of the planets are found to move in consequentia, but so very slowly that some have doubted whether they move at all. The motions of the aphelia of Saturn, Jupiter, Mars, the Earth, Venus, and Mercury, as deduced from the comparison of distant observations, are respectively 2° 30', 1° 45' 20", 1° 51' 40", 1° 49' 10", 4° 10', 1° 57' 40", in a century.

"The actions of the inferior planets on each other are very minute, on account of the smallness of their bulks; but those of Jupiter and Saturn are not altogether insensible. When Jupiter is between the Sun and Saturn, its whole attraction acts upon the latter, and increases the gravity of that planet towards the sun. This is found by comparing the respective masses of Jupiter and the Sun; and the respective squares of their distances from Saturn, to be equal to \( \frac{1}{2} \) of the Sun's action upon Saturn. That planet, on the other hand, at the conjunction, acts upon Jupiter and the Sun in the same direction; and therefore disturbs their relative position only so far as its actions on each are not equal. The difference of these actions is found, by the same principles, to be \( \frac{1}{2} \) of Jupiter's whole gravity."

Sect. VI. Of the Ebbing and Flowing of the Sea, and the Phenomena of the Harvest and Horizontal Moon.

The cause of the tides was discovered by Kepler, who, in his Introduction to the Physics of the Heavens, thus explains it: "The orb of the attracting power, which is in the moon, is extended as far as the earth; and draws the waters under the torrid zone, acting upon places where it is vertical, insensibly on confined seas and bays, but sensibly on the ocean, whose beds are large, and where the waters have the liberty of reciprocation, that is, of rising and falling." And in the 70th page of his Lunar Astronomy—"But the cause of the tides of the sea appears to be the bodies of the sun and moon drawing the waters of the sea." This hint being given, the immortal Sir Isaac Newton improved it, Ebbing and wrote so amply on the subject, as to make the theory of the tides in a manner quite his own, by discovering the cause of their rising on the side of the earth opposite to the moon. For Kepler believed that the presence of the moon occasioned an impulse which caused another in her absence.

It has been already observed, that the power of gravity diminishes as the square of the distance increases; and therefore the waters at Z on the side of the earth ABCDEFGH next the moon M, are more attracted than the central parts of the earth O by the moon, and the central parts are more attracted by her than the waters on the opposite side of the earth at N; and therefore the distance between the earth's centre and the waters on its surface under and opposite to the moon will be increased. For, let there be three bodies at H, O, and D: if they are all equally attracted by the body M, they will all move equally fast toward it, their mutual distances from each other continuing the same. If the attraction of M is unequal, then that body which is most strongly attracted will move fastest, and this will increase its distance from the other body. Therefore, by the law of gravitation, M will attract H more strongly than it does O, by which the distance between H and O will be increased; and a spectator on O will perceive H rising higher toward Z. In like manner, O being more strongly attracted than D, it will move farther towards M than D does; consequently, the distance between O and D will be increased; and a spectator on O, not perceiving his own motion, will see D receding farther from him towards N; all effects and appearances being the same, whether D recedes from O, or O from D.

Suppose now there is a number of bodies, as A, B, C, D, E, F, G, H, placed round O, so as to form a flexible or fluid ring; then, as the whole is attracted towards M, the parts at H and D will have their distance from O increased; whilst the parts at B and F being nearly at the same distance from M as O is, these parts will not recede from one another; but rather, by the oblique attraction of M, they will approach nearer to O. Hence, the fluid ring will form itself into an ellipse ZIBLmKFNZ, whose larger axis nOZ produced will pass through M, and its shorter axis BOF will terminate in B and F. Let the ring be filled with fluid particles, so as to form a sphere round O; then, as the whole moves towards M, the fluid sphere being lengthened at Z and n, will assume an oblong or oval form. If M is the moon, O the earth's centre, ABC DEFGH the sea covering the earth's surface, it is evident, by the above reasoning, that whilst the earth by its gravity falls toward the moon, the water directly below her at B will swell and rise gradually towards her; also the water at D will recede from the centre [strictly speaking, the centre recedes from D], and rise on the opposite side of the earth; whilst the water at B and F is depressed, and falls below the former level. Hence as the earth turns round its axis from the ebbing and flowing of the sea to the moon again in 24½ hours, there will be two tides of flood and two of ebb in that time, as we find by experience.

As this explanation of the ebbing and flowing of the sea is deduced from the earth's constantly falling towards the moon by the power of gravity, some may think it full find a difficulty in conceiving how this is possible, when the moon is full, or in opposition to the sun; since the earth revolves about the sun, and must continually fall towards it, and therefore cannot fall contrary ways at the same time: or if the earth is constantly falling towards the moon, they must come together at last. To remove this difficulty, let it be considered, that it is not the centre of the earth that describes the annual orbit round the sun, but the (§) common centre of gravity of the earth and moon together: and that whilst the earth is moving round the sun, it also describes a circle round that centre of gravity; going as many times round it in one revolution about the sun as there are lunations or courses of the moon round the earth in a year: and therefore the earth is constantly falling towards the moon from a tangent to the circle it describes round the said common centre of gravity. Let M be the moon, TW part of the moon's orbit, and C the centre of gravity of the earth and moon; whilst the moon goes round her orbit, the centre of the earth describes the circle d g e round C, to which circle g a k is a tangent; and therefore when the moon has gone from M to a little past W, the earth has moved from g to e; and in that time has fallen towards the moon, from the tangent at a to e: and so on, round the whole circle.

The sun's influence in raising the tides is but small in comparison of the moon's; for though the earth's diameter bears a considerable proportion to its distance from the moon, it is next to nothing when compared to its distance from the sun. And therefore the difference of the sun's attraction on the sides of the earth under and opposite to him, is much less than the difference of the moon's attraction on the sides of the earth under and opposite to her; and therefore the moon must raise the tides much higher than they can be raised by the sun.

On this theory, the tides ought to be highest directly under and opposite to the moon; that is, when the moon is due north and south. But we find, that in open seas, where the water flows freely, the moon is generally past the north and south meridian, as at p, the meridian when it is high water at Z and at n. The reason is plain: for though the moon's attraction was to cease altogether when she was past the meridian, yet the motion of ascent communicated to the water before that time would make it continue to rise for some time after; much more must it do so when the attraction is only diminished; as a little impulse given to a moving ball will cause it still to move farther than otherwise it could.

(§) This centre is as much nearer the earth's centre than the moon's as the earth is heavier, or contains a greater quantity of matter than the moon, namely, about 40 times. If both bodies were suspended on it, they would hang in equilibrium. So that dividing 240,000 miles, the moon's distance from the earth's centre, by 40, the excess of the earth's weight above the moon's, the quotient will be 6000 miles, which is the distance of the common centre of gravity of the earth and moon from the earth's centre. The tides answer not always to the same distance of the moon from the meridian at the same places; but are variously affected by the action of the sun, which brings them on sooner when the moon is in her first and third quarters, and keeps them back later when she is in her second and fourth: because, in the former case, the tide raised by the sun alone would be earlier than the tide raised by the moon; and, in the latter case, later.

The moon goes round the earth in an elliptic orbit; and therefore, in every lunar month, she approaches nearer to the earth than her mean distance, and recedes farther from it. When she is nearest, she attracts strongest, and so raises the tides most; the contrary happens when she is farthest, because of her weaker attraction. When both luminaries are in the equator, and the moon in perigee, or at her least distance from the earth, she raises the tides highest of all, especially at her conjunction and opposition; both because the equatorial parts have the greatest centrifugal force from their describing the largest circle, and from the concurring actions of the sun and moon. At the change, the attractive forces of the sun and moon being united, they diminish the gravity of the waters under the moon, and their gravity on the opposite side is diminished by means of a greater centrifugal force. At the full, whilst the moon raises the tide under and opposite to her, the sun, acting in the same line, raises the tide under and opposite to him; whence their conjoint effect is the same as at the change; and, in both cases, occasion what we call the Spring Tides. But at the quarters the sun's action on the waters at O and H diminishes the effect of the moon's action on the waters at Z and N; so that they rise a little under and opposite to the sun at O and H, and fall as much under and opposite to the moon at Z and N; making what we call the Neap Tides, because the sun and moon then act cross-wise to each other. But these tides happen not till some time after; because in this, as in other cases, the actions do not produce the greatest effect when they are at the strongest, but some time afterward.

The sun, being nearer the earth in winter than in summer, is of course nearer to it in February and October than in March and September; and therefore the greatest tides happen not till some time after the autumnal equinox, and return a little before the vernal.

The sea, being thus put in motion, would continue to ebb and flow for several times, even though the sun and moon were annihilated, or their influence should cease; as, if a basin of water were agitated, the water would continue to move for some time after the basin was left to stand still; or like a pendulum, which, having been put in motion by the hand, continues to make several vibrations without any new impulse.

When the moon is in the equator, the tides are equally high in both parts of the lunar day, or time of the moon's revolving from the meridian to the meridian again, which is 24 hours 50 minutes. But as the moon declines from the equator towards either pole, the tides are alternately higher and lower at places having north or south latitude. For one of the highest elevations, which is that under the moon, follows her towards the pole to which she is nearest, and the other declines towards the opposite pole; each elevation describing parallels as far distant from the equator, on opposite sides, as the moon declines from it to either side; and consequently the parallels described by these elevations of the water are twice as many degrees from one another as the moon is from the equator; increasing their distance as the moon increases her declination, till it be at the greatest, when the said parallels are, at a mean state, 47 degrees from one another: and on that day, the tides are most unequal in their heights. As the moon returns towards the equator, the parallels described by the opposite elevations approach towards each other, until the moon comes to the equator, and then they coincide. As the moon declines towards the opposite pole, at equal distances, each elevation describes the same parallel in the other part of the lunar day, which its opposite elevation described before. Whilst the moon has north declination, the greatest tides in the northern hemisphere are when she is above the horizon; and the reverse whilst her declination is south. Let NESQ be the earth, NSC its axis, EQ Fig. 192, the equator, T the tropic of Cancer, t the tropic of Capricorn, a b the arctic circle, c d the antarctic, N the north pole, S the south pole, M the moon, F and G the two eminences of water, whose lowest parts are, at a and d, at N and S, and at b and c, always 90 degrees from the highest. Now, when the moon is in her greatest north declination at M, the highest elevation G under her is on the tropic of Cancer T, and the opposite elevation F on the tropic of Capricorn t; and these two elevations describe the tropics by the earth's diurnal rotation. All places in the northern hemisphere ENQ have the highest tides when they come into the position b Q, under the moon; and the lowest tides when the earth's diurnal rotation carries them into the position a TE, on the side opposite to the moon; the reverse happens at the same time in the southern hemisphere ESQ, as is evident to sight. The axis of the tides a C d had now its poles a and d (being always 90 degrees from the highest elevations) in the arctic and antarctic circles; and therefore it is plain, that at these circles there is but one tide of flood, and one of ebb, in the lunar day. For, when the point a revolves half round to b in 12 lunar hours, it has a tide of flood; but when it comes to the same point a again in 12 hours more, it has the lowest ebb. In seven days afterward, the moon M comes to the equinoctial circle, and is over the equator EQ, when both elevations describe the equator; and in both hemispheres, at equal distances from the equator, the tides are equally high in both parts of the lunar day. The whole phenomena being reversed, when the moon has south declination, to what they were when her declination was north, require no farther description.

In the three last-mentioned figures, the earth is orthographically projected on the plane of the meridian; but in order to describe a particular phenomenon, we now project it on the plane of the ecliptic. Let HZON be the earth and sea, FED the equator, T Fig. 192, the tropic of Cancer, C the arctic circle, P the north pole, and the curves, 1, 2, 3, &c. 24 meridians or hour- Astronomy

AGM is the moon's orbit, S the sun, M the moon, Z the water elevated under the moon, and N the opposite equal elevation. As the lowest parts of the water are always 90 degrees from the highest, when the moon is in either of the tropics (as at M), the elevation Z is on the tropic of Capricorn, and the opposite elevation N on the tropic of Cancer; the low-water circle HCO touches the polar circles at C; and the high-water circle ETPG goes over the poles at P, and divides every parallel of latitude into two equal segments. In this case, the tides upon every parallel are alternately higher and lower; but they return in equal times: the point T, for example, on the tropic of Cancer, (where the depth of the tide is represented by the breadth of the dark shade), has a shallower tide of flood at T than when it revolves half round from thence to 6, according to the order of the numeral figures; but it revolves as soon from 6 to T as it did from T to 6. When the moon is in the equinoctial, the elevations Z and N are transferred to the equator at O and H, and the high and low-water circles are got into each other's former places; in which case the tides return in unequal times, but are equally high in both parts of the lunar day; for a place at 1 (under D) revolving as formerly, goes sooner from 1 to 11 (under F) than from 11 to 1, because the parallel it describes is cut into unequal segments by the high-water circle HCO; but the points 1 and 11 being equidistant from the pole of the tides at C, which is directly under the pole of the moon's orbit MGA, the elevations are equally high in both parts of the day.

And thus it appears, that as the tides are governed by the moon, they must turn on the axis of the moon's orbit, which is inclined 23° 27' degrees to the earth's axis at a mean state; and therefore the poles of the tides must be so many degrees from the poles of the earth, or in opposite points of the polar circles, going round these circles in every lunar day. It is true, that according to fig. 4, when the moon is vertical to the equator ECQ, the poles of the tides seem to fall in with the poles of the world N and S; but when we consider that FGH is under the moon's orbit, it will appear, that when the moon is over H, in the tropic of Capricorn, the north pole of the tides (which can be no more than 90 degrees from under the moon) must be at C in the arctic circle, not at P the north pole of the earth; and as the moon ascends from H to G in her orbit, the north pole of the tides must shift from c to a in the arctic circle, and the south poles as much in the antarctic.

It is not to be doubted, but that the earth's quick rotation brings the poles of the tides nearer to the poles of the world, than they would be if the earth were at rest, and the moon revolved about it only once a month; for otherwise the tides would be more unequal in their heights and times of their returns, than we find they are. But how near the earth's rotation may bring the poles of its axis and those of the tides together, or how far the preceding tides may affect those which follow, so as to make them keep up nearly to the same heights and times of ebbing and flowing, is a problem more fit to be solved by observation than by theory.

Those who have opportunity to make observations, and choose to satisfy themselves whether the tides are really affected in the above manner by the different positions of the moon, especially as to the unequal times of their returns, may take this general rule for knowing when they ought to be so affected. When the earth's axis inclines to the moon, the northern tides, if not retarded in their passage through shoals and channels, nor affected by the winds, ought to be greatest when the moon is above the horizon, least when she is below it, and quite the reverse when the earth's axis declines from her; but in both cases, at equal intervals of time. When the earth's axis inclines sidewise to the moon, both tides are equally high, but they happen at unequal intervals of time. In every lunation the earth's axis inclines once to the moon, once from her, and twice sidewise to her, as it does to the sun every year; because the moon goes round the ecliptic every month, and the sun but once in a year. In summer, the earth's axis inclines towards the moon when new; and therefore the day-tides in the north ought to be highest, and night-tides lowest, about the change: at the full, the reverse. At the quarters, they ought to be equally high, but unequal in their returns: because the earth's axis then inclines sidewise to the moon. In winter, the phenomena are the same at full moon as in summer at new. In autumn the earth's axis inclines sidewise to the moon when new and full; therefore the tides ought to be equally high and uneven in their returns at these times. At the first quarter, the tides of flood should be least when the moon is above the horizon, greatest when she is below it; and the reverse at her third quarter. In spring, the phenomena of the first quarter answer to those of the third quarter in autumn; and vice versa. The nearer any tide is to either of the seasons, the more the tides partake of the phenomena of these seasons; and in the middle between any two of them the tides are at a mean state between those of both.

In open seas, the tides rise but to very small heights in proportion to what they do in wide-mouthed rivers, opening in the direction of the stream of tide. For in channels growing narrower gradually, the water is accumulated by the opposition of the contracting bank; like a gentle wind, little felt on an open plain, but strong and brisk in a street; especially if the wider end of the street be next the plain, and in the way of the wind.

The tides are so retarded in their passage thro' different shoals and channels, and otherwise so variously affected by striking against capes and headlands, that to different places they happen at all distances of the moon from the meridian, consequently at all hours of the lunar day. The tide propagated by the moon in the German ocean, when she is three hours past the meridian, takes 12 hours to come from thence to London bridge, where it arrives by the time that a new tide is raised in the ocean. And therefore, when the moon has north declination, and we should expect the tide at London to be greatest when the moon is above the horizon, we find it is least; and the contrary when she has south declination. At several places it is high-water three hours before the moon comes to the meridian; but that tide which the moon pushes as it were before her, is only the tide opposite to that which was raised by her when she was nine hours past the opposite meridian.

There are no tides in lakes, because they are generally... Astronomy.

It is generally believed that the moon rises about 50 minutes later every day than on the preceding; but this is true only with regard to places on the equator. In places of considerable latitude there is a remarkable difference, especially in the harvest time. Here the autumnal full moon rises very soon after sun-set for several evenings together. At the polar circles, where the mild season is of very short duration, the autumnal full moon rises at sun-set from the first to the third quarter. And at the poles, where the sun is for half a year absent, the winter full-moons shine constantly without setting from the first to the third quarter.

All these phenomena are owing to the different angles made by the horizon and different parts of the moon's orbit; and may be explained in the following manner.

The plane of the equinoctial is perpendicular to the earth's axis; and therefore as the earth turns round its axis, all parts of the equinoctial make equal angles with the horizon both at rising and setting; so that equal portions of it always rise or set in equal times. Consequently, if the moon's motion were equable, and in the equinoctial, at the rate of 12 degrees 11 min. from the sun every day, as it is in her orbit, she would rise and set 50 minutes later every day than on the preceding: for 12 deg. 11 min. of the equinoctial rise or set in 50 minutes of time in all latitudes.

But the moon's motion is so nearly in the ecliptic, that we may consider her at present as moving in it. Now the different parts of the ecliptic, on account of its obliquity to the earth's axis, make very different angles with the horizon as they rise or set. Those flowing of parts or signs which rise with the smallest angles set with the greatest, and vice versa. In equal times, whenever this angle is least, a greater portion of the ecliptic rises than when the angle is larger; as may be seen by elevating the pole of a globe to any considerable latitude, and then turning it round its axis in the horizon. Consequently, when the moon is in those signs which rise or set with the smallest angles, she rises or sets with the least difference of time; and with the greatest difference in those signs which rise or set with the greatest angles.

Let FUP be the axis of a globe, TR the tropic of Cancer, LT the tropic of Capricorn, EU the ecliptic touching both the tropics, which are 47 degrees from each other, and AB the horizon. The equator, being in the middle between the tropics, is cut by the ecliptic in two opposite points, which are the beginnings of Aries and Libra; K is the hour-circle with its index, F the north pole of the globe elevated to a considerable latitude, suppose 40 degrees above the horizon; and P the south pole depressed as much below it. Because of the oblique position of the sphere in this latitude, the ecliptic has the high elevation N above the horizon, making the angle NU of 73½ degrees with it when Cancer is on the meridian, at which time Libra rises in the east. But let the globe be turned half round its axis, till Capricorn comes to the meridian and Aries rises in the east; and then the ecliptic will have the low elevation NL above the horizon, making only an angle NUL of 26½ degrees with it; which is 47 degrees less than the former angle, equal to the distance between the tropics.

In northern latitudes, the smallest angle made by the ecliptic and horizon is when Aries rises, at which time Libra sets; the greatest when Libra rises, at which time Aries sets. From the rising of Aries to the rising of Libra (which is twelve (a) sidereal hours) the angle increases; and from the rising of Libra to the rising of Aries, it decreases in the same proportion. By this article and the preceding, it appears, that the ecliptic rises fastest about Aries, and slowest about Libra.

On the parallel of London, as much of the ecliptic rises about Pisces and Aries in two hours as the moon goes through in six days: and therefore, whilst the moon is in these signs, she differs but two hours in rising for six days together; that is, about 20 minutes later every day or night than on the preceding, at a mean rate. But in 14 days afterwards, the moon comes to Virgo and Libra, which are the opposite signs to Pisces and Aries; and then she differs almost four times as much in rising; namely, one hour and about fifteen minutes later every day or night than the former, whilst she is in these signs.

As the moon can never be full but when she is opposite to the sun, and the sun is never in Virgo and Libra but in our autumnal months, it is plain that the moon

(a) The ecliptic, together with the fixed stars, make 366½ apparent diurnal revolutions about the earth in a year; the sun only 365¼. Therefore the stars gain 3 minutes 56 seconds upon the sun every day: so that a sidereal day contains only 23 hours 56 minutes of mean solar time, and a natural or solar day 24 hours. Hence 12 sidereal hours are 11 minutes 58 seconds shorter than 12 solar. moon is never full in the opposite signs, Pisces and Aries, but in these two months. And therefore we can have only two full moons in the year, which rise so near the time of sun-set for a week together, as abovementioned. The former of these is called the harvest moon, and the latter the hunter's moon.

Here it will probably be asked, why we never observe this remarkable rising of the moon but in harvest, seeing she is in Pisces and Aries twelve times in the year besides, and must then rise with as little difference of time as in harvest? The answer is plain: for in winter these signs rise at noon; and being then only a quarter of a circle distant from the sun, the moon in them is in her first quarter: but when the sun is above the horizon, the moon's rising is neither regarded nor perceived. In spring, these signs rise with the sun, because he is then in them; and as the moon changeth in them at that time of the year, she is quite invisible. In summer they rise about midnight; and the sun being then three signs, or a quarter of a circle, before them, the moon is in them about her third quarter; when rising so late, and giving but very little light, her rising passes unobserved. And in autumn, these signs, being opposite to the sun, rise when he sets, with the moon in opposition, or at the full, which makes her rising very conspicuous.

At the equator, the north and south poles lie in the horizon; and therefore the ecliptic makes the same angle southward with the horizon when Aries rises, as it does northward when Libra rises. Consequently, as the moon rises and sets nearly at equal angles with the horizon all the year round, and about 50 minutes later every day or night than on the preceding, there can be no particular harvest-moon at the equator.

The farther that any place is from the equator, if it be not beyond the polar circle, the more the angle is diminished which the ecliptic and horizon make when Pisces and Aries rise; and therefore when the moon is in these signs, she rises with a nearly proportionable difference later every day than on the former; and for that reason the more remarkable about the full, until we come to the polar circles, or 66 degrees from the equator; in which latitude the ecliptic and horizon become coincident every day for a moment, at the same sidereal hour (or 3 minutes 56 seconds sooner every day than the former), and the very next moment one half of the ecliptic containing Capricorn, Aquarius, Pisces, Aries, Taurus, and Gemini, rises, and the opposite half sets. Therefore, whilst the moon is going from the beginning of Capricorn to the beginning of Cancer, which is almost 14 days, she rises at the same sidereal hour; and in autumn just at sun-set, because all that half of the ecliptic, in which the sun is at that time, sets at the same sidereal hour, and the opposite half rises; that is, 3 minutes 56 seconds of mean solar time, sooner every day than on the day before. So, whilst the moon is going from Capricorn to Cancer, she rises earlier every day than on the preceding; contrary to what she does at all places between the polar circles. But during the above 14 days, the moon is 24 sidereal hours later in setting: for the six signs which rise all at once on the eastern side of the horizon are 24 hours in setting on the western side of it.

In northern latitudes, the autumnal full moons are in Pisces and Aries, and the vernal full moons in Virgo Harvast and Libra; in southern latitudes, just the reverse, because the seasons are contrary. But Virgo and Libra moon rise at small angles with the horizon in southern latitudes, as Pisces and Aries do in the northern; and therefore the harvest-moons are just as regular on one side of the equator as on the other.

As these signs, which rise with the least angles, set with the greatest, the vernal full moons differ as much in their times of rising every night as the autumnal full moons differ in their times of setting; and set with as little difference as the autumnal full moons rise; the one being in all cases the reverse of the other.

Hitherto, for the sake of plainness, we have supposed the moon to move in the ecliptic, from which the sun never deviates. But the orbit in which the moon really moves is different from the ecliptic; one half being elevated 5° 30' degrees above it, and the other half as much depressed below it. The moon's orbit therefore intersects the ecliptic in two points diametrically opposite to each other; and these intersections are called the Moon's Nodes. So the moon can never be in the ecliptic but when she is in either of her nodes, which is at least twice in every course from change to change, and sometimes thrice: For, as the moon goes almost a whole sign more than around her orbit from change to change; if she passes by either node about the time of change, she will pass by the other in about 14 days after, and come round to the former node two days again before the next change. That node from which the moon begins to ascend northward, or above the ecliptic, in northern latitudes, is called the Ascending Node; and the other the Descending Node, because the moon, when she passes by it, descends below the ecliptic southward.

The moon's oblique motion with regard to the ecliptic, causes some difference in the times of her rising and setting from what is already mentioned. For when she is northward of the ecliptic, she rises sooner and sets later than if she moved in the ecliptic: and when she is southward of the ecliptic, she rises later and sets sooner. This difference is variable, even in the same signs, because the nodes shift backward about 19° 30' degrees in the ecliptic every year; and so go round it contrary to the order of signs in 18 years 225 days.

When the ascending node is in Aries, the southern half of the moon's orbit makes an angle of 5° 30' degrees less with the horizon than the ecliptic does, when Aries rises in northern latitudes: for which reason the moon rises with less difference of time whilst she is in Pisces and Aries, than she would do if she kept in the ecliptic. But in 9 years and 112 days afterward, the descending node comes to Aries; and then the moon's orbit makes an angle 5° 30' degrees greater with the horizon when Aries rises, than the ecliptic does at that time; which causes the moon to rise with greater difference of time in Pisces and Aries than if she moved in the ecliptic.

To be a little more particular: When the ascending node is in Aries, the angle is only 9° 30' degrees on the parallel of London when Aries rises; but when the descending node comes to Aries, the angle is 20° 30' degrees. This occasions as great a difference of the moon's moon's rising in the same signs every nine years, as there would be on two parallels 10° degrees from one another, if the moon's course were in the ecliptic.

As there is a complete revolution of the nodes in 18° years, there must be a regular period of all the varieties which can happen in the rising and setting of the moon during that time. But this shifting of the nodes never affects the moon's rising so much, even in her quickest descending latitude, as not to allow us still the benefit of her rising nearer the time of full moon for a few days together about the full in harvest, than when the sun is full at any other time of the year.

At the polar circles, when the sun touches the summer-light meridian, he continues 24 hours above the horizon; in winter at 24 hours below it when he touches the winter tropic. For the same reason, the full moon neither rises in summer nor sets in winter, considering her as moving in the ecliptic. For the winter full moon being as high in the ecliptic as the summer full, must therefore continue as long above the horizon; and the summer full moon being as low in the ecliptic as the winter sun, can no more rise than he does. But these are only the two full moons which happen about the tropics, for all the others rise and set. In summer, the full moons are low, and their stay is short above the horizon, when the nights are short, and we have least occasion for moon-light; in winter they go high, and stay long above the horizon, when the nights are long, and we want the greatest quantity of moon-light.

At the poles, one half of the ecliptic never sets, and the other half never rises; and therefore, as the sun is always half a year in describing one half of the ecliptic, and as long in going through the other half, it is natural to imagine that the sun continues half a year together above the horizon of each pole in its turn, and as long below it; rising to one pole when he sets to the other. This would be exactly the case if there were no refraction; but by the atmosphere's refracting the sun's rays, he becomes visible some days sooner, and continues some days longer in sight, than he would otherwise do; so that he appears above the horizon of either pole before he has got below the horizon of the other. And, as he never goes more than 23° degrees below the horizon of the poles, they have very little dark night; it being twilight there as well as at other places, till the sun be 18 degrees below the horizon. The full moon, being always opposite to the sun, can never be seen while the sun is above the horizon, except when she is in the northern half of her orbit; for whenever any point of the ecliptic rises, the opposite point sets. Therefore, as the sun is above the horizon of the north pole from the 20th of March till 23d of September, it is plain that the moon, when full, being opposite to the sun, must be below the horizon during that half of the year. But when the sun is in the southern half of the ecliptic, he never rises to the north pole; during which half of the year, every full moon happens in some part of the northern half of the ecliptic which never sets. Consequently, as the polar inhabitants never see the full moon in summer, they have her always in the winter, before, at, and after, the full, shining for 14 of our days and nights. And when the sun is at his greatest depression below the horizon, being then in Capricorn, the moon is at her third quarter in Aries, full in Cancer, and at her first quarter in Libra. And as the beginning of Aries is the rising point and the setting point, the moon rises at her first quarter in Aries; is most elevated above the horizon, and full, in Cancer; and sets, at the beginning of Libra, in her third quarter, having continued visible for 14 diurnal rotations of the earth. Thus the poles are supplied one half of the winter-time with constant moon-light in the sun's absence; and only lose sight of the moon from her third to her first quarter, while she gives but very little light and could be but of little and sometimes of no service to them. A bare view of the figure will make this plain: in which let S be the sun; E, the earth in summer, when its north pole n inclines toward the sun; and E the earth in winter, when its north pole declines from him. SEN and NWS is the horizon of the north pole, which is coincident with the equator; and, in both these positions of the earth, VP is the moon's orbit, in which she goes round the earth, according to the order of the letters a b c d, A B C D. When the moon is at a, she is in her third quarter to the earth at e, and just rising to the north pole n; at b she changes, and is at the greatest height above the horizon, as the sun likewise is; at c she is in her first quarter, setting below the horizon; and is lowest of all under it at d, when opposite to the sun, and her enlightened side toward the earth. But then she is full in view to the fourth pole p; which is as much turned from the sun as the north pole inclines toward him. Thus, in our summer, the moon is above the horizon of the north pole whilst she describes the northern half of the ecliptic VP, or from her third quarter to her first; and below the horizon during her progress through the southern half VP; highest at the change, most depressed at the full. But in winter, when the earth is at E, and its north pole declines from the sun, the new moon at D is at her greatest depression below the horizon NWS, and the full moon at B at her greatest height above it; rising at her first quarter A, and keeping above the horizon till she comes to her third quarter C. At a mean state she is 23° degrees above the horizon at B and b, and as much below it at D and d, equal to the inclination of the earth's axis F. S VP, or S VP, are, as it were, a ray of light proceeding from the sun to the earth; and shows that when the earth is at e, the sun is above the horizon, vertical to the tropic of Cancer; and when the earth is at E, he is below the horizon, vertical to the tropic of Capricorn.

The sun and moon generally appear larger when near the horizon than when at a distance from it; for which there have been various reasons assigned. The following account is given by Mr. Ferguson: "These luminaries, although at great distances from the earth, appear floating as it were on the surface of our atmosphere, HGF&C, a little way beyond the clouds; of which, those about F, directly over our heads at E, are nearer us than those about H or C in the horizon HEc. Therefore, when the sun or moon appear in the horizon at e, they are not only seen in a part of the sky which is really farther from us than if they were at any considerable altitude, as about f; but they are also seen through a greater quantity of air and vapors at c than at f. Here we have two concurring appearances." ances which deceive our imagination, and cause us to refer the sun and moon to a greater distance at their rising or setting about c, than when they are considerably high, as at f; first, their seeming to be on a part of the atmosphere at c, which is really farther than f from a spectator at E; and, secondly, their being seen through a groser medium when at c than when at f, which, by rendering them dimmer, causes us to imagine them to be at a yet greater distance. And as, in both cases, they are seen much under the same angle, we naturally judge them to be largest when they seem farthest from us.

"Any one may satisfy himself that the moon appears under no greater angle in the horizon than on the meridian, by taking a large sheet of paper, and rolling it up in the form of a tube, of such a width, that, observing the moon through it when she rises, she may as it were just fill the tube: then tie a thread round it to keep it of that size; and when the moon comes to the meridian, and appears much less to the eye, look at her again through the same tube, and she will fill it just as much, if not more, than she did at her rising.

"When the full moon is in her perigee, or at her least distance from the earth, she is seen under a larger angle, and must therefore appear bigger than when she is full at other times: And if that part of the atmosphere where she rises be more replete with vapours than usual, she appears so much the dimmer; and therefore we fancy her to be still the bigger, by referring her to an unusually great distance, knowing that no objects which are very far distant can appear big unless they really be so."

To others this solution has appeared unsatisfactory; and accordingly Mr Dunn has given the following dissertation on this phenomenon, Phil. Trans. Vol. LXIV.

"1. The sun and moon when they are in or near the horizon, appear to the naked eye of the generality of persons, so very large in comparison with their apparent magnitudes when they are in the zenith, or somewhat elevated, that several learned men have been led to inquire into the cause of this phenomenon; and after endeavouring to find certain reasons, founded on the principles of physics, they have at last pronounced this phenomenon a mere optical illusion.

"2. The principal dissertations which I have seen conducing to give any information on this subject, or helping to throw any light on the same, have been those printed in the Transactions of the Royal Society, the Academy of Sciences at Paris, the German Acts, and Dr Smith's Optics; but as all the accounts which I have met with in these writings any way relative to this subject, have not given me that satisfaction which I have desired, curiosity has induced me to inquire after the cause of this singular phenomenon in a manner somewhat different from that which others have done before me, and by such experiments and observations as have appeared to me pertinent; some of which have been as follows, viz.

"3. I have observed the rising and setting sun near the visible horizon, and near rising grounds elevated above the visible horizon about half a degree, and found him to appear largest when near to the visible horizon; and particularly a considerable alteration of his magnitude and light has always appeared to me from the time of his being in the horizon at rising, to the time of his being a degree or two above the horizon, and Harvest the contrary at his setting; which property I have and Hori-endeavoured to receive as a prejudice, and an imposi- zontal Moon, on my sight and judgment, the usual reasons for this appearance.

"4. I have also observed that the sun near the horizon appears to put on the figure of a spheroid, having its vertical diameter appearing to the naked eye shorter than the horizontal diameter; and, by measuring those diameters in a telescope, have found the vertical one shorter than the other.

"5. I have made frequent observations and comparisons of the apparent magnitude of the sun's disk, with objects directly under him, when he has been near the horizon, and with such objects as I have found by measurement to be of equal breadth with the sun's diameter; but in the sudden transition of the eye from the sun to the object, and from the object to the sun, have always found the sun to appear least; and that when two right lines have been imaginarily produced by the sides of those equal magnitudes, they have not appeared to keep parallel, but to meet beyond the sun.

"6. From these and other like circumstances, I first began to suspect that a sudden dip of the sun into the horizontal vapours, might somehow or other be the cause of a sudden apparent change of magnitude, although the horizontal vapours had been disallowed to be able to produce any other than a refraction in a vertical direction; and, reducing things to calculation, found, that from the time when the sun is within a diameter or two of the horizon, to the time when he is a semidiameter below the horizon, the sun's rays become passable through such a length of medium, reckoning in the direction of the rays, that the total quantity of medium (reckoning both depth and density) through which the rays pass, being compared with the like total depth and density through which they pass at several elevations, it was proportionable to the difference of apparent magnitude, as appearing to the naked eye.

"7. This circumstance of sudden increase and decrease of apparent magnitude, and as sudden decrease and increase of light (for they both go together), seemed to me no improbable cause of the phenomenon, although I could not then perceive how such vapours might contribute toward enlarging the diameter of the sun in a horizontal direction.

"8. I therefore examined the sun's disk again and again, by the naked eye and by telescopes, at different altitudes; and, among several circumstances, found the solar maculae to appear larger and plainer to the naked eye, and through a telescope, the sun being near the horizon, than they had appeared the same days when the sun was on the meridian, and to appearance more strongly defined, yet obscured.

"9. A little before sun-setting, I have often seen the edge of the sun with such protuberances and indentures as have rendered him in appearance a very odd figure; the protuberances shooting out far beyond, and the indentures pressing into the disk of the sun; and always, through a telescope magnifying 55 times, the lower limb has appeared with a red glowing arch beneath it, and close to the edge of the sun, while the other parts have been clear.

"To At "10. At sun-setting, these protuberances and indentures have appeared to slide along the vertical limbs, from the lower limb to the higher, and there vanishing, so as often to form a segment of the sun's upper limb, apparently separated from the disk for a small space of time.

"11. At sun-rising I have seen the like protuberances, indentures, and slices, above described; but with this difference of motion, that at sun-rising they first appear to rise in the sun's upper limb, and slide or move downward to the lower limb; or, which is the same thing, they always appear at the rising and setting of the sun, to keep in the same parallels of altitude by the telescope. This property has been many times so discernible, even by the naked eye, that I have observed the sun's upper limb to shoot out towards right and left, and move downwards, forming the upper part of the disk an apparent portion of a lesser spheroid than the lower part at rising, and the contrary at setting. Through the telescope this has appeared more plain in proportion to the power of magnifying.

"12. These protuberances and indentures so easily measurable by the micrometer, whilst the telescope wires appeared straight, enabled me to conclude, that certain strata of the atmosphere have different refractive powers; and, lying horizontally across the conical or cycloidal space traced out by the rays between the eye and that part of the atmosphere first touched by the rays, must have been the cause of such apparent protuberances and indentures in a horizontal direction across the sun's vertical limbs; and also that the bottoms of those protuberances and indentures must be considerably enlarged, and removed to appearance farther from the centre of the disk than they would have been had there been no such strata to refract.

"13. Before sun-rising, when the sun has been near the tropic; and the sky, at the utmost extent of the horizon, hath appeared very clear; and when certain fogs have appeared in strata placed alternately between the hills, and over intervening rivers, valleys, &c. so as to admit a sight of the rising sun over those fogs; I have observed with admiration the most distant trees and bushes, which at other times have appeared small to the naked eye, but while the sun has been passing along a little beneath the horizon obliquely under them, just before sun-rising, when the sun has been thus approaching towards trees and bushes, they have grown apparently very large to the naked eye, and also through a telescope; and they have lost that apparent largeness as the sun has been passed by them. Thus a few trees standing together on the rising ground, at the distance of a few miles, have appeared to grow up into an apparent mountain. Such apparent mountains formed from trees put on all forms and shapes, as sloping, perpendicular, over-leaning, &c. but soon recover their natural appearance when the sun is passed by them, or got above the horizon.

"14. Mountains themselves, at a distance, sometimes appear larger than at other times. Beasts and cattle in the midst of, and being surrounded with, water, appear nearer to us than when no water surrounds them. Cattle, houses, trees, all objects on the summit of a hill, when seen through a fog, and at a proper distance, appear enlarged. All bodies admit of larger apparent magnitudes when seen through some mediums than others.

"But more particularly,

"15. I took a cylindrical glass-vessel about two feet high; and having graduated its sides to inches, I placed it upright on a table, with a piece of paper under the bottom of the glass, on which paper were drawn parallel right lines at proper distances from each other; and having placed a shilling at the bottom of the vessel, it was nearly as low as the paper. Pouring water into the vessel, and viewing the shilling through the medium of water with one eye, whilst I beheld with the other eye where the edges of the shilling were projected on the paper and its parallels, I found the shilling appear larger at every additional inch depth of the water; and this was the case if either eye was used; and the same when the eye was removed far from the surface or near to it, or in any proportion thereto.

"16. I took large vessels; and, filling them with water, placed different bodies at the bottoms of those vessels. It always followed, that the greater depth of water I looked through, in the direction from my eye to the objects in the water, the nearer those objects appeared to me. Thus light bodies appeared more mellow and faint, and dark bodies rather better defined, than out of the water, when they were not deeply immersed. And thus they appeared under whatever directions or positions I viewed the bodies.

"17. I placed different bodies in proper vessels of fair water, and immersed my face in the water; viewing the bodies in and through the water. They all appeared to me plain, when not too far from the eye; and although a little hazy at the edges, they appeared much enlarged, and always larger through a greater depth of water. Thus a shilling appeared nearly as large as half a crown, with a red glowing arch on that side opposite to the sun, when the sun shined on the water. From this experiment I concluded, that divers see light objects not only larger, but very distinctly, in the water."

From these experiments he draws a confirmation of his doctrine, that the appearances treated of arise from the different strata of the atmosphere; and then concludes, that the rays coming from the sun are by the horizontal vapours "first obstructed, and many of them totally absorbed; the rest proceeding with a retarded motion, are thereby first reflected, and then let refracted through the humours of the eye; and, lastly, that hereby the image on the retina becomes enlarged."

Sect. VII. Of drawing a Meridian Line. Of Solar and Sidereal Time, and of the Equation of Time.

The foundation of all astronomical observations is a knowledge of the exact time when the sun, or any other of the celestial bodies, comes to the meridian; and therefore astronomers have been very attentive to the most proper methods of drawing a meridian line, by which only this can be exactly known. The easiest method of doing this is the following, recommended by Mr Fergufson, and is found a very good method of placing a sun-dial horizontally on its pedestal.

Make four or five concentric circles (fig. 5.), about a quarter of an inch from one another, on a flat board about a foot in breadth, and let the outmost circle be but little line. little less than the board will contain. Fix a pin perpendicularly in the centre, and of such a length that its whole shadow may fall within the innermost circle for at least four hours in the middle of the day. The pin ought to be about the eighth part of an inch thick, and to have a round blunt point. The board being set exactly level in a place where the sun shines, suppose from eight in the morning till four in the afternoon, about which hours the end of the shadow should fall without all the circles; watch the times in the forenoon when the extremity of the shortening shadow just touches the several circles, and there make marks. Then, in the afternoon of the same day, watch the lengthening shadow; and where its end touches the several circles in going over them, make marks also. Lastly, with a pair of compasses, find exactly the middle point between the two marks on any circle, and draw a straight line from the centre to that point; which line will be covered at noon by the shadow of a small upright wire, which should be put in the place of the pin. The reason for drawing several circles is, that in case one part of the day should prove clear, and the other part somewhat cloudy, if you miss the time when the point of the shadow should touch one circle, you may perhaps catch it in touching another. The pin is usually about five inches in length. The best time for drawing a meridian line in this manner is about the summer solstice; because the sun changes his declination slowest, and his altitude fastest, in the longest days.

If the casement of a window on which the sun shines at noon be quite upright, you may draw a line along the edge of its shadow on the floor, when the shadow of the pin is exactly on the meridian line of the board; and as the motion of the shadow of the casement will be much more sensible on the floor than that of the shadow of the pin on the board, you may know to a few seconds when it touches the meridian line on the floor.

This method may suffice for ordinary purposes; but for astronomers the following is preferable. Take the gnomon of an horizontal dial for the latitude of the place, and to the hypotenuse fix two sights, whose centres may be parallel to the same; let the eye-sight be a small hole, but the other's diameter must be equal to the tangent of the double distance of the north-star from the pole; the distance of the sights being made radius, let the stile be rivetted to the end of a straight ruler; then when you would make use of it, lay the ruler on an horizontal plane, so that the end to which the stile is fixed may overhang; then look through the eye-sight, moving the instrument till the north-star appears to touch the circumference of the hole in the other sight, on the same hand with the girdle of Cassiopeia, or on the opposite side to that whereon the star in the Great Bear's rump is at that time; then draw a line by the edge of the ruler, and it will be a true meridian line.

A meridian line being by either of these methods exactly drawn, the time when the sun or any other of the celestial bodies is exactly in the meridian may be found by a common quadrant, placing the edge of it along the line, and observing when the sun or other luminary can be seen exactly through its two sights, and noting exactly the time; which, supposing the luminary viewed to be the sun, will be exactly noon, or 12 o'clock; but as the apparent diameter of the sun is pretty large, it ought to be known exactly when his centre is in the meridian, which will be some short space after his western limb has arrived at it, and before his eastern limb comes thither. It will be proper, therefore, to observe exactly the time of the two limbs being seen through the sights of the quadrant; and the half of the difference between these times added to the one or subtracted from the other, will give the exact time when the sun's centre is in the meridian. What we say with regard to the sun, is also applicable to the moon; but not to the stars, which have no sensible diameter. To render this more intelligible, the following short description of the quadrant, and method of taking the altitudes of celestial bodies by it, is subjoined.

Let HOX (fig. 195.) be a horizontal line, supposed to be extended from the eye at A to X, where altitudes of the sky and earth seem to meet at the end of a long celestial and level plain; and let S be the sun. The arc XY will be the sun's height above the horizon at X, and is found by the instrument EDC, which is a quadrantal board, or plate of metal, divided into 90 equal parts or degrees in its limb DPC; and has a couple of little brass plates, as a and b, with a small hole in each of them, called sight-holes, for looking through, parallel to the edge of the quadrant whereon they stand. To the centre E is fixed one end of a thread F, called the plumb-line, which has a small weight or plummet P fixed to its other end. Now, if an observer holds the quadrant upright, without inclining it to either side, and so that the horizon at X is seen through the sight-holes a and b, the plumb-line will cut or hang over the beginning of the degrees at 0, in the edge EC; but if he elevates the quadrant so as to look through the sight-holes at any part of the heavens, suppose to the sun at S; just so many degrees as he elevates the sight-hole b above the horizontal line HOX, so many degrees will the plumb-line cut in the limb CP of the quadrant. For, let the observer's eye at A be in the centre of the celestial arc XYV (and he may be said to be in the centre of the sun's apparent and diurnal orbit, let him be on what part of the earth he will), in which arc the sun is at that time, suppose 25 degrees high, and let the observer hold the quadrant so that he may see the sun through the sight-holes; the plumb-line freely playing on the quadrant will cut the 25th degree in the limb CP, equal to the number of degrees of the sun's altitude at the time of observation. (N.B. Whoever looks at the sun, must have a smoked glass before his eyes, to save them from hurt. The better way is, not to look at the sun through the sight-holes, but to hold the quadrant facing the eye, at a little distance, and so that, the sun shining through one hole, the ray may be seen to fall on the other.)

By observation made in the manner above directed, it is found, that the stars appear to go round the earth in 23 hours 56 minutes 4 seconds, and the sun in 24 hours; i.e. and sidereal, so that the stars gain three minutes 56 seconds upon real days the sun every day, which amounts to one diurnal revolution in a year; and therefore, in 365 days as measured by the returns of the sun to the meridian, there are 366 days as measured by the stars returning to it; the former are called solar days, and the latter sidereal.

If the earth had only a diurnal motion, without an annual, any given meridian would revolve from the sun to the sun again in the same quantity of time as from any star to the same star again; because the sun would never change his place with respect to the stars. Equation of as the earth advances almost a degree eastward in its orbit in the time that it turns eastward round its axis, whatever star passes over the meridian on any day with the sun, will pass over the same meridian on the next day when the sun is almost a degree short of it; that is, 3 minutes 56 seconds sooner. If the year contained only 360 days, as the ecliptic does 360 degrees, the sun's apparent place, so far as his motion is equable, would change a degree every day; and then the sidereal days would be just 4 minutes shorter than the solar.

Let ABCDEFGHIKLM (fig. 179.) be the earth's orbit, in which it goes round the sun every year, according to the order of the letters, that is, from west to east; and turns round its axis in the same way from the sun to the sun again in every 24 hours. Let S be the sun, and R a fixed star at such an immense distance, that the diameter of the earth's orbit bears no sensible proportion to that distance. Let Nm be any particular meridian of the earth, and N a given point or place upon that meridian when the earth is at A, the sun hides the star R, which would always be hid if the earth never removed from A; and consequently, as the earth turns round its axis, the point N would always come round to the sun and star at the same time. But when the earth has advanced, suppose a twelfth part of its orbit, from A to B, its motion round its axis will bring the point N a twelfth part of a natural day, or two hours, sooner to the star than to the sun; for the angle of NBn is equal to the angle ASB; and therefore any star, which comes to the meridian at noon with the sun when the earth is at A, will come to the meridian at 10 in the forenoon when the earth is at B. When the earth comes to C, the point N will have the star on its meridian at 8 in the morning, or four hours sooner than it comes round to the sun; for it must revolve from N to n before it has the sun in its meridian. When the earth comes to D, the point N will have the star on its meridian at 6 in the morning; but that point must revolve six hours more from N to n, before it has mid-day by the sun; for now the angle ASD is a right angle, and so is NDn; that is, the earth has advanced 90 degrees in its orbit, and must turn 90 degrees on its axis to carry the point N from the star to the sun; for the star always comes to the meridian when Nm is parallel to RSA; because DS is but a point in respect of RS. When the earth is at E, the star comes to the meridian at 4 in the morning; at F, at two in the morning; and at G, the earth having gone half round its orbit, N points to the star R at midnight, it being then directly opposite to the sun; and therefore, by the earth's diurnal motion, the star comes to the meridian 12 hours before the sun. When the earth is at H, the star comes to the meridian at 10 in the evening; at I, it comes to the meridian at 8, that is, 16 hours before the sun; at K, 18 hours before him; at L, 20 hours; at M, 22; and at A, equally with the sun again.

Thus it is plain, that an absolute turn of the earth on its axis (which is always completed when any particular meridian comes to be parallel to its situation at any time of the day before) never brings the same meridian round from the sun to the sun again; but that the earth requires as much more than one turn on its axis to finish a natural day, as it has gone forward in that time; which, at a mean rate, is a 365th part of a circle. Hence, in 365 days, the earth turns 360 times round its axis; and therefore, as a turn of the earth on its axis completes a sidereal day, there must be one sidereal day more in a year than the number of solar days, be the number what it will, on the earth or any other planet; one turn being lost with respect to the number of solar days in a year, by the planet's going round the sun; just as it would be lost to a traveller, who, in going round the earth, would lose one day by following the apparent diurnal motion of the sun; and consequently would reckon one day less at his return (let him take what time he would to go round the earth) than those who remained all the while at the place from which he set out. So if there were two earths revolving equally on their axes, and if one remained at A until the other had gone round the sun from A to A again, that earth which kept its place at A would have its solar and sidereal days always of the same length; and so would have one solar day more than the other at its return. Hence, if the earth turned but once round its axis in a year, and if that turn was made the same way as the earth goes round the sun, there would be continual day on one side of the earth, and continual night on the other.

The earth's motion on its axis being perfectly uniform and equal at all times of the year, the sidereal or solar days are always precisely of an equal length; and so would the solar or natural days be, if the earth's orbit were a perfect circle, and its axis perpendicular to its orbit. But the earth's diurnal motion on an inclined axis, and its annual motion in an elliptic orbit, cause the sun's apparent motion in the heavens to be unequal; for sometimes he revolves from the meridian to the meridian again in somewhat less than 24 hours, shown by a well-regulated clock; and at other times in somewhat more; so that the time shown by an equal going clock and a true sun-dial is never the same but on the 15th of April, the 16th of June, the 31st of August, and the 24th of December. The clock, if it goes equably and true all the year round, will be before the sun from the 24th of December till the 15th of April; from that time till the 16th of June, the sun will be before the clock; from the 16th of June till the 31st of August, the clock will be again before the sun; and from thence to the 24th of December, the sun will be faster than the clock.

As the equation of time, or difference between the time shown by a well-regulated clock and a true sun-dial, depends upon two causes, namely, the obliquity of the ecliptic, and the unequal motion of the earth in it, we shall first explain the effects of these causes separately considered, and then the united effects resulting from their combination.

The earth's motion on its axis being perfectly equable, or always at the same rate, and the plane of the equator being perpendicular to its axis, it is evident that in equal times equal portions of the equator pass over the meridian; and so would equal portions of the ecliptic, if it were parallel to, or coincident with, the equator. But, as the ecliptic is oblique to the equator, the equable motion of the earth carries unequal portions of the ecliptic over the meridian in equal times, the difference being proportionate to the obliquity; and as some parts of the ecliptic are much equation of more oblique than others, those differences are unequal among themselves. Therefore, if two suns should start from the beginning either of Aries or Libra, and continue to move through equal arcs in equal times, one in the equator and the other in the ecliptic, the equatorial sun would always return to the meridian in 24 hours time, as measured by a well-regulated clock; but the sun in the ecliptic would return to the meridian sometimes sooner and sometimes later than the equatorial sun; and only at the same moments with him on four days of the year; namely, the 20th of March, when the sun enters Aries; the 21st of June, when he enters Cancer; the 23rd of September, when he enters Libra; and the 21st of December, when he enters Capricorn; and to this fictitious sun the motion of a well-regulated clock always answers.

Let \( Z \varphi \omega \) be the earth; \( ZFRz \), its axis; \( abcd \), etc., the equator; \( ABCDE \), etc., the northern half of the ecliptic from \( \varphi \) to \( \omega \), on the side of the globe next the eye; and \( MNOP \), etc., the southern half on the opposite side from \( \omega \) to \( \varphi \). Let the points at \( A, B, C, D, E, F \), etc., quite round from \( \varphi \) to \( \varphi \) again bound equal portions of the ecliptic, gone through in equal times by the real sun; and those at \( a, b, c, d, e, f \), etc., equal portions of the equator described in equal times by the fictitious sun; and let \( Z \varphi \omega \) be the meridian.

As the real sun moves obliquely in the ecliptic, and the fictitious sun directly in the equator, with respect to the meridian; a degree, or any number of degrees, between \( \varphi \) and \( F \) on the ecliptic, must be nearer the meridian \( Z \varphi \omega \), than a degree, or any corresponding number of degrees, on the equator from \( \varphi \) to \( f \); and the more so, as they are the more oblique; and therefore the true sun comes sooner to the meridian every day whilst he is in the quadrant \( \varphi F \), than the fictitious sun does in the quadrant \( \varphi f \); for which reason, the solar noon precedes noon by the clock, until the real sun comes to \( F \); and the fictitious sun to \( f \); which two points, being equidistant from the meridian, both suns will come to it precisely at noon by the clock.

Whilst the real sun describes the second quadrant of the ecliptic \( FGHIKL \) from Cancer to \( \omega \), he comes later to the meridian every day than the fictitious sun moving through the second quadrant of the equator from \( f \) to \( \omega \); for the points at \( G, H, I, K, \) and \( L \), being farther from the meridian, their corresponding points at \( g, h, i, \) and \( l \), must be later of coming to it; and as both suns come at the same moment to the point \( \omega \), they come to the meridian at the moment of noon by the clock.

In departing from Libra, through the third quadrant, the real sun going through \( MNOPQ \) towards \( \varphi \) at \( R \), and the fictitious sun through \( mnopq \) towards \( r \), the former comes to the meridian every day sooner than the latter, until the real sun comes to \( \varphi \), and the fictitious to \( r \), and then they come both to the meridian at the same time.

Lastly, as the real sun moves equably thro' \( STUVW \), from \( \varphi \) towards \( \varphi \); and the fictitious sun thro' \( stuvw \), from \( r \) towards \( \varphi \); the former comes later every day to the meridian than the latter, until they both arrive at the point \( \varphi \), and then they make it noon at the same time with the clock.

Having explained one cause of the difference of time shown by a well-regulated clock and a true sun-dial, and considered the sun, not the earth, as moving in the ecliptic; we now proceed to explain the other cause of this difference, namely, the inequality of the sun's apparent motion; which is slowest in summer, when the sun is farthest from the earth, and swiftest in winter when he is nearest to it. But the earth's motion on its axis is equable all the year round, and is performed from west to east; which is the way that the sun appears to change his place in the ecliptic.

If the sun's motion were equable in the ecliptic, the whole difference between the equal time as shown by the clock, and the unequal time as shewn by the sun, would arise from the obliquity of the ecliptic. But the sun's motion sometimes exceeds a degree in 24 hours, though generally it is less; and when his motion is slowest, any particular meridian will revolve sooner to him than when his motion is quickest; for it will overtake him in less time when he advances a less space than when he moves through a larger.

Now, if there were two suns moving in the plane of the ecliptic, so as to go round it in a year; the one describing an equal arc every 24 hours, and the other describing sometimes a less arc in 24 hours, and at other times a larger, gaining at one time of the year what it lost at the opposite; it is evident, that either of these suns would come sooner or later to the meridian than the other, as it happened to be behind or before the other; and when they were both in conjunction, they would come to the meridian at the same moment.

As the real sun moves unequally in the ecliptic, let us suppose a fictitious sun to move equably in a circle coincident with the plane of the ecliptic. Let \( ABCD \) (fig. 181.) be the ecliptic or orbit in which the real sun moves, and the dotted circle \( abcd \) the imaginary orbit of the fictitious sun; each going round in a year according to the order of letters, or from west to east. Let \( HIKL \) be the earth turning round its axis the same way every 24 hours; and suppose both suns to start from \( A \) and \( a \), in a right line with the plane of the meridian \( EH \), at the same moment: the real sun at \( A \), being then at his greatest distance from the earth, at which time his motion is slowest; and the fictitious sun at \( a \), whose motion is always equable, because his distance from the earth is supposed to be always the same. In the time that the meridian revolves from \( H \) to \( H \) again, according to the order of the letters \( HIKL \), the real sun has moved from \( A \) to \( F \); and the fictitious with a quicker motion from \( a \) to \( f \); through a large arc; therefore, the meridian \( EH \) will revolve sooner from \( H \) to \( h \) under the real sun at \( F \), than from \( H \) to \( k \) under the fictitious sun at \( f \); and consequently it will then be noon by the sun-dial sooner than by the clock.

As the real sun moves from \( A \) towards \( C \), the swiftness of his motion increases all the way to \( C \), where it is at the quickest. But notwithstanding this, the fictitious sun gains so much upon the real, soon after his departing from \( A \), that the increasing velocity of the real sun does not bring him up with the equally moving fictitious sun till the former comes to \( C \), and the latter to \( e \), when each has gone half round its respective orbit; and then being in conjunction, the meridian \( EH \), revolving to \( LK \), comes to both suns at the same time. Calculating and therefore it is noon by them both at the same moment.

But the increased velocity of the real sun, now being at the quickest, carries him before the fictitious one; and therefore, the same meridian will come to the fictitious sun sooner than to the real: for whilst the fictitious sun moves from $c$ to $g$, the real sun moves through a greater arc from $C$ to $G$: consequently the point $K$ has its noon by the clock when it comes to $k$, but not its noon by the sun till it comes to $l$. And although the velocity of the real sun diminishes all the way from $C$ to $A$, and the fictitious sun by an equable motion is still coming nearer to the real sun, yet they are not in conjunction till the one comes to $A$ and the other to $a$, and then it is noon by them both at the same moment.

Thus it appears, that the solar noon is always later than noon by the clock whilst the sun goes from $C$ to $A$; sooner, whilst he goes from $A$ to $C$; and at these two points the sun and clock being equal, it is noon by them both at the same moment.

The point $A$ is called the sun's apogee, because when he is there he is at his greatest distance from the earth; the point $C$ his perigee, because when in it he is at his least distance from the earth: and a right line, as $AEC$, drawn through the earth's centre, from one of the points to the other, is called the line of the Ap-sides.

The distance that the sun has gone in any time from his apogee (not the distance he has to go to it, though ever so little) is called his mean anomaly, and is reckoned in signs and degrees, allowing 30 degrees to a sign. Thus, when the sun has gone suppose 174 degrees from his apogee at $A$, he is said to be 5 signs 14 degrees from it, which is his mean anomaly; and when he is gone suppose 355 degrees from his apogee, he is said to be 11 signs 25 degrees from it, although he be but 5 degrees short of $A$ in coming round to it again.

From what was said above, it appears, that when the sun's anomaly is less than 6 signs, that is, when he is anywhere between $A$ and $C$, in the half $ABC$ of his orbit, the solar noon precedes the clock noon; but when his anomaly is more than 6 signs, that is, when he is anywhere between $C$ and $A$, in the half $CDA$ of his orbit, the clock noon precedes the solar. When his anomaly is 0 signs 0 degrees, that is, when he is in his apogee at $A$; or 6 signs 0 degrees, which is when he is in his perigee at $C$; he comes to the meridian at the moment that the fictitious sun does, and then it is noon by them both at the same instant.

**Sect. VIII. Of calculating the Distances, Magnitudes, &c. of the Sun, Moon, and Planets.**

To find the moon's horizontal parallax of the body whose distance you desire to know; that is, the angle under which the semidiameter of the earth would appear provided we could see it from that body; and this is to be found out in the following manner.

Let $BAG$ (fig. 171.) be one half of the earth, $AC$ Calculating its semidiameter, $S$ the sun, $m$ the moon, and $EKOL$ the Dithys, &c., quarter of the circle described by the moon in revolving from the meridian to the meridian again. Let $CRS$ be the rational horizon of an observer at $A$, extended to the sun in the heavens; and $HAO$, his sensible horizon extended to the moon's orbit. $ALC$ is the angle under which the earth's semidiameter $AC$ is seen from the moon at $L$; which is equal to the angle $OAL$, because the right lines $AO$ and $CL$ which include both these angles are parallel. $ASC$ is the angle under which the earth's semidiameter $AC$ is seen from the sun at $S$; and is equal to the angle $OAF$, because the lines $AO$ and $CRS$ are parallel. Now, it is found by observation, that the angle $OAL$ is much greater than the angle $OAF$; but $OAL$ is equal to $ALC$, and $OAF$ is equal to $ASC$. Now as $ASC$ is much less than $ALC$, it proves that the earth's semidiameter $AC$ appears much greater as seen from the moon at $L$ than from the sun at $S$; and therefore the earth is much farther from the sun than from the moon. The quantities of these angles may be determined by observation in the following manner.

Let a graduated instrument, as $DAE$ (the larger the better), having a moveable index with sight-holes, be fixed in such a manner, that its plane surface may be parallel to the plane of the equator, and its edge $AD$ in the meridian: so that when the moon is in the equinoctial, and on the meridian $ADE$, she may be seen through the sight-holes when the edge of the moveable index cuts the beginning of the divisions at $o$, on the graduated limb $DE$; and when she is so seen, let the precise time be noted. Now as the moon revolves about the earth from the meridian to the meridian again in about 24 hours 48 minutes, she will go a fourth part round it in a fourth part of that time, viz. in 6 hours 12 minutes, as seen from $C$, that is, from the earth's centre or pole. But as seen from $A$, the observer's place on the earth's surface, the moon will seem to have gone a quarter round the earth when she comes to the sensible horizon at $O$; for the index through the sights of which she is then viewed will be at $d$, 90 degrees from $D$, where it was when she was seen at $E$. Now let the exact moment when the moon is seen at $O$ (which will be when she is in or near the sensible horizon) be carefully noted (c) that it may be known in what time she has gone from $E$ to $O$; which time subtracted from 6 hours 12 minutes (the time of her going from $E$ to $L$) leaves the time of her going from $O$ to $L$, and affords an easy method for finding the angle $OAL$ (called the moon's horizontal parallax, which is equal to the angle $ALC$) by the following analogy: As the time of the moon's describing the arc $EO$ is 90 degrees, so is 6 hours 12 minutes to the degrees of the arc $DdE$, which measures the angle $EAL$; from which subtract 90 degrees, and there remains the angle $OAL$, equal to the angle $ALC$, under which the earth's semidiameter $AC$ is seen from the moon. Now, since all the angles of a right-lined triangle are equal to 180 degrees, or to two right angles,

(c) Here proper allowance must be made for the refraction, which being about 34 minutes of a degree in the horizon, will cause the moon's centre to appear 34 minutes above the horizon when her centre is really in it. angles, and the sides of a triangle are always proportional to the sines of the opposite angles, say, by the Rule of Three. As the sine of the angle ALC at the moon L, is to its opposite side AC, the earth's semidiameter, which is known to be 3985 miles; so radius, viz. the sine of 90 degrees, or of the right angle ACL, to its opposite side AL, which is the moon's distance at L from the observer's place at A on the earth's surface; or, so is the sine of the angle CAL to its opposite side CL, which is the moon's distance from the earth's centre, and comes out at a mean rate to be 240,000 miles. The angle CAL is equal to what OAL wants of 90 degrees.

Other methods have been fallen upon for determining the moon's parallax; of which the following is recommended as the best, by Mr Ferguson, tho' hitherto it has not been put in practice. "Let two observers be placed under the same meridian, one in the northern hemisphere and the other in the southern, at such a distance from each other, that the arc of the celestial meridian included between their two zeniths may be at least 80 or 90 degrees. Let each observer take the distance of the moon's centre from his zenith, by means of an exceeding good instrument, at the moment of her passing the meridian; and these two zenith distances of the moon together, and their excess above the distance between the two zeniths, will be the distance between the two apparent places of the moon. Then, as the sum of the natural sines of the two zenith distances of the moon is to radius, so is the distance between her two apparent places to her horizontal parallax: which being found, her distance from the earth's centre may be found by the analogy mentioned above.

Thus, in fig. 199, let VECQ be the earth, M the moon, and Zbax an arc of the celestial meridian. Let V be Vienna, whose latitude EV is 48° 20' north; and C the Cape of Good Hope, whose latitude EC is 34° 30' south; both which latitudes we suppose to be accurately determined before-hand by the observers. As these two places are on the same meridian nVEC, and in different hemispheres, the sum of their latitudes 82° 50' is their distance from each other. Z is the zenith of Vienna, and z the zenith of the Cape of Good Hope; which two zeniths are also 82° 50' distant from each other, in the common celestial meridian Zz. To the observer at Vienna, the moon's centre will appear at a in the celestial meridian; and at the same instant, to the observer at the Cape, it will appear at b. Now suppose the moon's distance Za from the zenith of Vienna to be 38° 1' 53", and her distance zb from the zenith of the Cape of Good Hope, to be 46° 4' 41": the sum of these two zenith distances (Za + zb) is 84° 5' 34"; from which subtract 82° 50', the distance of Zz between the zeniths of these two places, and there will remain 1° 16' 34" for the arc ba, or distance between the two apparent places of the moon's centre, as seen from V and from C. Then, supposing the tabular radius to be 10,000,000, the natural sine of 38° 1' 53" (the arc Za) is 6,160,816, and the natural sine of 46° 4' 41" (the arc zb) is 7,202,821: the sum of both these sines is 13,363,637. Say therefore, As 13,363,637 is to 10,000,000, so is 1° 16' 34" to 57' 48", which is the moon's horizontal parallax.

If the two places of observation be not exactly under the same meridian, their difference of longitude Calculating must be accurately taken, that proper allowance may be made for the moon's declination whilst she is passing the Planets, &c., &c., from the meridian of the one to the meridian of the other.

The parallax, and consequently the distance and bulk, of any primary planet, might be found in the above manner, if the planet was near enough to the earth, so as to make the difference of its two apparent places sufficiently sensible: but the nearest planet is too remote for the accuracy required.

The sun's distance from the earth might be found Parallax of the same way, though with more difficulty, if his horizontal parallax, or the angle OAS equal to the angle ASC (fig. 171.), were not so small as to be hardly perceptible, being found in this way to be scarce 10 seconds of a minute, or the 360th part of a degree. Hence all astronomers, both ancient and modern have failed in taking the sun's parallax to a sufficient degree of exactness; but as some of the methods used are very ingenious, and shew the great acuteness and sagacity of the ancient astronomers, we shall here give an account of them. The first method was invented by Hipparchus; and has been made use of by Ptolemy and his followers, and many other astronomers. It depends on an observation of an eclipse of the moon: And the finding it, the principles on which it is founded are, 1/4, In a lunar eclipse, the horizontal parallax of the sun is equal to the difference between the apparent semidiameter of the sun, and half the angle of the conical shadow; which is easily made out in this manner. Let the circle AFG (fig. 87.) represent the sun, and DHC the earth; let DHM be the shadow, and DMC the half angle of the cone. Draw from the centre of the sun the right line SD touching the earth, and the angle DSC is the apparent semidiameter of the earth, seen from the sun, which is equal to the horizontal parallax of the sun; and the angle ADS is the apparent semidiameter of the sun seen from the earth: The external angle ADS is equal to the two internals DMS and DSM, by the 32d Prop. Elem. I. And therefore the angle DSM, or DSC, is equal to the difference of the angles ADS and DMS. Half the angle of the cone is equal to the difference of the horizontal parallax of the moon and the apparent semidiameter of the shadow, seen from the earth at the distance of the moon. For let CTE Fig. 178: be the earth, CME the shadow, which at the distance of the moon being cut by a plane, the section will be the circle FLK, whose semidiameter is FG, and is seen from the centre of the earth under the angle FTG. But by the 32d Prop. Elem. I. the angle CFT is equal to the two internals FMT and FIT. Wherefore the angle FMT is the difference of the two angles CFT and GTF; But the angle CFT is the angle under which the semidiameter of the earth is seen from the moon, and this is equal to the horizontal parallax of the moon; and the angle GTF is the apparent semidiameter of the shadow seen from the earth's centre. It is therefore evident that the half angle of the cone is equal to the difference of the horizontal parallax of the moon, and the apparent semidiameter of the shadow seen from the earth. Therefore, if to the apparent semidiameter of the sun there be added the apparent semidiameter of the shadow, and from the sum you take away the horizontal parallax of the moon, there will Calculating will remain the horizontal parallax of the sun; which the distance therefore, if these were accurately known, would be the Planets likewise known accurately: But none of them can be to exactly and nicely obtained, as to be sufficient for determining the parallax of the sun; for very small errors, which cannot be easily avoided in measuring these angles, will produce very great errors in the pa- rallax; and there will be a prodigious difference in the distances of the sun, when drawn from these parallaxes. For example, Suppose the horizontal parallax of the moon to be $60'15''$, the semidiameter of the sun $16'$, and the semidiameter of the shadow $44'30''$, we shall conclude from thence, that the parallax of the sun was $15''$, and its distance from the earth about $13,700$ semidiameters of the earth. But if there be an error committed, in determining the semidia- meter of the shadow, of $12''$ in defect (and certainly the semidiameter of the shadow cannot be had so pre- cisely as not to be liable to such an error), that is, if instead of $44'30''$ we put $44'18''$ for the apparent dia- meter of the shadow, all the others remaining as before, we shall have the parallax of the sun $3''$, and its distance from the earth almost $70,000$ semidiameters of the earth, which is five times more than what it was by the first position. But if the fault were in excess, or the diameter of the shadow exceeded the true by $12''$, so that we should put in $44'42''$ the parallax would arise to $27''$, and the distance of the sun only $7700$ of the earth's semidiameters; which is nine times less than what it comes to by a like error in defect. If an er- ror in defect was committed of $15''$, which is still but a small mistake, the sun's parallax would be equal to nothing, and his distance infinite. Therefore, since from so small mistakes the parallax and distance of the sun vary so much, it is plain that the distance of the sun cannot be obtained by this method.

Since therefore, the angle that the earth's semidia- meter subtends at the sun, is so small that it cannot be determined by any observation, Aristaarchus Samius, an ancient and great philosopher and astronomer, con- ceived a very ingenious way for finding the angle which the semidiameter of the moon's orbit subtends when seen from the sun: This angle is about 60 times big- ger than the former, subtended only by the earth's fe- midiameter. To find this angle, he lays down the fol- lowing principles.

From the phases of the moon, it hath been de- monstrated, that if a plane passed through the moon's centre, to which the line joining the sun and moon's centre was perpendicular, this plane would divide the illuminated hemisphere of the moon from the dark one: And therefore, if this plane should likewise pass through the eye of a spectator on the earth, the moon would appear bisected, or like a half circle; and a right line, drawn from the earth to the centre of the moon, would be in the plane of illumination, and consequent- ly would be perpendicular to the right line which joins the centres of the sun and moon. Let $S$ be the sun, and $T$ the earth, $ALq$ a quadrant of the moon's or- bit; and let the line $SL$, drawn from the sun, touch the orbit of the moon in $L$; the angle $TLS$ will be a right angle: And therefore, when the moon is seen in $L$, it will appear bisected, or just half a circle. At the same time take the angle $LTS$, the elongation of the moon from the sun, and then we shall have the angle $LST$, its compliment to a right angle. But we have Calcu- late the side $TL$, by which we can find the side $ST$, the dis- tance of the sun from the earth.

But the difficult point is to determine exactly the moment of time when the moon is bisected, or in its true dichotomy; for there is a considerable space of time both before and after the dichotomy, nay even in the quadrature, when the moon will appear bisected, or half a circle; so that the exact moment of biflection cannot be known by observation, as experience tells us: And consequently, the true distance of the sun from the earth cannot be obtained by this method.

Since the moment in which the true dichotomy hap- pens is uncertain, but it is certain that it happens be- fore the quadrature; Ricciolus takes that point of Riccio time which is in the middle, between the time that the meth- od begins to be doubtful whether it be bisected or not, and the time of quadrature: but he had done better, if he had taken the middle point between the time when it becomes doubtful whether the moon's side is concave or straight, and the time again when it is doubtful whether it is straight or convex; which point of time is after the quadrature: and if he had done this, he would have found the sun's distance a great deal more than he has made it.

There is no need to confine this method to the pha- This is a dichotomy or bisection, for it can be as well Dr K performed when the moon has any other phasis bigger or less than a dichotomy: for observe by a very good telescope, with a micrometer, the phasis of the moon, that is, the proportion of the illuminated part of the diameter to the whole; and at the same moment of time take her elongation from the sun: The illuminated part of the diameter, if it be less than the semidia- meter, is to be subducted from the semidiameter; but if it be greater, the semidiameter is to be subducted from it, and mark the residue: then say, As the semidia- meter of the moon is to the residue, so is the radius to the fine of an angle, which is therefore found: this angle added to, or subtracted from, a right angle, gives the exterior angle of the triangle at the moon: but we have the angle at the earth, which is the elongation observed; which therefore being subducted from the exterior angle, leaves the angle at the sun. And in the triangle $SLT$, having all the angles and one side $L'T$, we can find the other side $ST$, the distance of the sun from the earth. But it is almost impossible to All the determine accurately the quantity of the lunar phasis, methods so that there may not be an error of a few seconds committed; and consequently, we cannot by this me- thod find precisely enough the true distance of the sun. However, from such observations, we are sure that the sun is above $7000$ semidiameters of the earth distant from us. Since therefore the true distance of the sun can neither be found by eclipses nor by the phases of the moon, the astronomers are forced to have recourse to the parallaxes of the planets that are next to us, as Mars and Venus, which are sometimes much nearer to us than the sun is. Their parallaxes they endeavour to find by some of the methods above explained; and if these parallaxes were known, then the parallax and distance of the sun, which cannot directly by any ob- servations be attained, would easily be deduced from them. For from the theory of the motions of the earth and planets, we know at any time the proportion Calculating of the distances of the sun and planets from us; and the distances, &c., or the horizontal parallaxes are in a reciprocal proportion to these distances. Therefore, knowing the parallax of a planet, we may from thence find the parallax of the sun.

Mars, when he is in an achronical position, that is, opposite to the sun, is twice as near to us as the sun is; and therefore his parallax will be twice as great. But Venus, when she is in her inferior conjunction with the sun, is four times nearer to us than he is, and her parallax is greater in the same proportion: Therefore, though the extreme smallness of the sun's parallax renders it unobservable by our senses, yet the parallaxes of Mars or Venus, which are twice or four times greater, may become sensible. The astronomers have bestowed much pains in finding out the parallax of Mars; but some time ago Mars was in his opposition to the sun, and also in his perihelion, and consequently in his nearest approach to the earth: And then he was most accurately observed by two of the most eminent astronomers of our age, who have determined his parallax to have been scarce 30 seconds; from whence it was inferred, that the parallax of the sun is scarce 11 seconds, and his distance about 19,000 semidiameters of the earth.

As the parallax of Venus is still greater than that of Mars, Dr Halley proposed a method by it of finding the distance of the sun to within a 500th part of the whole. The times of observation were at her transits over the sun in 1761 and 1769. At these times the greatest attention was given by astronomers, but it was found impossible to observe the exact times of immersion and emersion with such accuracy as had been expected; so that the matter is not yet determined so exactly as could be wished. The method of calculating the sun's distance by means of these transits, is as follows.

In fig. 91, let DBA be the earth, V Venus, and TSR the eastern limb of the sun. To an observer at B, the point t of that limb will be on the meridian, its place referred to the heaven will be at E, and Venus will appear just within it at S. But at the same instant, to an observer at A, Venus is east of the sun, in the right line AVF; the point t of the sun's limb appears at e in the heaven; and if Venus were then visible, she would appear at F. The angle CVA is the horizontal parallax of Venus, which we seek; and is equal to the opposite angle FVE, whose measure is the arc FE. ASC is the sun's horizontal parallax, equal to the opposite angle e SE, whose measure is the arc e E; and FAe (the same as VAe) is Venus's horizontal parallax from the sun, which may be found by observing how much later in absolute time her total ingress on the sun is, as seen from A than as seen from B, which is the time she takes to move from V to v in her orbit OVv.

It appears by the tables of Venus's motion and the sun's, that at the time of her transit in 1761 she moved 4' of a degree on the sun's disk in 60 minutes of time, and consequently 4' of a degree in one minute of time.

Now let us suppose, that A is 90° west of B, so that when it is noon at B it will be six in the morning at A; that the total ingress as seen from B is at one minute past 13, but that as seen from A it is at seven minutes 30 seconds past six; deduct six hours for the difference of meridians of A and B, and the remainder will be six minutes 30 seconds for the time by which the total ingress of Venus on the sun at S, is later as the distance from A than as seen from B; which time being converted into parts of a degree, is 26", or the arc FE of Venus's horizontal parallax from the sun; for, as 1 minute of time is to 4 seconds of a degree, so is 6½ minutes of time to 26 seconds of a degree.

The times in which the planets perform their annual revolutions about the sun, are already known by observation.—From these times, and the universal power of gravity by which the planets are retained in their orbits, it is demonstrable, that if the earth's mean distance from the sun be divided into 100,000 equal parts, Mercury's mean distance from the sun must be equal to 38,710 of these parts—Venus's mean distance from the sun, to 72,333—Mars's mean distance, 152,369—Jupiter's, 520,096—and Saturn's, 954,006. Therefore, when the number of miles contained in the mean distance of any planet from the sun is known, we can by these proportions find the mean distance in miles of all the rest.

At the time of the above-mentioned transit, the earth's distance from the sun was 1015 (the mean distance being here considered as 1000), and Venus's distance from the sun 726 (the mean distance being considered as 723), which differences from the mean distances arise from the elliptical figure of the planets' orbits—Subtracting 726 parts from 1015, there remain 289 parts for Venus's distance from the earth at that time.

Now, since the horizontal parallaxes of the planets are inversely as their distances from the earth's centre, it is plain, that as Venus was between the earth and the sun on the day of her transit, and consequently her parallax at that time greater than the sun's, if her horizontal parallax was then ascertained by observation, the sun's horizontal parallax might be found, and consequently his distance from the earth.—Thus, suppose Venus's horizontal parallax was found to be 36".3480, then, As the sun's distance 1015 is to Venus's distance 289, so is Venus's horizontal parallax 36".3480 to the sun's horizontal parallax 10".3493 on the day of her transit. And the difference of these two parallaxes, viz. 25".0987 (which may be esteemed 26"), will be the quantity of Venus's horizontal parallax from the sun.

To find the sun's horizontal parallax at the time of his mean distance from the earth, say, As 1000 parts of the sun's mean distance from the earth's centre, is to 1015, his distance therefrom on the day of the transit, so is 10".3493, his horizontal parallax on that day, to 10".5045, his horizontal parallax at the time of his mean distance from the earth's centre.

The sun's parallax being thus (or any other way Method of supposed to be) found, at the time of his mean distance computing from the earth, we may find his true distance therefrom, in semidiameters of the earth, by the following analogy. As the fine (or tangent of so small an arc as that) of the sun's parallax 10".5045 is to radius, so is unity or the earth's semidiameter to the number of semidiameters of the earth that the sun is distant from its centre; which number, being multiplied by 3985, the number of miles contained in the earth's semidiameter, will give the number of miles by which the sun is distant from the earth's centre. Calculating Then, As 100,000, the earth's mean distance from the sun in parts, is to 38,710, Mercury's mean distance from the sun in parts, so is the earth's mean distance from the sun in miles to Mercury's mean distance from the sun in miles.—And,

As 100,000 is to 72,333, so is the earth's mean distance from the sun in miles to Venus's mean distance from the sun in miles.—Likewise,

As 100,000 is to 152,369, so is the earth's mean distance from the sun in miles to Mars's mean distance from the sun in miles.—Again,

As 100,000 is to 520,996, so is the earth's mean distance from the sun in miles to Jupiter's mean distance from the sun in miles.—Lastly,

As 100,000 is to 954,006, so is the earth's mean distance from the sun in miles to Saturn's mean distance from the sun in miles.

And thus, by having found the distance of any one of the planets from the sun, we have sufficient data for finding the distances of all the rest. And then from their apparent diameters at these known distances, their real diameters and bulks may be found. According to the calculations made from the transit in 1769, we have given the distance of each of the primary and secondary planets from one another, and from the sun, in fig. 119. In fig. 153, their proportional bulks are shown, according to former calculations by Mr Ferguson; and in fig. 18, their relative magnitudes according to the latest calculations by Mr Dunn. The proportional distances of the satellites of Jupiter and Saturn, with the magnitudes of the sun, and orbit of our moon, by Mr Ferguson, are represented fig. 186.

With regard to the fixed stars, no method of ascertaining their distance hath hitherto been found out. Those who have formed conjectures concerning them, have thought that they were at least 400,000 times farther from us than we are from the sun.

They are said to be fixed, because they have been generally observed to keep at the same distances from each other; their apparent diurnal revolutions being caused solely by the earth's turning on its axis. They appear of a sensible magnitude to the bare eye, because the retina is affected not only by the rays of light which are emitted directly from them, but by many thousands more, which falling upon our eye-lids, and upon the aerial particles about us, are reflected into our eyes so strongly as to excite vibrations not only in those points of the retina where the real images of the stars are formed, but also in other points at some distance round about. This makes us imagine the stars to be much bigger than they would appear if we saw them only by the few rays which come directly from them, so as to enter our eyes without being intermixed with others. Any one may be sensible of this, by looking at a star of the first magnitude through a long narrow tube; which, though it takes in as much of the sky as would hold 1000 such stars, yet scarce renders that one visible.

The more a telescope magnifies, the less is the aperture through which the star is seen; and consequently the fewer rays it admits into the eye. Now, since the stars appear less in a telescope which magnifies 200 times, than they do to the bare eye, inasmuch that they seem to be only indivisible points, it proves at once that the stars are at immense distances from us, and that they shine by their own proper light. If Calculi they shone by borrowed light, they would be as invisible without telescopes as the satellites of Jupiter are; the plane for these satellites appear bigger when viewed with a good telescope than the largest fixed stars do.

Dr Herschel has proposed a method of ascertaining the parallax of the fixed stars, something similar, but more complete, than that mentioned by Galileo and others; for it is by the parallax of the fixed stars that we should be best able to determine their distance. The method pointed out by Galileo, and first attempted by Hooke, Flamsteed, Molineux, and Bradley, of taking distances of stars from the zenith that pass very near it, has given us a much juster idea of the immense distance of the stars, and furnished us with an approximation to the knowledge of their parallax, that is much nearer the truth than we ever had before. But Dr Herschel mentions the insufficiency of their instruments, which were similar to the present zenith sectors, the method of zenith distances being liable to considerable errors on account of refraction, the change of position of the earth's axis arising from nutation, precession of the equinoxes, and other causes, and the aberration of light. The method of his own is by means of double stars; which is exempted from these errors, and of such a nature that the annual parallax, even if it should not exceed the tenth part of a second, may still become more visible, and be ascertained, at least to a much greater degree of approximation than it has ever been done. This method is capable of every improvement which the telescope and mechanism of micrometers can furnish. The method and its theory will be seen by the following investigation, extracted vol. lx. from his paper on the subject. Let O, E, (fig. 164), P, Q, be two opposite points in the annual orbit, taken in the same plane with two stars a, b, of unequal magnitudes. Let the angle a O b be observed when the earth is at O, and a E b be observed when the earth is at E. From the difference of these angles, if there should be any, we may calculate the parallax of the stars, according to the theory subjoined. These two stars ought to be as near each other as possible, and also to differ as much in magnitude as we can find them.

Dr Herschel's theory of the annual parallax of double stars, with the method of computing from thence what is generally called the parallax of the fixed stars, or of single stars of the first magnitude, such as are nearest to us, supposes, firstly, that the stars, one with another, are about the size of the sun; and, secondly, that the difference of their apparent magnitudes is owing to their different distances; so that the star of the second, third, or fourth magnitude, is two, three, or four times as far off as one of the first. These principles which he premises as postulates, have so great a probability in their favour, that they will hardly be objected to by those who are in the least acquainted with the doctrine of chances. Accordingly, let O (fig. 165) be the whole diameter of the earth's annual orbit; and let a, b, c, be three stars situated in the ecliptic, in such a manner that they may be seen all in one line O a b c, when the earth is at O. Let the line O a b c be perpendicular to OE, and draw PE parallel to cO; then, if O a, a b, b c, are equal to each other, a will be a star of the first magnitude, b of the second, and Calculating \( c \) of the third. Let us now suppose the angle \( OAE \), the diffraction or parallax of the whole orbit of the earth, to be \( 1'' \) of the Planets, &c., of a degree; then we have \( PEA = OAE = 1'' \); and because very small angles, having the same subtense \( OE \), may be taken to be in the inverse ratio of the lines \( OA, OB, OC, \) &c., we shall have \( OBE = \frac{1}{2}'' \), \( OCE = \frac{1}{4}'' \), &c. Now when the earth is removed to \( E \), we shall have \( PEB = EOB = \frac{1}{2}'' \), and \( PEA - PEB = \frac{1}{2}'' \); i.e., the stars \( a, b \), will appear to be \( \frac{1}{2}'' \) distant. We also have \( PEC = EOC = \frac{1}{4}'' \), and \( PEA - PEC = \frac{1}{4}'' \); i.e., the stars \( a, c \), will appear to be \( \frac{1}{4}'' \) distant when the earth is at \( E \). Now, since we have \( bPE = \frac{1}{2}'' \), and \( cPE = \frac{1}{4}'' \), therefore \( bPE - cPE = \frac{1}{4}'' \); i.e., the stars \( b, c \), will appear to be only \( \frac{1}{4}'' \) removed from each other when the earth is at \( E \). Whence we may deduce the following expression, to denote the parallax that will become visible in the change of distance between the two stars, by the removal of the earth from one extreme of its orbit to the other. Let \( P \) express the total parallax of a fixed star of the first magnitude, \( M \) the magnitude of the largest of the two stars, \( m \) the magnitude of the smallest, and \( p \) the partial parallax to be observed by the change in the distance of a double star; then

\[ P = \frac{m-M}{Mm} P \]

and \( p \), being found by observation, will give us

\[ P = \frac{pMm}{m-M} \]

E.G. Suppose a star of the first magnitude should have a small star of the twelfth magnitude near it; then will the partial parallax we are to expect to see be

\[ \frac{12 \times 1}{12 - 1} = 1.0909 \]

If the stars are of the third and twenty-fourth magnitude, the partial parallax will be

\[ \frac{3 \times 24}{3 \times 24} = \frac{2}{72} P \]

and if, by observation, \( p \) is found to be a tenth of a second, the whole parallax will come out

\[ \frac{1 \times 3 \times 24}{24 - 3} = 0.3428 \]

Farther, suppose the stars, being still in the ecliptic, to appear in one line, when the earth is in any other part of its orbit between \( O \) and \( E \); then will the parallax still be expressed by the same algebraic formula, and one of the maxima will still lie at \( O \), the other at \( E \); but the whole effect will be divided into two parts, which will be in proportion to each other as radius — fine to radius \( + \) fine of the stars distance from the nearest conjunction or opposition.

When the stars are anywhere out of the ecliptic, situated so as to appear in one line \( Oabc \) perpendicular to \( OE \), the maximum of parallax will still be expressed by

\[ \frac{m-M}{Mm} P \]

but there will arise another additional parallax in the conjunction and opposition, which will be to that which is found \( 90^\circ \) before or after the sun, as the fine \( S \) of the latitude of the stars seen at \( O \) is to the radius \( R \); and the effect of this parallax will be divided into two parts; half of it lying on one side of the large star, the other half on the other side of it. This latter parallax, moreover, will be compounded with the former, so that the distance of the stars in the conjunction and opposition will then be represented by the diagonal of a parallelogram, the Planets, whereof the two semiparallaxes are the sides; a general expression for which will be

\[ \sqrt{\frac{m-M}{2Mm} P} \times \frac{SS}{RR} + 1 \]

for the stars will apparently describe two ellipses in the heavens, whose transverse axis will be to each other in the ratio of \( M \) to \( m \) (fig. 167.), and \( Aa, Bb, Cc, Dd \) will be the cotemporary situations. Now, if \( Q \) be drawn parallel to \( AC \), and the parallelogram \( BQ \) be completed, we shall have \( bQ = \frac{1}{2} CA - \frac{1}{2} cA = \frac{1}{2} p \), or semiparallax \( 90^\circ \) before or after the sun, and \( Bb \) may be resolved into, or is compounded of, \( bQ \) and \( bq \); but \( bQ = \frac{1}{2} BD - \frac{1}{2} bd = \) the semiparallax in the conjunction or opposition. We also have \( R : S : : bQ : bq = \frac{pS}{2R} \); therefore the distance

\[ Bb \text{ (or } Dd) = \sqrt{\left(\frac{pS}{2R}\right)^2 + \frac{PS}{2R}} \]

and by substituting the value of \( p \) into this expression, we obtain

\[ \sqrt{\frac{m-M}{2Mm} P} \times \frac{SS}{RR} + 1, \text{ as above. When the stars are in the pole of the ecliptic, } bq \text{ will become equal to } bQ, \text{ and } Bb \text{ will be } 7071 P \frac{m-M}{Mm}. \]

Again, let the stars be at some distance, e.g., \( 5'' \) from each other, and let them both be in the ecliptic. This case is resolvable into the first; for imagine the star \( a \) (fig. 166.) to stand at \( x \), and in that situation the stars \( x, b, c \), will be in one line, and their parallax expressed by

\[ \frac{m-M}{Mm} P \]

But the angle \( aEx \) may be taken to be equal to \( aOx \); and as the foregoing formula gives us the angles \( xEb, xEc \), we are to add \( aEx \) or \( 5'' \) to \( xEb \), and we shall have \( aEb \). In general, let the distance of the stars be \( d \), and let the observed distance at \( E \) be \( D \), then will \( D = d + p \), and therefore the whole parallax of the annual orbit will be expressed by

\[ DMm - dMm = P \]

Suppose the two stars now to differ only in latitude, one being in the ecliptic, the other, e.g., \( 5^\circ \) north, when seen at \( O \). This case may also be resolved by the former; for imagine the stars \( b, c \) (fig. 165.) to be elevated at right angles above the plane of the figure, so that \( aOb \), or \( aOc \), may make an angle of \( 5^\circ \) at \( O \); then, instead of the lines \( Oab, Oac, Eab, Ec \), imagine them all to be planes at right angles to the figure; and it will appear that the parallax of the stars in longitude must be the same as if the small star had been without latitude. And since the stars \( b, c \), by the motion of the earth from \( O \) to \( E \), will not change their latitude, we shall have the following construction for finding the distance of the stars \( ab, ac \) at \( E \), and from thence the parallax \( P \). Let the triangle \( abP \) (fig. 168.) represent the situation of the stars; \( ab \) is the subtense of \( 5'' \), the angle under which they are supposed to be seen at \( O \). The quantity \( bP \) by the former theorem is found, \( \frac{m-M}{Mm} P \), which is the partial parallax that would have been seen by the earth's a Calculating earth's motion from O to E, if both stars had been in the ecliptic; but on account of the difference in latitude, it will be now represented by \(a\beta\), the hypotenuse of the triangle \(ab\beta\); therefore, in general, putting \(ab = d\), and \(a\beta = D\), we have

\[ DD = \frac{d^2 \times Mm}{m - M} \]

\(P\). Hence \(D\) being taken by observation, and \(d\), \(M\), and \(m\), given, we obtain the total parallax.

If the situation of the stars differs in longitude as well as latitude, we may resolve this case by the following method. Let the triangle \(ab\beta\) (fig. 169.) represent the situation of the stars, \(ab = d\) being their distance seen at O, \(a\beta = D\) their distance seen at E. That the change \(b\beta\), which is produced by the earth's motion will be truly expressed by

\[ \frac{m - M}{Mm} P \]

may be proved as before, by supposing the star \(a\) to have been placed at \(a'\). Now let the angle of position \(baa'\) be taken by a micrometer, or by any other method sufficiently exact; then, by solving the triangle \(aba'\), we shall have the longitudinal and latitudinal differences \(aa'\) and \(ba'\) of the two stars. Put \(aa' = x\), \(ba' = y\), and it will be \(x + b\beta = ag\), whence \(D = \sqrt{x + \frac{m - M}{Mm} P} + yy\); and

\[ \sqrt{D^2 - y^2 \times Mm - xMm} = P. \]

If neither of the stars should be in the ecliptic, nor have the same longitude or latitude, the last theorem will still serve to calculate the total parallax whose maximum will lie in E. There will, moreover, arise another parallax, whose maximum will be in the conjunction and opposition, which will be divided, and lie on different sides of the large star; but as we know the whole parallax to be exceedingly small, it will not be necessary to investigate every particular case of this kind; for by reason of the division of the parallax, which renders observations taken at any other time, except where it is greatest, very unfavourable, the formulae would be of little use. Dr Herschel closes his account of this theory with a general observation on the time and place where the maxima of parallax will happen.

When two unequal stars are both in the ecliptic, or, not being in the ecliptic, have equal latitudes, north or south, and the largest star has most longitude; the maximum of the apparent distance will be when the sun's longitude is 90 degrees more than the stars, or when observed in the morning; and the minimum when the longitude of the sun is 90 degrees less than that of the stars, or when observed in the evening. When the small star has most longitude, the maximum and minimum, as well as the time of observation, will be the reverse of the former. When the stars differ in latitudes, this makes no alteration in the place of the maximum or minimum, nor in the time of observation; i.e. it is immaterial whether the largest star has the least or the greatest distance of the two stars.

The stars, on account of their apparently various magnitudes, have been distributed into several classes, or orders. Those which appear largest are called stars of the first magnitude; the next to them in lustre, stars of the second magnitude; and so on to the sixth, which are the smallest that are visible to the bare eye. This distribution having been made long before the invention of telescopes, the stars which cannot be seen without the assistance of these instruments are distinguished by the name of telegraphic stars.

The ancients divided the starry sphere into particular constellations, or systems of stars, according as they lay near one another, so as to occupy those spaces which the figures of different sorts of animals or things would take up, if they were there delineated. And those stars which could not be brought into any particular constellation were called unformed stars.

This division of the stars into different constellations, serves to distinguish them from one another, so that any particular star may be readily found in the heavens by means of a celestial globe; on which the constellations are so delineated, as to put the most remarkable stars into such parts of the figures as are most easily distinguished. The number of the ancient constellations is 48, and upon our present globes about 70. On Senex's globes are inserted Bayer's letters; the first in the Greek alphabet being put to the biggest star in each constellation, the second to the next, and so on; by which means, every star is as easily found as if a name were given to it. Thus, if the star \(y\) in the constellation of the ram be mentioned, every astronomer knows as well what star is meant as if it were pointed out to him in the heavens. See figs. 205, 206, where the stars are represented with the figures of the animals from whence the constellations are marked.

There is also a division of the heavens into three divisions. 1. The zodiac (\(\tau\sigma\alpha\iota\alpha\kappa\)), from \(\tau\sigma\alpha\iota\alpha\kappa\), the heaven, "an animal," because most of the constellations in it, which are 12 in number, have the names of animals: As Aries the ram, Taurus the bull, Gemini the twins, Cancer the crab, Libra the balance, Scorpio the scorpion, Sagittarius the archer, Capricornus the goat, Aquarius the water-bearer, and Pisces the fishes. The zodiac goes quite round the heavens; it is about 16 degrees broad, so that it takes in the orbits of all the planets, and likewise the orbit of the moon. Along the middle of this zone or belt is the ecliptic, or circle which the earth describes annually as seen from the sun, and which the sun appears to describe as seen from the earth. 2. All that region of the heavens which is on the north side of the zodiac, containing 21 constellations. And, 3. That on the south side, containing 15.

The ancients divided the zodiac into the above 12 zodiacal constellations or signs in the following manner. They took a vessel with a small hole in the bottom, and, having filled it with water, suffered the same to distil drop by drop into another vessel set beneath to receive it; beginning at the moment when some star rose, and continuing till it rose the next following night. The water falling down into the receiver, they divided into twelve equal parts; and having two other small vessels in readiness, each of them fit to contain one part, they again poured all the water into the upper vessel; and, observing the rising of some star in the zodiac, they at the same time suffered the water to drop into one of the small vessels; and as soon as it was full, they shifted it, and set an empty one in its place. When each vessel was full, they took notice what star of the zodiac rose; and though this could not be done The names of the constellations, and the number of stars observed in each of them by different astronomers, are as follow.

| Constellation | Ptolemy | Tycho | Hevelius | Flamstead | |-------------------------------|---------|-------|----------|-----------| | Ursa minor | 8 | 7 | 12 | 24 | | Ursa major | 35 | 29 | 73 | 87 | | Draco | 31 | 32 | 40 | 80 | | Cepheus | 13 | 4 | 51 | 35 | | Bootes, Arctophila | 23 | 18 | 52 | 54 | | Corona Borealis | 8 | 8 | 8 | 21 | | Hercules, Engonafin | 29 | 28 | 45 | 113 | | Lyra | 10 | 11 | 17 | 21 | | Cygnus, Gallina | 10 | 18 | 47 | 81 | | Cassiopeia | 13 | 26 | 37 | 55 | | Perseus | 29 | 29 | 46 | 59 | | Auriga | 14 | 9 | 40 | 66 | | Serpentarius, Ophiuchus | 29 | 15 | 40 | 74 | | Serpens | 18 | 13 | 23 | 64 | | Sagitta | 5 | 5 | 5 | 18 | | Aquila, Vultur | 15 | 12 | 23 | 71 | | Antinous | 15 | 3 | 19 | 71 | | Delphinus | 10 | 10 | 14 | 18 | | Equuleus, Equi festis | 4 | 4 | 6 | 10 | | Pegasus, Equus | 20 | 19 | 38 | 89 | | Andromeda | 23 | 23 | 47 | 66 | | Triangulum | 4 | 4 | 12 | 16 | | Aries | 18 | 21 | 27 | 66 | | Taurus | 44 | 43 | 51 | 141 | | Gemini | 25 | 25 | 38 | 85 | | Cancer | 23 | 15 | 29 | 83 | | Leo | 35 | 30 | 49 | 95 | | Coma Berenices | 35 | 14 | 21 | 43 | | Virgo | 32 | 33 | 50 | 110 | | Libra, Chelie | 17 | 10 | 20 | 51 | | Scorpius | 24 | 10 | 20 | 44 | | Sagittarius | 31 | 14 | 22 | 69 | | Capricornus | 28 | 28 | 29 | 51 | | Aquarius | 45 | 41 | 47 | 103 | | Piscis | 38 | 36 | 39 | 113 | | Cetus | 22 | 21 | 45 | 97 | | Orion | 38 | 42 | 62 | 78 | | Eridanus, Fluvius | 34 | 10 | 27 | 84 | | Lepus | 12 | 13 | 10 | 19 | | Canis major | 29 | 13 | 21 | 31 | | Canis minor | 2 | 2 | 13 | 14 | | Argo Navis | 45 | 3 | 4 | 64 | | Hydra | 27 | 19 | 31 | 60 | | Crater | 7 | 3 | 10 | 31 | | Corvus | 7 | 4 | | 9 | | Centaurus | 37 | | | 35 | | Lupus | 19 | | | 24 | | Ara | 7 | | | 9 | | Corona Australis | 13 | | | 12 | | Piscis Australis | 18 | | | 24 |

The new Southern Constellations.

| Constellation | Apus, Avis Indica | Apis, Myoeca | Chamaeleon | Triangulum Australis | Piscis volans, Paffer | Dorado, Xiphias | Toucan | Hydrus | |-------------------------------|-------------------|--------------|------------|----------------------|----------------------|-----------------|--------|--------| | Columba Noachi | Noah's Dove | 10 | | | | | | | | Robur Carolinum | The Royal Oak | 12 | | | | | | | | Grus | The Crane | 13 | | | | | | | | Phœnix | The Phenix | 13 | | | | | | | | Indus | The Indian | 12 | | | | | | | | Pavo. | The Peacock | 14 | | | | | | | | | Apus, Avis Indica | Apis, Myoeca | Chamaeleon | Triangulum Australis | Piscis volans, Paffer | Dorado, Xiphias | Toucan | Hydrus | | | The Bird of Paradise | The Bee or Fly | The Chameleon | The South Triangle | The Flying Fish | The Sword Fish | The American Goose | The Water Snake | The obliquity of the ecliptic to the equinoctial is found at present to be above the third part of a degree less than Ptolemy found it. And most of the observers after him found it to decrease gradually down to Tycho's time. If it be objected, that we cannot depend on the observations of the ancients, because of the incorrectness of their instruments; we have to answer, that both Tycho and Flamstead are allowed to have been very good observers; and yet we find that Flamstead makes this obliquity 2½ minutes of a degree less than Tycho did about 100 years before him; and as Ptolemy was 1324 years before Tycho, so the gradual decrease answers nearly to the difference of time between these three astronomers. If we consider, that the earth is not a perfect sphere, but an oblate spheroid, having its axis shorter than its equatorial diameter; and that the sun and moon are constantly acting obliquely upon the greater quantity of matter about the equator, pulling it, as it were, towards a nearer and nearer coincidence with the ecliptic; it will not appear improbable that these actions should gradually diminish the angle between those planes. Nor is it less probable that the mutual attractions of all the planets should have a tendency to bring their orbits to a coincidence; but this change is too small to become sensible in many ages.

Sect. IX. Of calculating the periodical Times, Places, &c. of the Sun, Moon, and Planets; Delineation of the Phases of the Moon for any particular Time; and the Construction of Astronomical Tables.

This title includes almost all of what may be called the Practical part of Astronomy; and as it is by far the most difficult and abstruse, so the thorough investigation of it would necessarily lead us into very deep geometrical demonstrations. The great labours of former astronomers have left little for succeeding ones to do in this respect: tables of the motions of all the celestial bodies have been made long ago, the periodical times, eccentricities, &c. of the planets determined; and as we suppose few will desire to repeat these laborious operations, we shall here content ourselves with giving some general hints of the methods by which these things have been originally accomplished, that so the operations of the young astronomer who makes use of tables already formed to his hand may not be merely mechanical.

It hath been already observed, that the foundation of all astronomical operations was the drawing a meridian line. This being done, the next thing is to find out the latitude of the place where the observations are to be made, and for which the meridian line &c. is drawn. From what hath been said, no 39, it will easily be understood that the latitude of a place must always be equal to the elevation either of the north or fourth pole above the horizon; because when we are exactly on the equator, both poles appear on the horizon. There is, however, no star exactly in either of the celestial poles; therefore, to find the altitude of that invisible point called the Pole of the heavens, we must choose some star near it which does not set; and having by several observations, according to the directions given no 377, found its greatest and least altitudes, divide their difference by 2; and half that difference added to the least, or subtracted from the greatest, altitude of the star, gives the exact altitude of the pole or latitude of the place. Thus, suppose the greatest altitude of the star observed is 60° and its least 50°, we then know that the latitude of the place where the observation was made is exactly 55°.

The latitude being once found, the obliquity of the ecliptic, or the angle made by the sun's annual path with the earth's equator, is easily obtained by the following method. Observe, about the summer solstice, the sun's meridian distance from the zenith, which is easily done by a quadrant with a moveable index furnished with sights; if this distance is subtracted from the latitude of the place, provided the sun is nearer the equator than the place of observation, the remainder will be the obliquity of the ecliptic: But if the place of observation is nearer the equator than the sun at that time, the zenith distance must be added. By this method, the obliquity of the ecliptic hath been determined to be 23° 29'.

By the same method the declination of the sun from the equator for any day may be found; and thus a table of his declination for every day in the year might be constructed: thus also the declination of the stars might be found.

Having the declination of the sun, his right ascension and place in the ecliptic may be geometrically found by the solution of a case in spherical trigonometry. For let EQ represent the celestial equator, y the sun, and y X the ecliptic; then, in the right-angled spherical triangle ECy, we have the side Ey, equal to the sun's declination: the angle ECy is always 23° 29', being the angle of the ecliptic with the equator; and the angle yEC is 90°, or a right angle. From these data we can find the side EC the right ascension; and Cy the sun's place in the ecliptic, or his distance from the equinoctial point; and thus a table of the sun's place for every day in the year, answerable to his declination, may be formed.

Having the sun's place in the ecliptic, the right ascension of the stars may be found by the help of it, and right ascension of a good pendulum clock: For which purpose the motion of the clock must be so adjusted, that the hand may run through the 24 hours in the same time that a star leaving the meridian will arrive at it again; which time is somewhat shorter than the natural day, because of the space the sun moves through in the mean time eastward. The clock being thus adjusted, when the sun is in the meridian, fix the hand to the point from whence we are to begin to reckon our time; and then observe when the star comes to the meridian, and mark the hour and minute that the hand then shows: The hours and minutes described by the index, turned into degrees and minutes of the equator, will give the difference between the right ascension of the sun and stars; which difference, being added to the right ascension of the sun will give the right ascension of the star. Now, if we know the right ascension of any one star, we may from it find the right ascensions of all the others which we see, by marking the time upon the clock between the arrival of the star whose right ascension we know to the meridian, and another star whose ascension is to be found. This time converted into hours and minutes of the equator, will give the difference of right ascensions; from whence, by addition, we collect the right ascension of the star which was to be found out.

The right ascension and declination of a star being known, its longitude and latitude, or distance from the first star of Aries, and north or south from the ecliptic, may thence be easily found, from the solution of a case in spherical trigonometry, similar to that already mentioned concerning the sun's place; and the places of the fixed stars being all marked in a catalogue according to their longitudes and latitudes, it may thence be conceived how the longitude and latitude of a planet or comet may be found for any particular time by comparing its distance from them, and its apparent path may thus be traced; and thus the paths of Mercury and Venus were traced by M. Cassini, though Mr Ferguson made use of an error for that purpose.

With regard to the planets, the first thing to be done is to find out their periodical times, which is done by observing when they have no latitude. At that time the planet is in the ecliptic, and consequently in one of its nodes; so that, by waiting till it returns to the same node again, and keeping an exact account of the time, the periodical time of its revolution round the sun may be known pretty exactly. By the same observations, from the theory of the earth's motion we can find the position of the line of the nodes; and when once the position of this line is found, the angle of inclination of that planet's orbit to the earth may also be known.

The eccentricity of the earth's orbit may be determined by observing the apparent diameters of the sun at different times: when the sun's diameter is least, the earth is at the greatest distance; and when this diameter is greatest, the earth is at its least distance from him. But as this method must necessarily be precarious, another is recommended by Dr Keil, by observing the velocity of the earth in its orbit, or the apparent velocity of the sun, which is demonstrated to be always reciprocally as the square of the distance.

The eccentricities of the orbits of the other planets may be likewise found by observing their velocities at different times; for all of them observe the same proportions with regard to the increase or decrease of their velocity that the earth does; only, in this case, care must be taken to observe the real, not the apparent, velocities of the planets, the last depending on the motion of the earth at the same time. Their aphelia, or points of their orbits where they are farthest from the sun, may be known by making several observations of their distances from him, and thus perceiving when these distances cease to increase.

The position of the aphelion being determined, the &c. planet's distance from it at any time may also be found by observation, which is called its true or coequate anomaly; but by supposing the motion of the planet to be regular and uniform, tables of that motion may be easily constructed. From thence the planet's mean place in its orbit may be found for any moment of time; and one of these moments being fixed upon as an epocha or beginning of the table, it is easy to understand, that from thence tables of the planet's place in its orbit for any number of years either preceding or consequent to that period may be constructed. These tables are to be constructed according to the meridian of equal time, and not true or apparent time, because of the inequalities of the earth's motion as well as of that of the planet, and equations must be made to be added to or subtracted from the mean motion of the planet as occasion requires; which will be readily understood from what we have already mentioned concerning the unequal motion of the earth in its orbit. When all the necessary tables are constructed by this or similar methods, the calculating of the planetary places becomes a mere matter of mechanism, and consists only in the proper additions and subtractions according to the directions always given along with such tables.

It must be observed, however, that the accidental interference of the planets with one another by their mutual attractions, render it impossible to construct any of the mutual tables that shall remain equally perfect; and therefore frequent actual observations and corrections of the tables of the planets will be necessary. This disturbance, however, is considerable, except in the planets Jupiter and Saturn, and they are in conjunction only once in 800 years.

What hath been already mentioned with regard to difficulties of the planets, is also applicable to the moon; but with more difficulty, on account of the greater inequalities of her motions, the cause of which has been already explained. She indeed moves in an ellipse as the rest do, and its eccentricity may be better computed from observing her diameter at different times than that of the earth's orbit; but that eccentricity is not always the same. The reason of this, and indeed of all the other lunar inequalities, is, that the sun has a sensible effect upon her by his attraction, as well as the earth. Consequently, when the earth is at its least distance from the sun, her orbit is dilated, and she moves more slowly; and, on the contrary, when the earth is in its aphelion, her orbit contracts, and she moves more swiftly. The eccentricity is always greatest when the line of the apsides coincides with that of the syzygies, and the earth at its least distance from the sun. When the moon is in her syzygy, i.e., in the line that joins the centres of the earth and sun, which is either in her conjunction or opposition, she moves swifter, ceteris paribus, than in the quadratures. According to the different distances of the moon from the syzygies, she changes her motion: from the conjunction to her first quadrature, she moves somewhat slower; but recovers her velocity in the second quarter. In the third quarter she again loses, and in the last again recovers it. The apogee of the moon is also irregular; being found to move forward when it coincides with the line of the syzygies, and backwards when it cuts that line at right angles. Nor is this motion in any degree equal; in the conjunction or opposition, it goes briskly forwards, and in the quadratures moves either slowly forwards, stands still, or goes backwards. The motion of the nodes has been already taken notice of; but this motion is not uniform more than the rest; for when the line of the nodes coincides with that of the syzygies, they stand still; when their line cuts that at right angles, they go backwards with the velocity, as Sir Isaac Newton hath shown, of $16'' 19'' 24''$ an hour.

The only equable motion the moon has, is her revolution on her axis, which she always performs exactly in the space of time in which she moves round the earth. From hence arises what is called the moon's libration; for as the motion round her axis is equable, and that in her orbit unequal, it follows, that when the moon is in her perigee, where she moves swiftest, that part of her surface, which on account of the motion in her orbit would be turned from the earth, is not so, by reason of the motion on her axis. Thus some parts in the limb or margin of the moon sometimes recede from, and sometimes approach towards the centre of the disk. Yet this equable rotation produces an apparent irregularity; for the axis of the moon not being perpendicular, but a little inclined to its orbit, and this axis maintaining its parallelism round the earth, it must necessarily change its situation with respect to an observer on the earth, to whom sometimes the one and sometimes the other pole of the moon becomes visible; whence it appears to have a kind of wavering or vacillatory motion.

From all these irregularities it may well be concluded, that the calculation of the moon's place in her orbit is a very difficult matter; and indeed, before Sir Isaac Newton, astronomers in vain laboured to subject the lunar irregularities to any rule. By his labours, however, and those of other astronomers, these difficulties are in a great measure overcome; and calculations with regard to this luminary may be made with as great certainty as concerning any other. Her periodical time may be determined from the observation of two lunar eclipses, at as great a distance from one another as possible; for in the middle of every lunar eclipse, the moon is exactly in opposition to the sun. Compute the time between these two eclipses or oppositions, and divide this by the number of lunations that have intervened, and the quotient will be the synodical month, or time the moon takes to pass from one conjunction to another, or from one opposition to another. Compute the sun's mean motion in the time of the synodical month, and add this to the entire circle described by the moon. Then, As that sum is to $360^\circ$, so is the quantity of the synodical month to the periodical, or time that the moon takes to move from one point of her orbit to the same point again. Thus, Copernicus, in the year $1500$, November $6^{th}$, at $2$ hours $20$ minutes, observed an eclipse of the moon at Rome; and August $18^{th} 1523$, at $4$ hours $25$ minutes, another at Cracow: hence the quantity of the synodical month is thus determined:

| Year | Day | Hours | Minutes | |------|-----|-------|---------| | 1523 | 23 | 4 | 25 |

Add the intercalary days for leap years

Exact interval $22 297 2$

This interval divided by $282$, the number of months elapsed in that time, gives $29$ days $12$ hours $41$ minutes for the length of the synodical month. But from the observations of two other eclipses, the same author more accurately determined the quantity of the synodical month to be $29$ degrees $11$ hours $45$ minutes $3$ seconds; from whence the mean periodical time of the moon comes to be $27$ degrees $7$ hours $43$ minutes $5$ seconds, which exactly agrees with the observations of later astronomers.

The quantity of the periodical month being given, by the Rule of Three, we may find the moon's diurnal and horary motion; and thus may tables of the moon's mean motion be constructed; and if from the moon's mean diurnal motion that of the sun be subtracted, the remainder will be the moon's mean diurnal motion from the sun.

Having the moon's distance from the sun, her phasis for that time may be easily delineated by the following method laid down by Dr Keil. "Let the circle COBP represent the disk of the moon, which is turned towards the earth; and let OP be the line in fig. which the semicircle OMP is projected, which suppose to be cut by the diameter BC at right angles; and making LP the radius, take LF equal to the cosine of the elongation of the moon from the sun: And then upon BC, as the great axis, and LF the lesser axis, describe the semi-ellipse BFC. This ellipse will cut off from the disk of the moon the portion BFCP of the illuminated face, which is visible to us from the earth."

Since in the middle of a total eclipse the moon is exactly in the node, if the sun's place be found for that node time, and fix signs added to it, if the eclipse be a lunar one the sun will give the place of the node, or if the eclipse observed is a solar one, the place of the node and of the sun are the same. From comparing two eclipses together, the mean motion of the nodes will thus be found out. The apogee of the moon may be known from her apparent diameter, as already observed; and by comparing her place when in the apogee at different times, the motion of the apogee itself may also be determined.

These short hints will be sufficient to give a general knowledge of the methods used for the solution of some of the most difficult problems in astronomy. As for the proper equations to be added or subtracted, in order to find out the true motion and place of the moon, together with the particular methods of constructing tables for calculating eclipses, they are given from Mr Ferguson, in the following section. Every planet and satellite is illuminated by the sun; and casts a shadow towards that point of the heavens which is opposite to the sun. This shadow is nothing but a privation of light in the space hid from the sun by the opaque body that intercepts his rays.

When the sun's light is so intercepted by the moon, that to any place of the earth the sun appears partly or wholly covered, he is said to undergo an eclipse; though, properly speaking, it is only an eclipse of that part of the earth where the moon's shadow or penumbra falls. When the earth comes between the sun and moon, the moon falls into the earth's shadow; and having no light of her own, she suffers a real eclipse from the interception of the sun's rays. When the sun is eclipsed to us, the moon's inhabitants, on the side next the earth, see her shadow like a dark spot travelling over the earth, about twice as fast as its equatorial parts move, and the same way as they move. When the moon is in an eclipse, the sun appears eclipsed to her, total to all those parts on which the earth's shadow falls, and of as long continuance as they are in the shadow.

That the earth is spherical (for the hills take off no more from the roundness of the earth, than grains of dust do from the roundness of a common globe) is evident from the figure of its shadow on the moon; which is always bounded by a circular line, although the earth is incessantly turning its different sides to the moon, and very seldom shows the same side to her in different eclipses, because they seldom happen at the same hours. Were the earth shaped like a round flat plate, its shadow would only be circular when either of its sides directly faced the moon, and more or less elliptical as the earth happened to be turned more or less obliquely towards the moon when she is eclipsed. The moon's different phases prove her to be round; for as she keeps still the same side towards the earth, if that side were flat, as it appears to be, she would never be visible from the third quarter to the first; and from the first quarter to the third, she would appear as round as when we say she is full; because, at the end of her first quarter, the sun's light would come as suddenly on all her side next the earth, as it does on a flat wall, and go off as abruptly at the end of her third quarter.

If the earth and sun were equally large, the earth's shadow would be infinitely extended, and all of the same bulk; and the planet Mars, in either of its nodes and opposite to the sun, would be eclipsed in the earth's shadow. Were the earth larger than the sun, its shadow would increase in bulk the farther it extended, and would eclipse the great planets, Jupiter and Saturn, with all their moons, when they were opposite to the sun. But as Mars, in opposition, never falls into the earth's shadow, although he is not then above 42,000,000 miles from the earth, it is plain that the earth is much less than the sun; for otherwise its shadow could not end in a point of so small a distance. If the sun and moon were equally large, the moon's shadow would go on to the earth with an equal breadth, and cover a portion of the earth's surface more than 2000 miles broad, of calculation even if it fell directly against the earth's centre, as seen from the moon; and much more if it fell obliquely on the earth: But the moon's shadow is seldom 150 miles broad at the earth, unless when it falls very obliquely on the earth, in total eclipses of the sun. In annular eclipses, the moon's real shadow ends in a point at some distance from the earth. The moon's small distance from the earth, and the shortness of her shadow, prove her to be less than the sun. And, as the earth's shadow is large enough to cover the moon, if her diameter were three times as large as it is (which is evident from her long continuance in the shadow when she goes through its centre), it is plain that the earth is much bigger than the moon.

Though all opaque bodies, on which the sun shines, why there have their shadows, yet such is the bulk of the sun, and so few the distances of the planets, that the primary planets can never eclipse one another. A primary can eclipse only its secondary, or be eclipsed by it; and never but when in opposition or conjunction with the sun. The primary planets are very seldom in these positions, but the sun and moon are so every month: Whence one may imagine, that these two luminaries should be eclipsed every month. But there are few eclipses in respect of the number of new and full moons; the reason of which we shall now explain.

If the moon's orbit were coincident with the plane of the ecliptic, in which the earth always moves and the sun appears to move, the moon's shadow would fall upon the earth at every change, and eclipse the sun to some parts of the earth. In like manner, the moon would go through the middle of the earth's shadow, and be eclipsed at every full; but with this difference, that she would be totally darkened for above an hour and a half; whereas the sun never was above four minutes totally eclipsed by the interposition of the moon. But one half of the moon's orbit is elevated 5° degrees above the ecliptic, and the other half as much depressed below it; consequently, the moon's orbit intersects the ecliptic in two opposite points called the moon's nodes, as has been already taken notice of. When these points are in a right line with the centre of the sun at new or full moon, the sun, moon, and earth, are all in a right line; and if the moon be then new, her shadow falls upon the earth; if full, the earth's shadow falls upon her. When the sun and moon are more than 17 degrees from either of the nodes at the time of conjunction, the moon is then generally too high or too low in her orbit to cast any part of her shadow upon the earth; when the sun is more than 12 degrees from either of the nodes at the time of full moon, the moon is generally too high or too low in her orbit to go through any part of the earth's shadow; and in both these cases there will be no eclipse. But when the moon is less than 17 degrees from either node at the time of conjunction, her shadow or penumbra falls more or less upon the earth, as she is more or less within this limit. And when she is less than 12 degrees from either node at the time of opposition, she goes through a greater or lesser portion of the earth's shadow, as she is more or less within this limit. Her orbit contains 360 degrees; of which 17, the limit of solar eclipses on either side of the nodes, and 12, the limit of lunar eclipses, are but small portions; And as the sun commonly passes Of caleuk-by the nodes but twice in a year, it is no wonder that we have so many new and full moons without eclipses.

To illustrate this (fig. 195.) let ABCD be the ecliptic, RSTU a circle lying in the same plane with the ecliptic, and VXYZ the moon's orbit, all thrown into an oblique view, which gives them an elliptical shape to the eye. One half of the moon's orbit, as VWX, is always below the ecliptic, and the other half XYV above it. The points V and X, where the moon's orbit intersects the circle RSTU, which lies even with the ecliptic, are the moon's nodes; and a right line, as XEV, drawn from one to the other, through the earth's centre, is the line of the nodes, which is carried almost parallel to itself round the sun in a year.

If the moon moved round the earth in the orbit RSTU, which is coincident with the plane of the e- cliptic, her shadow would fall upon the earth every time she is in conjunction with the sun, and at every op- position she would go through the earth's shadow. Were this the case, the sun would be eclipsed at every change, and the moon at every full, as already mentioned.

But although the moon's shadow N must fall upon the earth at a, when the earth is at E, and the moon in conjunction with the sun at i, because she is then very near one of her nodes; and at her opposition n she must go through the earth's shadow J, because she is then near the other node; yet, in the time that she goes round the earth to her next change, according to the order of the letters XYVV, the earth advances from E to e, according to the order of the letters EFGH; and the line of the nodes VEX, being carried nearly parallel to itself, brings the point f of the moon's or- bit in conjunction with the sun, at that next change; and then the moon being at f, is too high above the ecliptic to cast her shadow on the earth: and as the earth is still moving forward, the moon at her next op- position will be at g, too far below the ecliptic to go through any part of the earth's shadow; for by that time the point g will be at a considerable distance from the earth as seen from the sun.

When the earth comes to F, the moon in conjunc- tion with the sun Z is not at k in a plane coincident with the ecliptic, but above it at Y in the highest part of her orbit; and then the point b of her shadow O goes far above the earth (as in fig. 2., which is an edge- view of fig. 1.). The moon at her next opposition, is not at a (fig. 1.), but at W, where the earth's shadow goes far above her, (as in fig. 2.). In both these cases, the line of the nodes VFX (fig. 1.) is about ninety de- grees from the sun, and both luminaries are as far as possible from the limits of the eclipses.

When the earth has gone half round the ecliptic from E to G, the line of the nodes VGX is nearly, if not exactly, directed towards the sun at Z; and then the new-moon l casts her shadow P on the earth G; and the full moon p goes through the earth's shadow L; which brings on eclipses again, as when the earth was at E.

When the earth comes to H, the new moon falls not at m in a plane coincident with the ecliptic CD, but at W in her orbit below it; and then her shadow Q (see fig. 197.) goes far below the earth. At the next full she is not at q (fig. 196.), but at Y in her orbit 57 degrees above q, and at her greatest height above the ecliptic CD; being then as far as possible, at any op- position, from the earth's shadow M, as in fig. 197.

So, when the earth is at E and G, the moon is a- bout her nodes at new and full, and in her greatest north and south declination (or latitude as it is gene- rally called) from the ecliptic at her quarters; but when the earth is at F or H, the moon is in her greatest north and south declination from the ecliptic at new and full, and in the nodes about her quarters.

The point X, where the moon's orbit crosses the ecliptic, is called the ascending node, because the moon ascends from it above the ecliptic; and the opposite point of intersection V is called the descending node, because the moon descends from it below the ecliptic. When the moon is at Y in the highest point of her or- bit, she is in her greatest north latitude; and when she is at W in the lowest point of her orbit, she is in her greatest south latitude.

If the line of the nodes, like the earth's axis, was Ap- plied parallel to itself round the sun, there would be an- just half a year between the conjunctions of the sun and nodes. But the nodes shift backwards, or contrary to the earth's annual motion, 10° 43' deg. every year; and therefore the same node comes round the sun 19 days sooner every year than on the year before. Conse- quently, from the time that the ascending node X (when the earth is at E) passes by the sun as seen from the earth, it is only 173 days (not half a year) till the descending node V passes by him. Therefore in what- ever time of the year we have eclipses of the luminaries about either node, we may be sure that in 173 days afterward we shall have eclipses about the other node. And when at any time of the year the line of the nodes is in the situation VGX, at the same time next year it will be in the situation rG3; the ascending node hav- ing gone backward, that is, contrary to the order of signs, from X to r, and the descending node from V to r; each 19° 43' deg. At this rate, the nodes shift through all the signs and degrees of the ecliptic in 18 years and 225 days; in which time there would al- ways be a regular period of eclipses, if any complete number of lunations were finished without a fraction. But this never happens: for if both the sun and moon should start from a line of conjunction with either of the nodes in any point of the ecliptic, the sun would perform 18 annual revolutions and 222 degrees over and above, and the moon 230 lunations and 85 degrees of the 231st, by the time the node came round to the same point of the ecliptic again; so that the sun would then be 138 degrees from the node, and the moon 85 degrees from the sun.

But in 223 mean lunations, after the sun, moon, and nodes, have been once in a line of conjunction, they return so nearly to the same state again, as that the same node, which was in conjunction with the sun and moon at the beginning of the first of these lu- nations, will be within 28° 12' of a degree of a line of conjunction with the sun and moon again, when the last of these lunations is completed. And therefore in that time there will be a regular period of eclipses, or return of the same eclipse, for many ages.—In this period, (which was first discovered by the Chaldeans) there are 18 Julian years 11 days 7 hours 43 minutes 20 seconds, when the last day of February in leap-years is four times included; but when it is five times inclu- the period consists of only 18 years 10 days 7 hours 43 minutes 20 seconds. Consequently, if to the mean time of any eclipse, either of the sun or moon, you add 18 Julian years 11 days 7 hours 43 minutes 20 seconds, when the last day of February in leap-years comes in four times, or a day less when it comes in five times, you will have the mean time of the return of the same eclipse.

But the falling back of the line of conjunctions or oppositions of the sun and moon 28° 12" with respect to the line of the nodes in every period, will wear it out in process of time; and after that, it will not return again in less than 12,492 years.—These eclipses of the sun, which happen about the ascending node, and begin to come in at the north pole of the earth, will go a little southerly at each return, till they go quite off the earth at the south pole; and those which happen about the descending node, and begin to come in at the south pole of the earth, will go a little northerly at each return, till at last they quite leave the earth at the north pole.

To exemplify this matter, we shall first consider the sun's eclipse (March 21st old style, April 1st new style), A.D. 1764, according to its mean revolutions, without equating the times, or the sun's distance from the node; and then according to its true equated times.

This eclipse fell in open space at each return, quite clear of the earth, ever since the creation, till A.D. 1295, June 13th old style, at 12 h. 52 m. 59 sec. p.m. meridian, when the moon's shadow first touched the earth at the north pole; the sun being then 70° 48' 27" from the ascending node. In each period since that time, the sun has come 28° 12" nearer and nearer the same node, and the moon's shadow has therefore gone more and more southerly.—In the year 1662, July 18th old style, at 10 h. 36 m. 21 sec. p.m. when the same eclipse will have returned 38 times, the sun will be only 24° 45" from the ascending node, and the centre of the moon's shadow will fall a little northward of the earth's centre.—At the end of the next following period, A.D. 1980, July 28th old style, at 18 h. 19 m. 41 sec. p.m. the sun will have receded back 3° 27" from the ascending node, and the moon will have a very small degree of southern latitude, which will cause the centre of her shadow to pass a very small matter south of the earth's centre.—After which, in every following period, the sun will be 28° 12" farther back from the ascending node than in the period last before; and the moon's shadow will go still farther and farther southward, until September 12th old style, at 23 h. 46 m. 22 sec. p.m. A.D. 2665; when the eclipse will have completed its 77th periodical return, and will go quite off the earth at the south pole (the sun being then 17° 55' 22" back from the node), and cannot come in at the north pole, so as to begin the same course over again, in less than 12,492 years afterwards.—And such will be the case of every other eclipse of the sun: For, as there is about 18 degrees on each side of the node within which there is a possibility of eclipses, their whole revolution goes through 36 degrees about that node, which, taken from 360 degrees, leaves remaining 324 degrees for the eclipses to travel in expansion. And as this 36 degrees is not gone through in less than 77 periods, which takes up 1388 years, the remaining 324 degrees cannot be so gone through in less than 12,492 years. For, as 36 is to 1388, so is 324 to 12,492.

To illustrate this a little farther, we shall examine some of the most remarkable circumstances of the returns of the eclipse which happened July 14th 1748, clip in about noon. This eclipse, after traversing the voids of space from the creation, at last began to enter the Terra Australis Incognita about 88 years after the conquest, which was the last of king Stephen's reign: every Chaldean period it has crept more northerly, but was still invisible in Britain before the year 1622; when, on the 30th of April, it began to touch the south parts of England about two in the afternoon; its central appearance rising in the American south seas, and traversing Peru and the Amazons country, through the Atlantic ocean into Africa, and setting in the Ethiopian continent, not far from the beginning of the Red sea.

Its next visible period was, after three Chaldean revolutions, in 1676, on the first of June, rising central in the Atlantic ocean, passing us about nine in the morning, with four digits eclipsed on the under limb, and setting in the gulf of Cochinchina in the East Indies.

It being now near the solstice, this eclipse was visible the very next return in 1694, in the evening; and in two periods more, which was in 1730, on the 4th of July, was seen about half eclipsed just after sunrise, and observed both at Würtemberg in Germany, and Pekin in China, soon after which it went off.

Eighteen years more afforded us the eclipse which fell on the 14th of July 1748.

The next visible return happened on July 25th 1766 in the evening, about four digits eclipsed; and, after two periods more, will happen on August 16th 1802, early in the morning, about five digits, the centre coming from the north frozen continent, by the capes of Norway, through Tartary, China, and Japan, to the Ladrones islands, where it goes off.

Again, in 1820, August 26th, between one and two, there will be another great eclipse at London, about 10 digits; but, happening so near the equinox, the centre will leave every part of Britain to the west, and enter Germany at Embden, passing by Venice, Naples, Grand Cairo, and set in the gulf of Baffora near that city.

It will be no more visible till 1874, when five digits will be obscured (the centre being now about to leave the earth) on September 28th. In 1892, the sun will go down eclipsed in London; and again, in 1918, the passage of the centre will be in the expansion, though there will be two digits eclipsed at London, October the 31st of that year, and about the year 2010 the whole penumbra will be wore off; whence no more returns of this eclipse can happen till after a revolution of 10,000 years.

From these remarks on the entire revolution of this period in eclipse, we may gather, that a thousand years more or less (for there are some irregularities that may protract phenomena or lengthen this period 100 years) complete the whole of an eclipse, or terrestrial phenomena of any single eclipse: and since 20 periods of 54 years each, and about 33 days, comprehend the entire extent of their revolution, it is evident, that the times of the returns will pass through a circuit of one year and ten months, every Chaldean period. being 10 or 11 days later, and of the equable appearances, about 32 or 33 days. Thus, though this eclipse happens about the middle of July, no other subsequent eclipse of this period will return till the middle of the same month again; but wear constantly each period 10 or 11 days forward, and at last appear in winter, but then it begins to cease from affecting us.

Another conclusion from this revolution may be drawn, that there will seldom be any more than two great eclipses of the sun in the interval of this period, and these follow sometimes next return, and often at greater distances. That of 1715 returned again in 1733 very great; but this present eclipse will not be great till the arrival of 1820, which is a revolution of four Chaldean periods; so that the irregularities of their circuits must undergo new computations to assign them exactly.

Nor do all eclipses come in at the south pole: that depends altogether on the position of the lunar nodes, which will bring in as many from the expanse one way as the other; and such eclipses will wear more foutherly by degrees, contrary to what happens in the present case.

The eclipse, for example, of 1736 in September, had its centre in the expanse, and let about the middle of its obscurity in Britain; it will wear in at the north pole, and in the year 2600, or thereabouts, go off into the expanse on the south side of the earth.

The eclipses therefore which happened about the creation are little more than half way yet of their etherial circuit; and will be 4000 years before they enter the earth any more. This grand revolution seems to have been entirely unknown to the ancients.

It is particularly to be noted, that eclipses which have happened many centuries ago will not be found by our present tables to agree exactly with ancient observations, by reason of the great anomalies in the lunar motions; which appears an incontestable demonstration of the non-eternity of the universe. For it seems confirmed by undeniable proofs, that the moon now finishes her period in less time than formerly, and will continue, by the centripetal law, to approach nearer and nearer the earth, and to go sooner and sooner round it; nor will the centrifugal power be sufficient to compensate the different gravitations of such an assemblage of bodies as constitute the solar system, which would come to ruin of itself, without some regulation and adjustment of their original motions.

We are credibly informed from the testimony of the ancients, that there was a total eclipse of the sun predicted by Thales to happen in the fourth year of the 48th Olympiad, either at Sardis or Miletus in Asia, where Thales then resided. That year corresponds to the 585th year before Christ; when accordingly there happened a very signal eclipse of the sun, on the 28th of May, answering to the present 10th of that month, central through North America, the south parts of France, Italy, &c. as far as Athens, or the isles in the Ægean sea; which is the farthest that even the Caroline tables carry it; and consequently make it invisible to any part of Asia, in the total character; though there are good reasons to believe that it extended to Babylon, and went down central over that city. We are not however to imagine, that it was set before it passed Sardis and the Asiatic towns, where the presiding dictator lived; because an invisible eclipse could have been of no service to demonstrate his ability in astronomical sciences to his countrymen, as it could give no proof of its reality.

For a further illustration, Thucydides relates, That a solar eclipse happened on a summer's day, in the afternoon, in the first year of the Peloponnesian war, so great, that the stars appeared. Rhodius was victor in the Olympic games the fourth year of the said war, being also the fourth year of the 87th Olympiad, or the 428th year before Christ. So that the eclipse must have happened in the 431st year before Christ; and by computation it appears, that on the third of August there was a signal eclipse which would have passed over Athens, central about six in the evening, but which our present tables bring no farther than the ancient Syrtes on the African coast, above 400 miles from Athens; which, suffering in that case but nine digits, could by no means exhibit the remarkable darkness recited by this historian; the centre therefore seems to have passed Athens about six in the evening, and probably might go down about Jerusalem, or near it, contrary to the construction of the present tables. These things are only mentioned by way of caution to the present astronomers, in recomputing ancient eclipses; and they may examine the eclipse of Nicias, so fatal to the Athenian fleet; that which overthrew the Macedonian army, &c.

In any year, the number of eclipses of both luminaries cannot be less than two, nor more than seven; the most usual number is four, and it is very rare to have more than five. For the sun passes by both the nodes but once a-year, unless he passes by one of them in the beginning of the year; and, if he does, he will pass by the same node again a little before the year be finished; because, as these points move 19° degrees backwards every year, the sun will come to either of them 173 days after the other. And when either node is within 17 degrees of the sun at the time of new moon, the sun will be eclipsed. At the subsequent opposition, the moon will be eclipsed in the other node, and come round to the next conjunction again ere the former node be 17 degrees past the sun, and will therefore eclipse him again. When three eclipses fall about either node, the like number generally falls about the opposite; as the sun comes to it in 173 days afterward; and six lunations contain but four days more. Thus, there may be two eclipses of the sun and one of the moon about each of her nodes. But when the moon changes in either of the nodes, she cannot be near enough the other node at the next full to be eclipsed; and in six lunar months afterward she will change near the other node; in these cases, there can be but two eclipses in a year, and they are both of the sun.

A longer period than the above mentioned, for comparing and examining eclipses which happen at long intervals of time, is 557 years, 21 days, 18 hours, 30 minutes, 11 seconds; in which time there are 6890 mean lunations; and the sun and node meet again so nearly as to be but 11 seconds distant; but then it is not the same eclipse that returns, as in the shorter period above-mentioned. Eclipses of the sun are more frequent than of the moon, because the sun's ecliptic limits are greater than the moon's; yet we have more visible eclipses of the moon than of the sun, because eclipses of the moon are seen from all parts of that hemisphere of the earth which is nearest her, and are equally great to each of those parts; but the sun's eclipses are visible only to that small portion of the hemisphere nearest him whereon the moon's shadow falls.

The moon's orbit being elliptical, and the earth in one of its focuses, she is once at her least distance from the earth, and once at her greatest, in every lunation. When the moon changes at her least distance from the earth, and so near the node that her dark shadow falls upon the earth, she appears big enough to cover the whole disk of the sun from that part on which her shadow falls; and the sun appears totally eclipsed there for some minutes; but when the moon changes at her greatest distance from the earth, and so near the node that her dark shadow is directed towards the earth, her diameter subtends a less angle than the sun's; and therefore she cannot hide his whole disk from any part of the earth, nor does her shadow reach it at that time; and to the place over which the point of her shadow hangs, the eclipse is annular, the sun's edge appearing like a luminous ring all around the body of the moon.

When the change happens within 17 degrees of the node, and the moon at her mean distance from the earth, the point of her shadow just touches the earth, and she eclipses the sun totally to that small spot whereon her shadow falls; but the darkness is not of a moment's continuance.

The moon's apparent diameter, when largest, exceeds the sun's, when least, only 1 minute 38 seconds of a degree; and in the greatest eclipse of the sun that can happen at any time and place, the total darkness continues no longer than whilst the moon is going 1 minute 38 seconds from the sun in her orbit, which is about 3 minutes and 13 seconds of an hour.

The moon's dark shadow covers only a spot on the earth's surface about 180 English miles broad, when the moon's diameter appears largest, and the sun's least; and the total darkness can extend no farther than the dark shadow covers. Yet the moon's partial shadow or penumbra may then cover a circular space 4900 miles in diameter, within all which the sun is more or less eclipsed, as the places are less or more distant from the centre of the penumbra. When the moon changes exactly in the node, the penumbra is circular on the earth at the middle of the general eclipse; because at that time it falls perpendicularly on the earth's surface; but at every other moment it falls obliquely, and will therefore be elliptical; and the more so, as the time is longer before or after the middle of the general eclipse; and then much greater portions of the earth's surface are involved in the penumbra.

When the penumbra first touches the earth, the general eclipse begins; when it leaves the earth, the general eclipse ends: from the beginning to the end the sun appears eclipsed in some part of the earth or other. When the penumbra touches any place, the eclipse begins at that place, and ends when the penumbra leaves it. When the moon changes in the node, the penumbra goes over the centre of the earth's disk as seen from the moon; and consequently, by describing the longest line possible on the earth, continues the longest upon it; namely, at a mean rate, 5 hours 50 minutes; more, if the moon be at her greatest distance from the earth, because she then moves slowest; less, if she be at her least distance, because of her quicker motion.

To make several of the above and other phenomena plainer, let S be the sun, E the earth, M the moon, Fig. 193, and AMP the moon's orbit. Draw the right line W e from the western side of the sun at W, touching the western side of the moon at e, and the earth at e; draw also the right line V d' from the eastern side of the sun at V, touching the eastern side of the moon at d', and the earth at e: the dark space c o d included between those lines is the moon's shadow, ending in a point at e, where it touches the earth; because in this case the moon is supposed to change at M in the middle between A the apogee, or farthest point of her orbit from the earth, and P the perigee, or nearest point to it. For, had the point P been at M, the moon had been nearer the earth; and her dark shadow at e would have covered a space upon it about 180 miles broad, and the sun would have been totally darkened, with some continuance: but had the point A been at M, the moon would have been farther from the earth, and her shadow would have ended in a point a little above e, and therefore the sun would have appeared like a luminous ring all around the moon. Draw the right lines WX d b and VX c g, touching the contrary sides of the sun and moon, and ending on the earth at a and b: draw also the right line SXM, from the centre of the sun's disk, through the moon's centre, to the earth; and suppose the two former lines WX d b and VX c g to revolve on the line SXM as an axis, and their points a and b will describe the limits of the penumbra TT on the earth's surface, including the large space a b a; within which the sun appears more or less eclipsed, as the places are more or less distant from the verge of the penumbra a b.

Draw the right line y 12 across the sun's disk, perpendicular to SXM the axis of the penumbra: then divide the line y 12 into twelve equal parts, as in the figure, for the twelve digits or equal parts of the sun's diameter; and at equal distances from the centre of the penumbra at e (on the earth's surface YY) to its edge a b, draw twelve concentric circles, marked with the numeral figures, 1 2 3 4 &c., and remember that the moon's motion in her orbit AMP is from west to east, as from s to t. Then,

To an observer on the earth at b, the eastern limb of the moon at d seems to touch the western limb of the sun at W, when the moon is at M; and the sun's eclipse begins at b, appearing as at A, fig. 203, at the left hand; but at the same moment of absolute time, to an observer at a in fig. 193, the western edge of the moon at c leaves the eastern edge of the sun at V, and the eclipse ends, as at the right hand C, fig. 203. At the very same instant, to all those who live on the circle marked 1 on the earth E, in fig. 193, the moon M cuts off or darkens a twelfth part of the sun S, and eclipses him one digit, as at 1 in fig. 203: to those who live on the circle marked 2 in fig. 198, the moon cuts off two twelfth parts of the sun, as at 2 in fig. 203: to those on the circle 3, three parts; and so Of calculation to the centre at 12 in fig. 198, where the sun is centrally eclipsed, as at B in the middle of fig. 203; under which figure there is a scale of hours and minutes, to show at a mean state how long it is from the beginning to the end of a central eclipse of the sun on the parallel of London; and how many digits are eclipsed at any particular time from the beginning at A to the middle at B, or the end at C. Thus, in 16 minutes from the beginning, the sun is two digits eclipsed; in an hour and five minutes, eight digits; and in an hour and 37 minutes, 12 digits.

By fig. 198, it is plain, that the sun is totally or centrally eclipsed but to a small part of the earth at any time, because the dark conical shadow e of the moon M falls but on a small part of the earth; and that the partial eclipse is confined at that time to the space included by the circle a b, of which only one half can be projected in the figure, the other half being supposed to be hid by the convexity of the earth E; and likewise, that no part of the sun is eclipsed to the large space Y Y of the earth, because the moon is not between the sun and any of that part of the earth; and therefore to all that part the eclipse is invisible.

The earth turns eastward on its axis, as from g to h, which is the same way that the moon's shadow moves; but the moon's motion is much swifter in her orbit from s to t; and therefore, although eclipses of the sun are of longer duration on account of the earth's motion on its axis than they would be if that motion was stopped, yet, in four minutes of time at most, the moon's swifter motion carries her dark shadow quite over any place that its centre touches at the time of greatest obscuration. The motion of the shadow on the earth's disk is equal to the moon's motion from the sun, which is about $30^\circ$ minutes of a degree every hour at a mean rate; but so much of the moon's orbit is equal to $30^\circ$ degrees of a great circle on the earth; and therefore the moon's shadow goes $30^\circ$ degrees, or $830$ geographical miles on the earth in an hour, or $30^\circ$ miles in a minute, which is almost four times as swift as the motion of a cannon-ball.

As seen from the sun or moon, the earth's axis appears differently inclined every day of the year, on account of keeping its parallelism throughout its annual course. In fig. 204, let EDON be the earth at the two equinoxes and the two solstices, NS its axis, N the north pole, S the south pole, AEQ the equator, T the tropic of Cancer, f the tropic of Capricorn, and ABC the circumference of the earth's enlightened disk as seen from the sun or new moon at these times. The earth's axis has the position NES at the vernal equinox, lying towards the right hand, as seen from the sun or new moon; its poles N and S being then in the circumference of the disk; and the equator and all its parallels seem to be straight lines, because their planes pass through the observer's eye looking down upon the earth from the sun or moon directly over E, where the ecliptic FG intersects the equator AE. At the summer solstice the earth's axis has the position NDS; and that part of the ecliptic FG, in which the moon is then new, touches the tropic of Cancer T at D. The north pole N at that time inclining $23^\circ$ degrees towards the sun, falls so many degrees within the earth's enlightened disk, because the sun is then vertical to D $23^\circ$ degrees north of the equator AEQ; and the equator, with all its parallels seem elliptic curves bending downward, or towards the south pole, as seen from the sun; which pole, together with $23^\circ$ degrees all round it, is hid behind the disk in the dark hemisphere of the earth. At the autumnal equinox, the earth's axis has the position NOS, lying to the left hand as seen from the sun or new moon, which are then vertical to O, where the ecliptic cuts the equator AEQ. Both poles now ly in the circumference of the disk, the north pole just going to disappear behind it, and the south pole just entering into it; and the equator with all its parallels, seem to be straight lines, because their planes pass through the observer's eye, as seen from the sun, and very nearly so as seen from the moon. At the winter solstice, the earth's axis has the position NNS when its south pole S inclining $23^\circ$ degrees towards the sun, falls $23^\circ$ degrees within the enlightened disk, as seen from the sun or new moon, which are then vertical to the tropic of Capricorn f, $23^\circ$ degrees south of the equator AEQ; and the equator, with all its parallels, seem elliptic curves bending upward; the north pole being as far hid behind the disk in the dark hemisphere as the south pole is come into the light. The nearer that any time of the year is to the equinoxes or solstices, the more it partakes of the phenomena relating to them.

Thus it appears, that from the vernal equinox to the autumnal, the north pole is enlightened; and the equator and all its parallels appear elliptical as seen from the sun, more or less curved as the time is nearer to, or farther from the summer solstice; and bending downwards, or towards the south pole; the reverse of which happens from the autumnal equinox to the vernal. A little consideration will be sufficient to convince the reader, that the earth's axis inclines towards the sun at the summer solstice; from the sun at the winter solstice; and sidewise to the sun at the equinoxes; but towards the right hand, as seen from the sun at the vernal equinox; and towards the left hand at the autumnal. From the winter to the summer solstice, the earth's axis inclines more or less to the right hand, as seen from the sun; and the contrary from the summer to the winter solstice.

The different positions of the earth's axis, as seen from the sun at different times of the year, affect solar eclipses greatly with regard to particular places; yea, of so far as would make central eclipses which fall at one earth's time of the year invisible if they fell at another, even though the moon should always change in the nodes, and at the same hour of the day; of which indefinitely various affections, we shall only give examples for the times of the equinoxes and solstices.

In the same diagram, let FG be part of the ecliptic, and IK, i k, i k, i k, part of the moon's orbit; both seen edgewise, and therefore projected into right lines; and let the intersections NODE be one and the same node at the above times, when the earth has the forementioned different positions; and let the spaces included by the circles P p p p be the penumbra at these times, as its centre is passing over the centre of the earth's disk. At the winter solstice, when the earth's axis has the position NNS, the centre of the penumbra P touches the tropic of Capricorn f in N at the middle of the general eclipse; but no part of the penumbra touches the tropic of Cancer T. At the summer solstice, when the earth's axis has the position ND (i D k being then part of the moon's orbit whose node is at D), the penumbra P has its centre at D, on the tropic of Cancer T, at the middle of the general eclipse, and then no part of it touches the tropic of Capricorn t. At the autumnal equinox, the earth's axis has the position NOS (i O k being then part of the moon's orbit), and the penumbra equally includes part of both tropics T and t at the middle of the general eclipse; at the vernal equinox it does the same, because the earth's axis has the position NES; but, in the former of these two last cases, the penumbra enters the earth at A, north of the tropic of Cancer T, and leaves it at m south of the tropic of Capricorn t; having gone over the earth obliquely southward, as its centre described the line AOM; whereas, in the latter case, the penumbra touches the earth at n, south of the equator EQ, and describing the line nEq (similar to the former line AOM in open space), goes obliquely northward over the earth, and leaves it at q, north of the equator.

In all these circumstances the moon has been supposed to change at noon in her descending node: Had she changed in her ascending node, the phenomena would have been as various the contrary way, with respect to the penumbra's going northward or southward over the earth. But because the moon changes at all hours, as often in one node as in the other, and at all distances from them both at different times as it happens, the variety of the phases of eclipses are almost innumerable, even at the same places; considering also how variously the same places are situated on the enlightened disk of the earth, with respect to the penumbra's motion, at the different hours when eclipses happen.

When the moon changes 17 degrees short of her descending node, the penumbra P just touches the northern part of the earth's disk, near the north pole N; and as seen from that place, the moon appears to touch the sun, but hides no part of him from sight. Had the change been as far short of the ascending node, the penumbra would have touched the southern part of the disk near the south pole S. When the moon changes 12 degrees short of the descending node, more than a third part of the penumbra P falls on the northern parts of the earth at the middle of the general eclipse: Had she changed as far past the same node, as much of the other side of the penumbra about P would have fallen on the southern part of the earth; all the rest in the expanse, or open space. When the moon changes 6 degrees from the node, almost the whole penumbra P falls on the earth at the middle of the general eclipse. And finally, when the moon changes in the node at N, the penumbra PN takes the longest course possible on the earth's disk; its centre falling on the middle thereof, at the middle of the general eclipse. The farther the moon changes from either node, within 17 degrees of it, the shorter is the penumbra's continuance on the earth, because it goes over a less portion of the disk, as is evident by the figure.

The nearer that the penumbra's centre is to the equator at the middle of the general eclipse, the longer is the duration of the eclipse at all those places where it is central; because, the nearer that any place is to the equator, the greater is the circle it describes by the earth's motion on its axis: and so, the place moving quicker, keeps longer in the penumbra, whose motion is the same way with that of the place, though faster, as has been already mentioned. Thus (see the earth at D and the penumbra at 12) whilst the point b in the polar circle abcd is carried from b to c by the earth's diurnal motion, the point d on the tropic of Cancer T is carried a much greater length from d to D; and therefore, if the penumbra's centre goes one time over c and another time over D, the penumbra will be longer in passing over the moving place d than it was in passing over the moving place b. Consequently, central eclipses about the poles are of the shortest duration; and about the equator, of the longest.

In the middle of summer, the whole frigid zone, included by the polar circle abcd, is enlightened; and if it then happens that the penumbra's centre goes over the north pole, the sun will be eclipsed much the same number of digits at a as at c; but whilst the penumbra moves eastward over c, it moves eastward over a; because, with respect to the penumbra, the motions of a and c are contrary: for c moves the same way with the penumbra towards d, but a moves the contrary way towards b; and therefore the eclipse will be of longer duration at c than at a. At a the eclipse begins on the sun's eastern limb, but at c on his western; at all places lying without the polar circles, the sun's eclipses begin on his western limb, or near it, and end on or near his eastern. At those places where the penumbra touches the earth, the eclipse begins with the rising sun, on the top of his western or uppermost edge; and at those places where the penumbra leaves the earth, the eclipse ends with the setting sun, on the top of his eastern edge, which is then the uppermost, just at its disappearing in the horizon.

If the moon were surrounded by an atmosphere of any considerable density, it would seem to touch the sun a little before the moon made her appulse to his edge, and we should see a little faintness on that edge before it were eclipsed by the moon: but as no such faintness has been observed, it seems plain, that the moon has no such atmosphere as that of the earth. The faint ring of light surrounding the sun in total eclipses, called by Cassini la chevelure du soleil, is said to be the atmosphere of the sun; because it has been observed to move equally with the sun, not with the moon. See n° 147.

Having been so prolix concerning eclipses of the sun, we shall drop that subject at present, and proceed to the doctrine of lunar eclipses; which, being more simple, may be explained in less time.

That the moon can never be eclipsed but at the time of her being full, and the reason why she is not eclipsed at every full, has been shown already. In fig. 198, let S be the sun, E the earth, RR the earth's shadow, and B the moon in opposition to the sun: In this situation the earth intercepts the sun's light in its way to the moon; and when the moon touches the earth's shadow at v, she begins to be eclipsed on her eastern limb x, and continues eclipsed until her western limb y leaves the shadow at w: at B she is in the middle of the shadow, and consequently in the middle of the eclipse.

The moon, when totally eclipsed, is not invisible if Of calculating eclipses, &c.

Why the moon is visible when eclipsed.

She be above the horizon and the sky be clear; but appears generally of a dusky colour, like tarnished copper, which some have thought to be the moon's native light. But the true cause of her being visible is the scattered beams of the sun, bent into the earth's shadow by going through the atmosphere; which, being more or less dense near the earth than at considerable heights above it, refracts or bends the sun's rays more inward, the nearer they are passing by the earth's surface, than those rays which go through higher parts of the atmosphere, where it is less dense according to its height, until it be so thin or rare as to lose its refractive power. Let the circle \( fgh \) be concentric to the earth, include the atmosphere whose refractive power vanishes at the heights \( f \) and \( i \); so that the rays \( Ww \) and \( Vv \) go on straight without suffering the least refraction; but all those rays which enter the atmosphere between \( f \) and \( k \), and between \( i \) and \( l \), on opposite sides of the earth, are gradually more bent inward as they go thro' a greater portion of the atmosphere, until the rays \( Wk \) and \( Vl \) touching the earth at \( m \) and \( n \), are bent so much as to meet at \( q \), a little short of the moon; and therefore the dark shadow of the earth is contained in the space \( mopn \), where none of the sun's rays can enter; all the rest \( RR \), being mixed by the scattered rays which are refracted as above, is in some measure enlightened by them; and some of those rays falling on the moon, give her the colour of tarnished copper, or of iron almost red hot. So that if the earth had no atmosphere, the moon would be as invisible in total eclipses as she is when new. If the moon were so near the earth as to go into its dark shadow, suppose about \( p \), she would be invisible during her stay in it; but visible before and after in the fainter shadow \( RR \).

When the moon goes thro' the centre of the earth's shadow, she is directly opposite to the sun; yet the moon has been often seen totally eclipsed in the horizon when the sun was also visible in the opposite part of it: for the horizontal refraction being almost 34 minutes of a degree, and the diameter of the sun and moon being each at a mean state but 32 minutes, the refraction causes both luminaries to appear above the horizon when they are really below it.

When the moon is full at 12 degrees from either of her nodes, she just touches the earth's shadow, but enters not into it. In fig. 204, let GH be the ecliptic, \( ef \) the moon's orbit where she is 12 degrees from the node at her fall, \( cd \) her orbit where she is 6 degrees from the node, \( ab \) her orbit where she is full in the node, \( AB \) the earth's shadow, and \( M \) the moon. When the moon describes the line \( ef \), she just touches the shadow, but does not enter into it; when she describes the line \( cd \), she is totally, though not centrally, immersed in the shadow; and, when she describes the line \( ab \), she passes by the node at \( M \) in the centre of the shadow, and takes the longest line possible, which is a diameter, through it; and such an eclipse being both total and central is of the longest duration, namely, 3 hours 57 minutes 6 seconds from the beginning to the end, if the moon be at her greatest distance from the earth; and 3 hours 37 minutes 26 seconds, if she be at her least distance. The reason of this difference is, that when the moon is farthest from the earth, she moves slowest; and when nearest to it, quickest.

The moon's diameter, as well as the sun's, is supposed to be divided into 12 equal parts, called digits; and of so many of these parts as are darkened by the earth's shadow, so many digits is the moon eclipsed. All that is eclipsed above 12 digits shows how far the shadow of the earth is over the body of the moon, on that edge to which she is nearest at the middle of the eclipse.

It is difficult to observe exactly either the beginning or ending of a lunar eclipse, even with a good telescope, because the earth's shadow is so faint and ill-defined about the edges, that when the moon is either just entering or leaving it, the obscuration of her limb is scarcely sensible; and therefore the nicest observers can hardly be certain to four or five seconds of time. But both the beginning and ending of solar eclipses are visibly instantaneous; for the moment that the edge of the moon's disk touches the sun's, his roundness seems a little broke on that part; and the moment she leaves it, he appears perfectly round again.

In astronomy, eclipses of the moon are of great use for ascertaining the periods of her motions; especially useful such eclipses as are observed to be alike in all her circumstances, and have long intervals of time between them. In geography, the longitudes of places are found by eclipses: but for this purpose eclipses of the moon are more useful than those of the sun, because they are more frequently visible, and the same lunar eclipse is of equal largeness and duration at all places where it is seen. In chronology, both solar and lunar eclipses serve to determine exactly the time of any past event; for there are so many particulars observable in every eclipse, with respect to its quantity, the places where it is visible (if of the sun), and the time of the day or night, that it is impossible there can be two solar eclipses in the course of many ages which are alike in all circumstances.

From the above explanation of the doctrine of eclipses, it is evident that the darkness at our Saviour's crucifixion was supernatural. For he suffered on the day on which the passover was eaten by the Jews, on which day it was impossible that the moon's shadow could fall on the earth; for the Jews kept the passover at the time of the sun last above four minutes in any place; whereas the darkness at the crucifixion lasted three hours, Matth. xxviii. 15, and overspread at least all the land of Judea.

The theory of eclipses being now, we hope, pretty plainly laid down, the construction of tables for their calculation will be understood from the following considerations.

The motions of the sun and moon are observed to be continually accelerated from the apogee to the perigee, and as gradually retarded from the perigee to the apogee; being slowest of all when the mean anomaly is nothing, and swiftest of all when it is six signs.

When the luminary is in its apogee or perigee, its place is the same as it would be if its motion were equable in all parts of its orbit. The supposed equable motions are called mean; the unequable are justly called the true.

The mean place of the sun or moon is always forwarder than the true place, whilst the luminary is moving from its apogee to its perigee; and the true place... place is always forwarder than the mean, whilst the luminary is moving from its perigee to its apogee. In the former case, the anomaly is always less than six signs; and in the latter case, more.

It has been found, by a long series of observations, that the sun goes through the ecliptic, from the ver- nal equinox to the same equinox again, in 365 days 5 hours 48 minutes 55 seconds; from the first star of Aries to the same star again, in 365 days 6 hours 9 minutes 24 seconds; and from his apogee to the same again, in 365 days 6 hours 14 minutes 0 seconds.— The first of these is called the solar year; the second the sidereal year; and the third the anomalistic year. So that the solar year is 20 minutes 29 seconds shorter than the sidereal; and the sidereal year is four minutes 36 seconds shorter than the anomalistic. Hence it ap- pears, that the equinoctial point, or intersection of the ecliptic and equator at the beginning of Aries, goes backward with respect to the fixed stars, and that the sun's apogee goes forward.

It is also observed, that the moon goes through her orbit, from any given fixed star to the same star again, in 27 days 7 hours 43 minutes 4 seconds at a mean rate; from her apogee to her apogee again, in 27 days 13 hours 18 minutes 43 seconds; and from the sun to the sun again, in 29 days 12 hours 44 minutes 3 seconds. This shows, that the moon's apogee moves forward in the ecliptic, and that at a much quicker rate than the sun's apogee does: since the moon is 5 hours 55 mi- nutes 39 seconds longer in revolving from her apogee to her apogee again, than from any star to the same star again.

The moon's orbit crosses the ecliptic in two oppo- site points, which are called her Nodes; and it is ob- served, that she revolves sooner from any node to the node again, than from any star to the star again, by 2 hours 38 minutes 27 seconds; which shows, that her nodes move backward, or contrary to the order of signs in the ecliptic.

The time in which the moon revolves from the sun to the sun again (or from change to change) is called a Lunation; which, according to Dr Pound's mean measures, would always consist of 29 days 12 hours 44 minutes 3 seconds 2 thirds 58 fourths, if the mo- tions of the sun and moon were always equable. Hence 12 mean lunations contain 354 days 8 hours 48 mi- nutes 36 seconds 35 thirds 40 fourths, which is 10 days 21 hours 11 minutes 23 seconds 24 thirds 20 fourths less than the length of a common Julian year, consis- ting of 365 days 6 hours; and 13 mean lunations con- tain 383 days 21 hours 32 minutes 39 seconds 38 thirds 38 fourths, which exceeds the length of a com- mon Julian year, by 18 days 15 hours 32 minutes 39 seconds 38 thirds 38 fourths.

The mean time of new moon being found for any given year and month, as suppose for March 1700, old style, if this mean new moon falls latter than the 11th day of March, then 12 mean lunations added to the time of this mean new moon will give the time of the mean new moon in March 1701, after having thrown off 365 days. But when the mean new moon happens to be before the 11th of March, we must add 13 mean lunations, in order to have the time of mean new moon in March the year following; always taking care to subtract 365 days in common years, and 366 Of calcula- days in leap-years, from the sum of this addition.

Thus, A. D. 1700, old style, the time of mean new moon in March was the 8th day, at 16 hours 11 minutes 25 seconds after the noon of that day (viz. at 11 minutes 25 seconds past four in the morning of the 9th day), according to common reckoning. To this we must add 13 mean lunations, or 383 days 21 hours 32 minutes 39 seconds 38 thirds 38 fourths, and the sum will be 392 days 13 hours 44 minutes 4 seconds 38 thirds 38 fourths: from which subtract 365 days, because the year 1701 is a common year, and there will remain 27 days 13 hours 44 minutes 4 seconds 38 thirds 38 fourths for the time of mean new moon in March, A. D. 1701.

Carrying on this addition and subtraction till A. D. 1703, we find the time of mean new moon in March that year to be on the 6th day, at 7 hours 21 mi- nutes 17 seconds 49 thirds 46 fourths past noon; to which add 13 mean lunations, and the sum will be 390 days 4 hours 53 minutes 57 seconds 28 thirds 20 fourths; from which subtract 365 days, because the year 1704 is a leap-year, and there will remain 24 days 4 hours 53 minutes 57 seconds 28 thirds 20 fourths, for the time of mean new moon in March, A. D. 1704.

In this manner was the first of the following tables constructed to seconds, thirds, and fourths; and then wrote out to the nearest seconds. The reason why we chose to begin the year with March, was to avoid the inconvenience of adding a day to the tabular time in leap-years after February, or subtracting a day there- from in January and February in those years; to which all tables of this kind are subject, which begin the year with January, in calculating the times of new or full moons.

The mean anomalies of the sun and moon, and the sun's mean motion from the ascending node of the moon's orbit, are set down in Table III. from 1 to 13 mean lunations. These numbers, for 13 lunations, be- ing added to the radical anomalies of the sun and moon, and to the sun's mean distance from the ascending node, at the time of mean new moon in March 1700 (Table I.), will give their mean anomalies, and the sun's mean dis- tance from the node, at the time of mean new moon in March 1701; and being added for 12 lunations to those for 1701, give them for the time of mean new moon in March 1702. And so on as far as you please to continue the table (which is here carried on to the year 1800), always throwing off 12 signs when their sum exceeds 12, and setting down the remainder as the proper quantity.

If the numbers belonging to A. D. 1700 (in Table I.) be subtracted from those belonging to 1800, we shall have their whole differences in 100 complete Julian years; which accordingly we find to be 4 days 8 hours 10 minutes 52 seconds 15 thirds 40 fourths, with respect to the time of mean new moon. These being added together 60 times (always taking care to throw off a whole lunation when the days exceed 29½) make up 60 centuries, or 6000 years, as in Table VI. which was carried on to seconds, thirds, and fourths; and then wrote out to the nearest seconds. In the same manner were the respective anomalies and the sun's distance from the node found, for these cen- turies. Of calculating years; and then (for want of room) wrote out only to the nearest minutes, which is sufficient in whole centuries. By means of these two tables, we may find the time of any mean new moon in March, together with the anomalies of the sun and moon, and the sun's distance from the node at these times, within the limits of 6000 years either before or after any given year in the 18th century; and the mean time of any new or full moon in any given month after March, by means of the third and fourth tables, within the same limits, as shown in the precepts for calculation.

Thus it would be a very easy matter to calculate the time of any new or full moon, if the sun and moon moved equably in all parts of their orbits. But we have already shown, that their places are never the same as they would be by equable motions, except when they are in apogee or perigee; which is, when their mean anomalies are either nothing, or six signs; and that their mean places are always forwarder than their true places, whilst the anomaly is less than six signs; and their true places are forwarder than the mean, whilst the anomaly is more.

Hence it is evident, that whilst the sun's anomaly is less than six signs, the moon will overtake him, or be opposite to him, sooner than she could if his motion were equable; and later whilst his anomaly is more than six signs. The greatest difference that can possibly happen between the mean and true time of new or full moon, on account of the inequality of the sun's motion, is 3 hours 43 minutes 28 seconds; and that is, when the sun's anomaly is either 3 signs 1 degree, or 8 signs 29 degrees; sooner in the first case, and later in the last. In all other signs and degrees of anomaly, the difference is gradually less, and vanishes when the anomaly is either nothing or six signs.

The sun is in his apogee on the 30th of June, and in his perigee on the 30th of December, in the present age: so that he is nearer the earth in our winter than in our summer.—The proportional difference of distance, deduced from the difference of the sun's apparent diameter at these times, is as 983 to 1017.

The moon's orbit is dilated in winter, and contracted in summer; therefore the lunations are longer in winter than in summer. The greatest difference is found to be 22 minutes 29 seconds; the lunations increasing gradually in length whilst the sun is moving from his apogee to his perigee, and decreasing in length whilst he is moving from his perigee to his apogee.—On this account, the moon will be later every time in coming to her conjunction with the sun, or being in opposition to him, from December till June, and sooner from June till December, than if her orbit had continued of the same size all the year round.

As both these differences depend on the sun's anomaly, they may be fitly put together into one table, and called The annual or first equation of the mean to the true syzygy, (see Table VII.) This equational difference is to be subtracted from the time of the mean syzygy when the sun's anomaly is less than six signs, and added when the anomaly is more.—At the greatest it is four hours 10 minutes 57 seconds, viz. 3 hours 48 minutes 28 seconds, on account of the sun's unequal motion, and 22 minutes 29 seconds, on account of the dilatation of the moon's orbit.

This compound equation would be sufficient for reducing the mean time of new or full moon to the true of time thereof, if the moon's orbit were of a circular form, and her motion quite equable in it. But the moon's orbit is more elliptical than the sun's, and her motion in it is so much the more unequal. The difference is so great, that she is sometimes in conjunction with the sun, or in opposition to him, sooner by 9 hours 47 minutes 54 seconds, than she would be if her motion were equable; and at other times as much later. The former happens when her mean anomaly is 9 signs 4 degrees, and the latter when it is 2 signs 26 degrees. See Table IX.

At different distances of the sun from the moon's apogee, the figure of the moon's orbit becomes different. It is longest of all, or most eccentric, when the sun is in the same sign and degree either with the moon's apogee or perigee; shortest of all, or least eccentric, when the sun's distance from the moon's apogee is either three signs or nine signs; and at a mean state when the distance is either 1 sign 15 degrees, 4 signs 15 degrees, 7 signs 15 degrees, or 10 signs 15 degrees. When the moon's orbit is at its greatest eccentricity, her apogal distance from the earth's centre is to her perigal distance therefrom, as 1067 is to 933; when least eccentric, as 1043 is to 957; and when at the mean state, as 1055 is to 945.

But the sun's distance from the moon's apogee is equal to the quantity of the moon's mean anomaly at the time of new moon, and by the addition of six signs it becomes equal in quantity to the moon's mean anomaly at the time of full moon. Therefore, a table may be constructed so as to answer to all the various inequalities depending on the different eccentricities of the moon's orbit, in the syzygies, and called The second equation of the mean to the true syzygy, (See Table IX.) and the moon's anomaly, when equated by Table VIII. may be made the proper argument for taking out this second equation of time; which must be added to the former equated time, when the moon's anomaly is less than six signs, and subtracted when the anomaly is more.

There are several other inequalities in the moon's motion, which sometimes bring on the true syzygy a little sooner, and at other times keep it back a little later, than it would otherwise be; but they are so small, that they may be all omitted except two; the former of which (see Table X.) depends on the difference between the anomalies of the sun and moon in the syzygies, and the latter (see Table XI.) depends on the sun's distance from the moon's nodes at these times.—The greatest difference arising from the former is 4 minutes 58 seconds; and from the latter, 1 minute 34 seconds.

The tables here inserted being calculated by Mr Fer-Dingon according to the methods already given, he gives the following directions for their use.

To calculate the true Time of New or Full Moon.

Precept I. If the required time be within the limits of the 18th century, write out the mean time of new moon in March, for the proposed year, from Table I. in the old style, or from Table II. in the new; together with the mean anomalies of the sun and moon, and the sun's mean distance from the moon's ascending node. If you want the time of full moon in March, add the half lunation at the foot of Table III. with... its anomalies, &c. to the former numbers, if the new moon falls before the 15th of March; but if it falls after, subtract the half lunation, with the anomalies, &c. belonging to it, from the former numbers, and write down the respective sums or remainders.

II. In these additions or subtractions, observe, that 60 seconds make a minute, 60 minutes make a degree, 30 degrees make a sign, and 12 signs make a circle. When you exceed 12 signs in addition, reject 12, and set down the remainder. When the number of signs to be subtracted is greater than the number you subtract from, add 12 signs to the lesser number, and then you will have a remainder to set down. In the tables signs are marked thus $^s$, degrees thus $^o$, minutes thus $'$, and seconds thus $''$.

III. When the required new or full moon is in any given month after March, write out as many lunations with their anomalies, and the sun's distance from the node from Table III. as the given month is after March, setting them in order below the numbers taken out for March.

IV. Add all these together, and they will give the mean time of the required new or full moon, with the mean anomalies and sun's mean distance from the ascending node, which are the arguments for finding the proper equations.

V. With the number of days added together, enter Table IV. under the given month; and against that number you have the day of mean new or full moon in the left-hand column, which set before the hours, minutes, and seconds, already found.

But (as it will sometimes happen) if the said number of days fall short of any in the column under the given month, add one lunation and its anomalies, &c. (from Table III.) to the foresaid sums, and then you will have a new sum of days wherewith to enter Table IV. under the given month, where you are sure to find it the second time, if the first falls short.

VI. With the signs and degrees of the sun's anomaly, enter Table VII. and therewith take out the annual or first equation for reducing the mean syzygy to the true; taking care to make proportions in the table for the odd minutes and seconds of anomaly, as the table gives the equation only to whole degrees.

Observe, in this and every other case of finding equations, that if the signs are at the head of the table, their degrees are at the left hand, and are reckoned downwards; but if the signs are at the foot of the table, their degrees are at the right hand, and are counted upward; the equation being in the body of the table, under or over the signs, in a collateral line with the degrees. The titles Add or Subtract at the head or foot of the tables where the signs are found, show whether the equation is to be added to the mean time of new or full moon, or to be subtracted from it. In this table, the equation is to be subtracted, if the signs of the sun's anomaly are found at the head of the table; but it is to be added, if the signs are at the foot.

VII. With the signs and degrees of the sun's mean anomaly, enter table VIII. and take out the equation of the moon's mean anomaly; subtract this equation from her mean anomaly, if the signs of the sun's anomaly be at the head of the table, but add it if they are at the foot; the result will be the moon's equated anomaly, with which enter Table IX. and take out the second equation for reducing the mean to the true time of new or full moon; adding this equation, if the signs of the moon's anomaly are at the head of the table, but subtracting it if they are at the foot; and the result will give you the mean time of the required new or full moon twice equated, which will be sufficiently near for common almanacs.—But when you want to calculate an eclipse, the following equations must be used: thus,

VIII. Subtract the moon's equated anomaly from the sun's mean anomaly, and with the remainder in signs and degrees enter Table X. and take out the third equation, applying it to the former equated time, as the titles Add or Subtract do direct.

IX. With the sun's mean distance from the ascending node enter Table XI. and take out the equation answering to that argument, adding it to, or subtracting it from, the former equated time, as the titles direct, and the result will give the time of new or full moon, agreeing with well regulated clocks or watches very near the truth. But to make it agree with the solar, or apparent time, you must apply the equation of natural days, taken from an equation-table, as it is leap-year, or the first, second, or third after. This, however, unless in very nice calculations, needs not be regarded, as the difference between true and apparent time is never very considerable.

The method of calculating the time of any new or full moon without the limits of the 18th century, will be shown further on. And a few examples compared with the precepts will make the whole work plain.

N.B. The tables begin the day at noon, and reckon forward from thence to the noon following.—Thus, March the 31st, at 22 ho. 30 min. 25 sec. of tabular time is April 1st (in common reckoning) at 30 min. 25 sec. after 10 o'clock in the morning. ### Example I

**Required the true time of New Moon in April 1764, New Style?**

| By the Precepts | New Moon | Sun's Anomaly | Moon's Anomaly | Sun from Node | |-----------------|----------|---------------|---------------|--------------| | D. H. M. S. | | | | | | March 1764, | 2 8 55 | 36 | 8 2 20 | 10 13 35 | | Add 1 Lunation, | 29 12 44 | 3 | 0 29 6 | 19 25 49 | | Mean New Moon, | 31 21 39 | 9 | 1 26 | 19 11 | | First Equation, | + 4 10 | 40 | 11 10 | 59 18 | | Time once equated, | 32 1 50 | 19 | 9 20 | 27 11 | | Second Equation,| - 3 24 | 49 | Arg. 3d equation | Arg. 2d equation |

So the true time is 22 h. 30 min. 25 sec. after the noon of the 31st March; that is, April 1st, at 30 min. 25 sec. after ten in the morning. But the apparent time is 26 min. 37 sec. after ten in the morning.

### Example II

**The true time of the Full Moon in May 1762, New Style?**

| By the Precepts | New Moon | Sun's Anomaly | Moon's Anomaly | Sun from Node | |-----------------|----------|---------------|---------------|--------------| | D. H. M. S. | | | | | | March 1762, | 24 15 18 | 24 | 8 23 48 | 16 1 23 59 | | Add 2 Lunations,| 59 1 28 6| 1 | 28 12 | 39 1 21 38 | | New Moon, May, | 22 16 46 | 30 | 10 22 | 0 55 3 15 37 | | Subt. ½ Lunation,| 14 18 22 | 2 | 0 14 33 | 10 6 12 54 | | Full moon, May, | 7 22 24 28| 10 | 7 27 | 45 9 2 42 | | First Equation, | + 3 16 | 36 | 9 3 57 | 18 1 14 36 | | Time once equated, | 8 1 41 | 4 | 1 3 30 | 27 9 3 57 | | Second Equation,| - 9 47 | 53 | Arg. 3d equation | Arg. 2d equation |

Anf. May 7th at 15 h. 50 min. 50 sec. past noon, viz. May 8th at 3 h. 50 min. 50 sec. in the morning.

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To calculate the time of New and Full Moon in a given year and month of any particular century between the Christian era and the 18th century.

**Precept I.** Find a year of the same number in the 18th century with that of the year in the century proposed, and take out the mean time of new moon in March, old style, for that year, with the mean anomalies and sun's mean distance from the node at that time, as already taught.

II. Take as many complete centuries of years from Table VI. as, when subtracted from the above-faid year in the 18th century, will answer to the given year; and take out the first mean new moon and its anomalies, &c.

belonging to the said centuries, and set them below those taken out for March in the 18th century.

III. Subtract the numbers belonging to these centuries from those of the 18th century, and the remainders will be the mean time and anomalies, &c. of new moon, in March, in the given year of the century proposed,—Then, work in all respects for the true time of new or full moon, as shown in the above precepts and examples.

IV. If the days annexed to these centuries exceed the number of days from the beginning of March taken out in the 18th century, add a lunation and its anomalies, &c. from Table III. to the time and anomalies of new moon in March, and then proceed in all respects as above.—This circumstance happens in Example V. **ASTRONOMY**

**EXAMPLE III.** Required the true time of Full Moon in April, Old Style, A.D. 30?

From 1730 subtract 1700 (or 17 centuries) and there remains 30.

By the Precepts.

| New Moon | Sun's Anomaly | Moon's Anomaly | Sun from Node | |----------|---------------|----------------|--------------| | D. H. M. S. | s. o' " | s. o' " | s. o' " | | March 1730, Add ½ Lunation, | 7 12 34 16 | 8 18 4 31 | 9 0 32 17 | 1 23 17 16 | | Full moon, 1700 years subtr. | 22 6 56 18 | 9 2 37 41 | 3 13 26 47 | 2 8 37 23 | | Full March A.D. 30, Add 1 Lunation, | 7 13 19 36 | 9 3 51 41 | 4 13 50 47 | 9 9 14 23 | | Full Moon, April, First Equation, | 6 2 3 39 | 10 2 58 0 | 5 9 39 47 | 9 9 54 37 | | Time once equated, Second Equation, | 6 5 31 43 | 4 21 59 20 | 5 10 58 40 | Arg. 3d equation, Arg. 2d equation, equation. | | Time twice equated, Third Equation, | 6 8 29 31 | — 2 54 | | | | Time thrice equated, Fourth Equation, | 6 8 26 37 | — 1 33 | | | | True Full Moon, April, | 6 8 25 4 | | | |

To calculate the true time of New or Full Moon in any given year and month before the Christian era.

**Precept I.** Find a year in the 18th century, which being added to the given number of years before Christ diminished by one, shall make a number of complete centuries.

II. Find this number of centuries in Table VI. and

**EXAMPLE IV.** Required the true time of New Moon in May, Old Style, the year before Christ 585?

The years 584 added to 1716, make 2300, or 23 centuries.

By the Precepts.

| New Moon | Sun's Anomaly | Moon's Anomaly | Sun from Node | |----------|---------------|----------------|--------------| | D. H. M. S. | s. o' " | s. o' " | s. o' " | | March 1716, 2300 years subtract | 11 17 33 29 | 8 22 50 39 | 4 4 14 2 | 4 27 17 5 | | March before Christ 585, Add 3 Lunations, | 11 5 57 53 | 11 19 47 0 | 1 5 59 0 | 7 25 27 0 | | May before Christ 585, First equation, | 0 11 35 36 | 9 3 3 39 | 2 28 15 2 | 9 1 50 5 | | Time once equated, Second Equation, | 8 14 12 9 | 2 27 18 58 | 2 17 27 1 | 3 2 0 42 | | Time twice equated, Third Equation, | 28 1 47 45 | 0 0 22 37 | 5 15 42 3 | 0 3 50 47 | | Time thrice equated, Fourth equation, | 28 4 1 9 | + 1 9 | | | | True new moon, | 28 4 2 18 | + 1 12 | | |

These Tables are calculated for the meridian of London; but they will serve for any other place, by subtracting four minutes from the tabular time, for every degree that the meridian of the given place is westward of London, or adding four minutes for every degree that the meridian of the given place is eastward: as in

So the true time was May 28th, at 2 minutes 30 seconds past four in the afternoon. **EXAMPLE V.**

Required the true time of Full Moon at Alexandria in Egypt in September, Old Style, the year before Christ 201.

The years 200 added to 1800, make 2000, or 20 centuries.

| By the Precepts. | New Moon. | Sun's Anomaly. | Moon's Anomaly. | Sun from Node. | |-----------------|-----------|----------------|-----------------|---------------| | | D. H. M. S.| s o ' " | s o ' " | s o ' " | | March 1800, | 13 0 22 | 8 23 19 | 5 10 7 52 | 3 11 3 58 | | Add 1 Lunation, | 29 12 44 | 0 29 6 | 19 0 25 49 | 0 1 0 40 | | From the sum, | 42 13 6 | 9 22 26 | 14 1 3 41 | 36 0 4 38 | | Subtract 2000 | 27 18 9 | 0 8 50 | 0 0 15 42 | 0 6 27 45 | | N. M. bef. Chr. | 14 18 57 | 9 13 36 | 14 10 17 59 | 36 5 6 53 | | Add 6 Lunations,| 177 4 24 | 5 24 37 | 56 5 4 54 | 3 6 4 1 24 | | Add half Lunations, | 14 18 22 | 0 14 33 | 10 6 12 54 | 30 0 15 20 | | Full Moon, | 22 17 43 | 3 22 47 | 20 10 5 48 | 9 11 26 15 | | September, | 3 52 | 6 10 4 19 | 55 1 28 | 14 | | First Equation, | | | | | | Time once equated, | 22 13 51 | 5 18 27 | 25 10 4 19 | 55 | | Second Equation,| 8 25 | Arg. 3d equation.| Arg. 2d equation.| |

Thus it appears, that the true time of Full Moon at Alexandria, in September, old style, the year before Christ 201, was the 22nd day, at 26 minutes 28 seconds after seven in the evening.

**EXAMPLE VI.**

Required the true time of Full Moon at Babylon in October, Old Style, the 4008 year before the first year of Christ, or 4007 before the year of his birth?

The years 4007 added to 1793, make 5800, or 58 centuries.

| By the Precepts. | New Moon. | Sun's Anomaly. | Moon's Anomaly. | Sun from Node. | |-----------------|-----------|----------------|-----------------|---------------| | | D. H. M. S.| s o ' " | s o ' " | s o ' " | | March 1793, | 30 9 13 | 9 10 16 | 11 8 7 37 | 58 7 6 18 | | Subtract 5800 | 15 12 38 | 7 10 21 | 35 0 6 24 | 43 0 9 13 | | N. M. bef. Chr. | 14 20 35 | 10 18 41 | 11 1 12 54 | 58 9 23 17 | | Add 7 Lunations,| 206 17 8 | 6 23 44 | 15 6 0 43 | 3 7 4 41 | | Add half Lunations, | 14 18 22 | 0 14 33 | 10 6 12 54 | 30 0 15 20 | | Full Moon, | 22 8 6 | 5 26 58 | 36 1 26 32 | 31 5 13 19 | | October, | 13 26 | 1 26 27 | 26 5 5 | | | First Equation, | | | | | | Time once equated, | 22 7 52 | 4 0 31 | 10 1 26 27 | 26 | | Second Equation,| 8 29 | Arg. 3d equation.| Arg. 2d equation.| |

So that, on the meridian of London, the true time was October 23rd, at 17 minutes 5 seconds past four in the morning; but at Babylon, the true time was October 23rd, at 42 minutes 46 seconds past six in the morning.—This is supposed by some to have been the year of the creation. To calculate the true time of New or Full Moon in any given year and month after the 18th century.

Precept I. Find a year of the same number in the 18th century with that of the year proposed, and take out the mean time and anomalies, &c., of new moon in March, old style, for that year, in Table I.

II. Take so many years from Table VI. as when added to the abovementioned year in the 18th century will answer to the given year in which the new or full moon is required; and take out the first new moon, with its anomalies for these complete centuries.

III. Add all these together, and then work in all respects as above shown, only remember to subtract a lunation and its anomalies, when the above said addition carries the new moon beyond the 31st of March; as in the following example.

**Example VII.**

Required the true time of New Moon in July, Old Style, A.D. 2180?

Four centuries (or 400 years) added to A.D. 1780, make 2180.

By the Precepts.

| New Moon | Sun's Anomaly | Moon's Anomaly | Sun from Node | |----------|---------------|----------------|--------------| | D. H. M. S. | s o ' " | s o ' " | s o ' " | | March 1780, Add 400 years, | 23 23 1 44 | 9 4 18 13 | 1 21 7 47 | 10 18 21 1 | | From the sun Subtract 1 Lunation, | 17 8 43 29 | 0 13 24 0 | 10 1 28 0 | 6 17 49 0 | | New Moon March 2180, Add 4 Lunations, | 41 7 45 13 | 9 17 42 13 | 11 22 35 47 | 6 10 1 | | New Moon July 2180, First Equation, | 29 12 44 3 | 0 29 6 19 | 0 25 49 0 | 0 40 14 | | Time once equated, Second Equation, | 11 19 1 10 | 8 18 35 54 | 10 26 46 47 | 4 5 29 47 | | Time twice equated, Third Equation, | 18 2 56 12 | 3 26 25 17 | 3 13 16 2 | 4 2 40 56 | | Time thrice equated, Fourth Equation, | 7 21 57 22 | 0 15 1 11 | 2 10 2 49 | 8 8 10 43 | | True time, July 8th, at 22 minutes 55 seconds past six in the evening. |

In keeping by the old style, we are always sure to be right, by adding or subtracting whole hundreds of years to or from any given year in the 18th century. But in the new style we may be very apt to make mistakes, on account of the leap year's not coming in regularly every fourth year; and therefore, when we go without the limits of the 18th century, we had best keep to the old style, and at the end of the calculation reduce the time to the new. Thus, in the 22d century there will be fourteen days difference between the styles; and therefore the true time of new moon in this last example being reduced to the new style, will be the 22d of July, at 22 minutes 55 seconds past six in the evening.

To calculate the true place of the Sun for any given moment of time.

Precept I. In Table XII. find the next lesser year in number to that in which the Sun's place is sought, and write out his mean longitude and anomaly answering thereto; to which add his mean motion and anomaly for the complete residue of years, months, days, hours, minutes, and seconds, down to the given time, and this will be the Sun's mean place and anomaly at that time, in the old style, provided the said time be in any year after the Christian era. See the first following example.

II. Enter Table XIII. with the Sun's mean anomaly, and making proportions for the odd minutes and seconds thereof, take out the equation of the Sun's centre; which, being applied to his mean place as the title Add or Subtract directs, will give his true place or longitude from the vernal equinox, at the time for which it was required.

III. To calculate the Sun's place for any time in a given year before the Christian era, take out his mean longitude and anomaly for the first year thereof, and from these numbers subtract the mean motions and anomalies for the complete hundreds or thousands next above the given year; and to the remainders, add those for the residue of years, months, &c., and then work in all respects as above. See the second example following. **EXAMPLE I.**

Required the Sun's true place, March 20th Old Style, 1764, at 22 hours 30 minutes 25 seconds past noon?

In common reckoning, March 21st, at 10 hours 30 minutes 25 seconds in the forenoon.

| Sun's Longitude | Sun's Anomaly | |-----------------|--------------| | 9° 0' 0" | 8° 0' 0" | | 9° 20' 43" | 6° 13' 1" | | 9° 27' 12" | 11° 29' 26" | | 11° 29' 17" | 11° 29' 14" | | 11° 28' 9" | 11° 28' 9" | | 20° 41' 55" | 20° 41' 55" | | 54° 13' | 54° 13' | | 1° 14' | 1° 14' | | 1° | 1° |

Sun's mean place at the given time

Equation of the Sun's centre, add

Sun's true place at the same time

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**EXAMPLE II.**

Required the Sun's true place, October 23rd, Old Style, at 16 hours 57 minutes past noon, in the 4008th year before the year of Christ; which was the 4007th before the year of his birth, and the year of the Julian period 706.

By the precepts.

| Sun's Longitude | Sun's Anomaly | |-----------------|--------------| | 9° 7' 53" | 6° 28' 48" | | 1° 7' 46" | 10° 13' 25" | | 8° 0' 6" | 8° 15' 23" | | 6° 48" | 11° 21' 37" | | 3° 36" | 11° 29' 15" | | 0° 5' | 11° 29' 53" | | 8° 29' 4" | 8° 29' 4" | | 22° 40' 12" | 22° 40' 12" | | 39° 26" | 39° 26" | | 2° 20" | 2° 20" |

Sun's mean place at the given time

Equation of the sun's centre subtract

Sun's true place at the same time

So that in the meridian of London, the sun was then just entering the sign Libra, and consequently was upon the point of the autumnal equinox.

If to the above time of the autumnal equinox at London, we add 2 hours 25 minutes 41 seconds for the longitude of Babylon, we shall have for the time of the same equinox, at that place, October 23rd, at 19 hours 22 minutes 41 seconds; which, in the common way of reckoning, is October 24th, at 22 minutes 41 seconds past seven in the morning.

And it appears by Example VI. that in the same year the true time of full moon at Babylon was October 23rd, at 42 minutes 46 seconds after six in the morning; so that the autumnal equinox was on the day next after the day of full moon.—The dominical letter for that year was G, and consequently the 24th of October was on a Wednesday.

To find the Sun's distance from the Moon's ascending node, at the time of any given new or full moon: and consequently, to know whether there is an eclipse at that time or not.

The sun's distance from the moon's ascending node is the argument for finding the moon's fourth equation in the lyzygics; and therefore it is taken into all the foregoing examples in finding the times thereof. Thus, at the time of mean new moon in April 1764, the sun's mean distance from the ascending node is 0° 5' 35" 2". See Example I. p. 562. The descending node is opposite to the ascending one, and they are just six signs distant from each other.

When the sun is within 17 degrees of either of the nodes at the time of new moon, he will be eclipsed at that time; and when he is within 12 degrees of either of the nodes at the time of full moon, the moon will be then eclipsed. Thus we find, that there will be an eclipse of the sun at the time of new moon in April 1764.

But the true time of that new moon comes out by the equations to be 50 minutes 46 seconds later than the mean time thereof, by comparing these times in the above example; and therefore we must add the sun's motion from the node during that interval to the above mean distance $0^\circ 5^\prime 35\frac{1}{2}''$, which motion is found in Table XII. for 50 minutes 46 seconds, to be $2^\prime 12\frac{1}{2}''$. And to this we must apply the equation of the sun's mean distance from the node in Table XV. found by the sun's anomaly, which, at the mean time of new moon in Example I. is $9^\circ 1^\prime 26\frac{1}{2}''$; and then we shall have the sun's true distance from the node, at the true time of new moon, as follows:

| Sun from Node. | s o n | |----------------|------| | At the mean time of new moon in April 1764 | 0 5 35 2 | | Sun's motion from the node for 50 minutes 46 seconds | 2 10 2 | | Sun's mean distance from node at true new moon | 0 5 37 14 | | Equation of mean distance from node, add | 2 5 0 | | Sun's true distance from the ascending node | 0 7 42 14 |

Which being far within the above limit of 17 degrees, shows that the sun must then be eclipsed.

And now we shall show how to project this, or any other eclipse, either of the sun or moon.

To project an Eclipse of the Sun.

In order to this, we must find the following elements by means of the tables.

1. The true time of conjunction of the sun and moon; and at that time. 2. The semidiameter of the earth's disk, as seen from the moon, which is equal to the moon's horizontal parallax. 3. The sun's distance from the solstitial colure to which he is then nearest. 4. The sun's declination. 5. The angle of the moon's visible path with the ecliptic. 6. The moon's latitude. 7. The moon's true horary motion from the sun. 8. The sun's semidiameter. 9. The moon's semidiameter of the penumbra.

We shall now proceed to find these elements for the sun's eclipse in April 1764.

To find the true time of new moon. This, by Example I. p. 562, is found to be on the first day of the said month, at 30 minutes 25 seconds after ten in the morning.

2. To find the moon's horizontal parallax, or semidiameter of the earth's disk, as seen from the moon. Enter Table XVII. with the signs and degrees of the moon's anomaly (making proportions, because the anomaly is calculated in the table only to every 6th degree), and thereby take out the moon's horizontal parallax; which for the above time, answering to the anomaly $11^\circ 9^\prime 24\frac{1}{2}''$, is $54\frac{1}{2}''$.

3. To find the sun's distance from the nearest solstice, viz. the beginning of Cancer, which is $3^\circ$ or $90^\circ$ from the beginning of Aries. It appears by Example I. on p. 566 (where the sun's place is calculated to the above time of new moon), that the sun's longitude from the beginning of Aries is then $0^\circ 12^\prime 10\frac{1}{2}''$; that is, the sun's place at that time is $9^\circ$ Aries, $12^\prime 10\frac{1}{2}''$.

Therefore from

| Subtracted the sun's longitude or place | 0 12 10 12 | |----------------------------------------|------------| | Remains the sun's distance from the solstice | 2 17 49 48 |

Or $7^\circ 49\frac{1}{2}''$; each sign containing 30 degrees.

4. To find the sun's declination. Enter Table XIV. with the signs and degrees of the sun's true place, viz. $0^\circ 12^\prime$, and making proportions for the $10^\prime 12\frac{1}{2}''$, take out the sun's declination answering to his true place, and it will be found to be $4^\circ 49\frac{1}{2}''$ north.

5. To find the moon's latitude. This depends on her distance from her ascending node, which is the same as the sun's distance from it at the time of new moon; and is thereby found in Table XVI.

But we have already found, that the sun's equated distance from the ascending node, at the time of new moon in April 1764, is $0^\circ 7^\prime 42\frac{1}{2}''$. See above.

Therefore, enter Table XVI. with 7 signs at the top, and 7 and 8 degrees at the left hand, and take out $36^\circ$ and $39^\circ$, the latitude for $7^\circ$; and $41^\circ 51\frac{1}{2}''$, the latitude for $8^\circ$; and by making proportions between these latitudes for the $42^\frac{1}{2}''$, by which the moon's distance from the node exceeds 7 degrees; her true latitude will be found to be $40^\circ 18\frac{1}{2}''$ north ascending.

6. To find the moon's true horary motion from the sun. With the moon's anomaly, viz. $11^\circ 9^\prime 24\frac{1}{2}''$, Table XVII. and take out the moon's horary motion; which, by making proportions in that Table, will be found to be $30^\prime 22\frac{1}{2}''$. Then, with the sun's anomaly, $9^\circ 1^\prime 26\frac{1}{2}''$, take out his horary motion $2^\prime 28\frac{1}{2}''$ from the same table; and subtracting the latter from the former, there will remain $27^\prime 54\frac{1}{2}''$ for the moon's true horary motion from the sun.

7. To find the angle of the moon's visible path with the ecliptic. This, in the projection of eclipses, may be always rated at $5^\circ 35\frac{1}{2}''$, without any sensible error.

8. To find the semidiameters of the sun and moon. These are found in the same table, and by the same arguments, as their horary motions. In the present case, the sun's anomaly gives his semidiameters $16^\circ 6\frac{1}{2}''$, and the moon's anomaly gives her semidiameter $14^\circ 57\frac{1}{2}''$.

9. To find the semidiameter of the penumbra. Add the moon's semidiameter to the sun's, and their sum will be the semidiameter of the penumbra, viz. $31^\prime 3\frac{1}{2}''$.

Now collect these elements, that they may be found the more readily when they are wanted in the construction of this eclipse. 1. True time of new moon in April, 1764

2. Semidiameter of the earth's disk

3. Sun's distance from the nearest solst.

4. Sun's declination, north

5. Moon's latitude, north ascending

6. Moon's horary motion from the sun

7. Angle of the moon's visible path with the ecliptic

8. Sun's semidiameter

9. Moon's semidiameter

10. Semidiameter of the penumbra

To project an Eclipse of the Sun geometrically.

Make a scale of any convenient length, as AC, and divide it into as many equal parts as the earth's semidisk contains minutes of a degree; which, at the time of the eclipse in April 1764, is 54' 53". Then, with the whole length of the scale as a radius, describe the semicircle AMB upon the centre C; which semicircle shall represent the northern half of the earth's enlightened disk, as seen from the sun.

Upon the centre C raise the straight line CH, perpendicular to the diameter ACB; so ACB shall be a part of the ecliptic, and CH its axis.

Being provided with a good sector, open it to the radius CA in the line of chords; and taking from thence the chord of 23½ degrees in your compasses, set it off both ways from H to g and to b, in the periphery of the semidisk; and draw the straight line gVb, in which the north pole of the disk will be always found.

When the sun is in Aries, Taurus, Gemini, Cancer, Leo, and Virgo, the north pole of the earth is enlightened by the sun; but whilst the sun is in the other five signs, the south pole is enlightened, and the north pole is in the dark.

And when the sun is in Capricorn, Aquarius, Pisces, Aries, Taurus, and Gemini, the northern half of the earth's axis CXII P lies to the right hand of the axis of the ecliptic, as seen from the sun; and to the left hand, whilst the sun is in the other five signs.

Open the sector till the radius (or distance of the two 90's) of the fines be equal to the length of Vb, and take the fine of the sun's distance from the folioce (77° 49' 48") as nearly as you can guess, in your compasses, from the line of the fines, and set off that distance from V to P in the line gVb, because the earth's axis lies to the right hand of the axis of the ecliptic in this case, the sun being in Aries; and draw the straight line CXII P for the earth's axis, of which P is the north pole. If the earth's axis had lain to the left hand from the axis of the ecliptic, the distance VP would have been set off from V towards g.

To draw the parallel of latitude of any given place, as suppose London, or the path of that place on the earth's enlightened disk as seen from the sun, from sunrise till sunset, take the following method.

Subtract the latitude of London, 51° 1' from 90°, and the remainder 38° 57' will be the co-latitude, which take in your compasses from the line of chords, making No 35.

CA or CB the radius, and set it from b (where the earth's axis meets the periphery of the disk) to VI and VI, and draw the occult or dotted line VI K VI. Then, from the points where this line meets the earth's disk, set off the chord of the sun's declination 4° 49' to D and F, and to E and G, and connect these points by the two occult lines FXII G and DLE.

Bisect IX XII in K, and through the point K draw the black line VI K VI. Then making CB the radius of a line of fines on the sector, take the colatitude of London 38° 57' from the fines in your compasses, and set it both ways from K to VI and VI. These hours will be just in the edge of the disk at the equinoxes, but at no other time in the whole year.

With the extent K VI taken into your compasses, set one foot in K (in the black line below the occult one) as a centre, and with the other foot describe the semicircle VI 7 8 9 10, &c., and divide it into 12 equal parts. Then from these points of division draw the occult lines 7p, 8q, 9r, &c., parallel to the earth's axis CXII P.

With the small extent K XII as a radius, describe the quadrant arc XII½, and divide it into five equal parts, as XII a, ab, bc, cd, de, ef; and through the division-points a, b, c, d, e, draw the occult lines VII e V, VII i d IV, IX c III, X b II, and XI a I, all parallel to VI K VI, and meeting the former occult lines 7p, 8q, &c., in the points VII VIII IX XXI, V IV III II and I: which points shall mark the several situations of London on the earth's disk, at these hours respectively, as seen from the sun; and the elliptic curve VI VII VIII, &c., being drawn through these points, shall represent the parallel of latitude, or path of London on the disk, as seen from the sun, from its rising to its setting.

N.B. If the sun's declination had been south, the diurnal path of London would have been on the upper side of the line VI K VI, and would have touched the line DLE in L. It is requisite to divide the horary spaces into quarters (as some are in the figure), and, if possible, into minutes also.

Make CB the radius of a line of chords on the sector, and taking therefrom the chord of 5° 35', the angle of the moon's visible path with the ecliptic, set it off from H to M on the left hand of CH, the axis of the ecliptic, because the moon's latitude is north ascending. Then draw CM for the axis of the moon's orbit, and bisect the angle MCH by the right line Cz. If the moon's latitude had been north descending, the axis of her orbit would have been on the right hand from the axis of the ecliptic.—N.B. The axis of the moon's orbit lies the same way when her latitude is south ascending as when it is north ascending; and the same way when south descending as when north descending.

Take the moon's latitude 40° 18' from the scale CA in your compasses, and set it from i to x in the bisecting line Cz, making ix parallel to Cy; and thro' x, at right angles to the axis of the moon's orbit CM, draw the straight line Nwxy S for the path of the penumbra's centre over the earth's disk.—The point w, in the axis of the moon's orbit, is that where the penumbra's centre approaches nearest to the centre of the earth's disk, and consequently is the middle of the general eclipse: the point x is that where the conjunction of Take the moon's true horary motion from the sun, 27°54', in your compasses, from the scale CA (every division of which is a minute of a degree), and with that extent make marks along the path of the penumbra's centre; and divide each space from mark to mark into sixty equal parts or horary minutes, by dots; and set the hours to every 60th minute in such a manner, that the dot signifying the instant of new moon by the tables, may fall into the point x, half way between the axis of the moon's orbit and the axis of the ecliptic; and then, the rest of the dots will show the points of the earth's disk, where the penumbra's centre is at the instants denoted by them, in its transit over the earth.

Apply one side of a square to the line of the penumbra's path, and move the square backwards and forwards until the other side of it cuts the same hour and minute (as at m and m) both in the path of London and in the path of the penumbra's centre; and the particular minute or instant which the square cuts at the same time in both paths, shall be the instant of the visible conjunction of the sun and moon, or greatest obscuration of the sun, at the place for which the construction is made, namely London, in the present example; and this instant is at 47½ minutes past ten o'clock in the morning; which is 17 minutes five seconds later than the tabular time of true conjunction.

Take the sun's semidiameter, 16°6', in your compasses, from the scale CA, and setting one foot in the path of London, at m, namely at 47½ minutes past ten, with the other foot describe the circle UY, which shall represent the sun's disk as seen from London at the greatest obscuration.—Then take the moon's semidiameter, 14°57', in your compasses from the same scale; and setting one foot in the path of the penumbra's centre at m, in the 47½ minute after ten, with the other foot describe the circle TY for the moon's disk, as seen from London, at the time when the eclipse is at the greatest, and the portion of the sun's disk which is hid or cut off by the moon's will show the quantity of the eclipse at that time; which quantity may be measured on a line equal to the sun's diameter, and divided into 12 equal parts for digits.

Lastly, take the semidiameter of the penumbra, 31°3', from the scale CA in your compasses; and setting one foot in the line of the penumbra's central path, on the left hand from the axis of the ecliptic, direct the other foot toward the path of London; and carry that extent backwards and forwards till both the points of the compasses fall into the same instants in both the paths; and these instants will denote the time when the eclipse begins at London.—Then, do the like on the right hand of the axis of the ecliptic; and where the points of the compasses fall into the same instants in both the paths, they will show at what time the eclipse ends at London.

These trials give 20 minutes after nine in the morning for the beginning of the eclipse at London, at the points N and O; 47½ minutes after ten, at the points m and n, for the time of greatest obscuration; and 18 minutes after twelve, at R and S, for the time when the eclipse ends; according to mean or equal time.

From these times we must subtract the equation of natural days, viz., 3 minutes 48 seconds, in leap-year, April 1, and we shall have the apparent times; namely, 9 hours 16 minutes 12 seconds for the beginning of the eclipse, 10 hours 43 minutes 42 seconds for the time of greatest obscuration, and 12 hours 14 minutes 12 seconds for the time when the eclipse ends.

But the best way is to apply this equation to the true equal time of new moon, before the projection be begun; as is done in Example I. For the motion or position of places on the earth's disk answer to apparent solar time.

In this construction it is supposed, that the angle under which the moon's disk is seen, during the whole time of the eclipse, continues invariably the same; and that the moon's motion is uniform and rectilineal during that time. But these suppositions do not exactly agree with the truth; and therefore, supposing the elements given by the tables to be accurate, yet the times and phases of the eclipse, deduced from its construction, will not answer exactly to what passeth in the heavens; but may be at least two or three minutes wrong, though done with the greatest care. Moreover, the paths of all places of considerable latitudes, are nearer the centre of the earth's disk, as seen from the sun, than those constructions make them: because the disk is projected as if the earth was a perfect sphere although it is known to be a spheroid. Consequently the moon's shadow will go farther northward in all places of northern latitude, and farther southward in all places of southern latitude, than it is shown to do in these projections.—According to Meyer's Tables, this eclipse was about a quarter of an hour sooner than either these tables, or Mr Flamsteed's, or Dr Halley's, make it; and was not annular at London. But M. de la Caille's make it almost central.

The projection of lunar eclipses.

When the moon is within 12 degrees of either of her nodes at the time when she is full, she will be eclipsed, otherwise not.

We find by example second, page 562, that at the time of mean full moon in May 1762, the sun's distance from the ascending node was only 4°49'35"; and the moon being then opposite to the sun, must have been just as near her descending node, and was therefore eclipsed.

The elements for constructing an eclipse of the moon, are eight in number, as follow:

1. The true time of full moon; and at that time, 2. The moon's horizontal parallax. 3. The sun's semidiameter. 4. The moon's. 5. The semidiameter of the earth's shadow at the moon. 6. The moon's latitude. 7. The angle of the moon's visible path with the ecliptic. 8. The moon's true horary motion from the sun.

Therefore,

1. To find the true time of new or full moon. Work as already taught in the precepts.—Thus we have the true time of full moon in May 1762 (see Example II. page 562) on the 8th day, at 50 minutes 50 seconds past three o'clock in the morning.

2. To find the moon's horizontal parallax. Enter Table XVII. with the moon's mean anomaly (at the above full) 9°2'42"42", and thereby take out her horizontal parallax; which by making the requisite proportions, will be found to be 57°23".

3. To find the semidiameters of the sun and moon. Enter Table XVI. with their respective anomalies, the sun's being 10°72'27"45" (by the above example) and the moon's 9°2'42"42"; and thereby take out their respective semidiameters; the sun's 15°56", and the moon's 15°38". 5. To find the semidiameter of the earth's shadow at the moon. Add the sun's horizontal parallax, which is always 10°, to the moon's, which in the present case is 57° 23′, the sum will be 57° 33′, from which subtract the sun's semidiameter 15° 56′, and there will remain 41° 37′ for the semidiameter of that part of the earth's shadow, which the moon then passes through.

6. To find the moon's latitude. Find the sun's true distance from the ascending node (as already taught in page 566) at the true time of full moon; and this distance increased by six signs, will be the moon's true distance from the same node; and consequently the argument for finding her true latitude, as shown in p. 566.

Thus, in Example II., the sun's mean distance from the ascending node was 8° 49′ 35″, at the time of mean full moon: but it appears by the example, that the true time thereof was six hours 33 minutes 38 seconds sooner than the mean time; and therefore we must subtract the sun's motion from the node (found in Table XII.) during this interval, from the above mean distance 8° 49′ 35″, in order to have his mean distance from it at the true time of full moon.

Then to this apply the equation of his mean distance from the node, found in Table XV. by his mean anomaly 10° 5° 27′ 45″; and lastly add six signs: so shall the moon's true distance from the ascending node be found as follows:

| Sun from node at mean full moon | 8° 49′ 35″ | |---------------------------------|------------| | His motion from it in | 6 hours | | | 33 minutes | | | 38 seconds |

Sum, subtract from the uppermost line

Remains his mean distance at true full moon

Equation of his mean distance, add

Sun's true distance from the node

To which add

And the sum will be

Which is the moon's true distance from her ascending node at the true time of her being full; and consequently the argument for finding her true latitude at that time.—Therefore, with this argument, enter Table XVI., making proportions between the latitudes belonging to the 6th and 7th degree of the argument at the left hand (the signs being at top) for 10° 32′, and it will give 32° 21′, for the moon's true latitude, which appears by the table to be south descending.

7. To find the angle of the moon's visible path with the ecliptic. This may be stated at 5° 35′, without any error of consequence in the projection of the eclipse.

8. To find the moon's true horary motion from the sun. With their respective anomalies take out their horary motions from Table XVII., and the sun's horary motion subtracted from the moon's, leaves remaining the moon's true horary motion from the sun: in the present case 30° 57′.

Now collect these elements together for use.

1. True time of full moon in May, 1762

| D H M S. | |----------| | 8° 3° 50′ 50″ |

2. Moon's horizontal parallax

3. Sun's semidiameter

4. Moon's semidiameter

5. Semidiameter of the earth's shadow at the moon

6. Moon's true latitude, south descending

7. Angle of her visible path with the ecliptic

8. Her true horary motion from the sun

These elements being found for the construction of the moon's eclipse in May 1762, proceed as follows:

Make a scale of any convenient length, as WX (fig. 201.), and divide it into 60 equal parts, each part standing for a minute of a degree.

Draw the right line ACB (fig. 202.) for part of the ecliptic, and CD perpendicular thereto for the southern part of its axis; the moon having south latitude.

Add the semidiameters of the moon and earth's shadow together, which in this eclipse will make 57° 15′; and take this from the scale in your compasses, and setting one foot in the point C as a centre, with the other foot describe the semicircle ADB; in one point of which the moon's centre will be at the beginning of the eclipse, and in another at the end thereof.

Take the semidiameter of the earth's shadow, 41° 37′, in your compasses from the scale, and setting one foot in the centre C, with the other foot describe the semicircle KLM for the southern half of the earth's shadow, because the moon's latitude is south in this eclipse.

Make CD equal to the radius of a line of chords on the sector, and set off the angle of the moon's visible path with the ecliptic, 5° 35′, from D to E, and draw the right line CFE for the southern half of the axis of the moon's orbit lying to the right hand from the axis of the ecliptic CD, because the moon's latitude is south descending.—It would have been the same way (on the other side of the ecliptic) if her latitude had been north descending; but contrary in both cases, if her latitude had been either north ascending or south ascending.

Bifect the angle DCE by the right line CG; in which line the true equal time of opposition of the sun and moon falls, as given by the tables.

Take the moon's latitude, 32° 21′ from the scale with your compasses, and set it from C to G, in the line CG; and through the point G, at right angles to CFE, draw the right line PHGFN for the path of the moon's centre. Then, F shall be the point in the earth's shadow, where the moon's centre is at the middle of the eclipse: G, the point where her centre is at the tabular time of her being full; and H, the point where her centre is at the instant of her ecliptical opposition.

Take the moon's horary motion from the sun, 30° 52′, in your compasses from the scale; and with that extent make marks along the line of the moon's path PGN: then divide each space from mark to mark, into 60 equal parts, or horary minutes, and set the hours to the proper dots in such a manner, that the dot figuring the instant of full moon (viz. 50 minutes 50 seconds after three in the morning) may be in the point G, where the line of the moon's path cuts the line that bifects the angle DCE.

Take the moon's semidiameter, 15° 38′, in your compasses from the scale, and with that extent, as a radius, upon the points N, F, and P, as centres, describe the circle Q for the moon at the beginning of the eclipse, when she touches the earth's shadow at V; the circle R for the moon at the middle of the eclipse; and the circle S for the moon at the end of the eclipse, just leaving the earth's shadow at W.

The point N denotes the instant when the eclipse began, namely, at 15 minutes 10 seconds after II in the morning; the point F the middle of the eclipse at 47 minutes 44 seconds past III.; and the point P the end of the eclipse, at 18 minutes after V.—At the greatest obscuration the moon was 10 digits eclipsed. Table I. The mean time of New Moon in March, Old Style; with the mean Anomalies of the Sun and Moon, and the Sun's mean Distance from the Moon's ascending Node, from A.D. 1700 to A.D. 1800 inclusive.

| A.D. | D.H.M.S | Mean New Moon in March | Sun's mean Anomaly | Moon's mean Anomaly | |------|---------|------------------------|-------------------|--------------------| | | | | | | | 700 | 8 16 | | | | | 701 | 27 | | | | | 702 | 26 | | | | | 703 | 6 | | | | | 704 | 24 | | | | | 705 | 13 | | | | | 706 | 22 | | | | | 707 | 20 | | | | | 708 | 10 | | | | | 709 | 29 | | | | | 710 | 18 | | | | | 711 | 7 | | | | | 712 | 17 | | | | | 713 | 15 | | | | | 714 | 4 | | | | | 715 | 3 | | | | | 716 | 11 | | | | | 717 | 1 | | | | | 718 | 19 | | | | | 719 | 8 | | | | | 720 | 27 | | | | | 721 | 16 | | | | | 722 | 5 | | | | | 723 | 4 | | | | | 724 | 3 | | | | | 725 | 2 | | | | | 726 | 21 | | | | | 727 | 10 | | | | | 728 | 28 | | | | | 729 | 18 | | | | | 730 | 7 | | | | | 731 | 6 | | | | | 732 | 14 | | | | | 733 | 3 | | | | | 734 | 23 | | | | | 735 | 12 | | | | | 736 | 10 | | | | | 737 | 19 | | | | | 738 | 9 | | | | | 739 | 27 | | | | | 740 | 16 | | | | | 741 | 5 | | | | | 742 | 4 | | | | | 743 | 13 | | | | | 744 | 12 | | | | | 745 | 2 | | | | | 746 | 10 | | | | | 747 | 29 | | | | | 748 | 17 | | | | | 749 | 5 | | | | | 750 | 26 | | | | | 751 | 15 | | | | | 752 | 3 | | | | | 753 | 20 | | | | | 754 | 10 | | | | | 755 | 2 | | | |

Astronomical Tables for calculating Eclipses

Table II.

| A.D. | D.H.M.S | Mean New Moon in March | Sun's mean Anomaly | Moon's mean Anomaly | |------|---------|------------------------|-------------------|--------------------| | | | | | | | 1753| 2 | | | | | 1754| 12 | | | | | 1755| 1 | | | | | 1756| 19 | | | | | 1757| 8 | | | | | 1758| 27 | | | | | 1759| 17 | | | | | 1760| 6 | | | | | 1761| 24 | | | | | 1762| 13 | | | | | 1763| 3 | | | | | 1764| 20 | | | | | 1765| 10 | | | | | 1766| 29 | | | | | 1767| 18 | | | | | 1768| 6 | | | | | 1769| 25 | | | | | 1770| 15 | | | | | 1771| 4 | | | | | 1772| 22 | | | | | 1773| 11 | | | | | 1774| 3 | | | | | 1775| 20 | | | | | 1776| 8 | | | | | 1777| 27 | | | | | 1778| 16 | | | | | 1779| 6 | | | | | 1780| 23 | | | | | 1781| 13 | | | | | 1782| 2 | | | | | 1783| 21 | | | | | 1784| 9 | | | | | 1785| 28 | | | | | 1786| 18 | | | | | 1787| 7 | | | | | 1788| 25 | | | | | 1789| 19 | | | | | 1790| 4 | | | | | 1791| 23 | | | | | 1792| 11 | | | | | 1793| 30 | | | | | 1794| 18 | | | | | 1795| 9 | | | | | 1796| 27 | | | | | 1797| 16 | | | | | 1798| 5 | | | | | 1799| 25 | | | | | 1800| 13 | | | |

4C2 ### TABLE II. Mean New Moon, &c. in March, New Style, from A.D. 1752 to A.D. 1800.

| Year | Mean New Moon in March | Sun's mean Anomaly | Moon's mean Anomaly | Sun's mean Diff. from the Node | |------|------------------------|--------------------|---------------------|-------------------------------| | 1752 | 14 20 16 | 8 14 44 | 3 2 42 | 3 25 40 | | 1753 | 4 5 42 | 8 4 0 | 1 12 30 | 4 3 43 | | 1754 | 23 2 37 | 8 22 22 | 0 18 7 | 5 12 26 | | 1755 | 12 11 25 | 8 11 38 | 10 27 55 | 5 20 29 | | 1756 | 30 8 50 | 9 0 24 | 10 3 32 | 6 29 12 | | 1757 | 19 17 47 | 8 19 16 | 8 13 20 | 7 7 14 | | 1758 | 9 2 35 | 8 8 32 | 6 23 8 | 7 15 17 | | 1759 | 28 0 8 | 8 26 54 | 5 28 45 | 8 24 0 | | 1760 | 16 8 57 | 8 8 16 | 4 8 34 | 9 2 3 | | 1761 | 5 17 45 | 8 5 26 | 2 18 22 | 9 10 6 | | 1762 | 24 15 18 | 8 23 48 | 1 23 59 | 10 18 49 | | 1763 | 14 0 7 | 8 1 13 | 4 8 37 | 10 26 52 | | 1764 | 2 8 53 | 8 2 20 | 0 13 35 | 11 4 54 | | 1765 | 21 6 28 | 8 20 42 | 9 19 12 | 10 3 37 | | 1766 | 10 15 16 | 8 9 58 | 7 29 0 | 9 21 40 | | 1767 | 29 12 49 | 8 28 20 | 7 4 37 | 2 0 23 | | 1768 | 21 3 38 | 9 17 36 | 5 14 25 | 2 8 26 | | 1769 | 7 6 26 | 8 6 52 | 3 24 13 | 2 16 29 | | 1770 | 26 8 59 | 8 25 14 | 2 29 50 | 3 25 12 | | 1771 | 15 12 48 | 8 14 30 | 1 9 38 | 4 3 15 | | 1772 | 3 21 36 | 8 3 45 | 11 19 27 | 4 11 17 | | 1773 | 22 19 9 | 8 2 22 | 10 25 4 | 5 20 0 | | 1774 | 12 3 57 | 8 11 24 | 9 4 52 | 5 28 3 | | 1775 | 1 12 46 | 8 10 39 | 7 14 40 | 6 6 6 | | 1776 | 19 10 19 | 8 19 2 | 6 20 17 | 7 14 49 | | 1777 | 8 19 7 | 8 17 57 | 5 0 5 | 7 22 52 | | 1778 | 27 16 40 | 8 26 40 | 4 5 42 | 9 1 35 | | 1779 | 17 29 4 | 8 15 56 | 2 15 30 | 9 3 38 | | 1780 | 5 10 17 | 8 5 11 | 0 25 18 | 9 17 40 | | 1781 | 24 7 50 | 8 23 34 | 0 0 55 | 10 26 23 | | 1782 | 13 16 38 | 8 12 49 | 10 10 43 | 11 4 26 | | 1783 | 1 27 33 | 8 2 50 | 8 20 32 | 11 12 29 | | 1784 | 20 23 0 | 8 20 28 | 9 26 9 | 0 21 12 | | 1785 | 10 7 48 | 8 9 43 | 6 5 57 | 0 29 15 | | 1786 | 9 5 21 | 8 28 6 | 5 11 34 | 1 7 58 | | 1787 | 18 14 10 | 8 17 21 | 3 21 22 | 2 16 0 | | 1788 | 6 22 58 | 8 6 37 | 2 1 10 | 2 24 3 | | 1789 | 25 20 31 | 8 25 0 | 1 6 47 | 4 2 46 | | 1790 | 15 5 19 | 8 14 15 | 11 16 35 | 4 10 49 | | 1791 | 4 14 8 | 8 3 31 | 9 26 23 | 4 18 52 | | 1792 | 22 11 41 | 8 21 53 | 9 2 0 | 5 27 55 | | 1793 | 11 20 29 | 8 11 9 | 7 11 48 | 6 3 38 | | 1794 | 30 18 2 | 8 29 32 | 6 17 26 | 7 14 21 | | 1795 | 20 2 51 | 8 18 47 | 4 27 14 | 7 22 24 | | 1796 | 8 11 39 | 8 3 47 | 3 7 2 | 8 0 26 | | 1797 | 27 9 12 | 8 26 25 | 2 12 39 | 9 9 9 | | 1798 | 16 18 1 | 8 15 41 | 0 22 27 | 9 17 12 | | 1799 | 6 2 49 | 8 4 57 | 1 2 15 | 9 25 15 | | 1800 | 2 0 22 | 8 23 19 | 10 7 52 | 11 3 58 |

### TABLE III. Mean Anomalies, and Sun's mean Distance from the Node, for 13½ mean Lunations.

| Year | Mean Lunations | Sun's mean Anomaly | Moon's mean Anomaly | Sun's mean Dist. from the Node | |------|----------------|--------------------|---------------------|-------------------------------| | 1752 | 1 29 12 | 3 0 29 | 0 25 49 | 0 1 0 40 | | 1753 | 2 59 1 | 1 28 6 | 1 28 12 | 1 21 38 | | 1754 | 3 88 14 | 2 27 9 | 2 17 58 | 2 17 27 | | 1755 | 4 118 2 | 3 26 5 | 3 13 16 | 3 2 4 20 | | 1756 | 5 147 15 | 4 25 3 | 4 9 5 | 5 3 21 | | 1757 | 6 177 4 | 5 24 3 | 5 4 54 | 6 4 1 | | 1758 | 7 206 17 | 6 23 4 | 6 0 43 | 7 4 41 | | 1759 | 8 236 5 | 7 22 5 | 6 26 3 | 8 5 21 | | 1760 | 9 265 18 | 8 21 5 | 7 22 21 | 9 6 2 | | 1761 | 10 295 7 | 9 21 3 | 8 18 10 | 10 6 42 | | 1762 | 11 324 20 | 10 20 9 | 9 13 59 | 11 7 22 | | 1763 | 12 354 8 | 11 19 15 | 10 9 48 | 12 0 8 | | 1764 | 13 383 21 | 10 18 22 | 11 5 37 | 13 0 15 | | 1765 | 14 18 22 | 10 14 33 | 6 12 54 | 15 0 8 |

### TABLE IV. The days of the Year, reckoned from the beginning of March.

| Month | Days | |-------|------| | March | 1 | | April | 32 | | May | 62 | | June | 93 | | July | 123 | | Aug. | 154 | | Sept. | 185 | | Oct. | 215 | | Nov. | 246 | | Decem.| 276 | | Jan. | 307 |

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**Note:** The table provides data for calculating eclipses based on mean new moons and anomalies, along with additional information about mean lunations and distances from the node. ### TABLE V. Mean Lunations from 1 to 100000.

| Lunat | Days. Decimal Parts | Days. Hou. M. S. Th. Fo. | |-------|---------------------|-------------------------| | | | |

| Lunations | Julian Years | Full New Moon | Sun's mean Anomaly | M.'s mean Anomaly | Sun from Node | |-----------|--------------|---------------|--------------------|-------------------|--------------| | | | D. H. M. S. | s o ' s o ' s o ' |

### TABLE VI. The first mean New Moon, with the mean Anomalies of the Sun and Moon, and the Sun's mean Distance from the Ascending Node, next after complete Centuries of Julian years.

| Lunat | Days. Decimal Parts | Days. Hou. M. S. Th. Fo. | |-------|---------------------|-------------------------| | | | |

| Lunations | Julian Years | Full New Moon | Sun's mean Anomaly | M.'s mean Anomaly | Sun from Node | |-----------|--------------|---------------|--------------------|-------------------|--------------| | | | D. H. M. S. | s o ' s o ' s o ' |

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**TABLE VII.** ### TABLE VII. The annual, or first Equation of the mean to the true Szygy.

**Argument. Sun's mean Anomaly.**

| Degrees | Signs | Sign | Signs | Signs | Signs | |---------|-------|------|-------|-------|-------| | H.M.S. | H.M.S.| H.M.S.| H.M.S.| H.M.S.| H.M.S.|

#### Subtract.

| Degrees | Signs | Sign | Signs | Signs | Signs | |---------|-------|------|-------|-------|-------| | | | | | | |

### TABLE VIII. Equation of the Moon's mean Anomaly.

**Argument. Sun's mean Anomaly.**

| Degrees | Signs | Sign | Signs | Signs | Signs | |---------|-------|------|-------|-------|-------| | | | | | | |

#### Subtract.

| Degrees | Signs | Sign | Signs | Signs | Signs | |---------|-------|------|-------|-------|-------| | | | | | | |

### TABLE IX. The second Equation of the mean to the true Szygy.

**Argument. Moon's equated Anomaly.**

| Degrees | Signs | Sign | Signs | Signs | Signs | |---------|-------|------|-------|-------|-------| | | | | | | |

Add

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**TABLE IX. The second Equation of the mean to the true Szygy.**

**Argument. Moon's equated Anomaly.**

| Degrees | Signs | Sign | Signs | Signs | Signs | |---------|-------|------|-------|-------|-------| | | | | | | |

Add ### TABLE IX. Concluded.

| Signs | Sign | Signs | Signs | Signs | Signs | |-------|------|-------|-------|-------|-------| | H.M.S. | H.M.S. | H.M.S. | H.M.S. | H.M.S. | H.M.S. |

### TABLE XII. The Sun's mean Longitude, Motion, and Anomaly:

#### Old style

| Year | Sun's mean Longitude | Sun's mean Anomaly | Sun's mean Motion | |------|----------------------|--------------------|-------------------| | | s o ' " s o | s o ' " s o | s o ' " s o |

#### New style

| Year | Sun's mean Longitude | Sun's mean Anomaly | Sun's mean Motion | |------|----------------------|--------------------|-------------------| | | s o ' " s o | s o ' " s o | s o ' " s o |

### TABLE XIII.

In Leap years, after February, add one day, and one day's motion. ### TABLE XIII. Equation of the Sun's centre, or the difference between his mean and true place.

| Argument | Sun's mean Anomaly | |----------|--------------------| | Degrees | Signs | | | Sign | | | Signs | | | Signs | | | Signs | | | Degrees |

### TABLE XIV. The Sun's Declination

| Argument | Sun's true place | |----------|------------------| | Degrees | Signs | | | Signs | | | Signs | | | Degrees |

### TABLE XV. Equation of the Sun's mean Distance from the Node.

| Argument | Sun's mean Anomaly | |----------|--------------------| | Degrees | Signs | | | Sign | | | Signs | | | Signs | | | Degrees |

### TABLE XVI. The Moon's Latitude in Eclipses.

| Arg. Moon's equated Distance from the Node. | |---------------------------------------------| | Degrees | | Signs | | Sign | | Signs | | Degrees |

### TABLE XVII. The Moon's horizontal Parallax, the Semidiameters and true Hourly Motions of the Moon, to every fifth degree of their mean diameter, the quantities for the intermediate degrees easily proportioned by sight.

| Moon's horizontal Parallax | |----------------------------| | Moon's semidiameter | | Moon's hourly motion | | Moon's semidiameter | | Moon's hourly motion |

This Table shows the Moon's Latitude a little beyond the utmost Limits of Eclipses. Sect. XI. The method of finding the Longitude by the Eclipses of Jupiter's Satellites; the amazing Velocity of Light demonstrated by these Eclipses; and of Cometary Eclipses.

In the former section, having explained at great length how eclipses of the sun and moon happen at certain times, it must be evident, that similar eclipses will be observed by the inhabitants of Jupiter and Saturn, which are attended by so many moons. These eclipses indeed very frequently happen to the satellites of Jupiter; and as they are of the greatest service in determining the longitudes of places on this earth, astronomers have been at great pains to calculate tables for the eclipses of these satellites by their primary, for the satellites themselves have never been observed to eclipse one another. The construction of such tables is indeed much easier for these satellites than of any other celestial bodies, as their motions are much more regular.

The English tables are calculated for the meridian of Greenwich, and by these it is very easy to find how many degrees of longitude any place is distant either east or west from Greenwich; for, let an observer, who has these tables, with a good telescope and a well-regulated clock at any other place of the earth, observe the beginning or ending of an eclipse of one of Jupiter's satellites, and note the precise moment of time that he saw the satellite either immerse into, or emerge out of, the shadow, and compare that time with the time shown by the tables for Greenwich: then 15 degrees difference of longitude being allowed for every hours difference of time, will give the longitude of that place from Greenwich; and if there be any odd minutes of time, for every minute a quarter of a degree, east or west, must be allowed, as the time of observation is later or earlier than the time shown by the tables. Such eclipses are very convenient for this purpose at land, because they happen almost every day; but are of no use at sea, because the rolling of the ship hinders all nice telecopic observations.

To explain this by a figure, let J be Jupiter, K, L, M, N, his four satellites in their respective orbits, 1, 2, 3, 4; and let the earth be at F (suppose in November, although that month is no otherwise material than to find the earth readily in this scheme, where it is shown in eight different parts of the orbit). Let Q be a place on the meridian of Greenwich, and R a place on some other meridian eastward from Greenwich. Let a person at R observe the instantaneous vanishing of the first satellite K into Jupiter's shadow, suppose at three o'clock in the morning; but by the tables he finds the immersion of that satellite to be at midnight at Greenwich; he then can immediately determine, that as there are three hours difference of time between Q and K, and that R is three hours forwarder in reckoning than Q, it must be 45 degrees of east longitude from the meridian of Q. Were this method as practicable at sea as at land, any sailor might almost as easily, and with equal certainty, find the longitude as the latitude.

Whilst the earth is going from C to F in its orbit, only the immersion of Jupiter's satellites into his shadow are generally seen; and their immersions out of it eclipses of while the earth goes from G to B. Indeed, both these Jupiter's appearances may be seen of the second, third, and fourth satellite when eclipsed, whilst the earth is between D and E, or between G and A; but never of the first satellite, on account of the smallness of its orbit and the bulk of Jupiter, except only when Jupiter is directly opposite to the sun, that is, when the earth is at G; and even then, strictly speaking, we cannot see either the immersions or emergences of any of his satellites, because his body being directly between us and his conical shadow, his satellites are hid by his body a few moments before they touch his shadow; and are quite emerged from thence before we can see them, as it were just dropping from him. And when the earth is at C, the sun, being between it and Jupiter, hides both him and his moons from us.

In this diagram, the orbits of Jupiter's moons are drawn in true proportion to his diameter; but in proportion to the earth's orbit, they are drawn vastly too large.

In whatever month of the year Jupiter is in conjunction with the sun, or in opposition to him, in the next year it will be a month later at least. For whilst the earth goes once round the sun, Jupiter describes a twelfth part of his orbit. And therefore, when the earth has finished its annual period, from being in a line with the sun and Jupiter, it must go as much forward as Jupiter has moved in that time, to overtake him again; just like the minute-hand of a watch, which must, from any conjunction with the hour-hand, go once round the dial-plate and somewhat above a twelfth part more, to overtake the hour-hand again.

It is found by observation, that when the earth is velocity of between the sun and Jupiter, as at C, his satellites are light eclipsed about 8 minutes sooner than they should be according to the tables; and when the earth is at B or C, these eclipses happen about 8 minutes later than the tables predict them. Hence it is undeniably certain, that the motion of light is not instantaneous, since it takes about $16\frac{1}{2}$ minutes of time to go through a space equal to the diameter of the earth's orbit, which is $180,000,000$ of miles in length; and consequently the particles of light fly almost $200,000$ miles every second of time, which is above a million of times swifter than the motion of a cannon bullet. And as light is $16\frac{1}{2}$ minutes in travelling across the earth's orbit, it must be $8\frac{1}{2}$ minutes in coming from the sun to us; therefore if the sun were annihilated, we should see him for $8\frac{1}{2}$ minutes after; and if he were again created, he would be $8\frac{1}{2}$ minutes old before we could see him.

To illustrate this progressive motion of light, let A and B be the earth in two different parts of its orbit, whose distance from each other is $95,000,000$ of miles, equal to the earth's distance from the sun S. It is Fig. 178. plain, that if the motion of light were instantaneous, the satellite 1 would appear to enter into Jupiter's shadow FF at the same moment of time to a spectator in A, as to another in B. But by many years observations it has been found, that the immersion of the satellite into the shadow is seen $8\frac{1}{2}$ minutes sooner when the earth is at B than when it is at A. And so, as Mr Romeur first discovered, the motion of light is thereby proved to be progressive, and not instantaneous. neous, as was formerly believed. It is easy to compute in what time the earth moves from A to B; for the chord of 60 degrees of any circle is equal to the semidiameter of that circle: and as the earth goes through all the 360 degrees of its orbit in a year, it goes through 60 of those degrees in about 61 days. Therefore, if on any given day, suppose the first of June, the earth is at A, on the first of August it will be at B; the chord, or straight line AB, being equal to DS the radius of the earth's orbit, the same with AS its distance from the sun.

As the earth moves from D to C, thro' the side AB of its orbit, it is constantly meeting the light of Jupiter's satellites sooner, which occasions an apparent acceleration of their eclipses; and as it moves through the other half H of its orbit, from C to D, it is receding from their light, which occasions an apparent retardation of their eclipses, because their light is then longer before it overtakes the earth.

That these accelerations of the immersions of Jupiter's satellites into his shadow, as the earth approaches towards Jupiter, and the retardations of their emersions out of his shadow, as the earth is going from him, are not occasioned by any inequality arising from the motions of the satellites in eccentric orbits, is plain, because it affects them all alike, in whatever parts of their orbits they are eclipsed. Besides, they go often round their orbits every year, and their motions are no way commensurate to the earth's. Therefore, a phenomenon not to be accounted for from the real motions of the satellites, but so easily deducible from the earth's motion, and so answerable thereto, must be allowed to result from it. This affords one very good proof of the earth's annual motion.

From what we have said in general concerning eclipses, it is plain that secondary planets are not the only bodies that may occasion them. The primary planets would eclipse one another, were it not for their great distances; but as the comets are not subject to the same laws with the planets, it is possible they may sometimes approach so near to the primary planets, as to cause an eclipse of the sun to those planets; and as the body of a comet bears a much larger proportion to the bulk of a primary planet than any secondary, it is plain that a cometary eclipse would both be of much longer continuance, and attended with much greater darkness, than that occasioned by a secondary planet. This behoved to be the case at any rate; but if we suppose the primary planet and comet to be moving both the same way, the duration of such an eclipse would be prodigiously lengthened; and thus, instead of four minutes, the sun might be totally darkened to the inhabitants of certain places for as many hours. Hence we may account for that prodigious darkness which we sometimes read of in history at times when no eclipse of the sun by the moon could possibly happen. It is remarkable, however, that no comet hath ever been observed passing over the disk of the sun like a spot, as Venus and Mercury are; yet this must certainly happen, when the comet is in its perihelion, and the earth on the same side of its annual orbit. Such a phenomenon well deserves the watchful attention of astrologers, as it would be a greater confirmation of the planetary nature of comets than anything hitherto observed.

Sect. XII. A Description of the Astronomical Machinery serving to explain and illustrate the foregoing part of this Treatise.

The machine represented by fig. 207. is the Grand Orrery, first made in this kingdom by Mr Rowley for King George I. The frame of it, which contains the wheel-work, &c. and regulates the whole machine, is made of ebony, and about four feet in diameter; the outside thereof is adorned with 12 pilasters. Between these the 12 signs of the zodiac are neatly painted with gilded frames. Above the frame is a broad ring supported with 12 pillars. This ring represents the plane of the ecliptic; upon which are two circles of degrees, and between these the names and characters of the 12 signs. Near the outside is a circle of months and days, exactly corresponding to the sun's place at noon each day throughout the year. Above the ecliptic stand some of the principle circles of the sphere, agreeable to their respective situations in the heavens: viz. No. 10. are the two colures, divided into degrees and half degrees; No. 11. is one-half of the equinoctial circle, making an angle of 23° 50' degrees. The tropic of cancer and the arctic circle are each fixed parallel at their proper distance from the equinoctial. On the northern half of the ecliptic is a brass semicircle, moveable upon two points fixed in η and α. This semicircle serves as a moveable horizon to be put to any degree of latitude upon the north part of the meridian, and the whole machine may be set to any latitude without disturbing any of the internal motions, by two strong hinges (No. 13.) fixed to the bottom-frame upon which the instrument moves, and a strong brass arch, having holes at every degree, through which a strong pin is put at every elevation. This arch and the two hinges support the whole machine when it is lifted up according to any latitude; and the arch at other times lies conveniently under the bottom-frame. When the machine is set to any latitude (which is easily done by two men, each taking hold of two handles conveniently fixed for the purpose), set the moveable horizon to the same degree upon the meridian, and hence you may form an idea of the respective altitude or depression of the planets both primary and secondary. The sun (No. 1.) stands in the middle of the whole system upon a wire, making an angle with the ecliptic of about 82 degrees. Next the sun is a small ball (2) representing Mercury. Next to Mercury is Venus (3) represented by a larger ball. The earth is represented (No. 4.) by an ivory ball, having some circles and a map sketched upon it. The wire which supports the earth makes an angle with the ecliptic of 66° 1' degrees, the inclination of the earth's axis to the ecliptic. Near the bottom of the earth's axis is a dial-plate (No. 9.), having an index pointing to the hours of the day as the earth turns round its axis. Round the earth is a ring supported by two small pillars, representing the orbit of the moon; and the divisions upon it answer to the moon's latitude. The motion of this ring represents the motion of the moon's orbit according to that of the nodes. Within this ring is the moon (No. 5.), having a black cap or cape, by which its motion represents the phases of the moon according to her age. Without the orbits of the the earth and moon is Mars (No. 6.) The next in order to Mars is Jupiter and his four moons (No. 7.) Each of these moons is supported by a wire fixed in a socket which turns about the pillar supporting Jupiter. These satellites may be turned by the hand to any position, and yet when the machine is put into motion, they will all move in their proper times. The outermost of all is Saturn, his five moons, and his ring (No. 8.). These moons are supported and contrived similar to those of Jupiter. The machine is put into motion by turning a small winch (No. 14.), and the whole system is also moved by this winch, and by pulling out and pushing in a small cylindrical pin above the handle. When it is pushed in, all the planets, both primary and secondary, will move according to their respective periods by turning the handle. When it is drawn out, the motions of the satellites of Jupiter and Saturn will be stopped while all the rest move without interruption. There is also a brass lamp, having two convex glasses to be put in room of the sun; and also a smaller earth and moon, made somewhat in proportion to their distance from each other, which may be put on at pleasure. The lamp turns round at the same time with the earth, and the glasses of it cast a strong light upon her. And when the smaller earth and moon are placed on, it will be easy to show when either of them will be eclipsed. When this machine is intended to be used, the planets must be duly placed by means of an ephemeris hereafter described; and you may place a small black patch or bit of wafer upon the middle of the sun. Right against the first degree of γ, you may also place patches upon Venus, Mars, and Jupiter, right against some noted point in the ecliptic. Put in the handle, and push in the pin which is above it. One turn of this handle answers to a revolution of the ball which represents the earth about its axis; and consequently to 24 hours of time, as shown by the hour-index (9.), which is marked and placed at the foot of the wire on which the ball of the earth is fixed. Again, when the index has moved the space of ten hours, Jupiter makes one revolution round its axis, and so of the rest. By these means the revolutions of the planets, and their motions round their own axes, will be represented to the eye. By observing the motions of the spots upon the surface of the sun and of the planets in the heavens, their diurnal was first discovered, after the same manner as we in this machine observe the motions of their representatives by that of the marks placed upon them.

The Orrery (fig. 208.) is a machine contrived by the late ingenious Mr James Ferguson. It shows the motions of the sun, Mercury, Venus, earth, and moon; and occasionally the superior planets, Mars, Jupiter, and Saturn, may be put on. Jupiter's four satellites are moved round him in their proper times by a small winch; and Saturn has his five satellites, and his ring which keeps its parallelism round the sun; and by a lamp put in the sun's place, the ring shows all its various phases already described.

In the centre, No. 1. represents the sun, supported by its axis, inclining almost 8 degrees from the axis of the ecliptic, and turning round in 25½ days on its axis, of which the north pole inclines toward the eighth degree of Pisces in the great ecliptic (No. 11.), whereon the months and days are engraven over the signs and degrees in which the sun appears, as seen from the Astronomical Machinery, on the different days of the year.

The nearest planet (No. 2.) to the sun is Mercury, which goes round him in 87 days 23 hours, or 87½ diurnal rotations of the earth; but has no motion round its axis in the machine, because the time of its diurnal motion in the heavens is not known to us.

The next planet in order is Venus (No. 3.), which performs her annual course in 224 days 17 hours, and turns round her axis in 24 days 8 hours, or in 24½ diurnal rotations of the earth. Her axis inclines 75 degrees from the axis of the ecliptic, and her north pole inclines towards the 20th degree of Aquarius, according to the observations of Bianchini. She shows all the phenomena described in Sect. ii.

Next, without the orbit of Venus, is the earth (No. 4.), which turns round its axis, to any fixed point at a great distance, in 23 hours 56 minutes 4 seconds of mean solar time; but from the sun to the sun again, in 24 hours of the same time. No. 6. is a sidereal dial-plate under the earth, and No. 7. a solar dial-plate on the cover of the machine. The index of the former shows sidereal, and of the latter, solar time; and hence the former index gains one entire revolution on the latter every year, as 365 solar or natural days contain 366 sidereal days, or apparent revolutions of the stars. In the time that the earth makes 365½ diurnal rotations on its axis, it goes once round the sun in the plane of the ecliptic; and always keeps opposite to a moving index (No. 10.) which shows the sun's daily change of place, and also the days of the months.

The earth is half covered with a black cap, for dividing the apparently enlightened half next the sun from the other half, which, when turned away from him, is in the dark. The edge of the cap represents the circle bounding light and darkness, and flows at what time the sun rises and sets to all places throughout the year. The earth's axis inclines 23½ degrees from the axis of the ecliptic; the north pole inclines toward the beginning of Cancer, and keeps its parallelism throughout its annual course; so that in summer the northern parts of the earth incline towards the sun, and in winter from him: by which means, the different lengths of days and nights, and the cause of the various seasons, are demonstrated to sight.

There is a broad horizon, to the upper side of which is fixed a meridian semicircle in the north and south points, graduated on both sides from the horizon to 90° in the zenith or vertical point. The edge of the horizon is graduated from the east and west to the south and north points, and within these divisions are the points of the compasses. From the lower side of this thin horizontal plate stand out four small wires, to which is fixed a twilight-circle 18 degrees from the graduated side of the horizon all round. This horizon may be put upon the earth (when the cap is taken away), and rectified to the latitude of any place; and then by a small wire called the solar ray, which may be put on so as to proceed directly from the sun's centre towards the earth's, but to come no farther than almost to touch the horizon. The beginning of twilight, time of sun-rising, with his amplitude, meridian altitude, time of setting, amplitude then, and end of twilight, are shown for every day of the year, at that place to which the horizon is rectified. The moon (No. 5.) goes round the earth, from between it and any fixed point at a great distance, in 27 days 7 hours 43 minutes, or through all the signs and degrees of her orbit, which is called her periodical revolution; but she goes round from the sun to the sun again, or from change to change, in 29 days 12 hours 35 minutes, which is her synodical revolution; and in that time she exhibits all the phases already described.

When the abovementioned horizon is rectified to the latitude of any given place, the times of the moon's rising and setting, together with her amplitude, are shown to that place as well as the sun's; and all the various phenomena of the harvest-moon are made obvious to sight.

The moon's orbit (No. 9.) is inclined to the ecliptic (No. 11), one-half being above, and the other below it. The nodes, or points at O and O', lie in the plane of the ecliptic, as before described, and shift backward through all its signs and degrees in 18½ years. The degrees of the moon's latitude to the highest at NL (north latitude) and lowest at SL (south latitude), are engraved both ways from her nodes at O and O', and as the moon rises and falls in her orbit according to its inclination, her latitude and distance from her nodes are shown for every day, having first rectified her orbit so as to set the nodes to their proper places in the ecliptic; and then as they come about at different and almost opposite times of the year, and then point towards the sun, all the eclipses may be shown for hundreds of years (without any new rectification), by turning the machinery backward for time past, or forward for time to come. At 17 degrees distance from each node, on both sides, is engraved a small sun; and at 12 degrees distance, a small moon, which show the limits of solar and lunar eclipses; and when, at any change, the moon falls between either of these suns and the node, the sun will be eclipsed on the day pointed to by the annual index (No. 10.); and as the moon has then north or south latitude, one may easily judge whether that eclipse will be visible in the northern or southern hemisphere: especially as the earth's axis inclines toward the sun or from him at that time. And when at any full the moon falls between either of the little moon's and node, she will be eclipsed, and the annual index shows the day of that eclipse. There is a circle of 29½ equal parts (No. 8.) on the cover of the machine, on which an index shows the days of the moon's age.

There are two semicircles (fig. 216.) fixed to an elliptical ring, which being put like a cap upon the earth, and the forked part F upon the moon, shows the tides as the earth turns round within them, and they are led round it by the moon. When the different places come to the semicircle AaEbB, they have tides of flood; and when they come to the semicircle CED, they have tides of ebb; the index on the hour-circle (fig. 208.) showing the times of these phenomena.

There is a jointed wire, of which one end being put into a hole in the upright stem that holds the earth's cap, and the wire laid into a small forked piece which may be occasionally put upon Venus or Mercury, shows the direct and retrograde motions of these two planets, with their stationary times and places, as seen from the earth.

The whole machinery is turned by a winch or handle (No. 12.); and is so easily moved, that a clock might turn it without any danger of stopping.

To give a plate of the wheel-work of this machine would answer no purpose, because many of the wheels lie so behind others as to hide them from sight in any view whatever.

The Planetarium (fig. 209.) is an instrument contrived by Mr William Jones of Holborn, London, mathematical instrument maker, who has paid considerable attention to those sort of machines, in order to reduce them to their greatest degree of simplicity and perfection. It represents in a general manner, by various parts of its machinery, all the motions and phenomena of the planetary system. This machine consists of, the Sun (in the centre), with the planets, Mercury, Venus, the Earth and Moon, Mars, Jupiter, and his four moons, Saturn and his five moons; and to it is occasionally applied an extra long arm for the Georgian planet and his two moons. To the earth and moon is applied a frame CD, containing only four wheels and two pinions, which serve to preserve the earth's axis in its proper parallelism in its motion round the sun, and to give the moon her due revolution about the earth at the same time. These wheels are connected with the wheel-work in the round box below, and the whole is set in motion by the winch H. The arm M that carries round the moon, points out on the plate C her age and phases for any situation in her orbit, and which accordingly are engraved thereon. In the same manner the arm points out her place in the ecliptic B, in signs and degrees, called her geocentric place; that is, as seen from the earth. The moon's orbit is represented by the flat rim A; the two joints of which, and upon which it turns, denoting her nodes. This orbit is made to incline to any desired angle. The earth of this instrument is usually made of a three inch or 1½ globe, papered, &c. for the purpose; and by means of the terminating wire that goes over it, points out the changes of the seasons, and the different lengths of days and nights more conspicuously. This machine is also made to represent the Ptolemaic System, or such as is vulgarly received; which places the earth in the centre, and the planets and sun revolving about it. (It is done by an auxiliary small sun and an earth, which change their places in the instrument.) At the same time, it affords a most manifest confutation of it: for it is plainly observed in this construction, (1.) That the planets Mercury and Venus being both within the orbit of the sun, cannot at any time be seen to go behind it; whereas in nature we observe them as often to go behind as before the sun in the heavens. (2.) It shows, that as the planets move in circular orbits about the central earth, they ought at all times to be of the same apparent magnitude; whereas, on the contrary, we observe their apparent magnitude in the heavens to be very variable, and so far different, that, for instance, Mars will sometimes appear as big as Jupiter nearly, and at other times you will scarcely know him from a fixed star. (3.) It shows that any of the planets might be seen at all distances from the sun in the heavens; or, in other words, that when the sun is setting, Mercury or Venus may be seen not only in the south but even in the east; which circumstances were never yet observed. (4.) You. You see by this planetarium that the motions of the planets should always be regular and uniformly the same; whereas, on the contrary, we observe them always to move with a variable velocity, sometimes faster, then slower, and sometimes not at all, as will be perfectly shown. (5.) By the machine you see the planets move all the same way, viz. from west to east continually; but in the heavens we see them move sometimes direct from west to east, sometimes retrograde from east to west, and at other times to be stationary. All which phenomena plainly prove this system to be a false and absurd hypothesis.

The truth of the Copernican or Solar System of the world is hereby most clearly represented. For taking the earth from the centre, and placing thereon the usual large brass ball for the sun, and restoring the earth to its proper situation among the planets, then everything will be right and agree exactly with celestial observations. For turning the winch H, (1.) You will see the planets Mercury and Venus go both before and behind the sun, or have two conjunctions. (2.) You will observe Mercury never to be more than a certain angular distance, 21°, and Venus 47°, from the sun. (3.) That the planets, especially Mars, will be sometimes much nearer to the earth than at others, and therefore must appear larger at one time than at another. (4.) You will see that the planets cannot appear at the earth to move with an uniform velocity; for when nearest they appear to move faster, and slower when most remote. (5.) You will observe the planets will appear at the earth to move sometimes directly from west to east, and then to become retrograde from east to west, and between both to be stationary or without any apparent motion at all. Which particulars all correspond exactly with observations, and fully prove the truth of this excellent system. Fig. 210 represents an apparatus to show these latter particulars more evidently. An hollow wire, with a slit at top, is placed over the arm of the planet Mercury or Venus at E. The arm DG represents a ray of light coming from the planet at D to the earth, and is put over the centre which carries the earth at F. The planets being then put in motion, the planet D, as seen in the heavens from the earth at F, will undergo the several changes of position as above described. The wire prop that is over Mercury at E, may be placed over the other superior planets, Mars, &c. and the same phenomena be exhibited.

By this machine you at once see all the planets in motion about the sun, with the same respective velocities and periods of revolution which they have in the heavens; the wheel-work being calculated to a minute of time, from the latest discoveries.

You will see here a demonstration of the earth's motion about the sun, as well as those of the rest of the planets: for if the earth were to be at rest in the heavens, then the time between any two conjunctions of the same kind, or oppositions, would be the same with the periodical time of the planets, viz. 88 days in Mercury, 225 in Venus, &c.; whereas you here observe this time, instead of being 225 days, is no less than 583 days in Venus, occasioned by the earth's moving in the meantime about the sun the same way with the planet. And this space of 583 days always passes between two like conjunctions of Venus in the heavens. Hence the most important point of astronomy is satisfactorily demonstrated.

The diurnal rotation of the earth about its axis, and a demonstration of the cause of the different lengths of the year, and the different lengths of days and nights, are here answered completely: for as the earth is placed on an axis inclining to that of the ecliptic in an angle of 23½ degrees, and is set in motion by the wheel-work, there will be evidently seen the different inclination of the sun's rays on the earth, the different quantity thereof which falls on a given space, the different quantity of the atmosphere they pass through, and the different continuance of the sun above the horizon at the same place in different times of the year; which particulars constitute the difference between heat and cold in the summer and winter seasons.

As the globe of the earth is moveable about its inclined axis, so by having the horizon of London drawn upon the surface of it, and by means of the terminating wire going over it, by which is denoted, that on that side of the wire next the sun is the enlightened half of the earth, and the opposite side the darkened half, you will here see very naturally represented the cause of the different lengths of day and night, by observing the unequal portions of the circle which the island of Great Britain, or the city of London, or any other place, describes in the light and dark hemispheres at different times of the year, by turning the earth on its axis with the hand. But in some of the better orries on this principle, the earth revolves about its axis by wheel-work.

As to the eclipses of the sun and moon, the true causes of them are here very clearly seen: for by placing the lamp (fig. 211.) upon the centre, in room of the brass ball denoting the sun, and turning the winch until the moon comes into a right line between the centres of the lamp (or sun) and earth, the shadow of the moon will fall upon the earth, and all who live on that part over which the shadow passes will see the sun eclipsed more or less. On the other side, the moon passes (in the aforementioned case) through the shadow of the earth, and is by that means eclipsed. And the orbit A (fig. 209.) is so moveable on the two joints called nodes, that any person may easily represent the due position of the nodes and intermediate spaces of the moon's orbit; and thence show when there will or will not be an eclipse of either luminary, and what the quantity of each will be.

While the moon is continuing to move round the earth, the lamp on the centre will so illumine the moon, that you will easily see all her phases, as new, dichotomized, gibbous, full, waning, &c. just as they appear in the heavens. You will moreover observe all the same phases of the earth as they appear at the moon.

The satellites of Jupiter and Saturn are moveable only by the hand; yet may all their phenomena be easily represented, excepting the true relative motions and distances. Thus, if that gilt globe which before represented the sun be made now to denote Jupiter, and four of the primary planets only be retained, then will the Jovian system be represented; and by candlelight only you will see (the machine being in motion) the immersions and emergences of the satellites into and out of Jupiter's shadow. You will see plainly the manner in which they transit his body, and their occultations. culations behind it. You will observe the various ways in which one or more of these moons may at times disappear. And if the machine be set by a white wall, &c. then by the projection of their shadows will be seen the reasons why those moons always appear on each side of Jupiter in a right line, why those which are most remote may appear nearest, and e contrario. And the same may be done for Saturn's five moons and his ring.

The method of Rectifying the Orrery, and the proper Manner of placing the Planets in their true Situation.

Having dwelled thus much on the description of orreries, it may be useful to young readers, to point out the method by which the orrery should be first rectified, previous to the exhibition or using of it; and the following is extracted from Mr William Jones's description of his new Portable Orrery. "The method of showing the places, and relative aspects of the planets on any day of the year in the planetarium, must be done by the affixture of an ephemeris or almanac, which among other almanacs is published annually by the Stationer's Company.

"This ephemeris contains a diary or daily account of the places places in the heavens, in signs, degrees, and minutes, both as they appear to the eye supposed to be at the sun, and at the earth, throughout the year. The first of these positions is called the heliocentric place, and the latter, the geocentric place. The heliocentric place is that made use of in orreries; the geocentric place, that in globes. As an example for finding their places, and setting them right in the orrery, we will suppose the ephemeris (by White, which for this purpose is considered the best) at hand, wherein at the bottom of the left-hand page for every month is the heliocentric longitudes (or places) of all the planets to every six days of the month; which is near enough for common use: A copy of one of these tables for March 1784 is here inserted for the information of the tyro."

| Day | Helioc. long. h | Helioc. long. y | Helioc. long. s | Helioc. long. Θ | Helioc. long. Φ | Helioc. long. χ | |-----|----------------|----------------|----------------|----------------|----------------|----------------| | 1 | 3 | 11 | 16 | 43 | 17 | 11 | | 7 | 3 | 35 | 16 | 56 | 17 | 43 | | 13 | 3 | 59 | 17 | 78 | 15 | 7 | | 19 | 4 | 23 | 17 | 17 | 18 | 47 | | 25 | 4 | 47 | 17 | 28 | 19 | 19 |

"Now, as an example, we will suppose, that in order to set the planets of the orrery, we want their heliocentric places for the 21st of this month. Looking into the table, we take the 19th day, which is the nearest to the day wanted; then, accordingly we find the place of Saturn (h) is in 17° 17', or 17 degrees (rejecting the minutes, being in this case useless); of Capricornus (y), of Jupiter (y), in 18° of Aquarius (w), Mars (Θ), in 10° of Cancer (ξ), the earth (Θ), in 20° of Virgo (π), Venus (Φ) in 20° of Sagittarius (λ), Mercury (χ) in 28° of the same sign; and in the same manner for any other day therein specified. Upon even this circumstance depends a very pleasing astronomical praxis, by which the young tyro may at any time be able to entertain himself in a most rational and agreeable manner, viz. he may in a minute or two represent the true appearance of the planetary system just as it really is in the heavens, and for any day he pleases, by affixing to each planet its proper place in its orbit; as in the following manner: For the 19th of March, as before, the place of Saturn is in 17° of Capricornus (y); now, laying hold of the arm of Saturn in the orrery, you place it over or against the 17° of Capricorn on the ecliptic circle, constantly placed on or surrounding the instrument; thus doing the same for the other planets, they will have their proper heliocentric places for that day.

"Now in this situation of the planets, we observe, that if a person was placed on the earth, he would see Venus and Jupiter in the same line and place of the ecliptic, consequently in the heavens they would appear together, or in conjunction; Mercury a little to the left or eastward of them, and nearer to the sun; Saturn to the right, or the westward, farther from the Sun; Mars directly opposite to Saturn; so that when Saturn appears in the west, Mars appears in the east, and vice versa. Several other curious and entertaining particulars as depending on the above, may be easily represented and shown by the learner; particularly the foregoing, when the winch is turned, and all the planets set into their respective motions."

We cannot close this detail on orreries more agreeably than by the following account of an instrument of that fort invented by Mr James Ferguson, to which he gives the name of a Mechanical Paradox, and which is actuated by means of what many, as he observes, even good mechanics, would be ready to pronounce impossible, viz. That the teeth of one wheel taking equally deep into the teeth of three others, should affect them in such a manner, that in turning it any way round its axis, it should turn one of them the same way, another the contrary way, and the third no way at all.

The solution of the paradox is given under the article Mechanics; after which our author proceeds to give the following account of its uses. "This machine is so much of an orrery, as is sufficient to show the different lengths of days and nights, the vicissitudes of the seasons, the retrograde motion of the nodes of the moon's orbit, the direct motion of the apogeeal point of her orbit, and the months in which the sun and moon must be eclipsed.

"On the great immovable plate A (see fig. 213.) are the months and days of the year, and the signs and degrees of the zodiac so placed, that when the annual index b is brought to any given day of the year, it will point to the degree of the sign in which the sun is on that day. The index is fixed to the moveable frame BC." BC, and is carried round the immoveable plate with it, by means of the knob n. The carrying this frame and index round the immoveable plate, answers to the earth's annual motion round the sun, and to the sun's apparent motion round the ecliptic in a year.

"The central wheel D (being fixt on the axis a, which is fixt in the centre of the immoveable plate) turns the thick wheel E round its own axis by the motion of the frame; and the teeth of the wheel E take into the teeth of the three wheels F, G, H, whose axes turn within one another, like the axes of the hour, minute, and second hands of a clock or watch, where the seconds are shown from the centre of the dial-plate.

"On the upper ends of these axes, are the round plates I, K, L; the plate I being on the axis of the wheel F, K on the axis of G, and L on the axis of H. So that whichever way these wheels are affected, their respective plates, and what they support, must be affected in the same manner; each wheel and plate being independent of the others.

"The two upright wires M and N are fixed into the plate I; and they support the small ecliptic OP, on which, in the machine, the signs and degrees of the ecliptic are marked. This plate also supports the small terrestrial globe e, on its inclining axis f, which is fixed into the plate near the foot of the wire N. This axis inclines $23\frac{1}{2}$ degree from a right line, supposed to be perpendicular to the surface of the plate I, and also to the plane of the small ecliptic OP, which is parallel to that plate.

"On the earth e is the crescent g, which goes more than half way round the earth, and stands perpendicular to the plane of the small ecliptic OP, directly facing the sun Z; its use is to divide the enlightened half of the earth next the sun from the other half which is then in the dark; so that it represents the boundary of light and darkness, and therefore ought to go quite round the earth; but cannot in a machine, because in some positions the earth's axis would fall upon it. The earth may be freely turned round on its axis by hand, within the crescent, which is supported by the crooked wire w, fixt to it, and into the upper plate of the moveable frame BC.

"In the plate K are fixed the two upright wires Q and R; they support the moon's inclined orbits ST in its nodes, which are the two opposite points of the moon's orbit where it intersects the ecliptic OP. The ascending node is marked O, to which the descending node is opposite below e, but hid from view by the globe e. The half S T e of this orbit is on the north-side of the ecliptic OP, and the other half S O is on the south side of the ecliptic. The moon is not in this machine; but when she is in either of the nodes of her orbit in the heavens, she is then in the plane of the ecliptic: when she is at T in her orbit, she is in her greatest north latitude; and when she is at S, she is in her greatest south latitude.

"In the plate L is fixed the crooked wire UU, which points downward to the small ecliptic OP, and shows the motion of the moon's apogee therein, and its place at any given time.

"The ball Z represents the sun, which is supported by the crooked wire XY, fixt into the upper plate of the frame at X. A straight wire W proceeds from the sun Z, and points always toward the centre of the Astronomical Ma-chinery e; but toward different points of its surface at different times of the year, on account of the obliquity of its axis, which keeps its parallelism during the earth's annual course round the sun Z; and therefore must incline sometimes toward the sun, at other times from him, and twice in the year neither toward nor from the sun, but sidewise to him. The wire W is called the solar ray.

"As the annual-index b shows the sun's place in the ecliptic for every day of the year, by turning the frame round the axis of the immoveable plate A, according to the order of the months and signs, the solar ray does the same in the small ecliptic OP: for as this ecliptic has no motion on its axis, its signs and degrees will keep parallel to those on the immoveable plate. At the same time, the nodes of the moon's orbit ST (or points where it intersects the ecliptic OP) are moved backward, or contrary to the order of signs, at the rate of $19\frac{1}{2}$ degrees every Julian year; and the moon's apogee wire UU is moved forward, according to the order of the signs of the ecliptic, nearly at the rate of $41$ degrees every Julian year; the year being denoted by a revolution of the earth e round the sun Z; in which time the annual-index b goes round the circles of mouths and signs on the immoveable plate A.

"Take hold of the knob n, and turn the frame round thereby; and in doing this, you will perceive that the north pole of the earth e is constantly before the crescent g, in the enlightened part of the earth toward the sun, from the 20th of March to the 23d of September; and the south pole all that time behind the crescent in the dark; and from the 23d of September to the 20th of March, the north pole as constantly in the dark behind the crescent, and the south pole in the light before it; which shows, that there is but one day and one night at each pole, in the whole year; and that when it is day at either pole, it is night at the other.

"From the 20th of March to the 23d of September, the days are longer than the nights in all those places of the northern hemisphere of the earth which revolve through the light and dark, and shorter in those of the southern hemisphere. From the 23d of September to the 20th of March the reverse.

"There are 24 meridian semicircles drawn on the globe, all meeting in its poles; and as one rotation or turn of the earth on its axis is performed in 24 hours, each of these meridians is an hour distant from the other, in every parallel of latitude. Therefore, if you bring the annual-index b to any given day of the year, on the immoveable plate, you may see how long the day then is at any place of the earth, by counting how many of these meridians are in the light, or before the crescent, in the parallel of latitude of that place; and this number being subtracted from 24 hours, will leave remaining the length of the night. And if you turn the earth round its axis, all those places will pass directly under the point of the solar ray, which the sun passes vertically over on that day, because they are just as many degrees north or south of the equator as the sun's declination is then from the equinoctial.

"At the two equinoxes, viz. on the 20th of March and 23d of September, the sun is in the equinoctial, and consequently has no declination. On these days, the solar ray points directly toward the equator, the earth's poles lie under the inner edge of the crescent, or boundary of light and darkness; and in every parallel of latitude there are 12 of the meridians or hour-circles before the crescent, and 12 behind it; which shows that the days and nights then are each 12 hours long at all places of the earth. And if the earth be turned round its axis, you will see that all places on it go equally through the light and the dark hemispheres.

"On the 21st of June, the whole space within the north polar circle is enlightened, which is 23½ degrees from the pole, all around; because the earth's axis then inclines 23½ degrees toward the sun; but the whole space within the south polar circle is in the dark; and the solar ray points toward the tropic of Cancer on the earth, which is 23½ degrees north from the equator. On the 20th of December the reverse happens, and the solar ray points toward the tropic of Capricorn, which is 23½ degrees south from the equator.

"If you bring the annual-index b to the beginning of January, and turn the moon's orbit ST by its supporting wires Q and R till the ascending node (marked N) comes to its place in the ecliptic OP, as found by an ephemeris, or by astronomical tables, for the beginning of any given year; and then move the annual-index by means of the knob n, till the index comes to any given day of the year afterward, the nodes will stand against their places in the ecliptic on that day; and if you move on the index till either of the nodes comes directly against the point of the solar ray, the index will then be at the day of the year on which the sun is in conjunction with that node. At the times of those new moons, which happen within seventeen days of the conjunction of the sun with either of the nodes, the sun will be eclipsed; and, at the times of those full moons, which happen within twelve days of either of these conjunctions, the moon will be eclipsed. Without these limits there can be no eclipses either of the sun or moon; because, in nature, the moon's latitude or declination from the ecliptic is too great for the moon's shadow to fall on any part of the earth, or for the earth's shadow to touch the moon.

"Bring the annual-index to the beginning of January, and let the moon's apogee wire UU to its place in the ecliptic for that time, as found by astronomical tables; then move the index forward to any given day of the year, and the wire will point on the small ecliptic to the place of the moon's apogee for that time.

"The earth's axis f inclines always toward the beginning of the sign Cancer on the small ecliptic OP. And if you set either of the moon's nodes, and her apogee wire to the beginning of that sign, and turn the plate A about, until the earth's axis inclines toward any side of the room (suppose the north side), and then move the annual index round and round the immovable plate A, according to the order of the months and signs upon it, you will see that the earth's axis and beginning of Cancer will still keep toward the same side of the room, without the least deviation from it; but the nodes of the moon's orbit ST will turn progressively towards all the sides of the room, contrary to the order of signs in the small ecliptic OP, or from east, by south, to west, and so on; and the mical apogee wire UU will turn the contrary way to the motion of the nodes, or according to the order of the signs in the small ecliptic, from west, by south, to east, and so on quite round. A clear proof that the wheel F, which governs the earth's axis and the small ecliptic, does not turn any way round its own centre; that the wheel G, which governs the moon's orbit OP, turns round its own centre backward, or contrary both to the motion of the frame BC and thick wheel E; and that the wheel H, which governs the moon's apogee wire UU, turns round its own centre forward, or in direction both of the motion of the frame and of the thick wheel E, by which the three wheels F, G, and H, are affected.

"The wheels D, E, and F, have each 39 teeth in the machine; the wheel G has 37, and H 44.

"The parallellism of the earth's axis is perfect in this machine; the motion of the apogee very nearly so; the motion of the nodes not quite so near the truth, though they will not vary sensibly therefrom in one year. But they cannot be brought nearer, unless larger wheels, with higher numbers of teeth, are used.

"In nature, the moon's apogee goes quite round the ecliptic in 8 years and 312 days, in direction of the earth's annual motion; and the nodes go round the ecliptic, in a contrary direction, in 18 years and 225 days. In the machine, the apogee goes round the ecliptic OP in eight years and four-fifths of a year, and the nodes in nineteen years and a half."

The Cometarium, (fig. 217.) This curious machine shows the motion of a comet or eccentric body moving round the sun, describing equal areas in equal times, and may be so contrived as to show such a motion for any degree of eccentricity. It was invented by the late Dr Delaguerie.

The dark elliptical groove round the letters a b c d e f g h i k l m is the orbit of the comet Y; this comet is carried round in the groove according to the order of letters, by the wire W fixed in the sun S, and slides on the wire as it approaches nearer to or recedes farther from the sun, being nearest of all in the perihelion a, and farthest in the aphelion g. The areas, a S b, b S c, c S d, &c. or contents of these several triangles, are all equal; and in every turn of the winch N, the comet Y is carried over one of these areas; consequently, in as much time as it moves from f to g, or from g to f, it moves from m to a, or from a to b; and so of the rest, being quickest of all at a, and slowest at g. Thus the comet's velocity in its orbit continually decreases from the perihelion a to the aphelion g; and increases in the same proportion from g to a.

The elliptic orbit is divided into 12 equal parts or signs, with their respective degrees, and so is the circle n o p q r s t u, which represents a great circle in the heavens, and to which the comet's motion is referred by a small knob on the point of the wire W. Whilst the comet moves from f to g in its orbit, it appears to move only about five degrees in this circle, as is shown by the small knob on the end of the wire W; but in as short time as the comet moves from m to a, or from a to b, it appears to describe the large space t n or n o in the heavens, either of which spaces contains Astronomy

contains 120 degrees, or four signs. Were the eccentricity of its orbit greater, the greater still would be the difference of its motion, and vice versa.

ABCDEF GHIKLM is a circular orbit for showing the equable motion of a body round the sun S, describing equal areas ASB, BSC, &c. in equal times with those of the body Y in its elliptical orbit above mentioned; but with this difference, that the circular motion describes the equal arcs AB, BC, &c. in the same equal times that the elliptical motion describes the unequal arcs ab, bc, &c.

Now suppose the two bodies Y and 1 to start from the points a and A at the same moment of time, and, each having gone round its respective orbit, to arrive at these points again at the same instant, the body Y will be forwarder in its orbit than the body 1 all the way from a to g, and from A to G: but 1 will be forwarder than Y through all the other half of the orbit; and the difference is equal to the equation of the body Y in its orbit. At the points a, A, and g, G, that is, that in the perihelion and aphelion they will be equal; and then the equation vanishes. This shows why the equation of a body moving in an elliptic orbit, is added to the mean or supposed circular motion from the perihelion to the aphelion, and subtracted from the aphelion to the perihelion, in bodies moving round the sun, or from the perigee to the apogee, and from the apogee to the perigee in the moon's motion round the earth.

This motion is performed in the following manner by the machine, fig. 218. ABC is a wooden bar (in the box containing the wheel-work), above which are the wheels D and E, and below it the elliptic plates FF and GG; each plate being fixed on an axis in one of its fociuses, at E and K; and the wheel E is fixed on the same axis with the plate FF. These plates have grooves round their edges precisely of equal diameters to one another, and in these grooves is the cat-gut string gg, gg crossing between the plates at b. On H, the axis of the handle or winch N in fig. 217, is an endless screw in fig. 218, working in the wheels D and E, whose numbers of teeth being equal, and should be equal to the number of lines aS, bS, cS, &c. in fig. 217, they turn round their axis in equal times to one another, and to the motion of the elliptic plates. For, the wheels D and E having equal numbers of teeth, the plate FF being fixed on the same axis with the wheel E, and turning the equally big plate GG by a cat-gut string round them both, they must all go round their axis in as many turns of the handle N as either of the wheels has teeth.

It is easy to see, that the end of b of the elliptic plate FF being farther from its axis E than the opposite end I is, must describe a circle so much the larger in proportion, and therefore move through so much more space in the same time; and for that reason the end b moves so much faster than the end I, although it goes no sooner round the centre E. But then the quick-moving end b of the plate FF leads about the short end b K of the plate GG with the same velocity; and the slow-moving end I of the plate FF coming half round as to B, must then lead the long end k of the plate GG as slowly about: so that the elliptic plate FF and its axis E move uniformly and equally quick in every part of its revolution; but the elliptical plate GG, together with its axis K, must move very unequally in different parts of its revolution; the difference being always inversely as the distance of any point of the circumference of GG from its axis at K: or in other words, to instance in two points, if the distance Kk be four, five, or six times as great as the distance Kb, the point b will move in that position, four, five, or six times as fast as the point k does, when the Plate GG has gone half round; and so on for any other eccentricity or difference of the distances Kk and Kb. The tooth I on the plate FF falls in between the two teeth at k on the plate GG; by which means the revolution of the latter is so adjusted to that of the former, that they can never vary from one another.

On the top of the axis of the equally moving wheel D in fig. 218, is the fun S in fig. 217: which fun, by the wire fixed to it, carries the ball t round the circle ABCD, &c. with an equable motion, according to the order of the letters: and on the top of the axis K of the unequally-moving ellipses GG, in fig. 218, is the fun S in fig. 217, carrying the ball Y unequally round in the elliptical groove a b c d, &c. N.B. This elliptical groove must be precisely equal and similar to the verge of the plate GG, which is also equal to that of FF.

In this manner machines may be made to show the true motion of the moon about the earth, or of any planet about the sun, by making the elliptical plates of the same eccentricities, in proportion to the radius, as the orbits of the planets are, whose motions they represent; and so their different equations in different parts of their orbits may be made plain to sight, and clearer ideas of these motions and equations acquired in half an hour, than could be gained from reading half a day about such motions and equations.

The Improved Celestial Globe, fig. 187. On the north pole of the axis, above the hour-circle, is fixed an arch MKH of $23\frac{1}{2}$ degrees; and at the end H is fixed an upright pin HG, which stands directly over the north pole of the ecliptic, and perpendicular to that part of the surface of the globe. On this pin are two moveable collets at D and H, to which are fixed the quadrantile wires N and O, having two little balls on their ends for the fun and moon, as in the figure. The collet D is fixed to the circular plate F, whereon the $29\frac{1}{2}$ days of the moon's age are engraven, beginning just under the fun's wire N; and as this wire is moved round the globe, the plate F turns round with it. These wires are easily turned, if the screw G be slackened: and when they are set to their proper places, the screw serves to fix them there, so as in turning the ball of the globe, the wires with the sun and moon go round with it; and these two little balls rise and set at the same times, and on the same points of the horizon, for the day to which they are rectified, as the sun and moon do in the heavens.

Because the moon keeps not her course in the ecliptic (as the fun appears to do), but has a declination of $5\frac{1}{2}$ degrees on each side from it in every lunation, her ball may be screwed as many degrees to either side of the ecliptic as her latitude or declination from the ecliptic amounts to at any given time.

The horizon is supported by two semicircular arches, because pillars would stop the progress of the balls. To rectify this globe. Elevate the pole to the latitude of the place; then bring the sun's place in the ecliptic for the given day to the brazen meridian, and set the hour index at 12 at noon, that is, to the upper 12 on the hour circle; keeping the globe in that situation, slacken the screw G, and set the sun directly over his place on the meridian; which done set the moon's wire under the number that expresses her age for that day on the Plate F, and she will then stand over her place in the ecliptic, and show what constellation she is in. Lastly, fasten the screw G, and adjust the moon to her latitude, and the globe will be rectified.

Having thus rectified the globe, turn it round, and observe on what points of the horizon the sun and moon balls rise and set, for these agree with the points of the compasses on which the sun and moon rise and set in the heavens on the given day: and the hour index shows the times of their rising and setting; and likewise the time of the moon's passing over the meridian.

This simple apparatus shows all the varieties that can happen in the rising and setting of the sun and moon; and makes the aforementioned phenomena of the harvest moon plain to the eye. It is also very useful in reading lectures on the globes, because a large company can see this sun and moon go round, rising above and setting below the horizon at different times, according to the seasons of the year; and making their apulses to different fixed stars. But in the usual way, where there is only the places of the sun and moon in the ecliptic to keep the eye upon, they are easily lost sight of, unless they be covered with patches.

The Trajectorium Lunare, fig. 208. This machine is for delineating the paths of the earth and moon, showing what sort of curves they make in the ethereal regions. S is the sun, and E the earth, whose centres are 95 inches distant from each other; every inch answering to 1,000,000 miles. M is the moon, whose centre is \( \frac{3}{4} \) parts of an inch from the earth's in this machine, this being in just proportion to the moon's distance from the earth. AA is a bar of wood, to be moved by hand round the axis g which is fixed in the wheel Y. The circumference of this wheel is to the circumference of the small wheel L (below the other end of the bar) as 365\(\frac{1}{4}\) days is to 29\(\frac{1}{2}\) or as a year is to a lunation. The wheels are grooved round their edges, and in the grooves is the cat-gut string GG crossing between the wheels at X. On the axis of the wheel L is the index F, in which is fixed the moon's axis M for carrying her round the earth E (fixed on the axis of the wheel L in the time that the index goes round a circle of 29\(\frac{1}{2}\) equal parts, which are the days of the moon's age. The wheel Y has the months and days of the year all round its limb; and in the bar AA is fixed the index I, which points out the days of the months answering to the days of the moon's age, shown by the index F, in the circle of 29\(\frac{1}{2}\) equal parts at the other end of the bar. On the axis of the wheel L is put the piece D, below the cock C, in which this axis turns round; and in D are put the pencils e and m directly under the earth E and moon M; so that m is carried round e as M is round E.

Lay the machine on an even floor, pressing gently on the wheel Y, to cause its spiked feet (of which two appear at P and P, the third being supposed to be hid from sight by the wheel) enter a little into the floor to secure the wheel from turning. Then lay a paper about four feet long under the pencils e and m, crookedwise to the bar; which done, move the bar slowly round the axis g of the wheel Y; and as the earth E goes round the sun S, the moon M will go round the earth with a duly proportioned velocity; and the friction wheel W running on the floor, will keep the bar from bearing too heavily on the pencils e and m, which will delineate the paths of the earth and moon. As the index I points out the days of the months, the index F shows the moon's age on these days, in the circle of 29\(\frac{1}{2}\) equal parts. And as this last index points to the different days in its circle, the like numeral figures may be set to those parts of the curves of the earth's path and moon's, where the pencils e and m are at those times respectively, to show the places of the earth and moon. If the pencil e be pushed a very little off, as if from the pencil m, to about \( \frac{1}{4} \) part of their distance, and the pencil m pushed as much towards e, to bring them to the same distances again, though not to the same points of space; then, as m goes round e, e will go as it were round the centre of gravity between the earth e and moon m; but this motion will not sensibly alter the figure of the earth's path or the moon's.

If a pin, as p, be put through the pencil m, with its head towards that of the pin q in the pencil e, its head will always keep thereto as m goes round e, or as the same side of the moon is still obverted to the earth. But the pin p, which may be considered as an equatorial diameter of the moon, will turn quite round the point m, making all possible angles with the line of its progress, or line of the moon's path. This is an ocular proof of the moon's turning round her axis.

Sect. XIII. A Description of the principal Astronomical Instruments by which Astronomers make the most accurate Observations.

By Practical astronomy is implied the knowledge of observing the celestial bodies with respect to their position and time of the year, and of deducing from those observations certain conclusions useful in calculating the time when any proposed position of these bodies shall happen.

For this purpose, it is necessary to have a room or place conveniently situated, suitably contrived, and furnished with proper astronomical instruments. It should have an uninterrupted view from the zenith down to (or even below) the horizon, at least towards its cardinal points; and for this purpose, that part of the roof which lies in the direction of the meridian in particular, should have moveable covers, which may easily be moved and put on again; by which means an instrument may be directed to any point of the heavens between the horizon and the zenith, as well to the northward as southward.

This place, called an Observatory, should contain some, if not all, of the following instruments.

1. A Pendulum Clock, for showing equal time. This should show time in hours, minutes, and seconds; and with which the observer, by hearing the beats of the pendulum, may count them by his ear, while his eye is employed on the motion of the celestial object he is observing. Just before the object arrives at the position described, the observer should look on the clock and remark the time, suppose it 9 hours 15 minutes 25 seconds; then saying, 25, 26, 27, 28, &c. responsive to the beat of the pendulum, till he sees through the instrument the object arrived at the position expected; which suppose to happen when he says 38, he then writes down 9 h. 15 min. 38 sec. for the time of observation, annexing the year and the day of the month. If two persons are concerned in making the observation, one may read the time audibly while the other observes through the instrument, the observer repeating the last second read when the desired position happens.

II. An Achromatic Refracting Telescope, or a reflecting one, of two feet at least in length, for observing particular phenomena. These instruments are particularly described under Optics.

III. A Micrometer, for measuring small angular distances. See Micrometer.

IV. Astronomical Quadrants, both mural and portable, for observing meridian and other altitudes of the celestial bodies.

1. The Mural Quadrant is in the form of a quarter of a circle, contained under two radii at right angles to one another, and an arch equal to one fourth part of the circumference of the circle. It is the most useful and valuable of all the astronomical instruments; and as it is sometimes fixed to the side of a stone or brick wall, and the plane of it erected exactly in the plane of the meridian, it in this case receives the name of mural quadrant or arch.

Tycho Brache was the first person who contrived this mural arch, viz. who first applied it to a wall; and Mr Flamsteed, the first in England who with indefatigable pains fixed one up in the royal observatory at Greenwich.

These instruments have usually been made from five to eight feet radius, and executed by those late celebrated artists Sisson, Graham, Bird, and other eminent mathematical instrument makers now in London. The construction of them being generally the same in all the sizes, we shall here describe one made by the late Jon. Sisson, under the direction of the late M. Graham. Fig. 214. represents the instrument as already fixed to the wall. It is of copper, and of about 5 feet radius. The frame is formed of flat bars, and strengthened by edge bars affixed underneath perpendicularly to them. The radii H.B. HA, being divided each into four equal parts, serves to find out the points D and E, by which the quadrant is freely suspended on its props or iron supports that are fastened securely in the wall.

One of the supports E is represented separately in e on one side of the quadrant. It is moveable by means of a long slender rod E.F or e f, which goes into a hollow screw in order to restore the instrument to its situation when it is discovered to be a little deranged. This may be known by the very fine perpendicular thread HA, which ought always to coincide with the same point A of the limb, and carefully examined to be so by a small magnifying telescope at every astronomical observation. In order to prevent the unsteadiness of the instrument, there should be placed behind the limb four copper ears with double cocks I, K, I, K. There are others along the radii HA and HB. Each of these cocks contains two screws, into which is fastened the ears that are fixed behind the quadrant.

Over the wall or stone which supports the instrument, and at the same height as the centre, is placed horizontally the axis PO, which is perpendicular to the plane of the instrument, and which would pass through the centre if it was continued. This axis turns on two pivots P. On this axis is fixed at right angles another branch ON, loaded at its extremity with a weight N capable of equilibrating with its weight that of the telescope LM; whilst the axis, by its extremity nearest the quadrant, carries the wooden frame PRM, which is fastened to the telescope in M. The counterpoise takes off from the observer the weight of the telescope when he raises it, and hinders him from either forcing or straining the instrument.

The lower extremity (V) of the telescope is furnished with two small wheels, which take the limb of the quadrant on its two sides. The telescope hardly bears any more upon the limb than the small friction of these two wheels; which renders its motion extremely easy and pleasant, that by giving it with the hand only a small motion, the telescope will run of itself over a great part of the limb, balanced by the counterpoise N.

When the telescope is to be stopped at a certain position, the copper hand T is to be made use of, which embraces the limb and springs at the bottom. It is fixed by setting a screw, which fastens it to the limb. Then, in turning the regulating screw, the telescope will be advanced; which is continued until the star or other object whose altitude is observing be on the horizontal fine thread in the telescope. Then on the plate X supporting the telescope, and carrying a vernier or nonius, will be seen the number of degrees and minutes, and even quarter of minutes, that the angular height of the object observed is equal to. The remainder is easily eliminated within two or three seconds nearly.

There are several methods of subdividing the divisions of a mural quadrant, which are usually from five to ten minutes each; but that which is most commonly adopted is by the vernier or nonius, the contrivance of Peter Vernier a Frenchman. This vernier consists of a piece of copper or brass, CDAB (fig. 215.), which is a small portion of X (fig. 214.), represented separately. The length CD is divided into 20 equal parts, and placed contiguously on a portion of the division of the limb of the quadrant containing 21 divisions, and thereby dividing this length into 20 equal parts. Thus the first division of the vernier piece marked 15, beginning at the point D, is a little matter backward, or to the left of the first division of the limb, equal to 15". The second division of the vernier is to the left of the second division of the limb double of the first difference, or 30"; and so on unto the twentieth and last division on the left of the vernier piece; where the 20 differences being accumulated each of the twentieth part of the division of the limb, this last division will be found to agree exactly with the 21st division on the limb of the quadrant.

The index must be pushed the 20th part of a division, or \( \frac{1}{5} \), to the right; for to make the second division on the vernier coincide with one of the divisions of the limb, in like manner is moving two 20ths, or \( \frac{3}{5} \), we must look at the second division of the index, and there will be a coincidence with a division of the limb. Thus may be conceived that the beginning D of the vernier, which is always the line of reckoning, has advanced two divisions, or \( \frac{3}{5} \), to the right, when the second division, marked 30 on the vernier, is seen to correspond exactly with one of the lines of the quadrant.

By means of this vernier may be readily distinguished the exactitude of \( \frac{1}{5} \) of the limb of a quadrant five feet radius, and simply divided into \( \frac{1}{5} \). By an estimation by the eye, afterwards the accuracy of two or three seconds may be easily judged. On the side of the quadrant is placed the plate of copper which carries the telescope. This plate carries two verniers. The outer line CD divides five minutes into 20 parts, or \( \frac{1}{5} \) each. The interior line AB answers to the parts of another division not having 90°, but 96 parts of the quadrant. It is usually adopted by English astronomers on account of the facility of its subdivisions. Each of the 96 portions of the quadrant is equivalent to \( \frac{5}{6} \) \( \frac{1}{5} \) of the usual divisions. It is divided on the limb into 16 parts, and the arch of the vernier AB contains 25 of these divisions; and being divided itself into 24, immediately gives parts, the value of each of which is \( \frac{8}{7} \) \( \frac{1}{5} \).

From this mode a table of reduction may easily be constructed, which will serve to find the value of this second mode of dividing in degrees, minutes, and seconds, reckoning in the usual manner, and to have even the advantage of two different modes; which makes an excellent verification of the divisions on the limb of the quadrant and observed heights by the vernier.

2. The Portable Astronomical Quadrant, is that instrument of all others which astronomers make the greatest use of, and have the most esteem for. They are generally made from 12 to 23 inches. Fig. 219. is a representation of the improved modern one as made by the late Mr Siffon and by the present mathematical instrument makers. This is capable of being carried to any part of the world, and put up for observation in an easy and accurate manner. It is made of brass, and strongly framed together by crossed perpendicular bars. The arch AC, and telescope EF, are divided and constructed in a similar manner to the mural quadrant, but generally without the division of 96 parts. The counterpoise to the telescope T is represented at P, and also another counterpoise to the quadrant itself at P. The quadrant is fixed to a long axis, which goes into the pillar KR. Upon this axis is fixed an index, which points to and subdivides by a vernier the divisions of the azimuth circle K. This azimuth circle is extremely useful for taking the azimuth of a celestial body at the same time its altitude is observed. The upper end of the axis is firmly connected with the adjusting frame GH; and the pillar is supported on the crossed feet at the bottom of the pillar KR with the adjusting screws a, b, c, d.

When this instrument is set up for use or observation, it is necessary that two adjustments be very accurately made: One, that the plane or surface of the instrument be truly perpendicular to the horizon; the other, that the line supposed to be drawn from the centre to the first line of the limb, be truly on a level or parallel with the horizon. The first of these particulars is done by means of the thread and plummet p; the thread of which is usually of very fine silver wire, and it is placed opposite to a mark made upon the end of the limb of the instrument. The four screws at the foot, a, b, c, d, are to be turned until a perfect coincidence is observed of the thread upon the mark, which is accurately observed by means of a small telescope T, that fits to the limb. The other adjustment is effected by means of the spirit-level L, which applies on the frame GH, and the small screws turned as before until the bubble of air in the level settles in the middle of the tube. The dotted tube EB is a kind of prover to the instrument: for by observing at what mark the centre of it appears against, or by putting up a mark against it, it will at any time discover if the instrument has been displaced. The screw S at the index, is the regulating or adjusting screw, to move the telescope and index, during the observation, with the utmost nicety.

V. Astronomical or Equatorial Sector.

This is an instrument for finding the difference in right ascension and declination between two objects, the difference of which is too great to be observed by the micrometer. It was the invention of the late ingenious Mr George Graham, F.R.S. and is constructed from the following particulars. Let AB (fig. 32.) represent an arch of a circle containing 10 or 12 degrees well divided, having a strong plate CD for its radius, fixed to the middle of the arch at D: let this radius be applied to the side of an axis HFI, and be moveable about a joint fixed to it at F, so that the plane of the sector may be always parallel to the axis HI; which being parallel to the axis of the earth, the plane of the sector will always be parallel to the plane of some hour-circle. Let a telescope CE be moveable about the centre C of the arch AB, from one end of it to the other, by turning a screw at G; and let the line of sight be parallel to the plane of the sector. Now, by turning the whole instrument about the axis HI, till the plane of it be successively directed, first to one of the stars and then to another, it is easy to move the sector about the joint F, into such a position, that the arch AB, when fixed, shall take in both the stars in their passage, by the plane of it, provided the difference of their declinations does not exceed the arch AB. Then, having fixed the plane of the sector a little to the westward of both the stars, move the telescope CE by the screw G; and observe by a clock the time of each transit over the cross hairs, and also the degrees and minutes upon the arch AB, cut by the index at each transit; then in the difference of the arches, the difference of the declinations, and by the difference of the times, we have the difference of the right ascensions of the stars.

The dimensions of this instrument are these: The length of the telescope, or the radius of the sector, is \( \frac{2}{3} \) feet; the breadth of the radius, near the end C, is \( \frac{1}{3} \) inch; and at the end D two inches. The breadth of the limb AB is \( \frac{1}{3} \) inch; and its length... six inches, containing ten degrees divided into quarters and numbered from either end to the other. The telescope carries a nonius or subdividing plate, whose length, being equal to fifteen quarters of a degree, is divided into fifteen equal parts; which, in effect, divides the limb into minutes, and, by estimation, into smaller parts. The length of the square axis HIF is eighteen inches, and of the part HI twelve inches; and its thickness is about a quarter of an inch: the diameters of the circles are each five inches: the thicknesses of the plates, and the other measures, may be taken at the direction of a workman.

This instrument may be rectified, for making observations, in this manner: By placing the intersection of the cross hairs at the same distance from the plane of the sector, as the centre of the object-glass, the plane described by the line of sight, during the circular motion of the telescope upon the limb, will be sufficiently true, or free from conical curvity; which may be examined by suspending a long plumb-line at a convenient distance from the instrument; and by fixing the plane of the sector in a vertical position, and then observing, while the telescope is moved by the screw along the limb, whether the cross hairs appear to move along the plumb-line.

The axis bfo may be elevated nearly parallel to the axis of the earth, by means of a small common quadrant; and its error may be corrected, by making the line of sight follow the circular motion of any of the circumpolar stars, while the whole instrument is moved about its axis bfo, the telescope being fixed to the limb: for this purpose, let the telescope k be directed to the star a, when it passes over the highest point of its diurnal circle, and let the division cut by the nonius be then noted: then, after twelve hours, when the star comes to the lowest point of its circle, having turned the instrument half round its axis, to bring the telescope into the position m n; if the cross hairs cover the same star supposed at b, the elevation of the axis bfo is exactly right; but if it be necessary to move the telescope into the position m n, in order to point to this star at c, the arch m u, which measures the angle mfu or bfc, will be known; and then the axis bfo must be depressed half the quantity of this given angle if the star passed below b, or must be raised so much higher if above it; and then the trial must be repeated till the true elevation of the axis be obtained. By making the like observations upon the same star on each side the pole, in the six-o’clock-hour-circle, the error of the axis, toward the east or west, may also be found and corrected, till the cross-hairs follow the star quite round the pole: for supposing aopbc to be an arch of the meridian (or in the second practice of the fix-o’clock hour-circle), make the angle afp equal to half the angle afc, and the line fp will point to the pole; and the angle of fp, which is the error of the axis, will be equal to half the angle bfc or mfu, found by the observation; because the difference of the two angles afb, afc, is double the difference of their halves afb, afp. Unless the star be very near the pole, allowance must be made for refractions.

VI. Transit and Equal Altitude Instruments.

1. The transit Instrument is used for observing objects as they pass over the meridian. It consists of a telescope fixed at right angles to an horizontal axis; which axis must be so supported that what is called the line of collimation, or line of sight of the telescope, may move in the plane of the meridian. This instrument was first made by the celebrated Mr. Romer in the year 1689, and has since received great improvements. It is made of various sizes, and of large dimensions in our great observatories; but the following is one of a size sufficiently large and accurate for all the useful purposes.

The axis AB (fig. 220.), to which the middle of the telescope is fixed, is about 2½ feet long, tapering gradually toward its ends, which terminate in cylinders well turned and smoothed. The telescope CD which is about four feet long and 1½ inch diameter, is connected with the axis by means of a strong cube or die G, and in which the two cones MQ, forming the axis, are fixed. This cube or block G serves as the principal part of the whole machine. It not only keeps together the two cones, but holds the two sockets KH, of 15 inches length, for the two telescopic tubes. Each of these sockets has a square base, and is fixed to the cube by four screws. These sockets are cut down in the sides about eight inches, to admit more easily the tube of the telescope; but when the tube is inserted, it is kept in firm by screwing up the tightening screws at the end of the sockets at K and H. These two sockets are very useful in keeping the telescope in its greatest possible degree of steadiness. They also afford a better opportunity of balancing the telescope and rectifying its vertical thread, than by any other means.

In order to direct the telescope to the given height that a star would be observed at, there is fixed a semicircle AN on one of the supporters, of about 8½ inches diameter, and divided into degrees. The index is fixed on the axis, at the end of which is a vernier, which subdivides the degrees into 12 parts or five minutes. This index is moveable on the axis, and may be closely applied to the divisions by means of a tightening screw.

Two upright posts of wood or stone YY, firmly fixed at a proper distance, are to sustain the supporters of this instrument. These supporters are two thick brass plates RR, having well smoothed angular notches in their upper ends, to receive the cylindrical arms of the axis. Each of these notched plates is contrived to be moveable by a screw, which slides them upon the surfaces of two other plates immovably fixed upon the two upright pillars; one plate moving in an horizontal, and the other in a vertical, direction; or, which is more simple, these two modes are sometimes applied only on one side, as at V and P, the horizontal motion by the screw P, and the vertical by the screw V. These two motions serve to adjust the telescope to the planes of the horizon and meridian: to the plane of the horizon by the spirit-level EF, hung by DC on the axis MQ, in a parallel direction; and to the plane of the meridian in the following manner:

Observe by the clock when a circumpolar star seen through this instrument transits both above and below the pole; and if the times of describing the eastern and western parts of its circuit are equal, the telescope is then in the plane of the meridian: otherwise the screw P must be gently turned that it may move the telescope. Astronomical Instruments.

Scope so much that the time of the star's revolution be bisected by both the upper and lower transits, taking care at the same time that the axis remains perfectly horizontal. When the telescope is thus adjusted, a mark must be set at a considerable distance (the greater the better) in the horizontal direction of the intersection of the cross wires, and in a place where it can be illuminated in the night-time by a lanthorn hanging near it; which mark being on a fixed object, will serve at all times afterwards to examine the position of the telescope by, the axis of the instrument being first adjusted by means of the level.

To adjust the Clock by the Sun's Transit over the Meridian. Note the times by the clock when the preceding and following edges of the sun's limb touch the cross wires. The difference between the middle time and 12 hours shows how much the mean, or time by the clock, is faster or slower than the apparent, or solar time, for that day; to which the equation of time being applied, will show the time of mean noon for that day, by which the clock may be adjusted.

2. The Equal Altitude Instrument is an instrument that is used to observe a celestial object when it has the same altitude on both the east and west sides of the meridian, or in the morning and afternoon. It principally consists of a telescope about 30 inches long fixed to a sextant or semicircular divided arch; the centre of which is fixed to a long vertical axis; but the particulars of this instrument the reader will see explained in Optics, Part III.

3. Compound Transit Instrument. Some instruments have been contrived to answer both kinds of observations, viz. either a transit or equal altitudes. Fig. 222 represents such an instrument, made first of all for Mr Le Monnier the French astronomer, by the late Mr Siffon, under the direction of Mr Opaham, mounted and fitted up ready for observation.

AB is a telescope, which may be 3, 4, 5, or 6 feet long, whose cylindrical tube fits exactly into another hollow cylinder ab, perpendicular to the axis; these several pieces are of the best hammered plate brass. The cylindrical extremity of this axis MN are of solid bell-metal, and wrought exquisitely true, and exactly the same size in a lathe; and it is on the perfection to which the cylinders or trunnions are turned that the justness of the instrument depends. In the common focus of the object-glass and eye-glass is placed a reticle (fig. 223), consisting of three horizontal and parallel fine-stretched silver wires, fixed by pins or screws to a brass circle, the middle one passing through its centre, with a fourth vertical wire likewise passing through the centre, exactly perpendicular to the former three.

The horizontal axis MN (fig. 222) is placed on a strong brass frame, into the middle of which a steel cylinder GH is fixed perpendicularly, being turned truly round, and terminating in a conical point at its lower extremity; where it is let into a small hole drilled in the middle of the dovetail slider; which slider is supported by a hollow tube fixed to the supporting piece IK, consisting of two strong plates of brass, joined together at right angles, to which are fixed two iron cramps L, L, by which it is fastened to the stone-wall of a south window.

The upper part G of the steel spindle is embraced by a collar def, being in contact with the blunt extremity of three screws, whose particular use will be explained by and by. O is another cylindrical collar closely embracing the steel spindle at about a third part of its length from the top; by the means of a small screw it may be loosened or pinched close as occasion requires. From the bottom of this collar proceeds an arm or lever acted upon by the two screws g h, whereby the whole instrument, excepting the supporting piece, may be moved laterally, so that the telescope may be made to point at a distant mark fixed in the vertical of the meridian. ik is a graduated semicircle of thin brass screwed to the telescope, whereby it may be elevated so as to point to a known celestial object in the daytime. lm is a spirit-level parallel to the axis of rotation on the telescope, on which two trunnions hang by two hooks at M and N. Along the upper side of the glass tube of the level slides a pointer to be set to the end of the air-bubble; and when the position of the axis of rotation is so adjusted by the screws that the air-bubble keeps to the pointer for a whole revolution of the instrument, the spindle GH is certainly perpendicular to the horizon, and then the line of collimation of the telescope describes a circle of equal altitude in the heavens. When the level is suspended on the axis, raise or depress the tube of the level by twisting the neck of the screw n till you bring either end of the air-bubble to rest at any point towards the middle of the tube, to which slide the index; then lift off the level, and, turning the ends of it contrary ways, hang it again on the trunnions; and if the air-bubble rests exactly again, the index as before, the axis of rotation is truly horizontal; if not, depress that end of the axis which lies on the same side of the pointer as the bubble does, by turning the neck of the screw at N, till the bubble returns about halfway towards the pointer; then having moved the pointer to the place where it now rests, invert the ends of the level again, and repeat the same practice till the bubble rests exactly at the pointer in both positions of the level. If, after the telescope is turned upside down, that is, after the trunnions are inverted end for end, you perceive that the same points of a remote fixed object is covered by the vertical wire in the focus of the telescope, that was covered by it before the inversion, it is certain that the line of sight or collimation is perpendicular to the transverse axis; but if the said vertical wire covers any other point, the brass circle that carries the hairs must be moved by a screw-key introduced through the perforation in the side of the tube at X, till it appears to bisect the line joining these two points, as near as you can judge; then, by reverting the axis to its former position, you will find whether the wires be exactly adjusted. N.B. The ball e is a counterpoise to the centre of gravity of the semicircle ik, without which the telescope would not rest in an oblique elevation without being fixed by a screw or some other contrivance.

The several beforementioned verifications being accomplished, if the telescope be elevated to any angle with the horizon and there stopped, all fixed stars which pass over the three horizontal wires of the reticle on the eastern side of the meridian in ascending, will have precisely the same altitudes when in descending they again cross the same respective wires on the west. well side, and the middle between the times of each respective equal altitude will be the exact moment of the star's culminating or passing the meridian. By the help of a good pendulum-clock, the hour of their true meridional transits will be known, and consequently the difference of right ascension of different stars. Now, since it will be sufficient to observe a star which has north declination two or three hours before and after its passing the meridian, in order to deduce the time of its arrival at that circle; it follows, that having once found the difference of right ascension of two stars about 60 degrees astern, and you again observe the first of these stand at the same altitude both in the east and west side, you infer with certainty the moment by the clock at which the second star will be on the meridian that same night, and by this means the transit instrument may be fixed in the true plane of the meridian till the next day; when, by depressing it to some distant land objects, a mark may be discovered whereby it may ever after be rectified very readily, so as to take the transits of any of the heavenly bodies to great exactness, whether by night or day.

When such a mark is thus found, the telescope being directed carefully to it, must be fixed in that position by pinching fast the end of the arm or lever between the two opposite screws g h; and if at any future time, whether from the effect of heat or cold on the wall to which the instrument is fixed, or by any settling of the wall itself, the mark appears no longer astronomically well bisected by the vertical wire, the telescope may easily be made to bisect it again, by giving a small motion to the pinching screws.

The transit-instrument is now considered as one of the most essential particulars of the apparatus of an astronomical observatory.

Besides the above may be mentioned,

The Equatorial or Portable Observatory; an instrument designed to answer a number of useful purposes in practical astronomy, independent of any particular observatory. It may be made use of in any steady room or place, and performs most of the useful problems in the science. The following is a description of one lately invented by Mr Ramsden, from whom it has received the name of the Universal Equatorial.

The principal parts of this instrument (fig. 211.) are, 1. The azimuth or horizontal circle A, which represents the horizon of the place, and moves on a long axis B, called the vertical axis. 2. The equatorial or hour-circle C, representing the equator placed at right angles to the polar axis D, or the axis of the earth, upon which it moves. 3. The semicircle of declination E, on which the telescope is placed, and moving on the axis of declination, or the axis of motion of the line of collimation F. These circles are measured and divided as in the following table:

| Measures of the several circles and divisions on them | Radius. | Limb divided to | Nominus of 30 gives seconds. | Divided on limb into parts of inch. | Divided by Nominus into parts of inch. | |------------------------------------------------------|---------|----------------|-------------------------------|---------------------------------|----------------------------------| | Azimuth or horizontal circle | 5 | 15' | 30" | 45th | 1350th | | Equatorial or hour circle | 5 | 15' | 30" | 45th | 1350th | | Vertical semicircle for declination or latitude | 5 | 15' | 30" | 42d | 1260th |

4. The telescope, which is an achromatic refractor with a triple object-glass, whose focal distance is 17 inches, and aperture 2.45 inches, and furnished with six different eye-tubes; so that its magnifying powers extend from 44 to 168. The telescope in this equatorial may be brought parallel to the polar axis, as in the figure, so as to point to the pole-star in any part of its diurnal revolution; and thus it has been observed near noon, when the sun has shone very bright.

5. The apparatus for correcting the error in altitude occasioned by refraction, which is applied to the eye-end of the telescope, and consists of a slide G moving in a groove or dove-tail, and carrying the several eyetubes of the telescope, on which slide there is an index corresponding to five small divisions engraved on the dove-tail; a very small circle, called the refraction circle H, moveable by a finger-screw at the extremity of the eye-end of the telescope; which circle is divided into half minutes, one entire revolution of it being equal to 3° 18', and by its motion raises the centre of the cross hairs on a circle of altitude; and likewise a quadrant I of 1½ inch radius, with divisions on each side, one expressing the degree of altitude of the object viewed, and the other expressing the minutes and seconds of error occasioned by refraction, corresponding to that degree of altitude: to this quadrant is joined a small round level K, which is adjusted partly by the pinion that turns the whole of this apparatus, and partly by the index of the quadrant; for which purpose the refraction circle is set to the same minute, &c., which the index points to on the limb of the quadrant; and if the minute, &c., given by the quadrant exceed the 3° 18' contained in one entire revolution of the refraction circle, this must be set to the excess above one or more of its entire revolutions; then the centre of the cross hairs will appear to be raised on a circle of altitude to the additional height which the error of refraction will occasion at that altitude.

This instrument stands on three feet L distant from each other 14½ inches; and when all the parts are horizontal is about 29 inches high: the weight of the equatorial and apparatus is only 59 lb. avoirdupois, which are contained in a mahogany case weighing 58 lb.

The principal adjustment in this instrument is that of making the line of collimation to describe a portion of an hour-circle in the heavens; in order to which, the azimuth circle must be truly level, the line of collimation... Astronomy or some corresponding line represented by the small brass rod M parallel to it, must be perpendicular to the axis of its own proper motion; and this last axis must be perpendicular to the polar axis: on the brass rod M there is occasionally placed a hanging level N, the use of which will appear in the following adjustments:

The azimuth-circle may be made level by turning the instrument till one of the levels is parallel to an imaginary line joining two of the feet screws; then adjust that level with these two feet screws; turn the circle half round, i.e. 180°; and if the bubble be not then right, correct half the error by the screw belonging to the level, and the other half error by the two foot screws; repeat this till the bubble comes right; then turn the circle 90° from the two former positions, and set the bubble right, if it be wrong, by the foot screw at the end of the level; when this is done, adjust the other level by its own screw, and the azimuth-circle will be truly level. The hanging level must then be fixed to the brass rod by two hooks of equal length, and made truly parallel to it: for this purpose make the polar axis perpendicular or nearly perpendicular to the horizon; then adjust the level by the pinion of the declination-femicircle; reverse the level, and if it be wrong, correct half the error by a small steel screw that lies under one end of the level, and the other half error by the pinion of the declination-femicircle; repeat this till the bubble be right in both positions.

In order to make the brass rod on which the level is suspended at right angles to the axis of motion of the telescope or line of collimation, make the polar axis horizontal, or nearly so; set the declination-femicircle to 0°, turn the hour circle till the bubble comes right; then turn the declination-circle to 90°; adjust the bubble by raising or depressing the polar axis (first by hand till it be nearly right, afterwards tighten with an ivory key the socket which runs on the arch with the polar axis, and then apply the same ivory key to the adjusting screw at the end of the said arch till the bubble comes quite right); then turn the declination-circle to the opposite 90°; if the level be not then right, correct half the error by the aforeaid adjusting screw at the end of the arch, and the other half error by the two screws which raise or depress the end of the brass rod. The polar axis remaining nearly horizontal as before, and the declination-femicircle at 0°, adjust the bubble by the hour-circle; then turn the declination-femicircle to 90°, and adjust the bubble by raising or depressing the polar axis; then turn the hour-circle 12 hours; and if the bubble be wrong, correct half the error by the polar axis, and the other half error by the two pair of capstan screws at the feet of the two supports on one side of the axis of motion of the telescope; and thus this axis will be at right angles to the polar axis. The next adjustment is to make the centre of crofs hairs remain on the same object, while you turn the eye-tube quite round by the pinion of the refraction apparatus: for this adjustment, set the index on the slide to the first division on the dove-tail; and set the division marked 18° on the refraction-circle to its index; then look through the telescope, and with the pinion turn the eye-tube quite round; and if the centre of the hairs does not remain on the same spot during that revolution, it must be corrected by the four small screws, two and two at a time (which you will find upon uncrewing the nearest calibrations).

end of the eye-tube that contains the first eye-glass); repeat this correction till the centre of the hairs remains on the spot you are looking at during an entire revolution. In order to make the line of collimation parallel to the brass rod on which the level hangs, set the polar axis horizontal, and the declination-circle to 90°; adjust the level by the polar axis; look through the telescope on some distant horizontal object, covered by the centre of the crofs hairs; then invert the telescope, which is done by turning the hour-circle half round; and if the centre of the crofs hairs does not cover the same object as before, correct half the error by the uppermost and lowermost of the four small screws at the eye-end of the large tube of the telescope; this correction will give a second object now covered by the centre of the hairs, which must be adopted instead of the first object; then invert the telescope as before; and if the second object be not covered by the centre of the hairs, correct half the error by the same two screws which were used before: this correction will give a third object, now covered by the centre of the hairs, which must be adopted instead of the second object; repeat this operation till no error remains; then set the hour-circle exactly to 12 hours (the declination circle remaining at 90° as before); and if the centre of the crofs hairs does not cover the last object fixed on, set it to that object by the two remaining small screws at the eye-end of the large tube, and then the line of collimation will be parallel to the brass rod.

For rectifying the nonius of the declination and equatorial circles, lower the telescope as many degrees, minutes, and seconds, below 0° or AE on the declination-femicircle as are equal to the complement of the latitude; then elevate the polar axis till the bubble be horizontal, and thus the equatorial circle will be elevated to the colatitude of the place; set this circle to 6 hours; adjust the level by the pinion of the declination-circle; then turn the equatorial circle exactly 12 hours from the last position; and if the level be not right, correct one half of the error by the equatorial circle, and the other half by the declination-circle; then turn the equatorial circle back again exactly 12 hours from the last position; and if the level be still wrong, repeat the correction as before till it be right, when turned to either position; that being done, set the nonius of the equatorial circle exactly to 6 hours, and the nonius of the declination-circle exactly to 0°.

The principal uses of this equatorial are,

1. To find your meridian by one observation only: for this purpose, elevate the equatorial circle to the co-latitude of the place, and set the declination-femicircle to the sun's declination for the day and hour of the day required; then move the azimuth and hour circles both at the same time, either in the same or contrary direction, till you bring the centre of the crofs hairs in the telescope exactly to cover the centre of the sun; when that is done, the index of the hour-circle will give the apparent or solar time at the instant of observation; and thus the time is gained, though the sun be at a distance from the meridian; then turn the hour-circle till the index points precisely at 12 o'clock, and lower the telescope to the horizon, in order to observe some point there. in the centre of your glass, and that point is your meridian mark found by one observation only; the best time for this operation is three hours before or three hours after 12 at noon.

2. To point the telescope on a star, though not on the meridian, in full day-light. Having elevated the equatorial circle to the co-latitude of the place, and set the declination-semicircle to the star's declination, move the index of the hour-circle till it shall point to the precise time at which the star is then distant from the meridian, found in tables of the right ascension of the stars, and the star will then appear in the glass. Besides these uses peculiar to this instrument, it is also applicable to all the purposes to which the principal astronomical instruments, viz. a transit, a quadrant, and an equal altitude instrument, are applied.

INDEX.

A

Aberration of light, discovered by Dr Bradley, no. 32. Furnishes an argument for the earth's motion, 337. Reason of the light's aberration particularly explained, ib.

Academy Royal at Paris founded by Merennus, 30.

Alhagiatus, a celebrated Arabian astronomer reforms the science, 17.

Alexander the Great obtains the astronomical observations of the Chaldeans, 6.

Alexandria: a famous astronomical school set up there by Ptolemy Philadelphus, 12.

Algol, or Medusa's head, a variable star, changes its lustre in four hours, 49.

Al Hazen an Arabian, shows the nature of refraction, &c. 17.

Almagest of Ptolemy, an astronomical treatise on the means of reviving the science in Europe, 18.

Alfonso king of Castile causes astronomical tables to be constructed, 18.

Altitudes of the celestial bodies, how taken, 379.

Anaxagoras foretells an eclipse, 10. His opinion concerning the sun, 121.

Anaximander introduces the gnomon into Greece, &c. 10.

Anaximenes improves the astronomy of the Greeks, 10. Absurd opinions of him and Anaximander concerning the sun, 121.

Angles: errors unavoidable in the measurement of small ones, 339.

Antecedence, motion in explained, 292.

Antediluvians supposed to have made considerable progress in astronomy, 1.

Vol. II. Part II.

Anticipation of the equinoxes explained, 349.

Aphelion of the planets defined, 254. Place of the aphelia of the different planets, 255. Motion of the aphelion accounted for, 268.

Apogee of the moon defined, 291. Its motion determined, 294. Inequalities in its motion, 295. Occasion an inequality in the moon's motion, 296. Apogee of the planets explained, 325.

Appian, one of the successors of Copernicus, his improvements in astronomy, 22.

Arabians cultivate the science of astronomy when it was neglected in the west, 17.

Archimedes makes great discoveries in astronomy, 13.

Aristaclitus determines the distance of the sun by the moon's dichotomy, 12. This method particularly explained, 389. Its insufficiency shown, 390.

Aristotle embraces the system of astronomy afterwards called the Ptolemaic, 11. His opinion concerning comets, 164.

Arystillus and Timocharis first cultivate the astronomical science in the school of Alexandria, 12.

Ascension, right, of the sun and stars, how to find it, 411, 412.

Aspects of the planets explained, 321.

Astrol zodiac explained, 381.

Astronomical knowledge of the Antediluvians, 1. Of the Chinese, 2. Of the Indians, 4. Of the Mexicans, 5. Of the Chaldeans and Egyptians, 6. Of the Phoenicians, 7. Of the Greeks, 8. Of the Arabs, 17. Revival of it in Europe, 18.

Astronomical instruments, description of those now in use, 495.

Astronomical sector delineated, 499.

Astronomical problems, a method invented by Appian of resolving them by instruments without tables or calculations, 22.

Astronomical quadrant invented by Nomus, 22. Divided by concentric circles and diagonals, ib. Method of dividing it shown by Appian, ib.

Atlas, a celestial one composed by Bayer, 26.

Atmosphere: Whether the planet Venus has one surrounding her, 78. Mr Herschel's account of the atmosphere of Mars, 90. Atmospheres of comets very large, 111. Whether the sun has an atmosphere, 123. Whether the moon has any, 147.

Attraction: Consequences of its action among the most distant stars, 215. Prevents them from remaining absolutely at rest, 234. Is diffused throughout the substance of all matter, 202. Acts equally through the whole universe, 305. Is a universal property of matter, 308. Solution of a difficulty concerning it derived from the moon's motion, 361.

Azoult applies a micrometer to telescopes, 31.

B

Babylon: Astronomical observations made there for 1903 years, 6.

Barometer: Why it is not affected by the moon, 369.

Bayer, John, his celestial atlas and nomenclature of the stars, 26.

Bear, Greater, Chinese delineation of that constellation, 3.

Unfit for making observations at sea, on account of its distance from the pole, 7.

Beam: The tail of a comet so called when it appears without the head being visible, 112.

Bear, Lesser: A proper guide for navigators, 7. Knowledge of this constellation introduced into Greece by Thales, 9.

Belts: One observed by Maraldi on the body of Mars, 80. Two others observed by him afterwards, ib. Belts of Jupiter discovered by Fontana, 89. Cannot be seen but by an excellent telescope, ib. Vary in their number from one to eight, ib. Sometimes seem to flow into one another, ib. Are extremely variable in the time of their continuance, ib. Black spots sometimes visible in them, ib. Two permanent belts discoverable on Saturn parallel to the edge of his ring, 106. Some bright belts discovered on this planet by Cassini and Fatio, 106.

Belus, temple of at Babylon, supposed to have been an astronomical observatory, 6.

Bergman's account of the transit of Venus over the sun, 78.

Bernouilli's conjectures concerning comets, 167.

Bianchini's observations on the planet Venus, 71. His dispute with Cassini concerning the time of her revolution on her axis, 73.

Bodin: His ridiculous opinion concerning comets, 165.

Bradley, Dr. succeeds Dr. Halley as astronomer-royal, 33. Discovers the lesser planetary inequalities, the aberration of light light, and nutation of the earth's axis, &c. 33. Corrects the lunar and other tables, and makes a vast number of celestial observations, ib.

Briggs, Henry, improves the logarithmic tables of Napier, and constructs much larger ones, 28.

Brightness of the sun and fixed stars compared, 199.

Brydone, Mr Patrick, his observation of the velocity of a comet, 184. His conjectures concerning comets without tails, 187.

C

Calippus invents a period of 76 years, 11.

Calini distinguishes himself as astronomer-royal at Paris, 33. Discovers the satellites, belts, &c. of Saturn, 31. His observations on the spots of Venus, 66. His conclusions concerning the revolution of that planet on her axis, 69. Dispute betwixt his son and Bianchini on this subject, 73. Has a view of the satellite of Venus, 74. Observes the spots on Mars, 80. An observation of his concerning the atmosphere of Mars doubted by Mr Herschel, 90. His observation on the tail of the comet of 1680, 115. His explanation of the zodiacal light, 124.

Cassiopeia: A new star appears in that constellation in 1572, 45. Conjectures concerning it by Mr Pigot, 47.

Celestial spaces void of all sensible matter, 261.

Centre of gravity: Sun and planets revolve about a common one, and why, 297. Place of the common centre of gravity betwixt the earth and moon, 306.

Centripetal forces explained, 257. How they may occasion the revolution of a body round a centre, 258. And cause it describe a curve of any kind, 259. Proved to extend throughout the solar system, 265. Differently denominated according to the bodies in which they act, 266.

Chaldea, a proper country for astronomical observations, 6.

Chaldeans dispute with the Egyptians the honour of being the first cultivators of astronomy, 6. Extreme antiquity of their observations, ib. Account of their astronomical knowledge from Petavius, ib.

Chancellor, an Englishman, discovers the diagonal method of dividing quadrants, 25.

Chappe's observations on the transit of Venus, 79.

China supposed to have been peopled by Noah, 2.

Chinese: Why so early instructed in the knowledge of astronomy, 2. Said to derive it from their first emperor Fohi, supposed to be Noah, ib. Their names for the signs of the zodiac, 3. Divide the heavens into 28 constellations, ib. Method of delineating these constellations, ib. Were early acquainted with the pole star, mariner's compass, &c. ib. Said to have calculated a great number of eclipses, ib. Enormous errors in some of their calculations, ib. P. Gaubil's account of their astronomical knowledge, ib. Du Halde's account of Tcheu-cong's observatory, ib. Have an astronomical tribunal, ib. Astronomy now in a very low state among them, and why, ib.

Chiron the Centaur makes a celestial sphere, 11.

Clairault, Mr, his accuracy in calculating the return of the comet in 1759, 192.

Clark, Dr: A star observed by his father through the space betwixt the ring and body of Saturn, 104.

Climate settled by Pytheas of Marseille, 11.

Clock, astronomical, how used, 496. How adjusted by the sun's transit over the meridian, 501.

Cluster of Stars: Mr Herschel's distinction betwixt this and a place of the heavens crowded with them, 223.

Cole's hypothesis concerning comets, 188.

Comets supposed by some of the Chaldeans to be permanent bodies revolving like the planets, 6. Supposed by others to be only meteors, ib. Called wandering stars by Pythagoras, 11. Comet of 1500 observed by Warner, 21. The return of one predicted by Dr Halley, 33. General account of their appearance to the naked eye, 43. Four hundred and fifty supposed to belong to our solar system, ib. Have a splendor inferior to the other heavenly bodies, and are apt to lose great part of it suddenly, ib. Their appearance through telescopes, 110. Are surrounded with atmospheres of a prodigious size, 111. Have sometimes different phases like the moon, ib. Dr Long's account of them, 112. Their heads seem to be composed of a solid nucleus and surrounding atmosphere, 110, 112. Nucleus sometimes breaks in pieces, 110, 112. Are generally furnished with luminous tails which distinguish them from other stars, 43, 112. Different appearances of their tails explained and accounted for, 113. Called bearded comets when the tail hangs downwards, 114. Extreme length of the tails of some comets, 114, 179. Hevelius observes the comet of 1665 to cast a shadow on its tail, 115. This observation disputed by Mr Hooke, ib. Comet of 1680 approaches within a sixth part of the sun's diameter to his surface, 116. Account of the comet of 1744, n° 117. Was thought capable of disturbing the motion of Mercury, but did not, ib. Shown to be incapable of doing so by Dr Betts of Oxford, ib. Account of the different changes it underwent, ib. Undulation of our atmosphere causes the tails of comets seem to sparkle, and lengthen, and shorten, ib. Why the comet of 1759 made such an inconsiderable appearance, 118. Hevelius's account of the appearances of different comets, 110, 115, 117, 118. Supposed to be the forerunners of various calamities, 162. Held by some of the ancient Greeks to be planets, 163. Supposed by Aristotle to be meteors, 164. This opinion contradicted by Seneca, who foretold that their true nature and motions would be discovered, ib. Only one species of them exists, 165. Their appearances vary by reason of their situation, &c. ib. Supposed by Kepler and Bodin to be animals or spirits, 166. By Bernouilli to be the satellites of a distant planet, 167. Tycho Brahe revives the true doctrine concerning them, 168. Few comets have a diurnal, but all an annual parallax, ib. Supposed by some to move in straight lines, ib. By Kepler, in parabolic trajectories, ib. Demonstrated by Newton to move in very eccentric eclipses, 169. Foundation of Dr Halley's prediction of the return of comets, 170, 190. Periodical times of different comets determined, 171. May sometimes be invisible during their perihelion, by reason of their being above our horizon in the daytime, 172. Ancient observation of a comet in this situation during an eclipse of the sun, ib. Why more comets are seen in the hemisphere turned towards the sun than in that removed from him, 173. Great differences in the eccentricities of the orbits of different comets, as well as of their velocities in them, 174. Planes of some of their orbits almost perpendicular to others, ib. Supposed by Hevelius to be transparent, 175. Demonstrated by Newton to be opaque bodies reflecting the sun's light, ib. Are of different magnitudes and at different distances, but most frequently smaller than the earth, 176. Computations of the distances and diameters of some of them, ib. Eclipses supposed to be occasioned by them, 177. Their tails supposed by Tycho-Brahe, Des Cartes, and others, to be occasioned by refraction, 178, 179. By Newton to be a vapour raised by the heat of the sun, ib. By Maran, to be formed of the sun's INDEX.

Atmosphere, 179. By de la Lande, to be occasioned by the rarefaction of their own atmospheres, ib. Objections to Newton's opinion by Mr Rouning, ib. By Dr Hamilton of Dublin, who supposes their tails to be streams of electric matter, 180. Sir Isaac's account defended, 181. Objections to Dr Hamilton's hypothesis, 182. Prodigious velocity of a comet observed by Mr Brydone, 184. Of that of 1680, according to Sir Isaac Newton, 179. Excessive heat of that comet, according to the same author, 185, 186. Doubted by Dr Long, 185. The deluge supposed by Whiston to be occasioned by a comet, ib. Apparent changes in the comets attributed by Newton to their atmospheres, 186. He supposes that the comet of 1680 must at last fall into the sun, ib. That water and a kind of vital spirit are derived from the tails of comets, ib. Comets without tails supposed by Mr Brydone always to fall into the sun, 187. Comets supposed by Mr Cole to be subject to the attraction of different centres, 188. How to calculate their periodical times, 189. Difference between the calculations of Mr Euler and those of Newton and Halley, ib. Comet of 1744 seemed to move in a parabola, ib. Periodical times of comets may be changed by the attraction of the planets or other comets, 191. Comets supposed by Des Cartes to be without the solar system, 298. Demonstrated by Newton to pass through the planetary regions, and to be generally invisible at a smaller distance than Jupiter, 299. His explanation of their motions and directions how to calculate them, 300, 301. Why they move in places so different from those of the planets, ib.

Cometarium, 492. Conflagration, general, supposed by Whiston to be occasioned by a comet, 185. Conjunction of Saturn and Jupiter, observed by Tycho in 1563, 24. Explanation of the conjunctions and oppositions of the planets, 320. Consequence, motion in, explained, 292. Constellations of the Chinese, how marked, 3. A number of stars arranged in them, by the Chaldeans, 6. Most of them supposed by Sir Isaac Newton to be invented about the time of the Argonautic expedition, 8. Uses of this division of the heavens, 403. Catalogue of them, 406. Copernicus revives the Pythagorean system of astronomy, 22. Determines the periodical time of the moon, 422. Curve of any kind may be described by a body acted upon by projectile and centripetal forces, 259.

D. Darkness at our Saviour's crucifixion supposed to be owing to an eclipse by a comet, 177. Could not be owing to a lunar eclipse by the moon, 450. Declination of the sun: Improved tables of it constructed by Mr Wright, 26. Declination of the celestial bodies explained, 315. How to compute the declination of the sun, 411. De la Lande objects to Dr Wilson's theory of the solar spots, 136. The Doctor's reply to his objections, 137. Remarks on La Lande's theory, 197. His opinion concerning the tails of comets, 185. Deluge supposed by Whiston to be occasioned by a comet, 185. Densities of the heavenly bodies how determined, 310. Des Cartes objects to the project of a reflecting telescope, 31. His opinion concerning the tails of comets, 179. His system overthrown by Newton, 261. Diagonals, method of dividing quadrants, &c. by them invented by Chanceker an Englishman, 25. Is not exact unless curved diagonals be made use of, ib. Dialling in use among the Chaldeans long before the Greeks had any knowledge of it, 6. Diameters, apparent, of the planets differ at different times, 326. Computation of their different diameters, ib. Dilatation of the moon's orbit how caused, 285. Direct motions of the planets explained, 323. Dolland's achromatic telescopes: Caution to be observed in using them, 144. Druids supposed to have been well skilled in astronomy, 11. Dunn's account of the solar spots, 62. His observations on the transit of Venus, 78. His hypothesis concerning variable stars, 196. His explanation of the horizontal moon, 375.

E. Earth supposed to be spherical by the Egyptians, 6. Its rotation on its axis taught by Hicetasthe Syracusian, 11. A great circle of it measured by Eratosthenes by means of a gnomon, 12. A degree of it measured again by order of the caliph Al Mamun, 17. True figure of the earth discovered by Newton, 34. A degree measured under the equator and near the poles by order of the king of France, ib. Revolves with the moon about a common centre of gravity, 278. Effects of its motion on the apparent motions of the planets, 324. Its distance from the sun, velocity in its orbit, &c. 333. Proofs of its motion, 334. From the proportional decrease of gravity, 335. Objections from the parallelism of its axis answered, 336. Proof of its motion from the aberration of light, 337. Objection from the apparent motion of the sun answered, 340. Proof from the spheroidal figure of the earth, 341. From the celestial appearances as viewed from different planets, 343. Objection from the perpendicular descent of bodies answered, 344. Its diurnal motion illustrated by experiment, 345. Appears a moon to the inhabitants of our moon, 352. Has a spherical figure, and casts a conical shadow, 427, 429.

Ebn Yunes observes three lunar eclipses with the utmost accuracy, by which the quantity of the moon's acceleration has since been determined, 17. Eclipses observed by the Chinese, and made use of by them to settle their chronology, 34. Of Jupiter's satellites, 97. At what time these eclipses are visible, 102. Moon sometimes disappears when eclipsed, 146. A luminous ring seen round the moon in the time of total eclipses of the sun, 149. A comet seen during the time of a solar eclipse, 172. Sometimes occasioned by comets, 177. Eclipses particularly defined, 426. Why there are so few eclipses, 430. Their appearances determined from the motion of the moon's nodes, 431. A complete revolution of eclipses in 223 lunations after the sun, moon, and nodes, have once been in a line of conjunction, ib. To know when the same eclipse returns again, 432. Eclipses of the sun which happen when the moon is in the ascending node come in at the north pole, and in the descending node at the south, ib. Exemplified from the solar eclipse in 1764, n° 433. And from that of 1748, n° 434. All the phenomena of a single eclipse completed in about 1000 years, 435. Seldom more than two remarkable appearances of the same eclipse during this period, 436. Very ancient eclipses coming in by the north pole, 436. Example of eclipses cannot be calculated by our tables, 438. Eclipses in a year cannot be fewer than two, or more than seven, 439. Eclipses of the sun more frequent than those of the moon, ib. Why more lunar eclipses are observed than solar, 440. Total and annular eclipses explained, 441. Illustration of the beginning and ending of a solar eclipse, 443. How the appearance of eclipses is affected by the position of the earth's axis, 444. Of the duration of eclipses in different parts of the earth; Astronomy

Equatorial or portable observatory described, 504.

Equinoxes, precession of them explained, 348. Cause of their anticipation, 349.

Eratosthenes attempts to measure a degree of the earth's circumference, 12.

Eternity of the world disproved from the system being evidently perishable, 272.

Eudoxus first introduces geometry into astronomy, 11.

Euler, difference between his calculations and those of Newton and Halley concerning the periodical return of the comet of 1680, n° 189.

Europe, astronomy revived in it, 18.

Eccentricity of the moon's orbit found out by Hipparchus, 14. Of the orbits of the different planets, 255. Inequality in the eccentricity of the moon's orbit occasioned by the motion of her apogon, 296. How to determine the eccentricity of the earth's orbit, 415.

Falling bodies, calculation of their velocity, 277.

Fatios observations on the solar eclipse in 1706, n° 149.

Ferguon's explanation of the horizontal moon, 374. His method of drawing a meridian line, 376.

Ferner's observations on the transit of Venus, 78.

Final causes, Nicholson's opinion on them, 160.

Fixed Stars. See Start.

Flamstead appointed astronomer-royal at Greenwich, 33. Makes a catalogue of the stars, ib. His computation of the height of the moon's atmosphere, 149.

Fouchy's observations on the transit of Venus, 78.

French astronomers, their improvements, 35.

Full moon, how to calculate the time of it by the tables, 454, 455, 456, 457.

Gaber, the Arabian, founds the present method of trigonometry, 17.

Galileo first brings telescopes to any perfection, 27. Discovers the uncommon shape of Saturn, 103. Discovers the spots on the sun, 58. His method of measuring the height of the lunar mountains, 141.

Gauging the heavens, Mr Herchel's method of doing so, 210, 220.

Geocentric latitude of a superior planet, how found, 316.

Geometry first introduced into astronomy by Eudoxus, 11.

Georgium Sidus, a new planet, 40. History of its discovery, 328, &c. Computations of its distance, 330.

Globe: the first made by Alexander, 10. The course of the sun, &c. exhibited in a globe by Pythagoras, 11.

Globe, improved celestial one, 493.

Graham makes great improvements in mathematical instruments, 32.

Gravity first assumed as a moving principle by Kepler, 26. Moon retained in her orbit by it, 275. Its properties demonstrated by pendulums, 303. To determine its power on any planet, 309.

Greeks: their knowledge in astronomy, 8. Improved by Thales, 9. By Anaximander, &c. 10.

Gregory, James, of Aberdeen, first shows how a reflecting telescope might be constructed, 31.

H.

Hadley improves the reflecting telescope, and invents the reflecting quadrant, 32.

Halley, Dr. succeeds Mr Flamstead as astronomer-royal, 33. Discovers the acceleration of the moon, ib. His history of new stars, 45. His account of a solar eclipse in 1715, 150. Predicts the return of comets, 170, 190.

Hamilton, Dr. of Dublin, his hypothesis concerning the tails of comets, 182.

Harwood accounted for, 370. Why the same phenomenon is not observed at other times, 371.

Heavens, Mr Wollaston's method of making a map of them, 55. Mr Herchel's opinion concerning their construction, 202, 214. They cannot be properly represented by a sphere, ib. His method of measuring their dimensions, 220.

Heliocentric circles of the planets, the same with their orbits round the sun, 311. Heliocentric latitude of any body defined, 315.

Herchel's account of the spots on the planet Mars, 82. Makes great discoveries in the starry regions, 119. And a vast number of nebulas, 117. His observations on the height of the lunar mountains, 143. Discovers volcanoes in the moon, 145. Solution of the celestial phenomena on his hypothesis of the construction of the heavens, 208. His method of gauging the heavens, 210. His account of the internal construction of the heavens, 214. Length of the line by which he measures the dimensions of the heavens, 212. Arguments in favour of his hypothesis from the observations of Mr Mayer, 241.

Hevelius, a celebrated astronomer of Dantzig, 30. Account of his observations, ib. His dispute with Dr Hooke about telescopic sights, ib. Dr Halley pays him a visit in order to decide it, ib. Extreme accuracy of the instruments of Hevelius, ib. His account of the four spots, 59, 61. His method of measuring the height of the lunar mountains, 141. His opinions concerning comets, 175, 186.

Hipparchus discovers the motions of the stars from the observations of Timocharis and Argyllus, 12. First applies himself to the study of all the parts of astronomy, 14. Discovers the orbits of the planets to be eccentric, ib. Collects accounts of eclipses, forms theories of the celestial motions, and attempts to calculate the fundamental, ib. This method particularly explained, 387. Its insufficiency shown, 388.

Hirsh's observations on the atmosphere of Venus, 78.

Hooke's dispute with Hevelius about the superiority of telescopic sights, 30. Observes the motion of a spot on the disk.

Orbit of their primary, ib. Their greatest latitude only $2^\circ 55'$, ib. Their distances and periodical times, 96. At what times they appear direct, retrograde, &c. ib. Of their occultations and eclipses, 97. Appear like lucid spots on the body of their primary when passing between us and it, ib. Sometimes appear as dark spots in that case, 98. This supposed to be owing to spots on their bodies, ib. Vary also in their light and apparent magnitude from the same cause, 99. Sometimes seem less than their shadows, ib. Appear always round, without putting on any of the phases of our moon, ib. In what cases their shadows may be seen on the disk of Jupiter, 100. Three of them eclipsed in every revolution, 101. At what times the eclipses and occultations of these moons become visible to us, 102. Of the light enjoyed by Jupiter and the other superior planets, 162. His day equivalent to 3330 times our moon-light, 162. Comets retarded in their course by the attraction of this planet, 191. Effect of this attraction on the comet of 1681, ib. The motion of Saturn likewise influenced by it, 269. The power of his attraction discovered by the revolutions of his satellites, 274. This planet contains $158\frac{1}{2}$ times as much matter as the earth, 309. But is $4\frac{1}{2}$ times less dense, 310. How to find the longitude of places on earth by the eclipses of Jupiter's satellites, 482. et seq. Frequency of these eclipses, ib. The satellites never eclipse one another, ib. At what times the immersions or emersions are to be observed, 483. See Belts and Eclipses.

Keill's method of finding the distance of the sun by observing exactly the phases of the moon, 192. Insufficiency of it showed, 393.

Kepler, his discoveries concerning the planetary motions, 26. Imagines the planets and comets to be animated beings, 166. Shows that the comets do not move in straight lines, 168.

La Caille constructs excellent solar tables, allowing for the attractions of the planets, 36. Determines the parallax of the sun not to be above ten seconds, ib.

Latitude of any phenomenon, its distance from the ecliptic, 315. How to find the latitude of any place on earth, 408. To find the latitudes of the stars, 413.

Lexell, professor at Petersburg, his observations on the Georgian Sidus, 330.

Libration of the moon, an apparent inequality in her motion, owing to her equable revolution on her axis, 420.

Lightning, supposed to be frequent in the moon, 153.

Light, its progressive motion discovered by the eclipses of Jupiter's satellites, 33. Why it is not refracted by the moon's atmosphere, 148. That enjoyed by the superior planets compared with our day, 162. In what cases it may be supposed to return to the body that emits it, 198. Its aberration, 337. Its velocity, 338, 484.

Local zodiac explained, 318.

Logarithmic tables, by whom composed, 28.

Logarithms, said to be known by the Prince of Hesse before they were discovered by Napier, 23. Of their discovery by the latter, 26.

Long, Dr. his opinion of the observations concerning the atmosphere of Venus, 79. His explanation of the disappearance of the fifth satellite of Saturn, 108. His account of comets, 112. Of the spots of the sun, 59. His conjectures concerning them, 127. Supposes that some of the dark spots of the moon may be seas, 146. His answer to the arguments against a lunar atmosphere, 147. His opinion on the plurality of worlds, 157. His conjectures why comets may sometimes be invisible in their perihelion, 172.

Longitude, the method of discovering it at sea by the apulses of the moon to fixed stars, proposed by Warner, 21. Of discovering the longitudes of places on earth by the eclipses of Jupiter's satellites, 27, 482. What satellites are most proper for this purpose, ib. Warner's method brought to perfection by Nevil Maskelyne, 36. Longitudes of the celestial bodies defined, 319. Longitude easily found by the lunar inhabitants, 354. How to find the longitudes of the stars, 413.

Leverrier's observations on a luminous ring round the moon in a solar eclipse, 152.

Lunar inhabitants, how they can measure the year, 353.

Mairan's opinion concerning the tails of comets, 179.

Map of the heavens, Mr Wolfson's method of constructing one, 55.

Maraldi's observations on the spots of Mars, 80. Observes belts on his disk, ib.

Mariners compass early known to the Chinese, 3.

Mars first observed on his disk by Cassini, who determines his revolution in 24 hours 40', no 80. Erroneously supposed by others to revolve only in 13 hours, ib. Maraldi's observations on them, ib. His belts sometimes parallel to his equator, and sometimes not, ib. Bright spots about his poles, 81. Mr Herschel's account of these spots, 82. The circle of motion of the north polar spot at some distance from the pole itself, 83. Not so with the south polar spot, ib.

Exact position of the poles of this planet, 84. Of the seasons in Mars, 85. How to compute the declination of the sun on this planet, ib.

A considerable resemblance betwixt the earth and Mars, 86. The white spots about the poles supposed to be occasioned by snow, 87. Has a spheroidal form, 88. Difference betwixt his equatorial and polar diameters, 89. Of his atmosphere, 90. Density of Montaigne observes the satellite of Venus at the transit in 1761, no. 76.

Montanere's account of changes among the fixed stars, 46.

Moon comes into the same situation with regard to the nodes, apogee, &c., once in 600 years, i. Her motion discovered by the Chaldeans not to be uniform, 6. Cause of her eclipses known to the Egyptians, ib. Held to be an opaque body reflecting the rays of the sun by Thales, 9. And by Pythagoras, 11. Said to be habitable by him and Anaxagoras, 10, 11. The spots and phases of the moon observed by Helvetius, 30. Occultation of Jupiter observed by him and Dr Halley, ib. Her apparent motion, 38. Appearance through a telescope, 63. Great inequalities on her surface, 140. Method of measuring the height of her mountains, 141. Their height greatly over-rated, 142. Mr Herschel's observations on them, 143. His method of measuring their height, ib. Volcanoes discovered in the moon, 145. Conjectures concerning her substance, 146. Supposed by some to have an obscure light of her own, ib. Sometimes disappears entirely in the time of a lunar eclipse, ib. Remarkable appearances in 1703, ib. Spots on her disk generally supposed to be hollow, ib. Some of them thought by Dr Long and others to be water, ib. Whether the moon has any atmosphere, 147. The light cannot be refracted by this atmosphere on account of its rarity, 148. Existence of an atmosphere argued from the appearance of a luminous ring about the moon in a total eclipse of the sun, 149. Observations on an eclipse of this kind in 1766, ib. Cap. Stanys's observation at Bern in Switzerland, ib. Fatio's observations at Geneva, ib. Flamsteed from these observations concludes the lunar atmosphere to be 180 geographical miles high, ib. Cassini's general account of this eclipse from a number of relations, ib. Dr Halley's account of a solar eclipse in 1715, no. 150. Observes flashes like lightning to proceed from the dark edge of the moon, 151. Louville's observations on this eclipse, ib. Supposes the flashes to be real lightnings, 153. Great height of the moon's atmosphere accounted for, 154. All the phenomena otherwise solved by the French academicians, 155. Whether the light of the fixed stars is refracted in passing by the moon, 156. Maraldi's conclusion against a lunar atmosphere, ib. Moon moves round the earth in an ellipse, 273. Retained in her orbit by the power of gravity, 275. Her motion particularly explained, 276. Moves with the earth about a common centre of gravity, 278. Inequalities in her motion explained, 282. Comes nearest the earth when least attracted by it, 284. Cause of the dilatation of her orbit, 285. Plane of her orbit changed by the action of the sun, 286. Nodes of her orbit explained, 287. Inclination of her orbit, 288. Irregularities arising from her motion in an ellipse, 290. Her apogon and perigon explained, 291. Motion of her apogon determined, 294. Inequality of it occasions an inequality of the eccentricity of her orbit, 296. Newton's computation of the lunar irregularities, 297. General account of her motions, &c., 352, et seq. Her phases explained, 351, 355. Delineated, 424. Never appears perfectly round, 356. Agreeable representation of her phases, 357. Delineation of her path round the sun, 359. Her path always concave towards the sun, 360. Why her edge always appears even, 362. Revolution of hernodes, 372. To find her horizontal parallax, 384. The best method as recommended by Mr Ferguson, 385. Difficulties in the calculation of her place in her orbit, 419.

Moves equably on her axis, which occasions an apparent inequality called her libration, 420. Irregularities in her motion explained by Sir Isaac Newton, 421. Periodical time discovered by Copernicus, 422. Her diurnal and horary motion, ib. Found, 423. Place of the nodes now found, 425. Figure spherical, 428. Figure of her shadow conical, 429. Why more eclipses of the moon than of the sun are observed, 440. Excess of the lunar above the solar diameter, 441. Extent of the shadow and penumbra, 442. Why she is visible when eclipsed, 447. Projection of the eclipses of the moon, 470. Moon light long in winter at poles, 373. Motions, apparent, of the planets explained, 323. Mountains of the moon, 340. Moon. Attraction of the refrial mountains discovered by the mathematicians to measure a degree of the earth, 34. Mural arch described, 497.

Napier, baron of Merchiston, invents logarithms, 26.

Navigation: Astronomy first applied to its purposes by the Phoenicians, 7.

Nebulae, whith specks in our heavens so called, 120. Great numbers of them discovered by Mr Herschel, ib. Planetary nebulae, why so called, ib. Refolved into a multitude of small stars by Mr Herschel's telescope, ib. Are arranged into flat, circular, Variety of shapes affixed to them, 214. Observations made on them by Mr Herschel, ib. Conjectures concerning their formation, 216. Arguments drawn from observations on them that the universe is composed of nebulae, 218. Extent of the nebulae in which we live, 224.

New moon, how to calculate the time of it, 452, 453.

Newton; Sir Isaac, makes the first reflecting telescope, ib. Reforms the theoretical part of astronomy, ib. His opinion concerning the fun... Astronomy

Stars, fixed, their motion in longitude discovered by Hipparchus, 12. Catalogue of fixed stars made by him, 15. Another by Ulug Beg, 17. Places of 400 of them settled by the Landgrave of Hesse, 23. Another catalogue made by Tycho Brahe, 24. These stars seemingly destructible and generable, 44. Dr Halley's history of new ones, 45. Montanare's account of changes among them, 46. Mr Pigot's remarks on those which change their lustre, 47, et seq. Number of fixed stars greatly increased by telescopes, 119. Of their occultations by the moon, 156. Supposed to be suns, 192. Conjectures concerning the variable stars, 194, 195, 196, 201, 235. Mr Michell's conjectures concerning their nature, 197. Comparison of their brightness with that of the sun, 199. Some stars supposed to be satellites of others, 197. Their supposed motion, and that of the solar system, 237, et seq. Their distance immeasurable, 398. Why they seem so big to the naked eye, 399. Their different magnitudes, 400. Telescopic stars, 401. Uniformed stars, 402.

Star-gauge. See Gauging.

Strata. See Nebulae.

Sun, his distance determined by Aristarchus, 12. By Eratosthenes, ib. His apparent motion, 37. Account of his appearance through telescopes, his spots, and conjectures concerning them, 58, et seq., 121, et seq. See also Moon, Ecliptic, Declination, &c.

Superior planets. See Mars, Jupiter, Saturn, and Georgium Sidus.

Swans, variable star in that constellation, 51, 52, 53.

T.

Telescopes introduced into astronomy by Galileo, 27. Their improvement, 31.

Telescopic sights invented by Hooke, 30.

Thales