ASTRONOMY is that science which treats of the motions of the heavenly bodies, and explains the laws by which these motions are regulated.
It is the most sublime and the most perfect of all the sciences. No subject has been longer studied, or has made greater progress. There is a vast interval between the rude observations of the earlier astronomers, and the precision and general views which direct our present observers. To ascertain the apparent motions of the heavenly bodies was a difficult task, and required the united observations of ages. To unravel these intricate mazes, and detect and demonstrate the real motions, demanded the most patient perseverance, judgment, and dexterity. To ascertain the laws of these motions, and to resolve the whole of them into one general fact, required the exertions of a sagacity scarcely to be expected in human nature. Yet all this has been accomplished; and even the most minute movement of the heavenly bodies has been shewn to depend upon the same general law with all the rest, and even to be a consequence of that law. Astronomy, therefore, is highly interesting, were it only because it exhibits the finest instance of the length that the reasoning faculties can go. It is the triumph of philosophy and of human nature. But this is not all. It has conferred upon mankind the greatest benefits, and may truly be considered as the grand improver and conductor of navigation.
The following treatise will be divided into four parts. In the first part, we shall give a sketch of the history of astronomy; in the second, we shall treat of the apparent motions of the heavenly bodies; in the third, of their real motions; and in the fourth, of gravitation, or of that general fact to which all their motions may be referred, and from which they proceed.
PART I. HISTORY OF ASTRONOMY.
History.
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 Cassini), whereof 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 Antedilu- vians had such a period of 600 years, they must have known the motion 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 names. 1. The mouse. 2. The ox or cow. 3. The tiger. 4. The hare. 5. The dragon. 6. The serpent. 7. The horse. 8. The sheep. 9. The monkey. 10. The cock or hen. 11. The dog; and, 12. The bear. They divide the heavens into 28 constellations, four of which are assigned to each of the seven planets; so that the year always begins with the same planet; and their constellations answer to the 28 mansions of the moon used by the Arabian astronomers. These constellations, in the Chinese books of astronomy, are not marked by the figures of animals, as was in use among the Greeks, and from them derived to the other European nations, but by connecting the stars by straight lines; and Dr Long informs us, that in a Chinese book in thin 4to, shown him by Lord Pembroke, the stars were represented by small circles joined by lines; so that the Great Bear would be marked thus,
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 assures us, that 36 are recorded by Confucius himself, who lived 551 years before Christ; and P. Trigault, who went to China in 1619, 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 light 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 skillful 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 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 Pekin in the same manner as in most of the capital cities of 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 observers; 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 Part I.
History. The instruction of the astronomical pupils; and the pre- fidents 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 expence, 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 millionaries have never been the authors of any of these ephemerides: their employment is to revive 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 happens; the eclipse must also be calculated for the longitude and latitude of the capital city of 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 upon 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 a noise they prevented the luminary from being devoured by the celestial dragon; and though 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 observed. 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 affiduous to observe the eclipse with accuracy. A ceremonial of this kind is observed through 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 the 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. Cassini in 1689. 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 not less a distance of time that 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 brahmins 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 spaces through which that luminary had passed in 1,600,984 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 American 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 even here of this favourite num- ber) 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 been 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 Ozymandias, 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 helical 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 assures 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 6,854\( \frac{1}{2} \) 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 Seothros, 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 Phenicia; 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, steering 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 diluent 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 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 Part I.
History. 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 defiring 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 coistical 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-720th 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 professedly on the history of ancient times, may have induced them to ascribe full as much knowledge to those who lived in them as was really their due."
The successors of Thales, Anaximander, Anaxime- nes, 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 cotemporary 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 cotemporary, 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, affected 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 cotemporary 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 Agathias to Neclanebus 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 Pythas, who lived at Marseilles 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, science flourished in Egypt more than in any other part of the world; and a famous school was set up at Alexandria under the auspices of Ptolemy Philadelphus, a prince instructed in all kinds of learning, and the patron of 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. Timochares and Arystillus, 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 13th year of Philadelphus, when Venus hid the former star of the four in the left wing of Virgo.
From this time the science 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 by him keeper of the royal library at that place. At his instigation the same prince set up those armillas or spheres, which Hipparchus and Ptolemy the astronomer 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 Berofus, a native of Chaldea, flourished at Athens. He is by some 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 observations 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 Timochares 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, Part I.
History.
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 Europe were immersed in deep ignorance and barbarity. However, several learned men arose among the Arabians. The caliph Al Manfur 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 cotemporary with the caliph Al Mamun. But the most celebrated of all their astronomers is Albategnius, 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 \(23.35'\). Finding that the tables of Ptolemy required much correction, he composed new ones of his own fitted to the meridian of Aracca, 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 Ebn 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 ecliptic, and constructed tables of sines, or half chords of double arcs, dividing the diameter into 300 parts; and Alhazen, his cotemporary, 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 Samarcand 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 Samarcand to be \(39^\circ 37' 23''\).
Besides these improvements, we are indebted to the Arabians 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 shew 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 Arabians also made the practice still more simple, by using lines 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, Alferganus, 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. Vitellio, a Polander, wrote a treatise on Optics about 1270, in which he shewed 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 astronomy. He wrote a commentary on Ptolemy's Almagest, 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 Regiomontanus afterwards extended to every single minute, making the whole sine 69, with 6 ciphers 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 \(23^\circ 33' 17''\), 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 Montegio (Königsberg), a town of Franconia, from whence he was called Regiomontanus, History. nus, 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 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.
Of Werner. John Werner, a clergyman, succeeded Walther as astronomer at Nuremberg; having applied himself with great affiduity 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 in 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, Alphonius, 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 hypothesis, 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 Nicholas Copernicus next makes his Pythagorean appearance, and is undoubtedly the great 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 assisted 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 thence of constructing tables of their motions which might be more agreeable to truth than those of Ptolemy and Alphonius. 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 hypothenuse 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 Otfander, 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 with a bloody flux, succeeded by a palsy; so that he received a copy only a few hours before his death, which happened on the 23d of May 1543.
After the death of Copernicus, the astronomical science was greatly improved by Schoner, Nonius, Appian, 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 cosmography. 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. circles. The first of these was divided into 90 equal parts, the second into 80, 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 proportionals betwixt two right lines. Appian's chief work was entitled The Cezarean 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 Appian entitled his Cosmography, 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 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 Wittemberg. 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 astronomer, 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 was built. The first stone of this building, afterwards called Uraniburg, was laid in the year 1576. It was Account 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 larger and more solid than had ever been seen before by any astronomer. They consisted of quadrants, sextants, circles, semicircles, armillae both equatorial and zodiacal, parallactic rulers, rings, astrolabes, globes, clocks, and sun-dials. These instruments were so 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 Chanceler, 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 Purback, 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, Discoveries invited thither John Kepler, afterwards so famous of Kepler. for his discoveries. Under the tuition of so great an astronomer, the latter quickly made an amazing progres. 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 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, entitled, "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, entitled, 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, Homelius, 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 sort 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 so 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 conjunctions 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 Logarithmic 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 Goude, 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 Venus first over the disk of the sun on the 24th of November, observed by 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 Mersennus, Royal Scientist at Paris 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, Part I.
History.
house, and furnishing it with excellent instruments of his own construction; particularly sextants and sextants of bras 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 entitled 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, entitled Machina Celestis; 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 cotemporary 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 southern 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 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 39', 33''.
In 1671 the royal observatory in Paris was finished, 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 123 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 Azout 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 Gascoigne, 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 Dollond, the excellence of whose achromatic telescopes is too well known to need any encomium.
About the beginning of the 18th century, the practical part of astronomy seemed to languish for want of proper instruments. Roemer, indeed, had invented some new ones, and Dr Hooke had turned his attention towards this subject in a very particular manner; but either through want of skill in the arts, 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 so long a 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 must 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 every one. 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 124 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 in 1721. 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 and 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 shewed 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 said 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 Cometicae, 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 Almanack, 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. Caffini, 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 flattened at the poles. To determine this point, Louis 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. Mauquertuis, Clairault, Camus, and Le Monier, to Lapland. They were accompanied by the abbé Outhier, a correspondent of the fame 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 Part I.
History.
fully accomplished their errand. On the southern expedition were despatched M. Godin, Condamine, 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 1733; 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 talk 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 appeared 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 betwixt 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 mean time 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; though 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 Göttingen 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 300l.
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 affluity by Signior Bianchini, Father Boscovich, Frifii, Manfredi, Zanotti, and many others; in Sweden by Wargentin already mentioned, Blingenstern, Mallet, and Planman; and in Germany, by Euler elder and younger, Mayer, Lambert, Grifichow, &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 Blis, 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 affluity and success in bringing the lunar method of determining the longitude at sea into general practice.
Such was the general state of astronomy, when Dr 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 fix 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 vations are equally surprising; of which we shall only say with Dr Priestley*, "Mr Herschel's late discoveries in and beyond the bounds of the solar system, 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."
PART II. OF THE APPARENT MOTIONS OF THE HEAVENLY BODIES.
WHEN we cast our eyes up towards the heavens, we perceive a vast hollow hemisphere at an unknown distance, of which our eyes seem to constitute the centre. The earth stretches at our feet like an immense plain, and at a certain distance appears to meet and to bound the heavenly hemisphere. Now the circle all around, where the earth and the heavens seem to meet and touch each other, is called the horizon. We can scarcely avoid supposing, that besides the hemisphere which we perceive, there is another, exactly similar, concealed from our view by the earth, and that the earth, therefore, is somehow or other suspended in the middle of this heavenly sphere, with all its inhabitants. A little observation turns this suspicion into certainty. For in a clear evening the heavenly hemisphere is seen studded with stars, and its appearance is changing every instant. New stars are continually rising in the east, while others in the mean time are setting in the west. Those stars, that, towards the beginning of the evening, were just seen above the eastern horizon, late at night are seen in the middle of the starry hemisphere, and may be traced moving gradually westward, till at last they sink altogether under the horizon. If we look to the north, we soon perceive that many stars in that quarter never set at all, but move round and round, describing a complete circle in 24 hours. These stars describe their circles round a fixed point in the heavens; and the circles are the smaller, the nearer the star is to the fixed point. This fixed point is called the north pole. There must be a similar fixed point in the southern hemisphere, called the south pole. Thus the heavenly sphere appears to turn round two fixed points, called the poles, once every 24 hours. The imaginary line which joins the points is called the axis of the world.
In order to have precise notions of the motions of the heavenly bodies, it is necessary to be able to assign precisely the place in which they are. This is done by means of several imaginary lines, or rather circles, supposed described upon the surface of the sphere; and these circles, as is usual with mathematicians, are divided into 360 equal parts called degrees. Every degree is divided into 60 minutes; every minute into 60 seconds, and so on. That great circle of the sphere, which is perpendicular to the axis of the world, and of course 90° distant from either pole, is called the equator. The smaller circles, which the stars describe in consequence of their diurnal motions, are called parallels, because they are obviously parallel to the equator.
The equator divides the heavenly sphere into two equal parts, the north and the south; but to be able to assign the position of the stars, it is necessary to have another circle, passing through the poles, and cutting the equator perpendicularly. This circle is called a meridian. It is supposed, not only to pass through the poles, but to pass also through the point directly over the head of the observer, and the point of the sphere exactly opposite to that. The first of these points is called the zenith, the second is called the nadir.
The meridian divides the circles described by the stars into two equal parts; and when they reach it they are either at their greatest height above the horizon, or they are at their least height. The situation of the pole is easily determined; for it is precisely half way between the greatest and least height of those stars which never set. When we advance towards the north we perceive that the north pole does not remain stationary, but rises towards the zenith, nearly in proportion to the space we pass over. On the other hand it sinks just as much when we travel towards the south. Hence we learn that the surface of the earth is not plane, as one would at first suppose, but curved.
All the heavenly bodies appear to describe a complete circle round the earth in 24 hours. But besides these motions which are common to them all, there are several of them which possess motions peculiar to themselves. The sun, the most brilliant of all the heavenly bodies, is obviously much farther to the south during winter than during summer. He does not, therefore, keep the same station in the heavens, nor describe the same circle every day. The moon not only changes her form, diminishes, and increases; but if we observe the stars, near which she is situated one evening, the next evening we shall find her considerably to the eastward of them; and every day she removes to a still greater distance, till in a month, she makes a complete tour of the heavens, and approaches them from the west. There are eight other stars, besides, which are continually changing their place; sometimes we observe them moving to the westward, sometimes to the eastward, and sometimes they appear stationary for a considerable time. These stars are called planets. There are other bodies which appear only occasionally, move for some time with immense celerity, and afterwards vanish. The bodies are called comets. But the greater number of the heavenly bodies always retain nearly the same relative distance from each other, and are therefore called fixed stars. It will be necessary for us to consider the nature and apparent motions of all these bodies. We shall, therefore, divide this first part of our treatise, into the following heads:
1. Of the Sun. 4. Of the Comets. 2. Of the Moon. 5. Of the Fixed Stars. 3. Of the Planets. 6. Of the figure of the Earth.
These topics shall be the subjects of the following chapters. Part II.
CHAP. I. Of the Sun.
The sun, as the most conspicuous and most important of all the heavenly bodies, would naturally claim the first place in the attention of astronomers. Accordingly its motions were first studied, and they have had considerable influence on all the other branches of the science. We shall subdivide this part of our subject into three parts. In the first, we shall give an account of the apparent motions of the sun; in the second, we shall treat of the division of time, which is regulated by these apparent motions; and in the third, we shall consider the figure and structure of the sun, as far as they have been determined by astronomers. These shall be the subjects of the following sections.
SECT. I. Apparent Motions of the Sun.
That the sun has a peculiar motion of its own, independent of the diurnal motion common to all the heavenly bodies, and in a direction contrary to that motion, is easily ascertained, by observing with care the changes which take place in the starry hemisphere during a complete year. If we note the time at which any particular star rises, we shall find that it rises somewhat sooner every successive day, till at last we lose it altogether in the west. But if we note it after the interval of a year, we shall find it rising precisely at the same hour as at first. Those stars which are situated nearly in the track of the sun, and which set soon after him, in a few evenings lose themselves altogether in his rays, and afterwards make their appearance in the east before sunrise. The sun then moves towards them in a direction contrary to his diurnal motion. It was by observations of this kind that the ancients ascertained his orbit. But at present this is done with greater precision, by observing every day the height of the sun when it reaches the meridian, and the interval of time which elapses between his passing the meridian and that of the stars. The first of these observations gives us the sun's daily motion northward or southward, in the direction of the meridian; and the second gives us his motion eastward in the direction of the parallels; and by combining the two together, we obviously obtain his orbit: But it will be necessary to be somewhat more particular.
These observations cannot be made without drawing a meridian line, or a line, which, if produced, would pass through both the poles of the earth, and the spot where the observer is placed. It is obvious, that such a line is in the same plane with the meridian as the the heavenly hemisphere. A meridian line may be found thus: On an horizontal plane describe three or four concentric circles, as E, G, H, fig. 1. Plate LIX., and in the common centre fix perpendicularly a wire CB, having a well-defined point. When the sun shines in the morning, observe where the shadow of the top of the wire, as CD, touches one of the circles; and in the afternoon mark where the extremity of the shadow CF just touches the same circle: then through the centre C draw the line CE, bisectiong the arc DF, and CE will be a meridian, as required. If the same be done with as many of the circles as the shining of the sun will admit of, and the mean of all the bisecting lines CE be chosen as a meridian, there will be no doubt of its accuracy, particularly if the observations be made about midsummer, which is the best time. After a meridian line is thus found, another parallel to it may be readily drawn at any convenient distance: the method is this: Hang a thread and plummet exactly over the fourth end of the known meridian line, and let another thread and plummet be hung over the fourth end of the plane upon which a meridian is to be drawn; then let a person observe when the shadow of the thread falls on the given meridian, and immediately give a signal to another person, who must at that moment mark two points on the shadow of the second thread, through which two points the new meridian must be described.
The height of the sun from the horizon, when it passes the meridian, or the arch of the meridian between the sun and the horizon, is called the sun's altitude. The ancients ascertained the sun's altitude in the following manner: They erected an upright pillar at the south end of a meridian line, and when the shadow of it exactly coincided with that line, they accurately measured the shadow's length, and then, knowing the height of the pillar, they found, by an easy operation in plane trigonometry, the altitude of the sun's upper limb: whence, after allowing for the apparent semidiameter, the altitude of the sun's centre was known. But the methods now adopted are much more accurate. In a known latitude, a large astronomical quadrant, of six, eight, or ten feet radius, is fixed truly upon the meridian; the limb of this quadrant is divided into minutes, and smaller subdivisions, by means of a vernier; and it is furnished with a telescope (having cross-bars, &c. turning properly upon the centre). By this instrument the altitude of the sun's centre is very carefully measured, and the proper deductions made.
With a similar instrument we may ascertain the apparent motions of the sun in the following manner, be-ascertaining our observations about the 20th of March. On this day we must note some fixed star which comes to the meridian exactly at the same time as the sun does; for the stars may be seen in the daytime with an astronomical telescope. On the following day, both the altitude of the sun, and the situation of the stars when the sun is on the meridian, must be observed; the sun's meridian altitude will be about 23° 45' greater than on the former day, and the star will be found on the meridian about 3 m. 39 sec. in time before the sun. Make similar observations for a few days, and it will be found, at the end of a week, that the sun's meridian altitude will be increased 2° 46', and the star will be on the meridian 25 m. 26 sec. in time before the sun, or it will be 6° 21' westward of the meridian when the sun is upon it. During this period of seven days, therefore, the sun has been moving towards the east, and has increased his altitude by regular gradations. In fig. 2. let EQ represent a portion of the equator, QS the meridian on which the sun is, QS his altitude above the equator, E the place of the star, and ES part of the path of the sun: then, in the spherical triangle EQS, right-angled at Q, there are given EQ = 6° 21' 46", and QS = 2° 46', to find the angle E. By the rules of spherical trigonometry, we have, tangt. of E = \frac{\text{tangt. of SQ}}{\text{fine of QE}} = \frac{0.083250}{1107463} = .0000756 4364479 = tangt. of 23° 34' 43", the angle E required.
The orbit in which the sun moves is called the ecliptic. It does not coincide with the equator, but cuts it, forming with it an angle, which in the year 1769 was determined by Dr Malkelyne, at 23° 28' 10", or 23° .46944. This angle is called the obliquity of the ecliptic.
The different seasons of the year are occasioned by the combination of this proper motion of the sun with his diurnal motion. The two points in which the ecliptic cuts the equator, are called the equinoxes, or equinoctial points; because on the days that the sun is in them, he describes by his diurnal motion the equator, which being divided into two equal parts by the horizon, the day is then equal to the night in every part of the earth. One of these equinoxes is called the vernal, because the sun is in it about the 2oth of March, or the beginning of the spring. As the sun advances in his orbit from that point, his meridian altitude becomes greater and greater every day. The visible arches of the parallels which it describes, become continually greater; and with them the length of the day increases, till the sun reaches his greatest altitude, or distance from the equator: then the day is the longest of the year. And as at that period the variations in the sun's altitude are scarcely sensible for some time, as far at least as it affects the length of the day; the point of the orbit, where the sun's altitude is a maximum, has for that reason been called the summer solstice. The parallel which the sun describes when in that point, is called the tropic of Cancer. From the solstice the sun descends again towards the equator, crosses it again at the autumnal equinox, and goes southward till its altitude becomes a minimum. This point of the orbit is called the winter solstice. The day is then the shortest of the year, and the parallel which the sun describes, is called the tropic of Capricorn. From the winter solstice the sun again approaches the equator, and returns to the vernal equinox.
Such is the constant course of the sun and of the seasons. The interval between the vernal equinox and the summer solstice, is called the spring; the interval between this solstice and the autumnal equinox, is called summer; that between the autumnal equinox and the winter solstice, is autumn; and that between this solstice and the vernal equinox, is winter.
The different altitudes of the pole in different climates, occasion remarkable peculiarities in the seasons, with which it is proper to be acquainted. At the equator the poles are situated in the horizon, which last circle cuts all the parallels into two equal parts. Hence the day and the night are constantly of the same length all the year round. On the equinoxes the sun is in the zenith at noon. His altitude is the least possible at the solstices, and is then equal to the complement of the inclination of the ecliptic. During the summer solstice, the shadows of bodies illuminated by the sun are directed towards the south; but they are directed towards the north at the winter solstice; changes which never take place in our northern climates. Under the equator then there are in reality two summers and two winters. The same thing takes place in all countries lying between the tropics. Beyond them there is only one summer and one winter in the year. The sun is never in the zenith. The Apparent length of the longest day increases, and that of the Motions of shortest day diminishes, as we advance toward the poles; and when the distance between the zenith and the pole is only equal to the inclination of the ecliptic, the sun does not set at all on the days of the summer solstice, nor rise on that of the winter solstice. Still nearer the pole, the period in which he never sets in summer, and never rises in winter, gradually increases from a few days to several months; and, under the pole itself, the equator then coinciding with the horizon, the sun never sets when it is upon the same side of the equator with the pole, and never rises while it is in the opposite side.
The intervals of time between the equinoxes and the solstices are not equal. There are about seven days more uniform between the vernal and autumnal equinox, than between the autumnal and vernal. Hence we learn, that the motion of the sun in its orbit is not uniform. Numerous observations, made with precision, have ascertained, that the sun moves fastest in a point of his orbit situated near the winter solstice, and slowest in the opposite point of his orbit near the summer solstice. When in the first point, the sun moves in 24 hours 1°.01943; in the second point, he moves only 6°.95319. The daily motion of the sun is constantly varying in every place of its orbit, between these two points. The medium of the two is 6°.98632, or 59' 11", which is the daily motion of the sun about the beginning of October and April. It has been ascertained, that the variation in the angular velocity of the sun, is very nearly proportional to the mean angular distance of it from the point of its orbit where its velocity is greatest.
It is natural to think, that the distance of the sun from the earth varies as well as its angular velocity. This is demonstrated by measuring the apparent diameter of the sun. Its diameter increases and diminishes in the same manner, and at the same time, with its angular velocity; but in a ratio twice as small. About the beginning of January, his apparent diameter is about 32' 39", and at the beginning of July it is about 31' 34", or more exactly, according to De la Place, 32' 3.3" = 1955" in the first case, and 31' 18" = 1878" in the second.
Opticians have demonstrated, that the distance of Sun's diameter of any body is always reciprocally as its apparent diameter. The sun must follow the same law; therefore, its distance from the earth increases in the same proportion that its apparent diameter diminishes. That point of the orbit in which the sun is nearest the earth, is called perigee, or perigee; and the point of the orbit in which that luminary is farthest distant from the earth, is called apogee. When the sun is in the first of these points, his apparent diameter is greatest, and his motion swiftest; but when he is in the other point, both his diameter and the rapidity of his motion are the smallest possible.
From these remarks it is obvious that if the orbit of the sun be a circle, the earth is not situated in the centre of that circle, otherwise the distance of the sun from the earth would remain always the same, which is contrary to fact. It is possible therefore, that the variation in his angular velocity may not be real, but only apparent. Thus in fig. 3. let AMPN be the orbit orbit of the sun, C the centre of that orbit, and E the motions of position of the earth at some distance from the centre. It is obvious that P is the sun's perigee, and A its apogee. Now as the sun's apparent orbit is a circle having the earth in its centre, it is evident that this orbit must be AMPN, and that the angular motion of the sun will be measured upon that circle. Suppose now that the sun in his apogee moves from A to A', it is obvious that his apparent or angular motion will be the segment \( a a' \) of the apparent orbit, considerably smaller than AA', so that at the apogee the angular motion of the sun will be less than his real motion. Again: let the sun in his perigee move from P to P', describing a segment precisely equal to the segment AA'. This segment as seen from the earth will be referred to \( p'p' \), which in that case will be the sun's angular motion, evidently considerably greater than his real motion.
Hence it is obvious that even on the supposition that the sun moved equably in his orbit, his angular motion as seen from the earth would still vary, that is, would be smallest at the apogee, and greatest at the perigee; and that the angular and real motion would only coincide in the points M and N, where the real and apparent orbits cut each other. From the figure it is obvious also, that the angular velocity would increase gradually from the apogee to the perigee, and diminish gradually from the perigee to the apogee, which likewise corresponds with observation. Now the line EC, which is the distance of the earth from the centre of the sun's orbit, is called the eccentricity of that orbit. The variation in the angular motion of the sun may be owing to this eccentricity.
But if it were owing to this cause alone, it is easy to demonstrate that in that case the diminution of his angular velocity would follow the same ratio as the diminution of his diameter. The fact however is, that the angular velocity diminishes in a ratio twice as great as the diameter of the sun does. The variation of the angular velocity cannot then be owing to the eccentricity alone. Hence it follows, that the variation of the motion of the sun is not merely apparent, but real; and that its velocity in its orbit actually diminishes, as his distance from the earth increases. Two causes then combine to produce the variation in the sun's angular velocity; namely, 1. The increase and diminution of his distance from the earth; and, 2. The real increase and diminution of his velocity in proportion to this variation of distance. These two causes combine in such a manner that the daily angular motion of the sun diminishes as the square of his distance increases, so that the product of the angular velocity multiplied into the square of the distance is a constant quantity. But this law is so important that it will be necessary to be more particular.
The observation that the sun's angular motion in his orbit is inversely proportional to the square of his distance from the earth, was first made by Kepler. The discovery was made by a careful comparison of the sun's diurnal motion with his apparent diameter, which were found to follow that law; and it is evident that the one is the angular motion of the sun, and the other his distance from the earth, which is inversely proportional to his apparent diameter. Let ASB (fig. 4.) be the sun's orbit, E the earth, and S the fun. Suppose a line ES joining the centres of the earth and sun to move round along with the sun. This line is called the radius vector. It is obvious, that when S moves to S', ES, moving along with it, is now in the situation ES', having described the small sector ESS'. In the same time that S performs one revolution in its orbit, the radius vector ES will describe the whole area ABS, enclosed within the sun's orbit. Let SS' be the sun's angular motion during one day. It is obvious that the small sector ESS' is proportional to the square of ES, multiplied by SS': for the radius vector is the sun's distance from the earth, and SS' his angular motion. Hence this sector is a constant quantity, whatever the angular motion of the sun be; and the whole area SEA increases as the number of days which the sun takes in moving from S to A. Hence results that remarkable law, first pointed out by Kepler, that the areas described by the radius vector are proportional to the times. Suppose the sun to describe SS' in one day, and SA in 20 days, then the area SES' is to the area SEA as 1 to 20; or the area SEA is 20 times greater than the area SES'.
The knowledge of these facts enables us to draw upon paper, from day to day, lines proportional to the length of the radius vector of the solar orbit, and having the same relative position as these lines. If we join the extremity of these lines, by making a curve pass through them, we shall perceive that this curve is not exactly circular. Let E in fig. 5. represent the earth, and E a, E b, E c, E d, E f, &c. the position and length of the radius vector during every day of the year: if we join together the points a, b, c, d, e, f, g, h, i, k, l, m, n, o, by drawing the curve a e i m, through them, it is obvious that this curve is not a circle, but elongated towards a and i, the points which represent the sun's greatest and least distance from the earth. The resemblance of this curve to the ellipse induced Kepler to compare them together, and he ascertained their identity. Hence it follows, that the orbit of the sun is an ellipse, having the earth in one of its foci. The centre C of the ellipse is the point where its greater axis is cut perpendicularly by its smaller axis. The distance CE, between the earth and that centre, is the eccentricity of the sun's orbit. The eccentricity of this orbit is not great. Let the earth's mean distance from the sun be represented by 10,000; it has been ascertained that the eccentricity is equal to 168 of these parts. Hence the sun's orbit does not differ much from a circle.
To form a precise notion of the elliptical motion of the sun, let us suppose a point to move uniformly in the circumference of a circle, whose centre coincides with the centre of the earth, and whose radius is equal to the sun's distance from the earth when in his perigee. Let us suppose also, that the sun and the point set out together from the perigee, and that the motion of the point, is equal to the sun's mean angular motion. While the radius vector of the point moves uniformly round the earth, the radius vector of the sun moves with unequal velocity, describing always areas proportional to the times. At first it gets before the radius vector of the point, and forms with it an angle, which after having increased till it reached a certain limit, diminishes again and becomes equal to zero, when the sun is in apogee; then the radius vector of the sun and of the Apparent point coincide both with each other, and with the Motions of greater axis of the ellipse. After passing the apogee the radius vector of the point gets before that of the sun, and forms with it angles exactly equal to the angles formed by the same lines in the former half of the ellipse, at the same distance from the perigee. At the perigee, the radius vector of the sun and of the point again coincide with each other, and with the greater axis of the ellipse. The angle which the radius vector of the sun makes with that of the point, which indicates how much the one precedes the other, is called the equation of the centre. It is always greatest when the motions of the point and of the sun are equal, and it vanishes altogether when there is the greatest difference between these motions. The angular motion of the point is called the mean motion, and that of the sun the real motion. The place of the point in the orbit is called the mean place. Now, if to the mean place in the orbit, we add or subtract the equation of the centre, it is obvious that we have the sun's real place for any given time. The angular motion of the point is known with precision for a given time, a day for instance, by ascertaining the exact length of time which the sun takes in making a complete revolution round its orbit. For if we ascertain how many days that revolution requires, we have only to divide the whole orbit by that number to prove the portion of it traversed by the point in one day. The equation of the centre can only be found by approximation. Its maximum in the year 1750 was 1°.9268.
In computations we begin always at that part of the orbit where the motion of the sun is slowest. The distance of the imaginary point from that part, is called the mean anomaly. A table is made of the equation of the centre, corresponding to each degree of the mean anomaly. By adding or subtracting these equations from the mean anomaly, we obtain the true anomaly or place of the sun for any given time.
The ecliptic is usually divided, by astronomers, into 12 equal parts, called signs, each of which of course contains 30 degrees. They are usually called the signs of the zodiac; and beginning at the equinox, where the sun intersects and rises above the equator, have these names and marks: Aries \( \alpha \), Taurus \( \gamma \), Gemini \( \Pi \), Cancer \( \Xi \), Leo \( \Omega \), Virgo \( \nu \), Libra \( \Delta \), Scorpio \( \eta \), Sagittarius \( \iota \), Capricornus \( \psi \), Aquarius \( \Xi \), Pisces \( \kappa \). Of these signs, the first six are called northern, lying on the north side of the equator; the last six are called southern, being situated to the south of the equator. The signs from Capricornus to Gemini are called ascending, the sun approaching or rising to the north pole while it passes through them; and the signs from Cancer to Sagittarius are called descending, the sun, as it moves through them, receding or descending from the north pole.
The longitude of the sun is his distance in the ecliptic from the first point of Aries. His right ascension is the arch of the equator intercepted between the first point of Aries, and the meridian circle which passes through his longitude. The distance of the sun from the equator, measured upon a meridian circle, is called his declination, and it is either north or south according to the situation of the sun.
It has been observed that the position of the larger axis of the elliptical orbit of the sun, is not constant. The angular distance of the perigee from the vernal equinox, counted according to the sun's movement, was Motions of 278°.6211 at the beginning of 1750; but it has, relative to the stars, an annual motion of about 11".89 in the same direction as the sun.
The orbit of the sun is gradually approaching to the equator. Its obliquity diminishes in a century at the rate of about 1".50.
The precision of modern astronomers has enabled them to ascertain small irregularities in the sun's elliptical motion, which observation alone would scarcely have been able to bring under precise laws. These irregularities will be considered afterwards.
To determine the distance of the sun from the earth, Distance of has always been an interesting problem to astronomers, the sun, and they have tried every method which astronomy or geometry possesses in order to resolve it. The amplest and most natural, is that which mathematicians employ to measure distant terrestrial objects. From the two extremities of a base whose length is known, the angles which the visual rays from the object, whose distance is to be measured, make with the base, are measured by means of a quadrant; their sum subtracted from 180°, give the angle which these rays form at the object where they intersect. This angle is called the parallax, and when it is once known, it is easy, by means of trigonometry, to ascertain the distance of the object. Let AB, in fig. 6. be the given base, and C the object whose distance we wish to ascertain. The angles CAB and CBA, formed by the rays CA and CB with the base, may be ascertained by observation; and their sum subtracted from 180° leaves the angle ACB, which is the parallax of the object C. It gives us the apparent size of the base AB as seen from C.
When this method is applied to the sun, it is necessary to have the largest possible base. Let us suppose two observers on the same meridian, observing at the same instant the meridian altitude of the centre of the sun, and his distance from the same pole. The difference of the two distances observed, will be the angle under which the line which separates the observers will be seen from the centre of the sun. The position of the observers gives this line in parts of the earth's radius. Hence, it is easy to determine, by observation, the angle at which the semidiameter of the earth would be seen from the centre of the sun. This angle is the sun's parallax. But it is too small to be determined with precision by that method. We can only conclude from it, that the sun's distance from the earth is at least equal to 10,000 diameters of the earth. We shall find afterwards, that other methods have been discovered for finding the parallax with much greater precision. It amounts very nearly to 8".8: hence it follows, that the distance of the sun from the earth amounts to 23,405 semi diameters of the earth.
Sect. II. Of the Division of Time.
Motion is peculiarly adapted for measuring time. For, as a body cannot be in different places in the same time, it can only arrive from one part to another, by passing successively through all the intermediate spaces. And if it be possible to ascertain, that in every point of the line which it describes it is actuated by the very same force, we can conclude with confidence, that it Part II.
Apparent will describe the line with a uniform motion. Of course Motions of the different parts of the line will be a measure of the time employed to traverse them. When a pendulum at the end of every oscillation is precisely in the same circumstances, the length of the oscillations is the same, and time may be measured by their number. We might employ also, for the same purpose, the revolutions of the heavenly sphere, which appear perfectly uniform. But all nations have agreed to employ the revolutions of the sun for that purpose.
In common language, the day is the interval of time which elapses from the rising to the setting of the sun; the night is the interval that the sun continues below the horizon. The astronomical day embraces the whole interval which passes during a complete revolution of the sun. It is the interval of time which passes from 12 o'clock at noon, till the next succeeding noon. It begins when the sun's centre is on the meridian of that place. It is divided into 24 hours, reckoning in a numerical succession from 1 to 24: the first 12 are sometimes distinguished by the mark P. M. signifying post meridiem, or after noon; and the latter 12 are marked A. M. signifying ante meridiem, or before noon. But astronomers generally reckon through the 24 hours, from noon to noon; and what are by the civil or common way of reckoning, called morning hours, are by astronomers reckoned in the succession from 12, or midnight, to 24 hours. Thus 9 o'clock in the morning of February 14th, is, by astronomers, called February the 13th at 21 hours.
An astronomical day is somewhat greater than a complete revolution of the heavens, which forms a sidereal day. For if the sun crosses the meridian at the same instant with a star, the day following it will come to the meridian somewhat later than the star, in consequence of its motion eastward, which causes it to leave the star; and after a whole year has elapsed, it will have crossed the meridian just one time less than the star. A sidereal day is less than the solar day, for it is measured by 360°, whereas the mean solar day is measured by 360° 59' 8" nearly. If an astronomical day be = 1, then a sidereal day is = 0.997269722; or the difference between the measures of a mean solar day, and a sidereal day, viz. 59' 8", reduced to time, at the rate of 24 hours to 360°, gives 3' 56"; from which we learn that a star which was on the meridian with the sun on one noon, will return to that meridian 3' 56" previous to the next noon: therefore, a clock which measures mean days by 24 hours, will give 23 h. 56 m. 4 sec. for the length of a sidereal day.
Astronomical or solar days, as they are also called, are not equal. Two causes confpire to produce their inequality, namely, the unequal velocity of the sun in his orbit, and the obliquity of the ecliptic. The effect of the first cause is sensible. At the summer solstice, when the sun's motion is slowest, the astronomical day approaches nearer the sidereal, than at the winter solstice when his motion is most rapid.
To conceive the effect of the second cause, it is necessary to recollect that the excess of the astronomical day above the sidereal is owing to the motion of the fun, referred to the equator. The fun describes every day a small arch of the ecliptic. Through the extremities of this arch suppose two meridian great circles drawn, the arc of the equator, which they intercept, is the fun's motion for that day referred to the equator; and the time which that arc takes to pass the meridian is equal to the excess of the astronomical day above the sidereal. But it is obvious, that at the equinoxes, the arc of the equator is smaller than the corresponding arc of the ecliptic in the proportion of the cofine of the obliquity of the ecliptic to radius: at the solstices, on the contrary, it is greater in the proportion of radius to the cofine of the same obliquity. The astronomical day is diminished in the first case, and lengthened in the second.
To have a mean astronomical day, independent of these causes of inequality, astronomers have supposed a nautical second fun to move uniformly on the ecliptic, and to pass over the extremities of the axis of the fun's orbit, at the same instant with the real fun. This removes the inequality arising from the inequality of the fun's motion. To remove the inequality arising from the obliquity of the ecliptic, astronomers suppose a third fun passing through the equinoxes at the same instant with the second fun, and moving along the equator in such a manner that the angular distances of the two funs at the vernal equinox shall be always equal. The interval between two consecutive returns of this third fun to the meridian forms the mean astronomical day. Mean time is measured by the number of the returns of this third fun to the meridian; and true time is measured by the returns of the real fun to the meridian. The arc of the equator, intercepted between two meridian circles drawn through the centres of the true fun, and the imaginary third fun, reduced to time, is what is called the equation of time. This will be rendered plainer by the following diagram.
Let Zγας (fig. 7.) be the earth; ZFRξ its axis; a b c d e, &c. the equator; ABCDE, &c. the northern half of the ecliptic from γ to α, on the side of the globe next the eye; and MNOP, &c. the southern half on the opposite side from W to ρ. Let the points at A, B, C, D, E, F, &c. quite round from γ to ρ again bound equal portions of the ecliptic, gone through in equal times by the real fun; and those at a, b, c, d, e, f, &c. equal portions of the equator described in equal times by the fictitious fun; and let Zγας be the meridian.
As the real fun moves obliquely in the ecliptic, and the fictitious fun directly in the equator, with respect to the meridian; a degree, or any number of degrees, between γ and F on the ecliptic, must be nearer the meridian Zγας, than a degree, or any corresponding number of degrees, on the equator from γ to f; and the more so, as they are the more oblique: and therefore the true fun comes sooner to the meridian every day whilst he is in the quadrant γ F, than the fictitious fun does in the quadrant ρ f; for which reason, the solar noon precedes noon by the clock, until the real fun comes to F, and the fictitious to f; which two points, being equidistant from the meridian, both funs will come to it precisely at noon by the clock.
Whilst the real fun describes the second quadrant of the ecliptic FGHIKL from Cancer to ας, he comes later to the meridian every day than the fictitious fun moving through the second quadrant of the equator from f to ας; 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: Apparent and as both suns come at the same moment to the point Motions of W, they come to the meridian at the moment of noon the Heaven by the clock.
In departing from Libra, through the third quadrant, the real sun going through MNOPQ towards r9 at R, and the fictitious sun through m n o p q towards r, the former comes to the meridian every day sooner than the latter, until the real sun comes to O, and the fictitious to r; and then they come both to the meridian at the same time.
Lastly, As the real sun moves equably through STUVW, from O towards r9; and the fictitious sun through s t u v w, from r towards r9, the former comes later every day to the meridian than the latter, until they both arrive at the point r9, 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, supposing the fun, 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.
If the sun's motion were equal in the ecliptic, the whole difference between the equal time as shewn 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. 8.) be the ecliptic or orbit in which the real sun moves, and the dotted circle a b c d 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 HE 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 fun 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 fun 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 EK, comes to both suns at the same time, 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 a to g, the real fun 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 fun, 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.
True time is obtained by adding or subtracting this equation to the mean time. The mean and apparent solar days are never equal, except when the sun's daily motion in right ascension is 59' 8"; this is nearly the case about April 15th, June 15th, September 1st, and December 24th: on these days the equation is nothing, or nearly so; it is at the greatest about November 1st, when it is 16 m. 14 sec.
The return of the fun to the same equinox marks the years, in the same way as his return to the same meridian indicates the days. It has been ascertained, that before the fun returns again to the same equinox, an interval of 365.242222 days elapses, or 365 days, 5 hours, 48 minutes, and 47 seconds. This is called the tropical year: The fun takes a larger interval of time to return again to the same star. The sidereal year is the interval which the fun employs to return from one star to another. It is greater than the tropical year by 0.014162 days, or 20 m. 23 sec.; therefore the length of the sidereal year is 365 days, 6 h. 9 m. and 10 sec. From this it follows, that the equinoxes do not retain the same place in the ecliptic, but that they have a retrograde motion, or contrary to that of the fun, in consequence of which they describe every year an arc equal to the mean space which the fun passes over in 20' 23", or about 50"; so that they would make a complete revolution in 25972 years. This is called the precession of the equinoxes.
Dr Mafkeleyne has invented a rule for computing Method of the equation of time, in which the precession of the computing equinoxes, as well as the two causes mentioned above, the equation are included. Let APLQ, fig. 9. be the ecliptic, ALQ the equator, A the first point of Aries, P the point where the fun's apparent motion is slowest, S any place of the sun; draw Sv perpendicular to the Part II.
Apparent equator, and take A n = AP. When the sun begins Motions of to move from P, suppose a star to begin to move from the Heaven- g, with the sun's mean motion in right ascension or ly Bodies. longitude, viz. at the rate of 50' 8'' in a day, and when n passes the meridian let the clock be adjusted to 12. Take n m = P s, and when the star comes to m, if the sun moved uniformly with his mean motion, he would be found at s; but at that time let S be the place of the sun. Let the sun S, and consequently v, be on the meridian; and then as m is the place of the imaginary star at that instant, m v must be the equation of time. The sun's mean place is at s, and as A n = AP, and n m = P s, we have A m = AP s, consequently m = Av - Am = Av - AP s. Let a be the mean equinox, or the point where it would have been if it had moved with its mean velocity, and draw a x perpendicular to AQ; then A m = A x + x m = A a × cosine x A a + x m: or because the cosine of x A a the obliquity of the ecliptic, 23° 28', is \( \frac{11}{12} \) very nearly, \( A m = \frac{11}{12} A a + x m \): hence \( m v = A v - x m - \frac{11}{12} A a \). Here A v is the sun's true right ascension, x m the mean right ascension or mean longitude; and \( \frac{11}{12} A a \) (viz. A x) is the equation of the equinoxes in right ascension; therefore the equation of time is equal to the difference of the sun's true right ascension and his mean longitude, corrected by the equation of the equinoxes in right ascension.—When A m is less than A v, mean or true time precedes apparent; when it is greater, apparent time precedes mean. That is, when the sun's true right ascension is greater than his mean longitude corrected as above shewn, we must add the equation of time to the apparent to obtain the mean time; and when it is less, we must subtract. To convert mean time into apparent, we must subtract in the former case, and add in the latter.
Tables of the equation of time are computed by this rule, for the use of astronomers: they are either calculated for the noon of each day, as given in the Nautical and some other almanacks; or for every degree of the sun's place in the ecliptic. But a table of this kind will not answer accurately for many years, on account of the precession and other causes, which render a frequent revival of the calculations necessary.
The smaller divisions of time were anciently measured by the phases of the moon. It is well known that the moon changes once every 29 or 30 days, and that the interval from one new moon to another is called a lunation, or, in common language, a month. There are about twelve lunations in a year. Hence the year was divided into twelve months. In ancient times people were placed upon eminences on purpose to watch the first appearance of the new moon, when their month began. It was customary for these persons to proclaim the first appearance of the moon. Hence the first day of every month was called Calends; from which term the world calendar is derived. Almost all nations have divided the year into twelve months, because the seasons nearly return in that period. But they soon perceived that twelve lunar months were far from making a complete year or revolution of the sun. They were anxious, however, to be able to divide the solar year into a precise number of lunar months, because many of the feasts depended upon particular new moons. Various contrivances were fallen upon for this purpose without much success, till at last Meton, a Greek philosopher, announced that 19 years contained exactly 235 lunations: an affirmation which is within 2\( \frac{7}{12} \) hours of being exact. To make every year correspond as nearly as possible to the lunar, he divided the year into 12 months, consisting alternately of 30 and 29 days each; at the end of every three years an intercalary month of 30 days was added, and at the end of the 19th year there was added an intercalary month of 29 days. So that at the end of 19 years the solar and lunar years began again on the same day their cycle of 19 years. This discovery of Meton appeared so admirable to the Greeks, that they engraved it in letters of gold in their public places. Hence the number which denotes the current year of that cycle is denominated golden number.
As the moon changes its appearance in a very remarkable degree every seven days, almost all nations have subdivided the month into periods of seven days, called weeks; the ancient Greeks were almost the only people who did not employ that division.
The Roman year in the time of Romulus consisted of 10 months only, of 30 or 31 days each, so that its length was 304 days only. Numa added 50 days to that year, and thus made it 354 days; and he added two additional months of 29 and 28 days, by shortening some of the ancient months. He made the year commence on the first of January. Numa's year was still more than 11 days shorter than a complete revolution of the sun. To make it correspond with the seasons, it was necessary to intercalate three days; and these intercalations being left entirely to the priests, were converted into a state engine; being omitted, inserted, altered, and varied, as it suited the purposes of those magistrates whose views they favoured. The consequence was, what might have been expected, the most complete confusion and want of correspondence between the year and the seasons.
Julius Caesar undertook to remedy this inconvenience. He was both dictator and high pontiff, and of course the reformation of the calendar was his peculiar province. That the undertaking might be properly executed, he invited Sofigenes, an Egyptian mathematician, to come to his assistance. It was agreed upon to abandon the motions of the moon altogether, and to make the year correspond with those of the sun.
The reformation was made in the year 47 before the Christian era. Ninety days were added to that year, which was from that circumstance called the year of confusion, consisting of 445 days. Instead of 354 days, the year of Numa, Sofigenes made the year to consist of 365 days, differing the additional days among those months which had only 29 days. As the revolution of the sun employs nearly fix hours more than 365 days, an additional day was intercalated every fourth year, so that every such year was to consist of 366 days. The additional day was inserted after the 23d of February, or the 7th before the calends of March; the day before the annual feast celebrated in commemoration of the flight of Tarquin from Rome. That feast was held the 6th before the calends of March. The intercalated day was also called the 6th before the Motions of calends of March. So that every fourth year there were two days denominated the 6th before the calends of March. Hence that year was called bissextille. In Britain it is denominated leap year. After the death of Julius Caesar there was a degree of confusion respecting the intercalations, from the ignorance of the priests. Augustus corrected the mistake, and after that time the Julian period went on without any interruption.
It is obvious that the Julian year, though a great improvement upon the ancient Roman, was still imperfect. It went on the supposition that the revolution of the sun occupied precisely 365 days and 6 hours, which is about 11 minutes more than the truth. This error in the interval which elapsed between the reformation of Julius Cesar and the year 1582, had accumulated till it amounted to 10 days; of course the year began 10 days later than it ought to have begun; and the same error had taken place respecting the seasons and the equinoctial points. Various attempts had been made to correct this error; at last it was corrected by Pope Gregory XIII. The Gregorian calendar commenced in the year 1582; the changes which he introduced were two in number. He ordered, that after the 4th of October 1582, ten days should be omitted, so that the day which succeeded the 4th was reckoned not the 5th but the 15th of the month. This corrected the error which bad crept into the Julian year. To prevent any such error from accumulating again, he ordered that the secular years 1700, 1800, 1900, should not be bissextille but common years; that the secular year 2000 should be bissextille, the next three secular years common, the fourth again bissextille, and so on, as in the following table.
<table> <tr> <th>1600</th> <th>2100</th> <th>2600</th> </tr> <tr> <td>bissextile.</td> <td>common.</td> <td>common.</td> </tr> <tr> <th>1700</th> <th>2200</th> <th>2700</th> </tr> <tr> <td>common.</td> <td>ib.</td> <td>ib.</td> </tr> <tr> <th>1800</th> <th>2300</th> <th>2800</th> </tr> <tr> <td>ib.</td> <td>ib.</td> <td>bissextile.</td> </tr> <tr> <th>1900</th> <th>2400</th> <th>2900</th> </tr> <tr> <td>ib.</td> <td>bissextile.</td> <td>common.</td> </tr> <tr> <th>2000</th> <th>2500</th> <th>3000</th> </tr> <tr> <td>bissextile.</td> <td>common.</td> <td>ib.</td> </tr> </table>
In short these secular years only are bissextille whose number, omitting the cyphers, is divisible by 4.
The Gregorian calendar is sufficiently exact for the purposes of common life, though it does not correspond precisely with the revolution of the sun. The error will amount to a day in 3600 years, so that in the year 5200 it will be necessary to omit the additional day which ought to be added according to the rule laid down above.
The Gregorian calendar was immediately adopted by all the Roman Catholic kingdoms in Europe, but the Protestant states refused at first to accede to it. It was adopted by most of them on the continent about the beginning of the 18th century; but in England the change did not take place till 1752. From that year 11 days were omitted; the omission of the additional day in 1700 having made the difference between the Julian and Gregorian calendar amount to 11 days. The Julian calendar is called the old style, the Gregorian, the new style. At present the difference between them is 12 days, in consequence of the omission of the additional day in 1800.
Sect. III. Of the Nature of the Sun.
The smallness of the sun's parallax is a demonstration of its immense size. We are certain that at the distance at which the sun appears to us under an angle of 0°.53424, the earth would be seen under an angle not exceeding 0°.009. Now, as the sun is obviously a spherical body as well as the earth; and as spheres are to each other as the cubes of their diameters, it follows from this, that the sun is at least 200,000 times bigger than the earth. By the exactest observations it has been ascertained, that the diameter of the sun is nearly 883,000 miles.
Dark spots are very frequently observed upon the surface of the sun. 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 Solar spots in the year 1611; and the honour of the discovery is when first disputed betwixt Galileo and Scheiner, a German Jesuit discovered 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.
There is great variety in the magnitudes of the Dr Long's solar spots; the difference is chiefly in superficial extent of length and breadth; their depth or thickness them 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 shapes, 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 estimation 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. Part II.
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 its 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 25 years, from 1650 to 1675, 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 1749, 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 diffused; while those that arise suddenly, are for the most part suddenly diffused. 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 macules) sometimes turn to spots.
The solar spots appear to have a motion which carries them across the sun's disk. Every spot, if it continues long enough without being diffused, appears to enter the sun's disk on the east side, to go from thence with the 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 track, 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 equally 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 its 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 our earth; for then we view the full extent of it 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 n, it appears as small as a thread, the thin edge being then all that we see.
These spots have made us acquainted with a very important phenomenon, namely the rotation of the sun upon its axis. Amidst the changes which these spots are continually undergoing, regular motions may be detected, agreeing exactly with the motion of the surface of the sun, on the supposition that this luminary revolves round an axis almost perpendicular to the ecliptic in the same direction with its motion in its orbit round the earth. By a careful examination of the motion of these spots, it has been ascertained that the sun turns round its axis in about 25 days and a half, and that its equator is inclined to the ecliptic about 7° 5'.
The spots on the sun's disk are almost always confined to a zone, extending about 30°.5 on each side of the equator. Sometimes, however, they have been observed at the distance of 39°.5 from the equator of the sun.
Bouguer demonstrated, by a number of curious experiments on the sun's light, that the intensity of the light is much greater toward the centre of the sun's disk than towards its circumference. Now, when a portion of the sun's surface is transported by the rotation of that luminary from the centre to the circumference of his disk, as it is seen under a smaller angle, the intensity of its light, instead of diminishing, ought to increase. Hence it follows, that part of the light which issues from the sun towards the circumference of his disk, must be somehow or other prevented from making its way to the earth. This cannot be accounted for, without supposing that the sun is surrounded by a dense atmosphere, which, being traversed obliquely by the rays from the circumference, intercepts more of them than of those from the centre which pass it perpendicularly.
The phenomena of the solar spots, as delivered by Scheiner and Hevelius, may be summed up in the following particulars. 1. Every spot which hath a nucleus, or considerably dark part, hath also an umbra, observers or fainter shade, surrounding it. 2. The boundary between 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 ever 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, insomuch that in a small space of time new encroachments are discernible, whereby the boundary between 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 umbræ 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 ceeded by a facula, or spot brighter than the rest of the disk, the place where it was is soon after not distinguishable from the rest.
In the Philosophical Transactions, vol. lxiv. Dr Wilfon, 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 to 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 mentioned; 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 Wilfon are not constant. He positively affirms, that the faculae 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 shewn 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 faculae 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 the 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 umbra; and gives the figure of one seen in November 13th 1773, which is a remarkable instance to 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 (lays 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 the Heaven- ly Bodies. 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."
Dr Herschel, with a view of ascertaining more accurately the nature of the sun, made frequent observations upon it from the year 1779 to the year 1794. He imagines that the dark spots on the sun are mountains on its surface, which, considering the great attraction exerted by the sun upon bodies placed at its surface, and the slow revolution it has upon its axis, he thinks may be more than 300 miles high, and yet stand very firmly. He says, that in August 1792, he examined the sun with several powers from 90 to 500; and it evidently appeared that the dark spots are the opaque ground or body of the sun; and that the luminous part is an atmosphere, which, being interrupted or broken, gives us a view of the sun itself. Hence he concludes, that the sun has a very extensive atmosphere, which consists of elastic fluids that are more or less lucid and transparent; and of which the lucid ones furnish us with light. This atmosphere, he thinks, is not less than 1843, nor more than 2765 miles in height; and, he supposes, that the density of the luminous solar clouds need not be much more than that of our aurora borealis, in order to produce the effects with which we are acquainted. The sun then, if this hypothesis be admitted, is similar to the other globes of the solar system, with regard to its solidity—its atmosphere—its surface diversified with mountains and valleys—its rotation on its axis—and the fall of heavy bodies on its surface; it therefore appears to be a very eminent, large, and lucid planet, the primary one in our system, diffusing its light and heat to all the bodies with which it is connected.
Dr Herschel has lately given up the use of the old terms such as spots, nuclei, penumbrae, &c. and has introduced a number of new terms, which he considers as more precise. It will be necessary, before we proceed farther, to insert his explanation of these terms.
"The expressions," says he, "which I have used Explan- tion of his are openings, shallows, ridges, nodules, corrugations, and pores.
"Openings are those places where, by the accidental removal of the luminous clouds of the sun, its own solid body may be seen; and this not being lucid, the openings through which we see it may, by a common telescope, be mistaken for mere black spots, or their nuclei.
"Shallows are extensive and level depressions of the luminous solar clouds, generally surrounding the openings to a considerable distance. As they are less luminous than the rest of the sun, they seem to have some distant, though very imperfect resemblance to penum- brae; which might occasion their having been called so formerly.
"Ridges are bright elevations of luminous matter, extended in rows of an irregular arrangement.
"Nodules are also bright elevations of luminous mat- Part II.
Apparent Motions of the Heavenly Bodies.
ter, but confined to a small space. These nodules, and Motions of ridges, on account of their being brighter than the general surface of the fun, and also differing a little from it in colour, have been called faculae, and luculi.
"Corrugations, I call that very particular and remarkable unevenness, ruggedness, or asperity, which is peculiar to the luminous solar clouds, and extends all over the surface of the globe of the sun. As the depressed parts of the corrugations are less luminous than the elevated ones, the disk of the sun has an appearance which may be called mottled.
Indentations are the depressed or low parts of the corrugations; they also extend over the whole surface of the luminous solar clouds.
Pores are very small holes or openings, about the middle of the indentations.
From the numerous observations of this philosopher he has drawn the following conclusions:—
1. Openings are places where the luminous clouds of the fun are removed: large openings have generally shallows about them; but small openings are generally without shallows. They have generally ridges and nodules about them, and they have a tendency to run into each other. New openings often break out near other openings. Hence he supposes that the openings are occasioned by an elastic but not luminous gas, which comes up through the pores and incipient openings, and spreads itself on the luminous clouds, forcing them out of its way, and widening its passage. Openings sometimes differ in colour; they divide when decayed; sometimes they increase again; but when divided they usually decrease and vanish; sometimes they become large indentations, and sometimes they turn into pores.
2. Shallows are depressed below the general surface of the fun, and are places from which the luminous solar clouds of the upper regions are removed. Their thickness is visible; sometimes they exist without openings in them. Incipient shallows come from the openings, or branch out from shallows already formed, and go forward. He supposes that the shallows are occasioned by something coming out of the openings, which, by its propelling motion, drives away the luminous clouds from the place where it meets with the least resistance; or which, by its nature, dissolves them as it comes up to them. If it be an elastic gas, its levity must be such as to make it ascend through the inferior region of the solar clouds, and diffuse itself among the superior luminous matter.
3. Ridges are elevations above the general surface of the luminous clouds of the fun. One of them, which he measured, extended over an angular space of 2' 45''9, which is nearly 75,000 miles.
Ridges generally accompany openings: but they often also exist in places where there are no openings. They usually disperse very soon. He supposes, that the openings permit a transparent elastic fluid to come out, which disturbs the luminous matter on the top, so as to occasion ridges and nodules; or, more precisely, that some elastic gas, acting below the luminous clouds, lifts them up, or increases them; and at last forces itself a passage through them, by throwing them aside.
4. Nodules are small, but highly elevated luminous places. He thinks that they may be ridges foreshortened.
5. Corrugations consist of elevations and depressions. They extend all over the surface of the sun; they change their shape and situation; they increase, diminish, divide, and vanish quickly. Dispersed ridges and nodules form corrugations.
6. The dark places of corrugations are indentations, Pores. Indentations are usually without openings, though in some places they contain small ones. They change to openings, and are of the same nature as shallows. They are low places, which often contain very small openings. They are of different sizes, and are extended all over the fun. With low magnifying powers they appear like points. The low places of indentations are pores. Pores increase sometimes, and become openings: they vanish quickly.
"It must be sufficiently evident," says Dr Herschel, "from what we have shewn of the nature of openings, shallows, ridges, nodules, corrugations, indentations, and pores, that these phenomena could not appear, if the shining matter of the fun were a liquid; since, by the laws of hydrostatics, the openings, shallows, indentations, and pores, would instantly be filled up; nor could ridges and nodules preserve their elevation for a single moment. Whereas, many openings have been known to last for a whole revolution of the fun; and extensive elevations have remained supported for several days. Much less can it be an elastic fluid of an atmospheric nature: this would be still more ready to fill up the low places, and to expand itself to a level at the top. It remains, therefore, only for us to admit this shining matter to exist in the manner of empyreal, luminous, or phosphoric clouds, residing in the higher regions of the solar atmosphere."
From his observations, Dr Herschel concludes, that there are two different regions of solar clouds; that the inferior clouds are opaque, and probably not unlike those of our planet; while the superior are luminous, and emit a vast quantity of light: that the opaque inferior clouds probably suffer but little of the light of the self-luminous superior clouds to come to the body of the sun. "The shallows about large openings," he observes, "are generally of such a size, as hardly to permit any direct illumination from the superior clouds to pass over them into the openings; and the great height and closeness of the sides of small ones, though not often guarded by shallows, must also have nearly the same effect. By this it appears, that the planetary clouds are indeed a most effectual curtain, to keep the brightness of the superior regions from the body of the fun.
"Another advantage arising from the planetary clouds of the fun, is of no less importance to the whole solar system. Corrugations are everywhere dispersed over the fun; and their indentations may be called shallows in miniature. From this we may conclude, that the immense curtain of the planetary solar clouds is everywhere closely drawn; and, as our photometrical experiments have proved that these clouds reflect no less than 469 rays out of 1000, it is evident that they must add a most capital support to the splendour of the fun, by throwing back so great a share of the apparent Motions of the Heavenly Bodies.
Vol. III. Part I. Apparent brightness coming to them from the illumination of the motions of whole superior regions."
These observations are sufficient to prove, that the sun has an atmosphere of great density, and extending to a great height. Like our atmosphere, it is obviously subject to agitations, similar to our winds; and it is also transparent. The following is Dr Herschel's theoretical explanation of the solar phenomena.
"We have admitted," says he, "that a transparent elastic gas comes up through the openings, by forcing itself a passage through the planetary clouds. Our observations seemed naturally to lead to this supposition, or rather to prove it; for, in tracing the shallows to their origin, it has been shewn, that they always begin from the openings, and go forwards. We have also seen, that in one cafe, a particular bias given to incipient shallows, lengthened a number of them out in one certain direction, which evidently denoted a propelling force acting the same way in them all. I am, however, well prepared to distinguish between facts observed, and the consequences that in reasoning upon them we may draw from them; and it will be easy to separate them, if that should hereafter be required.
If, however, it be now allowed, that the cause we have assigned may be the true one, it will then appear, that the operations which are carried on in the atmosphere of the sun are very simple and uniform.
"By the nature and construction of the sun, an elastic gas, which may be called empyreal, is constantly formed. This ascends everywhere, by a specific gravity less than that of the general solar atmospheric gas contained in the lower regions. When it goes up in moderate quantities, it makes itself small passages among the lower regions of clouds: these we have frequently observed, and have called them pores. We have shewn that they are liable to continual and quick changes, which must be a natural consequence of their fleeting generation.
"When this empyreal gas has reached the higher regions of the sun's atmosphere, it mixes with other gales, which, from their specific gravity, have their residence there, and occasions decompositions which produce the appearance of corrugations. It has been shewn, that the elevated parts of the corrugations are small self-luminous nodules, or broken ridges; and I have used the name of self-luminous clouds, as a general expression for all phenomena of the sun, in what shape ever they may appear, that shine by their own light. These terms do not exactly convey the idea affixed to them; but those of meteors, corrugations, inflammations, luminous wiffs, or others, which I might have selected, would have been liable to still greater objections. It is true, that when speaking of clouds, we generally conceive something too gross, and even too permanent, to permit us to apply that expression properly to luminous decompositions, which cannot float or swim in air, as we are used to see our planetary clouds do. But it should be remembered, that, on account of the great compression arising from the force of the gravity, all the elastic solar gales must be much condensed; and that, consequently, phenomena in the sun's atmosphere, which in ours would be mere transitory corrugations, such as those of the aurora borealis, will be so compressed as to become much more efficacious and permanent.
"The great light occasioned by the brilliant superior regions, must scatter itself on the tops of the inferior planetary clouds, and, on account of their great density, bring on a very vivid reflection. Between the interstices of the elevated parts of the corrugations, or self-luminous clouds, which, according to the observations that have been given, are not closely connected, the light reflected from the lower clouds will be plainly visible, and, being considerably less intense than the direct illumination from the upper regions, will occasion that faint appearance which we have called indentations.
"This mixture of the light reflected from the indentations, and that which is emitted directly from the higher parts of the corrugations, unless very attentively examined by a superior telescope, will only have the resemblance of a mottled surface.
"When a quantity of empyreal gas, more than what produces only pores in ascending, is formed, it will make itself small openings; or, meeting perhaps with some resistance in passing upwards, it may exert its actions in the production of ridges and nodules.
"Lastly, If still further an uncommon quantity of this gas should be formed, it will burst through the planetary regions of clouds, and thus will produce great openings; then, spreading itself above them, it will occasion large shallows, and, mixing afterwards gradually with other superior gales, it will promote the increase, and assist in the maintenance, of the general luminous phenomena.
"If this account of the solar appearances should be well founded, we shall have no difficulty in ascertaining the actual state of the sun, with regard to its energy in giving light and heat to our globe; and nothing will now remain, but to decide the question which will naturally occur, whether there be actually any considerable difference in the quantity of light and heat emitted from the sun at different times." This question he decides in the affirmative, considering the great number of spots as a proof that the sun is emitting a great quantity of light and heat, and the want of spots as the contrary. The first is connected with a warm and good season; the second, on the contrary, produces a bad one*.
Phil. Trans. 1801 part ii. p. 265.
CHAP. II. Of the Moon.
Next to the sun, the most conspicuous of all the heavenly bodies is the moon. The changes which it undergoes are more striking and more frequent than those of the sun, and its apparent motions much more rapid. Hence they were attended to even before those of the sun were known; a fact which explains why the first inhabitants of the earth reckoned their time by the moon's motions, and of course followed the lunar instead of the solar year. In considering the moon, we shall follow the same plan that we observed with respect to the sun. We shall first give an account of her apparent motions; and, secondly, of her nature as far as it has been ascertained. These topics shall occupy the two following sections.
SECT. Part II.
Apparent Motions of the Heavenly Bodies.
Sect. I. Of the Apparent Motions of the Moon.
The moon, like the sun, has a peculiar motion from east to west. If we observe her any evening when she is situated very near any fixed star, we shall find her, in 24 hours, about 13° to the east of that star; and her distance continually increases, till at last, after a certain number of days, she returns again to the same star from the west, having performed a complete revolution in the heavens. By a continued series of observations it has been ascertained, that the moon makes a complete revolution in 27.32166118036 days, or 27 days 7 hours 43' 11" 31'' 35''''. Such at least was the duration of its revolution at the commencement of 1700. But it does not remain always the same. From a comparison between the observations of the ancients and those of the moderns, it appears, that the mean motion of the moon in her orbit is accelerating. This acceleration, but just sensible at present, will gradually become more and more obvious. It is a point of great importance to discover, whether it will always continue to increase, or whether, after arriving at a certain maximum, it will again diminish. Observations could be of no service for many ages in the resolution of this question; but the Newtonian theory has enabled astronomers to ascertain that the acceleration is periodical.
The moon's motion in her orbit is still more unequal than that of the sun. In one part of her orbit she moves faster, in another slower. By knowing the time of a complete revolution, we can easily calculate the mean motion for a day, or any given time; and this mean motion is called the mean anomaly. The true motion is called the true anomaly; the difference between the two is called the equation. Now the moon's equation sometimes amounts to 6° 18' 32''.
Her apparent diameter varies with the velocity of her angular motion. When she moves fastest, her diameter is largest; it is smallest when her angular motion is slowest. When smallest, the apparent diameter is 0.489420°; when biggest, it is 0.58830°. Hence it follows, that the distance of the moon from the earth varies. By following the same mode of reasoning, which we have detailed in the last chapter, Kepler ascertained that the orbit of the moon is an ellipse, having the earth in one of its foci. Her radius vector describes equal areas in equal times; and her angular motion is inversely proportional to the square of her distance from the earth.
The eccentricity of the elliptic orbit of the moon, has been ascertained to amount to 0.0550368, (the mean distance of the earth being represented by unity); or the greater axis is to the smaller, nearly as 100,000 to 99,848.
That point of the moon's orbit which is nearest the earth, is called the perigee; the opposite point is the apogee. The line which joins these opposite points, is called the line of the moon's apsides. It moves slowly eastward, completing a sidereal revolution in 3232.46643 days, or nearly 9 years.
The inclination of the moon's orbit is also variable; the greatest inequality is proportional to the cosine of twice the sun's angular distance from the ascending node, and amounts when a maximum to 0.14679°.
Even the elliptical orbit of the moon represents but imperfectly her real motion round the earth; for that luminary is subjected to a great number of irregularities, evidently connected with the positions of the sun, which considerably alter the figure of her orbit. The three following are the principal of these.
1. The greatest of all, and the one which was first ascertained, is called by astronomers the moon's evection. It is proportional to the sine of twice the mean angular distance of the moon from the sun, minus the mean angular distance of the moon from the perigee of its orbit. Its maximum amounts to 1.3410°. In the oppositions and conjunctions of the sun and moon it coincides with the equation of the centre, which it always diminishes. Hence the ancients, who determined that equation by means of the eclipses, found that equation smaller than it is in reality.
2. There is another inequality in the motion of the moon, which disappears during the conjunctions and oppositions of the sun and moon; and likewise when these bodies are 90° distant from each other. It is at its maximum when their mutual distance is about 45°, and then amounts to about 0.594°. Hence it has been concluded to be proportional to the sine of twice the mean angular distance of the moon from the sun. This inequality is called the variation. It disappears during the eclipses.
3. The moon's motion is accelerated when that of the sun is retarded, and the contrary. This occasioned an irregularity called the annual equation. It follows exactly the same law with that of the equation of the centre of the sun, only with a contrary fine. At its maximum it amounts to 0.18576°. During eclipses, it coincides with the equation of the sun.
The moon's orbit is inclined to the ecliptic at an angle of 6.14602°. The points where it intersects the ecliptic are called the nodes. Their position is not fixed in the heavens. They have a retrograde motion, that is to say, a motion contrary to that of the sun. This motion may be easily traced by marking the successive stars which the moon passes when she crosses the ecliptic. They make a complete revolution of the heavens in 6793.3009 days. The ascending node is that in which the moon rises above the ecliptic towards the north pole, the descending node that in which she sinks below the equator towards the south pole. The motion of the nodes is subjected to several irregularities, the greatest of which is proportional to the sine of twice the angular distance of the sun from the ascending node of the lunar orbit. When at a maximum, it amounts to 1.62945°. The inclination of the orbit itself is variable. Its greatest inequality amounts to 0.14679°. It is proportional to the cosine of the same angle on which the irregularity in the motion of the nodes depends.
The apparent diameter of the moon varies as well as that of the sun, and in a more remarkable manner. When smallest, it measures 29.5'; when largest, 34'. This must be owing to the distance of the moon from the earth being subject to variations.
The great distance of the sun from the earth renders it difficult to determine its parallax, on account of its minuteness. This is not the case with the moon. The distance of that luminary from the earth may be determined without much difficulty. Let BAG (fig. 10.) be one half of the earth, AC Motions of its semidiameter, S the sun, m the moon, and EKOL the Heavenly Bodies. a 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 those 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 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 (g) 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 to 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, and the sides of a triangle are always proportional to the sines of the opposite angles, lay, 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 is radius, viz. the fine 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 fine 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 242,000 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, though 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 the 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. 11. let VECQ be the earth, M the moon, and Zbaa 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 beforehand by the observers. As these two places are on the same meridian nVECq, 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° 6' 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
(g) 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. Part II.
Apparent supposing the tabular radius to be 10,000,000, the natures of tural fine of 38° 1' 53'' (the arc Za) is 6,160,816, and the Heaven-the natural fine of 46° 4' 41'' (the arc zb) is 7,204,821: ly Bodies. The sum of both these fines is 13,365,637. Say therefore, As 13,365,637 is to 10,000,000, so is 1° 16' 34'' to 57' 18'', which is the moon's horizontal parallax.
If the two places of observation be not exactly under the same meridian, their difference of longitude must be accurately taken, that proper allowance may be made for the moon's declination whilst she is passing from the meridian of the one to the meridian of the other.
Moon's size.
From the theory of the parallax we know, that at the distance of the moon from the earth the apparent size of the earth would be to that of the moon as 21:352 to 5823. Their respective diameters must be proportional to these numbers, or almost as 11 to 3. Hence the bulk of the moon is 49 times less than that of the earth.
Her phases explained.
The different appearances, or phases, of the moon constitute some of the most striking phenomena of the heavens. When she emerges from the rays of the sun in an evening, she appears after sunset as a small crescent just visible. The size of this crescent increases continually as she separates to a greater distance from the sun, and when she is exactly in opposition to that luminary, she appears under the form of a complete circle. This circle changes into a crescent as she approaches nearer that luminary, exactly in the same manner it had increased, till at last she disappears altogether, plunging into the sun's rays in the morning at sunrise. The crescent of the moon being always directed towards the sun, indicates obviously that she borrows her light from that luminary; while the law of the variation of her phases, almost proportional to the verified side of the angular distance of the moon from the sun, demonstrates that her figure is spherical. Hence it follows, that the moon is an opaque spherical body.
These different phases of the moon are renewed after every conjunction. They depend upon the excess of the synodical movement of the moon above that of the sun, an excess which is usually termed the synodical motion of the moon. The duration of the synodical revolution of the moon in the mean period between two conjunctions is 29,530,588 days. It is to the tropical year nearly in the ratio of 19 to 235, that is to say, that 19 solar years consist of about 235 lunar months.
The points of the lunar orbit, in which the moon is either in conjunction or opposition to the sun are called syzygies. In the first point the moon is said to be new, in the second to be full. The quadratures are those points in which the moon is distant from the sun 90° or 270°. When in these points, the moon is said to be in her first and third quarter. One half only of the moon is then illuminated or seen from the earth. As a more particular account of these phases may be deemed necessary, we subjoin the following explanation, which will perhaps be better understood by the generality of readers.
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. Apparent Hence the disappears when he comes between us and Motions of the fun; because her dark side is then towards us. the Heavenly Bodies. When the 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 fun; and then her whole enlightened side is towards the earth, and she appears with a round illuminated orb, which we call the full moon; her dark side being then turned away from the earth. From the full the 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 fun, and then the disappears as before.
The moon has scarce any difference of leasons; 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 fun'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; Earth 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 the changes to us, the earth appears full to her; and when the 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 so 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 fun: 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 lunar equator; and the obliquity of her orbit, which is next to nothing as seen from the fun, cannot cause the fun to decline sensibly from her equator. Yet her inhabitants are not delitute 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 tracks 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! Apparent Motions of natural days, and the lunarians only 12\( \frac{1}{3} \), every day and night in the moon being as long as 29\( \frac{1}{2} \) on the earth.
The moon's inhabitants on the side next the earth 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 sunshine to the blackest darkness. For, let \( tr \hat{F} sw \) 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 take 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 disappearance, 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 Apparent equal distances from A to E, but as seen from the sun Motions of the Heavenly Bodies.
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 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 Agreeable frequently visible in the forenoon, even when the sun represents the moon; and then she affords us an opportunity of seeing her 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 day 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 nonagefimal degree; Nonagefimal which is the highest point of the ecliptic above the homal degree rizon at that time, and is 90° 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.
The explanation of the phases of the moon leads us Eclipses of to that of the eclipses; those phenomena which formerly, the moon, were the subjects of dread and error, but which philosophers have converted to the purposes of utility and information. The moon can only become eclipsed by the interposition of an opaque body, which intercepts from it the light of the sun; and it is obvious that this opaque body is the earth, because the eclipses of the moon never happen except when the moon is in opposition, and consequently when the earth is interposed between her and the sun. The globe of the earth projects behind it relatively to the motion of the sun a conical shadow, whose axis is the straight line that joins the centres of the earth and sun, and which terminates at the point when the apparent diameters of these two bodies become equal. The diameters of these bodies seen from the centre of the moon in oppo- Part II.
Apparent fition, are nearly in the proportion of 3 for the sun and Motions of 11 for the earth. Therefore, the conical shadow of the Heaven-earth is at least thrice as long as the distance between the earth and moon, and its breadth at the point where it is traversed by the moon more than double the diameter of that luminary.
The moon, therefore, would be eclipsed every time that it is in opposition, if the plane of its orbit coincided with the ecliptic. But in consequence of the mutual inclination of these two planes, the moon, when in opposition, is often elevated above the earth's conical shadow, or depressed below it; and never can pass through that shadow unless when it is near the nodes. If the whole of the moon's disk plunges into the shadow, the eclipse is said to be total; if only a part of the disk enter the shadow, the eclipse is said to be partial.
The mean duration of a revolution of the sun relatively to the nodes of the lunar orbit is 346.61963 days, and is to the duration of a synodical revolution of the moon nearly as 223 to 19. Consequently, after a period of 223 lunar months, the sun and moon return nearly to the same situation relatively to the order of the lunar orbit. Of course the eclipses must return in the same order after every 223 lunations. This gives us an easy method of predicting them. But the inequalities in the motions of the sun and moon occasion sensible differences; besides the return of the two luminaries to the same points relatively to the nodes not being rigorously true, the deviations occasioned by this want of exactness alter at last the order of the eclipses observed during one of these periods.
The following explanation of the lunar eclipses being more particular, may be acceptable to some of our readers.
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. 13. 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 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 f, g, h, i, concentric to the earth, include the atmosphere whose refractive power vanishes at the height f and i; so that the rays W f w and V i v 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 through a greater portion of the atmosphere, until the rays W k and V l 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 m o p q n, where none of the sun's rays can enter; all the rest R, R, 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 o, she would be invisible during her stay in it; but visible before and after in the fainter shadow RR.
When the moon goes through 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. 14. let GH be the ecliptic, e f the moon's orbit where she is 12 degrees from the node at her full, c d her orbit where she is 6 degrees from the node, a b her orbit where she is full in the node, AB the earth's shadow, and M the moon. When the moon describes the line e f, she just touches the shadow, but does not enter into it; when she describes the line c d, she is totally, though not centrally, immersed in the shadow; and when she describes the line a b, 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 h. 57. m. 6 sec. from the beginning to the end, if the moon be at her greatest distance from the earth; and 3 h. 37 m. 26 sec. if she be at her least distance. The reason of this difference is, that when the moon is farther'd from the earth, she moves slower; and when nearest to it, quicker.
The moon's diameter, as well as the sun's, is supposed to be divided into 12 equal parts, called digits; and so many of these parts as are darkened by the earth's shadow, so many digits is the moon eclipsed. All that the moon 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 touching or leaving it, the obliteration of her limb is scarcely sensible; and therefore the nicest observers can hardly be certain to four or five seconds of time. But Apparent Motions of the Heavenly Bodies.
Eclipses of the sun.
The eclipses of the sun only take place during the conjunctions of the sun and moon, or when the moon is placed between the sun and the earth. They are owing to the moon concealing the sun from the earth, or to the earth being plunged in the shadow of the moon. The moon is indeed much smaller than the sun; but it is so much nearer to the earth that its apparent diameter does not differ much from the diameter of that luminary: and, in consequence of the changes which take place in the apparent diameter of these bodies, it happens that sometimes the apparent diameter of the moon is greater than that of the sun. If we suppose the centres of the sun and moon in the same straight line with the eye of a spectator placed on the earth, he will see the sun eclipsed. If the apparent diameter of the moon happens to surpass that of the sun, the eclipse will be total: but if the moon's diameter be smallest, the observer will see a luminous ring, formed by that part of the sun's disk which exceeds that of the moon's, and the eclipse will in that case be annular. If the centre of the moon is not in the same straight line which joins the observer and the centre of the sun, the eclipse can only be partial, as the moon can only conceal a part of the sun's disk. Hence there must be a great variety in the appearance of the solar eclipses. We may add also to these causes of variety the elevation of the moon above the horizon, which changes its apparent diameter considerably. For it is well known, that the moon's diameter appears larger when she is near the horizon than when she is elevated far above it. Now, as the moon's height above the horizon varies according to the longitude of the observer, it follows, that the solar eclipses will not have the same appearance to the observers situated in different longitudes. One observer may see an eclipse which does not happen relatively to another. In this respect the solar differ from the lunar eclipses, which are the same to all the inhabitants of the earth.
Number of eclipses in a year.
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 six. 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 1/2 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 afterwards; and fix 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 Motions of to be eclipsed; and in six lunar months afterwards the moon will change nearer the other node; in these cases, there can be but two eclipses in a year, and they are both of the fun.
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 fun and node meet again fo 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 fun's ecliptic limits are greater than eclipses of the moon's; yet we have more visible eclipses of the moon than of the fun, because eclipses of the moon are observed from all parts of that hemisphere of the earth which is next her, and are equally great to each of those parts: but the fun's eclipses are visible only to that small portion of the hemisphere next him whereon the moon's shadow falls.
The moon's orbit being elliptical, and the earth in one of its fociuses, 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 annular upon the earth, she appears big enough to cover the whole disk of the fun from that part on which her shadow falls; and the fun 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 fun'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 fun's edge appearing like a luminous ring all round 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 the eclipseth the fun 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 fun's, when least, only 1 minute 38 seconds of a degree; and in the greatest eclipse of the fun than 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 fun 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 extent of the earth's surface about 180 English miles broad, when the moon's diameter appears largest, and the fun'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 fun is more or less eclipsed, as the places are less or more distant from the centre of the penumbra. When the moon changes exactly Part II.
Apparent exactly in the node, the penumbra is circular on the Motions of earth at the middle of the general eclipse; because at the Heavenly Bodies, 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.
Beginning, &c. 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; lest 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, and AMP the moon's orbit. Draw the right line We from the western side of the sun at W, touching the western side of the moon at c, and the earth at e: draw also the right line Vd 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 ccd 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 continuation: 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 WXdh and Vxcg, 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 WXdh and Vxcg 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 abab; within which the sun appears more or less eclipsed, as the places are more or less distant from the verge of the penumbra ab.
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 ab, 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. 15, at the left hand; but at the same moment of absolute time, to an observer at a in fig. 14, 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. 15. At the very same instant, to all those who live on the circle marked 1 on the earth A, in fig. 14, the moon M cuts off or darkens a twelfth part of the sun S, and eclipses him one digit, as at I in fig. 15.: to those who live on the circle marked 2 in fig. 14, the moon cuts off two twelfth parts of the sun, as at 2 in fig. 15.: to those on the circle 3, three parts; and so on to the centre at 12 in fig. 14, where the sun is centrally eclipsed, as at B in the middle of fig. 15.; 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. 14. 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 ab, 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 YY 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 no 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\( \frac{1}{2} \) minutes of a degree every hour at a mean rate: but so much of the moon's orbit is equal to 30\( \frac{1}{2} \) degrees of a great circle on the earth; and therefore the moon's shadow goes 30\( \frac{1}{2} \) degrees, or 1830 geographical miles on the earth in an hour, or 30\( \frac{1}{2} \) 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 parallellism throughout its annual course. In fig. 16. 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, t the tropic of Capricorn, and ABC the circumference of the earth's enlightened disk as seen from the fun 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 fun 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 fun 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 1/2 degrees towards the fun, falls so many degrees within the earth's enlightened disk, because the fun is then vertical to D 23 1/2 degrees north of the equator or AEQ; and the equator, with all its parallels, seem elliptic curves bending downward, or towards the south pole, as seen from the fun; which pole, together with 23 1/2 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 fun or new moon, which are then vertical to O, where the ecliptic cuts the equator AEQ. Both poles now lie 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 fun, 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 1/2 degrees towards the fun, falls 23 1/2 degrees within the enlightened disk, as seen from the fun or new moon, which are then vertical to the tropic of Capricorn t, 23 1/2 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 fun, 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 fun at the summer solstice; from the fun at the winter solstice; and likewise to the fun at the equinoxes: but towards the right hand, as seen from the fun 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 fun; and the contrary from the summer to the winter solstice.
The different positions of the earth's axis, as seen from the fun at different times of the year, affect solar eclipses greatly with regard to particular places: yea, so far as would make central eclipses which fall at one 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, iK, iK, iK, part of the moon's orbit; affected by both seen edgewise, and therefore projected into right lines; and let the intersections NODE be one and the same node at the above time; 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 t 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 NDS (iDk 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 (iO 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 AO m: whereas, in the latter case, the penumbra touches the earth at n, south of the equator AEQ, and describing the line n Eg (similar to the former line AO m 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 defending 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 18 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 fun, 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 defending node, more than a third part of the penumbra P 12 falls on the northern parts of the earth at the middle of the general eclipse: Had she changed as far past the same node, Part II.
Apparent as much of the other side of the penumbra about P would have fallen on the southern parts of the earth; the Heaven- all the rest in the expanse, or open space. When the moon changes 6 degrees from the node, almost the whole penumbra P 6 falls on the earth at the middle of the general eclipse. And lastly, 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) when the point b in the polar circle a b c d 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 a b c d, 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.
About the new moon, that part of the lunar disk which is not illuminated by the sun is perceptible, owing to the feeble light reflected on it by the hemisphere of the earth that is illuminated.
SECT. II. Of the Nature of the Moon.
We have seen that the moon is about 39 times smaller than the earth. Her diameter is generally reckoned about 2180 miles. This is to the diameter of the earth nearly as 20 to 73; therefore, the surface of the moon is to that of the earth (being as the squares of their diameters) nearly as 1 to \( \frac{14^3}{73} \). And, admitting the moon's density to be to that of the earth as 5 to 4, their respective quantities of matter will be as 1 to 39 very nearly.
Bouguer has shown, by a set of curious experiments, that the light emitted by the full moon is 300,000 times less intense than that of the sun. Even when concentrated by the most powerful mirrors, it produces no effect on the thermometer.
Many dusky spots may be seen upon the moon's disk, even with 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, Caffini, 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 Ricciolus, 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 Etna. Fig. 17. is a tolerably exact representation of the full moon in her mean libration, with the numbers to the principal spots according to Ricciolus, Caffini, 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 Gaffendus. 6 Shikardus. 7 Harpalus. 8 Heraclides. 9 Lanbergius. 10 Reinoldus. 11 Copernicus. 12 Helicon. 13 Capuanus. 14 Bullialdus. 15 Eratosthenes. 16 Timocharis. 17 Plato. 18 Archimedes. 19 Inula Sinus Medii. 20 Pitatus. 21 Tycho. 22 Eudoxus. 23 Aristoteles. 24 Manilius. 25 Menelaus. 26 Hermes. 27 Poffidonus. 28 Dionyfus. 29 Plinius. 30 Catharina Cyrillus. 31 Theophilus. Apparent Motions of the Heavenly Bodies.
31 Fracastorius. 32 Promontorium acutum. 33 Cenforinus. 34 Promontorium Somnii. 35 Proclus. 36 Cleomedes. 37 Snellius et Furnerius. 38 Petavius. 39 Langrenus.
Great inequalities on the surface of the Moon.
That there are prodigious inequalities on her surface, is proved by looking at her through a telescope, at any other time than when she is full; for then there is no regular line bounding light and darknes: 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 darknes 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 darknes 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 darknes, 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.
Now content with perceiving the bare existence of these lunar mountains, astronomers have endeavoured to measure their height in the following manner. Let EGD be the hemisphere of the moon illuminated by the sun, ECD the diameter of the circle bounding light and darknes, 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 subducting BC or EC, there will remain AB the height of the mountain. Ricciolus affirms, that upon the fourth day after new moon he has observed the top of the hill called St Catharine's to be illuminated, and that it was distant from the confines of the lucid surface about a sixteenth part of the moon's diameter. Therefore, if CF=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 deducting BC=8, there will remain AC=8,062. So that CB or CE is therefore to AB as 8 is to 8,062, 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 Keill, 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 1000 miles, the height of the mountain will be nearly 8\frac{1}{8} miles.
In the former edition of this work, we could not help making some remarks on the improbability that the lunar mountains of the moon, a planet so much inferior mountains 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 Dr Herchel. 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 5\frac{1}{2} miles; and according to Hevelius 3\frac{3}{4} miles, admitting the moon's diameter to be 2180 miles. Mr Fergulon, 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 Keill, 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, Dr Herchel 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 s l m, (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, l m, will not appear of the same length, though the mountain should be of an equal height; for LM will be projected into o n, and l m 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 o n, when the moon is not in her quadrature, to find LM, we have the following analogy. The triangles o OL, r ML, are similar; therefore L o : LO :: L r : LM, or \( \frac{LO + on}{L o} = LM \): but
LO is the radius of the moon, and L r or o n is the observed distance of the mountain's projection; and L o is the sine of the angle ROL = o LS; which we may take to be the distance of the sun from the moon without any material error, and which therefore we may find at any given time from the ephemeris."
The telescope used in these observations was a Newtonian Part II.
Apparent Newtonian reflector of six feet eight inches focal length, Motions of the Heavenly Bodies.
Motions of the Heavenly Bodies.
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 shewn 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 Dr Herfchel, "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 centifemals; then it will be in general, \( R : 1090 :: Q : \frac{1090\ Q}{R} = o\ n \) 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' 1"; the sine of which to the radius 1 is .9985, &c. and \( \frac{o\ n}{Lo} \) 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 \( o\ n \) 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, Dr Herfchel 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 overrated; 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 Riccioli found to project a fifteenth part of the moon's diameter. If Keill had calculated the height of this last-mentioned hill according to the theorem I had 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 would 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 3½ 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, Dr Herfchel informs us that he had measured the height of one of the mountains that had been measured by Hevelius. "Antitaurus (says he), the mountain measured by Hevelius, was badly situated: because Mount Moschus 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 Thrasimenes and Pontus Euxinus, measured a mile and a quarter; Mons Armenia, near Taurus, two-thirds of a mile; Mons Leucoptera 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 \( o\ n \) is 50.193 miles, LM=64 miles, and the perpendicular height above one mile and three-fourths.'
As the moon has on its surface mountains and volcanoes 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 the moon 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 Apparent motions of the heavenly bodies have induced astronomers to attribute appearances of this kind to the eruption of volcanic fire; and Dr Herfchel 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. 36 m. fide real 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. Meehan discovered at Paris the 10th of this month.
"April 20. 10 h. 2 m. fide real time. The volcano burns with greater violence than last night. Its diameter cannot be less than 3 sec. 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 nebulae, 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, 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 sunbeams. 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 glases, 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 Marcilles, 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 darkest spots, he says, cannot be seas, nor anything 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 Part II.
Apparent Motions of the Heavenly Bodies.
Whether the moon has any atmosphere.
121 Whether the moon has any atmosphere.
manent 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 appulse of the moon to a star, when she comes so near it that part of her atmosphere is interposed between our eye and the star, refraction would cause the latter to 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 sixth 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, therefore, surrounding the moon, being pressed together, by a weight, or being attracted towards the centre 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 stars light. Other astronomers have contended that such refraction was sometimes very apparent. M. 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 own, that in other occultations no such change could Apparent 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 1980th. The mean apparent diameter of the moon is 31' 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 ring observed about the moon time of solar eclipses. In the eclipse of May 1. 1766, in total Captain Stanyan, from Bern in Switzerland, writes, eclipses, 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 used 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 fix seconds shows that its height is not more than the five or fix 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 Apparent I think must be that luminous appearance (the zodiacal Motions of light) mentioned from Cassini." Flamsteed, 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 the 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 thinning, 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 refract the sun's light.
A total eclipse of the sun was observed on the 22d 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 dews 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, 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 interposition 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 whole 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, the ring appeared much brighter and whiter near the body of the moon than at a distance from it; and its outward circumference, which pear to dart 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 emersion. But this instantly vanished after the appearance of the sun, as did also the aforesaid luminous ring."
Mr Louville relates, that a luminous ring of a silver Mr Lou- colour appeared round the moon as soon as the sun was ville's ob- entirely covered by her disk, and disappeared the mo- servations. ment he recovered his light; and this ring was bright- est near the moon, and grew gradually fainter towards its outer circumference, where it was, however, de- fined; that it was not equally bright all over, but had several breaks in it: but he makes no doubt of its be- ing 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, to- wards 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 redness did not proceed from the glaies of his telescope, he took care to bring the red part into the middle of his glaies.
He lays great stres on the streaks of light which he saw dart instantaneously from different places of the moon during the time of total darkness, but chiefly near the eastern edge of the disk: thefe he takes to be lightning, such as a spectator would fee flashing from the dark hemisphere of the earth, if he were placed upon the moon, and faw the earth come between himself and the fun. "Now (fays 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 fee lightning in fome part of it or other." Louville farther obferves, that the moft mountainous countries are moft liable to tempefts; and that mountains being more frequent in the moon, and higher, than on earth*, thunder and lightning muft be more frequent there than with us; and that the eastern fide of the moon would be moft fubject to thunder and lightning, thoſe parts having been heated by the fun for half the month immediately preceding. It muft here be obferved, that Halley, in mentioning thefe fhares, fays they feemed to come from behind the moon; and Louville, though he fays they came fometimes from one part and fometimes from another, owns, that he himfelf only faw them near the eaftern part of the disk; and that, not knowing at that time what it was that he faw, he did not take notice whether the fame appearance was to be seen on other parts of the moon or not. He tells us, however, of an Englifh astronomer, who prefented the Royal Society with a draught of what he faw in the moon at the time of this eclipse; from which Louville feems to conclude, that lightning had been obferved by that astronomer near the centre of the moon's disk. "Now (fays Dr Long) thunder and lightning would be a demonftration of the moon having an atmosphere fimilar to ours, wherein vapours and exhalations may be supported, and furnifh materials for clouds, storms, and tempefts. But the moft ftrongeft proof brought by Louville of the moon having an atmosphere is this, that as foon as the eclipse began, thoſe parts of the fun which were going to be hid by the moon grew fensibly palilh as the former came near them, fuffering beforehand a kind of imperfect eclipse or diminution of light; this would be owing to nothing elfe but the atmosphere of the moon, the eaftern part whereof going before he reached the fun 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 a calculation come out 180 miles, above three times as high as the atmosphere of the earth, Louville thinks that no objection; fince if the moon were furronded with an atmosphere of the fame 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 fame luminous ring has been obferved in other total eclipses, and even in fuch as are annular, though without the luminous streaks or fhares of lightning Apparent above-mentioned; it is even taken notice of by Plutarch: Motions of however, fome members of the academy at Paris have endeavoured to account for both thefe phenomena without having recourse to a lunar atmosphere; and for this purpofe they made the following experiments: The image of the fun coming through a small hole in a darkened room, was received upon a circle of wood or metal of a diameter a good deal larger than that of the fun's image; then the shadow of this opaque circle was caft upon white paper, and there appeared round it, on the paper, a luminous circle fuch as that which surrounds the moon. The like experiment being made with a globe of wood and with another of flone not poftilhed, the shadows of both thefe caft upon paper were furronded with a palish light, moft vivid near the shadows, and gradually more diluted at a diftance from them. They obferve also, that the ring round the moon was feen in the eclipse of 1756 by Wurzelbaur, who caft her shadow upon white paper. The fame appearance was obferved on holding an opaque globe in the fun, fo as to cover his whole body from the eye; for, looking at it through a fmoaked glafs, in order to prevent the eye from being hurt by the glare of light it would otherwife be expofed to, the globe appeared with a light resembling that round the moon in a total eclipse of the fun.
Thus they folve the phenomenon of the ring feen round the moon by the inflection, or diffraction as they call it, of the solar rays pafting near an opaque fubflance. As for the fmall streaks of light above-mentioned, and which are fupposed to be lightning, they explain thefe by an hypothes concerning the cavities of the moon themfelves; which they confider as concave mirrors reflecting the light of the fun nearly to the fame point; and as thefe are continually changing their situation with great velocity by the moon's motion from the fun, the light which any one of them fends to our eye is feen but for a moment. This, however, will not account for the fhares, if any fuch there are, feen near the centre of the disk, though it does, in no very satisfactory manner, account for thefe at the edges.
It has already been obferved, 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 foetimes been obferved, and the stars have been feen manifeftly to change their shape and colour on going behind the moon's disk. An inftance 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 day-time. Some astronomers in France obferving this with a telescope, faw Venus change colour for about a minute before she was hid by the moon; and the fame change of colour was obferved immediately after her emerfion from behind the disk. At both times the edge of the disk of Venus that was nearest the moon appeared reddifh, and that which was moft diftant of a bluih colour. Thefe appearances, however, which might have been taken for proofs of a lunar atmosphere, were fuppoed to be owing to the observers having directed the axis of their telefopes towards the moon. This would neceffarily caufe any planet or star near the edge of the moon's disk to be feen through thefe parts of the glaies which are near their circumference, Apparent Motions of the Heavenly Bodies.
ference, 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?
We have related all these opinions at full length, in order to put our readers in possession of the arguments that have been advanced upon this subject; but it is now generally admitted, and indeed, scarcely can be denied, that the atmosphere of the moon, if it really has any, is almost entirely insensible.
From the spots upon the moon's disk it has been ascertained, that the same hemisphere of that luminary is always directed towards the earth. Hence it follows that she turns round her axis once during every revolution round the earth.
Exact observations have ascertained that slight varieties take place respecting the appearances of the moon's disk. The spots are observed alternately to approach towards and recede from the edge of the moon. Those that are very near the edge appear and disappear alternately, making periodical oscillations, which are distinguished by the name of the libration of the moon. To form a precise idea of the nature of this libration, we must consider that the disk of the moon, seen from the centre of the earth, is terminated by the circumference of a great circle of the moon, perpendicular to a line drawn from the earth's centre to that of the moon. The lunar hemisphere is projected upon the plane of this circle turned towards the earth, and its appearances are due to the movements of rotation of that body relative to its radius vector. If the moon did not revolve round her axis, this radius vector would describe a great circle on the moon's surface, all the points of which would present themselves successively to us. But the moon, revolving in the same time that this radius vector describes the great circle, always keeps the same point of the circle nearly upon the radius, and of course the same hemisphere turned towards the earth. The inequalities of her motion produce the flight variations in her appearance: for the rotation Apparent of the moon does not partake sensibly of these irregularities. Hence it varies somewhat relatively to the radius vector, which accurately cuts successively different points of the surface. Of course the globe of the moon makes oscillations relatively to that radius corresponding to the inequalities of her motions, which alternately conceal from our view and discover to us some parts of her surface.
Farther: the axis of rotation of the moon is not exactly perpendicular to the plane of her orbit. If we suppose the position of this axis fixed, during a revolution of the moon it inclines more or less to the radius vector, so that the angle formed by these two lines is acute during one part of her revolution, and obtuse during another part of it. Hence the poles of rotation are alternately visible from the earth, and those parts of her surface that are near these poles.
Besides all this, the observer is not placed at the centre of the earth, but at its surface. It is the radius drawn from his eye to the centre of the moon, which determines the middle point of her visible hemisphere. But in consequence of the lunar parallax, it is obvious that this radius must cut the surface of the moon in points sensibly different according to the height of that luminary above the horizon. All these causes concur to produce the libration of the moon, a phenomenon which is merely optical, and not connected with her rotation, which relatively to us is perfectly equable, or at least if it be subjected to any irregularities, they are too small to be observed.
This is not the case with the variations in the plane Theory of the moon's equator. While endeavouring to determine its position by the lunar spots, Cassini was led to this remarkable conclusion, which includes the whole astronomical theory of the real libration of that luminary. Conceive a plane passing through the centre of the moon perpendicular to her axis of rotation, and of course coinciding with the plane of her equator; conceive a second plane, parallel to the ecliptic, to pass through the same centre; and also a third plane, which is the mean plane of the lunar orbit: these three planes have a common intersection; the second, placed between the two others, forms with the first an angle of 1°.503, and with the third an angle of 5°.14692; therefore the intersections of the lunar equator with the ecliptic coincide always with the mean nodes of the lunar orbit, and like them have a retrograde motion, which is completed in the period of 6793 3009 days. During that interval the two poles of the equator and lunar orbit describe small circles parallel to the ecliptic, enclosing between them the pole of the ecliptic, so that these three poles are constantly upon a great circle of the heavenly sphere.
CHAP. III. Of the Planets.
AMIDST the infinite variety of stars which occupy a place in the sphere of the heavens, and which occupy nearly the same relative position with respect to each other, there are eight which may be observed to move in a very complicated manner, but following certain precise laws, for they always commence the same motions again after every period. The motions of these Part II.
Apparent Motions of Objects of Astronomy. These planets are called the Heavenly Bodies.
1. Mercury. 2. Venus. 3. Mars. 4. Ceres. 5. Pallas. 6. Jupiter. 7. Saturn. 8. Herschel.
Mercury and Venus never separate from the sun farther than certain limits; the rest separate to all the possible angular distances. The movements of all these bodies are included in a zone of the heavenly sphere called the zodiac. This zone is divided into two equal parts by the ecliptic. Its breadth was formerly considered as only about 16°; but it must be much increased if the orbits of Ceres and Pallas, the two newly discovered planets, are to be comprehended in it. It will be proper to consider the motions and appearances of each of these planets. This will be the subject of the following sections.
SECT. I. Of Mercury.
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.
Mercury never goes to a greater distance from the sun than about 27° 15'; so that he is never longer in getting after the sun than an hour and 50 minutes; nor does he ever rise sooner than 1 hour and 50 minutes before that luminary. Very frequently, he goes so near the sun as to be lost altogether in his rays. When he begins to make his appearance in the evening after sunset, he can scarcely at first be distinguished in the rays of the twilight. But the planet disengages itself more and more, and is seen at a greater distance from the sun every successive evening; and having got to the distance of about 22° 15', it begins to return again. During this interval, the motion of Mercury referred to the stars is direct; but when it approaches within 18° of the sun it appears for some time stationary; and then its motion begins to be retrograde. The planet continues to approach the sun, and at last plunges into his rays in the evening, and disappears. Soon after, it may be perceived in the morning, before sunrise, separating farther and farther from the sun, his motion being retrograde, as before he disappeared. At the distance of 18° it becomes stationary, and assumes a direct motion, continuing, however, to separate till it comes to 22.5° of distance; then it returns again to the sun, plunges into his rays, and appears soon after in the evening, after sunset, to repeat the same career. The angular distance from the sun, which the planet reaches on both sides of that luminosity, varies from 16° to nearly 28°.
The duration of a complete oscillation, or the interval of time that elapses before the planet returns again to the point from which it set out, varies also from 100 to 130 days. The mean arc of his retrogradation is about 13° 15'; its mean duration 23 days. But the quantity differs greatly in different retrogradations. In general, the laws of the movements of Mercury are very complicated; he does not move exactly in the plane of the ecliptic; sometimes he deviates from it more than 5°.
Some considerable time must have elapsed before astronomers suspected that the stars which were seen approaching the sun in the evening and in the morning were one and the same. The circumstance, however, of the one never being seen at the same time with the other would gradually lead them to the right conclusion.
The apparent diameter of Mercury varies as well as that of the sun and moon, and this variation is obviously connected with his position relatively to the sun, and with the direction of his movement. The diameter is at its minimum when the planet plunges into the solar rays in the morning, or when it disengages itself from them: it is at its maximum when the planet plunges into the solar rays in the evening, or when it disengages itself from them in the evening; that is to say, when the planet passes the sun in its retrograde motion, its diameter is the greatest possible; when it passes the sun in its direct motion, it is the smallest possible;—and the mean length of the apparent diameter of Mercury is 11''.
Sometimes, when the planet disappears during its retrograde motion, that is to say, when it plunges into the sun's rays in the evening, it may be seen crossing the sun under the form of a black spot, which describes a chord along the disk of the sun. This black spot is recognised to be the planet by its position, its apparent diameter, and its retrograde motion. These transits of Mercury, as they are termed, are real annular eclipses of the sun: they demonstrate that the planet is an opaque body, and that it borrows its light from the sun. When examined by means of telescopes magnifying about 200 or 300 times, he appears equally luminous throughout his whole surface, without the least dark spot. But he exhibits 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. Like the moon, the crescent is always turned towards the sun. These different phases throw considerable light on the orbit of Mercury.
SECT. II. Of Venus.
VENUS, the most beautiful star in the heavens, known by the names of the morning and evening star, likewise keeps near the sun, though he 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 western; but always seems to attend him in the evening, or to give notice of his approach in the morning.
The planet Venus presents the same phenomena with Mercury; but her different phases are much more sensible, her oscillations wider, and of longer duration. Her greatest distance from the sun varies from 45° to nearly 48°, and the mean duration of a complete oscillation is 584 days.
Venus has been sometimes seen moving across the Her apparent sun's disk in the form of a round black spot, with an apparent diameter of about 59''. A few days after this has been observed, Venus is seen in the morning, west of the sun, in the form of a fine cresent, with the convexity turned toward the sun. She moves gradually westward with a retarded motion, and the cresent becomes more full. In about ten weeks she has moved 46° west of the sun, and is now a semicircle, and her diameter is 26''. She is now stationary. She then moves eastward with a motion gradually accelerated, and overtakes the sun about 9 1/2 months after having been seen on his disk. Some time after, she is seen in the evening, east of the sun, round, but very small. She moves eastward, and increases in diameter, but looses of her roundness, till she gets about 46° east of the sun, when she is again a semicircle. She now moves westwards, increasing in diameter, but becoming a cresent like the waning moon; and, at last, after a period of nearly 384 days, comes again into conjunction with the sun with an apparent diameter of 59''.
The mean arc of her retrogradation is about 16°, and its mean duration is 42 days. She does not move exactly in the plane of the ecliptic, but deviates from it several degrees. Like Mercury, she sometimes crosses the sun's disk. The duration of these transits, as observed from different parts of the earth's surface, are very different: this is owing to the parallax of Venus, in consequence of which different observers refer to different parts of the sun's disk, and see her describe different chords on that disk. In the transit which happened in 1769, the difference of its duration, as observed at Otaheite and at Wardhuys in Lapland, amounted to 23 m. 10 sec. This difference gives us the parallax of Venus, and of course her distance from the earth during a conjunction. The knowledge of this parallax enables us, by a method to be afterwards described, to ascertain that of the sun, and consequently to discover its distance from the earth.
The great variations of the apparent diameter of Venus demonstrates that her distance from the earth is exceedingly variable. It is largest when the planet passes over the surface of the sun. Her mean apparent diameter is 58''.
From the movement of certain spots upon the surface of Venus, it has been concluded that she revolves round her axis once in 24 hours; but this requires to be corrected by future observations. It is extremely difficult to perceive or examine these spots in our climate. The subject merits the attention of astronomers farther to the south, in more favourable circumstances. The following detail will show the uncertainty which has prevailed among astronomers respecting these spots.
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 Burratini 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 fame 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 spot 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 farther 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 further 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 kind of 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 axis 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 globe, 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 account of part of the diameter distant from the southern horn. When the sun was eight degrees fix minutes high, it seemed to be got beyond the centre, and was cut different 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 Part II.
Apparent that of each of them from the nearest angular point or Motions of horn of the planet. The weather being at that time clear, he observed for an hour and half 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 spots, 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. 19. to 25.
But though, from the appearances just now related, M. Cassini was of opinion that Venus revolved on her axis, he was by no means so positive in this matter as with regard to Mars and Jupiter. "The spots on these (says he) I could attentively observe for a whole night, when the planets were in opposition to the sun: 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, M. Cassini again observed Venus through a telescope, but could not then perceive any spots upon her surface; the reason of which Du 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 observation 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° the time 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, ABC, being situated as in fig. 26. 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 behind the Barbarini 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 conclusion of 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 their 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 Burratini, 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 of Venus, by which he was enabled to ascertain her revolution, Apparent volution on an axis, had also a view of her satellite or Motions of moon, of which he gives the following account.—the Heaven: "A. D. 1686, August 28th, at 15 minutes after four ly Bodies.
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. 28th, A. D. 1672, from 52 minutes after fix 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 consistence 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 suspended 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 Caffini 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 23 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 the Heavens 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 Montaigne 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, 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 phases 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. 27. represents the three observations of Mr Montaigne. V is the planet Venus; ZN the vertical. EC, a parallel to the ecliptic, making them 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 his 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 difficult crustled over with spots, or otherwise unfit to reflect to be seen, 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 Part II.
Apparent equal to that in our earth; in which case it must have Motions of considerable influence in changing the obliquity of the the Heavenly ecliptic, the latitudes and longitudes of stars, &c.
It is now known that this supposed satellite of Cassini was merely an optical deception.
In the Philosophical Transactions for 1761, Mr Hirt 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, 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 fix-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 waterish 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 shewed itself with a faint light during almost the whole time of emersion. 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. 28.: 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. 29. As more of the planetary disk went off that of the sun, however, that part of the crescent which was farthest from the sun grew fainter, and vanished, until at last only the horns could be seen, as in fig. 30. 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. 31.; 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. 32.: he saw the like appearance at going off, but not so distinct, fig. 33. Mr Chappe likewise took notice, that the part of Venus which was not upon the sun was visible during part of the time of ingresses and egresses; 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 Fernier, 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 glaies, 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, as 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, when the whole planet was upon the sun, they saw a theffering of light round it, distinct from the light of the sun; or whether they mean only the light which surrounded that part of Venus that was not upon the sun." 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. 34, 35.
Sect. III. Of Mars.
The two planets which we have just described, appear to accompany the sun like satellites, and their mean motion round the earth is the same with that luminary. The remaining planets go to all the possible angular distances from the sun. But their motions have obviously a connection with the sun's position.
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 Dr Herschel thinks this has been exaggerated by former astronomers. Mars appears to move from west to east round the Motions of earth. The mean duration of his fideal revolution is 686.979579 days. His motion is very unequal. When we begin to perceive this planet in the morning when he begins to separate from the sun, his motion is direct and the most rapid possible. This rapidity diminishes gradually, and the motion ceases altogether when the planet is about 137° distant from the sun; then his motion becomes retrograde, and increases in rapidity till he comes into opposition with the sun. It then gradually diminishes again, and becomes nothing when Mars approaches within 137° of the sun. Then the motion becomes direct after having been retrograde for 73 days, during which interval the planet described an arch of about 16°. Continuing to approach the sun, the planet at last is lost in the evening in the rays of that luminary. All these different phenomena are renewed after every opposition of Mars; but there are considerable differences both in the extent and duration of his retrogradations.
Mars does not move exactly in the plane of the ecliptic, but deviates from it several degrees. His apparent diameter varies exceedingly. His mean apparent diameter is 27", and it increases so much, that when the planet is in opposition, the apparent diameter is 81". Then the parallax of Mars becomes sensible, and about double that of the sun.
The disk of Mars changes its form relatively to its position with regard to the sun, and becomes oval. Its phases show that it derives its light from that luminary. The spots observed on its surface have informed astronomers that it moves round its axis from west to east in 1.02733 days, and its axis is inclined to the ecliptic at an angle of about 59.7°.
They were first observed in 1666 by Caffini at Bologna with a telescope of Campani about 16½ 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 Eustachio Divini; but they assigned to it a rotation in 13 hours only. This, however, was afterwards shown by Mr Caffini 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 Caffini had determined it to be. See 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 21° 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, as seen by Maraldi through a telescope of 34 feet long, is represented in fig. 37. There was then a 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. 38. 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 30, 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. 39, 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 in fig. 40. 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 Bright notice that a segment of his globe about the south pole spots about 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 Herschel, who has 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 Dr Her- with a view to determine the figure of the planet, the schel's ac- position of his axis, &c. A very particular account count of them is given in the 74th volume of the Philosophi- cal Transactions, but which our limits will not allow us to insert. Fig. 41. to 64. show the particular appear- ances of Mars, as viewed on the days there marked. The magnifying powers he used were sometimes as high as 932; and with this the south polar spot was found to be in diameter 41''. Fig. 65. shows the connection of the other figures marked 56, 57, 58, 59, 60, 61, 62, which complete the whole equatorial succession of spots on the disk of the planet. The centre of the circle marked 57 is placed on the circumference of the inner circle, by making its distance from the circle marked 59 answer to the interval of time between the two ob- servations, properly calculated and reduced to fideal measure. The same is done with regard to the circles marked 58, 59, &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 fi- Part II.
Apparent gures in the single representation, which bears the Motions of 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, Dr Herschel informs us, that the poles of the planets 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 sun, June 25th, 12h. 15m. of our time was about 9° 56' 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 fourth 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 10h. 54 minutes, perhaps a little farther.
"From the appearances of the fourth 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 92° 35' and 85° 70' 15'; because the change of the situation of the pole from left to right, which happened in the 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 discovered 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 Pices, 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 EC (fig. 66.) be a part of the ecliptic, OM part of the orbit of Mars, PEO 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 M. de la Lande, we put the node of the orbit of Mars for 1783 in 18° 17' 58', we have from the place of the node of the axis, that is, 118° 17' 47' to the place of the node of the orbit, an arch EN of 65° 11'. In the triangle NEO, right-angled 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 Pices."
Our author next proceeds to show how the seasons of the year 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 Martian seasons. Thus, let it be required to compute the declination of the sun on Mars, June 25, 1781, at midnight of our time. If \( \varphi, \gamma, \Pi, \omega, \&c. \) (fig. 67.) represent the ecliptic of Mars, and \( \varphi \omega \Delta \psi \) the ecliptic of our planet, A, a, b B the mutual intersection of the Martial and terrestrial ecliptics; then there is given the heliocentric longitude of Mars, \( \varphi = 93^{\circ} 10' 30'' \); then taking away six signs, and \( \Delta b \) or \( \varphi a = 1^{\circ} 17' 58'' \), there remains \( b m = 1^{\circ} 22' 32'' \). 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 \( BM = 1^{\circ} 22' 33'' \). And taking away \( B \gamma = 1^{\circ} 10' 29'' \), which is the complement to \( \psi B \) (or \( \omega A \), already shewn to be \( 1^{\circ} 28' 31'' \)), there will remain \( \varphi M = 0^{\circ} 21' 4'' \), the place of Mars in its own orbit; that is, on the time above mentioned, the sun's longitude on Mars will be 68° 21' 4''; and the obliquity of the Martial ecliptic, 28° 42', being also given, we find, by the usual method, the sun's declination 9° 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: twixt the all the superior planets, the distance of Mars from the earth and the sun is by far the nearest alike to that of the earth; nor will the length of the Martial year appear very different." Apparent different from what we enjoy, when compared to the Motions of surprising duration of the years of Jupiter, Saturn, the Heavenly Bodies.
White spots melt when alternately exposed to the sun, I may well be permitted to furnish, 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 reduction 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 twelve-month'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 admittance, when two familiar instances in Jupiter and 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 9 h. 50 m. 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 shown, on the 28th of the same month, to Mr Wilfon 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, 10 h. 52 m. the equatorial diameter of the Heavenly Bodies was 22'' 9'', 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 first of October, at 10 h. 50 m. the equatorial diameter measured 103 by the micrometer; and the polar 98; the value of the divisions in seconds and thirds not being well 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'' 35''." In a great number of succeeding observations, the same appearance occurred; but on account of the quick changes in the appearance of this planet, Dr 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-foot telescope. This would indicate an atmosphere of a very extraordinary size and density; but the following observations of Dr Herschel seem to show 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 218 they appear considerably unequal. The distance from Mars of the nearest, which is also the largest, with 227 measured 3' 26'' 20''." 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 ascribe to any other cause than the variable disposition of clouds and vapours floating in the atmosphere of the planet." Part II.
Sect. IV. Of Jupiter.
JUPITER is the brightest of all the planets except Venus. He moves from west to east in a period of 4332.602298 days, exhibiting irregularities similar to those of Mars. Before he comes into opposition, and when distant from the sun about 115°, his motion becomes retrograde, and increases in swiftness till he comes into opposition. The motion then becomes gradually slower, and becomes direct when the planet advances within 115° of the sun. The duration of the retrograde motion is about 121 days, and the arch of retrogradation described is about 10°. But there is a considerable difference both in the amount and in the duration of this retrograde motion.
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. 68—71; 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 number is very variable, as sometimes only one, and at others no fewer than eight, may be perceived. They are generally parallel to one another, but not always so; 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, and returned in a short time to the same place; from whence 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 9 h. 56 m." This principal, or ancient spot as it is called, is the largest, and of the longest continuation 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 that spot. Besides this ancient spot, Cassini, in the year 1669, 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 9 h. 51 m.; and two other spots that revolved in 9 h. 52\( \frac{1}{2} \) m. 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 scarcely be any difference of seasons in that planet. His rotation has been observed to be somewhat quicker in his aphelion than his perihelion. The axis of rotation is nearly perpendicular to the plane of the ecliptic, and the planet makes one revolution in 0.41377 day, or about 9 h. 35' and 37''. The changes in the appearance of these spots, and the difference in the time of their rotation, make it probable that they do not adhere to Jupiter, but are clouds transported by the winds with different velocities in an atmosphere subject to violent agitations.
The apparent diameter of this planet is a maximum during his opposition to the sun, it is then equal to about 46''; when in conjunction it is smaller, being only about 31'': his mean apparent diameter is equal to 36''.
Four little stars are observed around Jupiter, which is attended constantly accompany him. Their relative situation by four moons. is continually changing. They oscillate on both sides of the planet, and their relative rank is determined by the length of these oscillations. That one in which the oscillation is shortest is called the first satellite, and so on. These satellites are analogous to our moon, See fig. 18. and 186. They 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 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° 55'. The first of these satellites revolves at the distance of 5,697 of Jupiter's femidiameters, or 1' 51'' as measured by proper-instruments; its periodical time is 1 d. 18 h. 27' 34''. The next satellite revolves at the distance of 9,017 femidiameters, or 2' 56'', in 3 d. 13 h. 13' 43''; the third at the distance of 14,384 femidiameters, or 4' 42'', in 7 d. 3 h. 42' 36''; and the fourth at the distance of 25,266, or 8' 16'', in 16 d. 16 h. 32' 09''.
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 alternate- Apparent ly from their greatest distance on one side to the Motions of greatest distance on the other, sometimes in a straight the Heaven-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, flower 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 through 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 satellite, though generally the smallest, sometimes appears bigger than any of the rest: the third sometimes seems least, though usually the largest; nay, a satellite may be so covered with spots as to appear less than its shadow-passing over the disk of the primary, though we are certain that the shadow must be smaller than the body which casts it. To a spectator placed on the surface of Jupiter, each of these satellites would put on the phases of the moon; but as the distance of any of them from Jupiter is but small when compared with the distance of that planet from the sun, the satellites are therefore illuminated by the sun very nearly in the same manner with the primary itself; hence they appear to us always round, having constantly the greatest part of their enlightened half turned towards the earth: and indeed they are so small, that were they to put on the phases of the moon, these phases could scarce be discerned through the best telescopes.
When the satellites pass through their inferior semicircles, they may cast a shadow upon their primary, and thus cause an eclipse of the sun to his inhabitants if there are any; and in some situations this shadow 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 piter shadow of Jupiter: and this is actually the case with the first, second, and third of these bodies; but the fourth, by reason of the largeness of its orbit, passes Jupiter's moons 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 the satellites, by going behind or coming before their primary, are observable; and from the apposition to the conjunction, only the emergions, 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 its perihelion and quadrature with the sun. With regard to the third, when Jupiter is more than 46 degrees from conjunction with, or apposition to, the sun, both its immersions and immediately subsequent emergions 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 egrets are both visible.
Sect. V. Of Saturn.
Saturn is likewise a very conspicuous planet, though not so brilliant as Jupiter. The period of his sidereal revolution round the earth, is 10759.077213 days. He moves from west to east nearly in the plane of the ecliptic, and exhibits irregularities similar to those of Jupiter and Mars. He becomes retrograde both before and after his opposition, when at the distance of about 109° from the sun. His retrograde motion continues about 139 days, and during its continuance he describes Apparent Motions of the Heavenly Bodies.
Telescopic appearance of Saturn.
His ring first discovered by Huygens.
an arc of about 6°. His diameter is a maximum at his opposition, and his mean apparent diameter is 18''.
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 anse 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. 72.; 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. 73. and 74. 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 Whitston 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 fight so rare, as the opening, 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 Apparent next the body of the planet appears brightest; when Motions of the ring appears of an elliptical form, the parts about the ends of the largest axis are called the anse, as has been already mentioned. These, a little before and after the disappearing of the ring, are of unequal magnitude: the largest anse is longer visible before the planet's round phase, and appears again sooner, than the other. On the first of October 1774, the largest anse was on the east side, and on the 12th on the west side; but the disk of the planet, which makes it probable that revolution the ring has a rotation round an axis. Herschel has demonstrated, that it revolves in its own plane in 10 hours 32' 15.4''. The observations of this philosopher have added greatly to our knowledge of Saturn's ring. According to him there is one single, dark, considerably broad line, belt, or zone, which he has constantly found on the north side of the ring. As this dark belt is subject to no change whatever, it is probably owing to some permanent construction of the surface of the ring: this construction cannot be owing to the shadow of a chain of mountains, since it is visible all round on the ring; for there could be no shade at the ends of the ring: a similar argument will apply against the opinion of very extended caverns. It is pretty evident that this dark zone is contained between two concentric circles; for all the phenomena correspond with the projection of such a zone. The nature of the ring Dr Herschel thinks no less solid than that of Saturn itself, and it is observed to cast a strong shadow upon the planet. The light of the ring is also generally brighter than that of the planet; for the ring appears sufficiently bright when the telescope affords scarcely light enough for Saturn. The doctor concludes that the edge of the ring is not flat, but spherical or spheroidal. The dimensions of the ring, or of the two rings with the space between them, Dr Herschel gives as below:
<table> <tr> <th></th> <th>Miles.</th> </tr> <tr> <td>Inner diameter of smaller ring</td> <td>146345</td> </tr> <tr> <td>Outside diam. of ditto</td> <td>184393</td> </tr> <tr> <td>Inner diam. of larger ring</td> <td>190248</td> </tr> <tr> <td>Outside diam. of ditto</td> <td>204883</td> </tr> <tr> <td>Breadth of the inner ring</td> <td>20000</td> </tr> <tr> <td>Breadth of the outer ring</td> <td>7200</td> </tr> <tr> <td>Breadth of the vacant space, or dark zone</td> <td>2839</td> </tr> </table>
There have been various conjectures relative to the nature of this ring. Some persons have imagined that the diameter of the planet Saturn was once equal to the present diameter of the outer ring, and that it was hollow; the present body being contained within the former surface, in like manner as a kernel is contained within its shell: they suppose that, in consequence of some concussion, or other cause, the outer shell all fell down to the inner body, and left only the ring at the greater distance from the centre, as we now perceive it. This conjecture is in some measure corroborated by the consideration, that both the planet and its ring perform their rotations about the same common axis, and in very nearly the same time. But from the observations of Dr Herschel, he thus concludes: "It does not appear to me that there is sufficient ground for admitting the ring of Saturn to be of a very changeable nature, and I guess that its phenomena will hereafter be so fully explained, as to reconcile reconcile all observations. In the meanwhile we must withhold a final judgment of its construction, till we have more observations. Its division, however, into two very unequal parts, can admit of no doubt."
The diameters of Saturn are not equal: that which is perpendicular to the plane of his ring appears less by one-eleventh than the diameter situated in that plane. If we compare this form with that of Jupiter, we have reason to conclude that Saturn turns rapidly round his shorter axis, and that the ring moves in the plane of his equator. Herschel has confirmed this opinion by actual observation. He has ascertained the duration of a revolution of Saturn round his axis to amount to 0.428 day. Huygens observed five belts upon this planet nearly parallel to the equator.
Saturn is still better attended than Jupiter (see fig. 18. and 186.) ; having, besides the ring above-mentioned, no fewer than seven 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' 57"; the second, at 2.686 semidiameters of the ring, and 6.268 of Saturn, revolves in 2 d. 17 h. 41' 22"; the third, at the distance of 8.754 semidiameters of Saturn, and 3.752 of the ring, in 4 d. 12 h. 25' 12"; 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' 12"; while the fifth, placed at the vast distance of 39.154 semidiameters of Saturn, or 25.348 of his ring, does not perform its revolution in less than 79 d. 7 h. 47' 00". 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 be 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 splendour.—"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 towards us. In the year 1705, 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 of the primary planets, are not permanent, but subject to change."
The two other satellites were discovered by Dr Herschel in 1787 and 1788. They are nearer to Saturn than any of the other five. But in order to prevent confusion, they have been called the 6th and 7th satellites. The fifth satellite has been observed by Dr Herschel to turn once round its axis, exactly in the time in which it revolves round Saturn. In this respect it resembles our moon.
Sect. VI. Of Herschel.
The planets hitherto described have been known from the remotest antiquity; but the planet Herschel, called also the Georgium Sidus, and Uranus, escaped the attention of the ancient astronomers. Flamsteed, Mayer, and Le Monnier had observed it as a small star; but in 1781 Dr Herschel discovered its motion, and ascertained it to be a planet. Like Mars, Jupiter, and Saturn, it moves from west to east round the sun. The duration of its sidereal revolution is 30689 days. Its motion, which is nearly in the plane of the ecliptic, begins to be retrograde before and after the opposition, when the planet is 103.5° from the sun; its retrograde motion continues for about 151 days; and the arc of retrogradation amounts to 3.6°. If we judge of the distance of this planet by the slowness of its motions, it ought to be at the very confines of the planetary system.
The apparent magnitude of this planet is so small that it can seldom be seen with the naked eye. It is, however, accompanied by fix satellites: two of them, which were discovered by Dr Herschel in 1787, revolve about that planet in periods of 8 d. 17 h. 1 m. 19. sec. and 13 d. 11 h. 5 m. 15 sec. respectively, the angular distances from the primary being 33" and 44.5"; their orbits are nearly perpendicular to the plane of the ecliptic. The history of the discovery of the other four, with such elements as could then be ascertained, are given in the Philosophical Transactions for 1798, Part I. The precise periods of these additional satellites cannot be ascertained without a greater number of observations than had been made when Dr Herschel sent the account of their discovery to the Royal Society; but he gave the following estimates as the most probable which could be formed by means of the data then determined. Admitting the distance of the interior satellite to be 25".5, its periodical revolution will be 5 d. 21 h. 25 m. If the intermediate satellite be placed at an equal distance between the two old satellites, or at 38".57, its period will be 10 d. 23 h. 4 m. The nearest exterior satellite is about double the distance of the farthest old one; its periodical time will therefore be about 38 d. 1 h. 49 m. The most distant satellite is full four times as far from the planet as the old second satellite; it will therefore take at least 107 d. 16 h. 40 m. to complete one revolution. All these satellites perform their revolutions in their orbits contrary to the order of the signs; that is, their real motion is retrograde. Part II.
SECT. VII. Of Ceres and Pallas.
These two planets, lately discovered by Piazzi and Olbers, two foreign astronomers, ought to have followed Mars in the order of description, as their orbits are placed between those of Mars and Jupiter; but as they have been observed only for a very short time, we judged it more proper to reserve the account of them till we came to the words CERES and PALLAS, when the elements of their orbits will in all probability be determined with more precision than at present. They are invisible to the naked eye; and Dr Herschel has ascertained that their size is extremely small. For that reason, together with the great obliquity of their orbits, he has proposed to distinguish them from the planets, and to call them asteroids.
CHAP. IV. Of the Comets.
The planets are not the only moving bodies visible in the heavens. 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 recess 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 splendour, and at last totally disappear. Their apparent magnitude is very different; sometimes they appear only of the brightness 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 splendour, but appeared with a pale and dim light, and had a dismal aspect. These bodies will also sometimes lose their splendour 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.
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 humid 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.—February 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 rounds a little impaired, and the edges lacerated.—March 28th. Its matter much dispersed; and no distinct nucleus at all appearing.
Wiegelius, 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 account of 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 cluster 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, na of their tails. Apparent and shortens as it recedes from, that luminary. If the Motions of 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. 75.; 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. 76. and 77. 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 through the eye of the spectator, the tail appears straight, as in fig. 78, 79.
"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 the 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. 77. 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. 13. 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. 23d, this comet appeared exceedingly distinct and bright, and the diameter of its nucleus nearly equal to that of Jupiter. Its tail extended above 26 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. February 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 shewed 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. 76. is a view of this comet, taken 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 met 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."
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 Part II.
Apparent his own hand from the views he had of them through Motions of a very long and excellent telescope. In these we find the Heaven- 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.
CHAP. V. Of the Fixed Stars.
The parallax of the stars is insensible. When viewed through the best telescopes, they 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 eyelids 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 smallness of their apparent diameter is proved by the fuddleness with which they disappear on their occultations by the moon. The time which they take does not amount to one second, which shows their apparent diameter not to exceed 4''. The vivacity of their light, compared with their small diameter, leads us to suppose them at a much greater distance than the planets, and to consider them as luminous bodies like our sun, instead of borrowing their light from that luminary like the planets.
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 naked 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 telescopic 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, Apparent or asterisms, 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 γ in the constellation of the Ram be mentioned, every astronomer knows as well what star it meant as if it were pointed out to him in the heavens. See fig. 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 parts. 1. The zodiac (ζωδιακός), from ζώον, zōon, the "an animal," because most of the constellations in it, which are 12 in number, have the names of animals: Fig. 26, 29. As Aries the ram, Taurus the bull, Gemini the twins, Cancer the crab, Leo the lion, Virgo the virgin, 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 fourth side, containing 15.
The ancients divided the zodiac into the above 12 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 arose, and continuing till it rose the next following night. The water falling down into the receiver they divided into 12 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 in one night, yet in many they observed the rising of 12 stars or points, by which they divided the zodiac into 12 parts.
The names of the constellations, and the number of stars observed in each of them by different astronomers, are as follows. <table> <tr> <th></th> <th></th> <th>Ptolemy.</th> <th>Tycho.</th> <th>Hevelius.</th> <th>Flamsteed.</th> </tr> <tr><td>Ursa minor</td><td>The Little Bear</td><td>8</td><td>7</td><td>12</td><td>24</td></tr> <tr><td>Ursa major</td><td>The Great Bear</td><td>35</td><td>29</td><td>73</td><td>87</td></tr> <tr><td>Draco</td><td>The Dragon</td><td>31</td><td>32</td><td>40</td><td>80</td></tr> <tr><td>Cepheus</td><td>Cepheus</td><td>13</td><td>4</td><td>51</td><td>35</td></tr> <tr><td>Bootes, Arctophila</td><td></td><td>23</td><td>18</td><td>52</td><td>54</td></tr> <tr><td>Corona Borealis</td><td>The Northern Crown</td><td>8</td><td>8</td><td>8</td><td>21</td></tr> <tr><td>Hercules, Engonasin</td><td>Hercules kneeling</td><td>29</td><td>28</td><td>45</td><td>113</td></tr> <tr><td>Lyra</td><td>The Harp</td><td>10</td><td>11</td><td>17</td><td>21</td></tr> <tr><td>Cygnus, Gallina</td><td>The Swan</td><td>10</td><td>18</td><td>47</td><td>81</td></tr> <tr><td>Cassiopeia</td><td>The Lady in her chair</td><td>13</td><td>26</td><td>37</td><td>55</td></tr> <tr><td>Perseus</td><td>Perseus</td><td>29</td><td>29</td><td>46</td><td>59</td></tr> <tr><td>Auriga</td><td>The Waggoner</td><td>14</td><td>9</td><td>40</td><td>66</td></tr> <tr><td>Serpentarius, Ophiuchus</td><td>Serpentarius</td><td>29</td><td>15</td><td>40</td><td>74</td></tr> <tr><td>Serpens</td><td>The Serpent</td><td>18</td><td>13</td><td>22</td><td>64</td></tr> <tr><td>Sagitta</td><td>The Arrow</td><td>5</td><td>5</td><td>5</td><td>18</td></tr> <tr><td>Aquila, Vultur</td><td>The Eagle</td><td>15</td><td>12</td><td>23</td><td>71</td></tr> <tr><td>Antinous</td><td>Antinous</td><td>3</td><td>3</td><td>19</td><td></td></tr> <tr><td>Delphinus</td><td>The Dolphin</td><td>10</td><td>10</td><td>14</td><td>18</td></tr> <tr><td>Equus, Equi sectio</td><td>The Horse's Head</td><td>4</td><td>4</td><td>6</td><td>10</td></tr> <tr><td>Pegasus, Equus</td><td>The Flying Horse</td><td>20</td><td>19</td><td>38</td><td>89</td></tr> <tr><td>Andromeda</td><td>Andromeda</td><td>23</td><td>23</td><td>47</td><td>66</td></tr> <tr><td>Triangulum</td><td>The Triangle</td><td>4</td><td>4</td><td>12</td><td>16</td></tr> <tr><td>Aries</td><td>The Ram</td><td>18</td><td>21</td><td>27</td><td>66</td></tr> <tr><td>Taurus</td><td>The Bull</td><td>44</td><td>43</td><td>51</td><td>141</td></tr> <tr><td>Gemini</td><td>The Twins</td><td>25</td><td>25</td><td>38</td><td>85</td></tr> <tr><td>Cancer</td><td>The Crab</td><td>23</td><td>15</td><td>29</td><td>83</td></tr> <tr><td>Leo</td><td>The Lion</td><td>30</td><td>30</td><td>49</td><td>95</td></tr> <tr><td>Coma Berenices</td><td>Berenice's Hair</td><td>35</td><td>14</td><td>21</td><td>43</td></tr> <tr><td>Virgo</td><td>The Virgin</td><td>32</td><td>33</td><td>50</td><td>110</td></tr> <tr><td>Libra, Chelae</td><td>The Scales</td><td>17</td><td>10</td><td>20</td><td>51</td></tr> <tr><td>Scorpio</td><td>The Scorpion</td><td>24</td><td>10</td><td>20</td><td>44</td></tr> <tr><td>Sagittarius</td><td>The Archer</td><td>31</td><td>14</td><td>22</td><td>69</td></tr> <tr><td>Capricornus</td><td>The Goat</td><td>28</td><td>28</td><td>29</td><td>51</td></tr> <tr><td>Aquarius</td><td>The Water-bearer</td><td>45</td><td>41</td><td>47</td><td>108</td></tr> <tr><td>Pisces</td><td>The Fishes</td><td>38</td><td>36</td><td>39</td><td>113</td></tr> <tr><td>Cetus</td><td>The Whale</td><td>22</td><td>21</td><td>45</td><td>97</td></tr> <tr><td>Orion</td><td>Orion</td><td>38</td><td>42</td><td>62</td><td>78</td></tr> <tr><td>Eridanus, Fluvius</td><td>Eridanus, the River</td><td>34</td><td>10</td><td>27</td><td>84</td></tr> <tr><td>Lepus</td><td>The Hare</td><td>12</td><td>13</td><td>16</td><td>19</td></tr> <tr><td>Canis major</td><td>The Great Dog</td><td>29</td><td>13</td><td>21</td><td>31</td></tr> <tr><td>Canis minor</td><td>The Little Dog</td><td>2</td><td>2</td><td>13</td><td>14</td></tr> <tr><td>Argo Navis</td><td>The Ship</td><td>45</td><td>3</td><td>4</td><td>64</td></tr> <tr><td>Hydra</td><td>The Hydra</td><td>27</td><td>19</td><td>31</td><td>60</td></tr> <tr><td>Crater</td><td>The Cup</td><td>7</td><td>3</td><td>10</td><td>31</td></tr> <tr><td>Corvus</td><td>The Crow</td><td>7</td><td>4</td><td>-</td><td>9</td></tr> <tr><td>Centaurus</td><td>The Centaur</td><td>37</td><td>-</td><td>-</td><td>35</td></tr> <tr><td>Lupus</td><td>The Wolf</td><td>19</td><td>-</td><td>-</td><td>24</td></tr> <tr><td>Ara</td><td>The Altar</td><td>7</td><td>-</td><td>-</td><td>9</td></tr> <tr><td>Corona Australis</td><td>The Southern Crown</td><td>13</td><td>-</td><td>-</td><td>12</td></tr> <tr><td>Piscis Australis</td><td>The Southern Fish</td><td>18</td><td>-</td><td>-</td><td>24</td></tr> </table>
The New Southern Constellations.
<table> <tr> <th>Columba Noachi</th><th>Noah's Dove</th><th>10</th><th>Apis, Musca</th><th>The Bee or Fly</th><th>4</th> </tr> <tr> <th>Robur Carolinum</th><th>The Royal Oak</th><th>12</th><th>Chamaeleon</th><th>The Chameleon</th><th>10</th> </tr> <tr> <th>Grus</th><th>The Crane</th><th>13</th><th>Triangulum Australe</th><th>The South Triangle</th><th>5</th> </tr> <tr> <th>Phoenix</th><th>The Phenix</th><th>13</th><th>Piscis volans, Paffer</th><th>The Flying Fish</th><th>8</th> </tr> <tr> <th>Indus</th><th>The Indian</th><th>12</th><th>Dorado, Xiphias</th><th>The Sword Fish</th><th>6</th> </tr> <tr> <th>Pavo</th><th>The Peacock</th><th>14</th><th>Toucan</th><th>The American Goose</th><th>9</th> </tr> <tr> <th>Apus, Avis Indica</th><th>The Bird of Paradise</th><th>11</th><th>Hydrus</th><th>The Water Snake</th><th>10</th> </tr> </table> Part II.
Hevelius's Constellations made out of the unformed Stars.
<table> <tr> <th>Constellation</th> <th>Star</th> <th>Right Ascension</th> <th>Declination</th> </tr> <tr> <td>Lynx</td> <td>The Lynx</td> <td>19</td> <td>44</td> </tr> <tr> <td>Leo minor</td> <td>The Little Lion</td> <td>53</td> <td></td> </tr> <tr> <td>Asterion & Chara</td> <td>The Greyhounds</td> <td>23</td> <td>25</td> </tr> <tr> <td>Cerberus</td> <td>Cerberus</td> <td>4</td> <td></td> </tr> <tr> <td>Vulpesula & Anser</td> <td>The Fox and Goose</td> <td>27</td> <td>35</td> </tr> <tr> <td>Scutum Sobielki</td> <td>Sobielki's Shield</td> <td>7</td> <td></td> </tr> <tr> <td>Lacerta</td> <td>The Lizard</td> <td>10</td> <td>16</td> </tr> <tr> <td>Camelopardalus</td> <td>The Camelopard</td> <td>32</td> <td>58</td> </tr> <tr> <td>Monoceros</td> <td>The Unicorn</td> <td>19</td> <td>31</td> </tr> <tr> <td>Sextans</td> <td>The Sextant</td> <td>11</td> <td>41</td> </tr> </table>
Hevel. Flam.
Several flares 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 it 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 cotemporary with him: which put him on the fame 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 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 splendour 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 0° 9' 17" a 1ma * 90is, with 53° 45' north latitude.
"Such another star was seen and observed by the scholars of Kepler, to begin to appear on Sep. 30. β. vet. 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 footstepsthereof in January 1605. This was near the ecliptic, following the right leg of Serpentinus; and by the observations of Kepler and others, was in 7° 20' 00" a 1ma * 90, 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, Apparent seen by David Fabricius on the third of Augus β. vet. Motions of as bright as a star of the 3d magnitude, which has been since found to appear and disappear periodically; its period being precisely enough seven revolutions in fix 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 fixt feet tube. This was singular in its kind, till that in Collo Cygni was discovered. It precedes the first star of Aries 1° 40', with 15° 57' south latitude.
"Another new star was first discovered by William Jansenius in the year 1600, in pedlore, 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 rose to the third magnitude; though soon after it decayed by degrees to the fifth or sixth magnitude, and at this day is to be seen as such in 9s 18° 38' a 1ma * 90, with 55° 29' north latitude.
"A fifth new star was first seen by Hevelius in the year 1670, on July 15, β. vet. 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 fought for its return; its place is 9s 3° 17' a 1ma * 90, and has lat. north 47° 28'.
"The sixth and last is that discovered by Mr G. Kirch in the year 1686, and its period determined to be of 404½ 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 fixt feet tube, that, bearing a very great aperture, discovers most minute stars. And on June 15. 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 till the month of September. After that, it again died away by degrees: and on the 8th of December, at night, was scarcely discernible by the tube; and, as near as could be guessed, equal to what it was at its first appearance on June 25th: 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 10th of September."
Concerning the changes which happen among the fixed stars, Mr Montanere, professor of mathematics at nere's count of Royal Society, dated April 30th 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 β and γ. 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 10th of April, there was not the least glimpse of them to be seen; and yet the Apparent 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 a hundred, though none of them are so great as those I have showed."
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 Transactions, Mr Edward Pigot gives a dissertation on the 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; 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 magnitude; including the new one which appeared in Cassiopeia in 1572, and that in Serpentarius in 1624: 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; but he never found it exceed the sixth magnitude; though Mr Goodricke had observed it on the 9th of August to be of the second magnitude, and on the 3d 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 2d. 20h. 49m. and 3 sec. 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 420th 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 419th of the same catalogue. In February that year, it was of the same brightness with the Heavenly Bodies 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 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 so, 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 sixth 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 fix 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 α Antinoi, whose variation and period he discovered in 1785. From his corrected observations, he concludes that it continues at its greatest brightness 49 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 Caffini 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. Part II.
Apparent "Perhaps (fays he) its period is irregular; to determinations of mine which several intervals of 15 years ought to be taken; and I am much inclined to believe that it will be found only 396 days 21 hours." The particulars relating to this star are, 1. When at its full brightness it undergoes no perceptible change for a fortnight. 2. It is about three months and a 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 fix months. 3. It does not always attain the same degree of lustre, being sometimes of the fifth and sometimes of the seventh magnitude.
In 1600, G. Jansonus discovered a variable star in the breast of the Swan, which was afterwards observed by different astronomers, and supposed to have a period of about 10 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 sixth, magnitude. "I am entirely ignorant (fays 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 δ 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 Cassiopeiae was missing in 1782, nor could Mr Pigot find it in 1783 and 1784.
2. ξ or 46 Andromedae, said to be variable, but the evidence is not convincing to Mr Pigot.
3. Flamsteed's 50, 52, τ Andromedae, and Hevelius's 41 Andromedae. 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 (fays 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. σ, or the 17th Eridani of Ptolemy and Ulug Beigh. Flamstead 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 new (fays he) is evident, since it is Ulug Beigh's 26th and Tycho's 43d.
7. A star about 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 & β Geminorum. "If any of these stars (fays our author) have changed in brightness, it is probably the β. In 1783, 1784, and 1785, the β was undoubtedly brighter than α."
10. § Leonis. According to Montanari, this star was hardly visible in 1693. In 1783, 1784, and 1785, it was of the fifth magnitude. By Tycho, Flamstead, Mayer, Bradley, &c. it is marked of the fourth.
11. ψ 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. 25th 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 1790, 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. δ Ursae Majoris. This star is supposed 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 Flamstead, 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 sixth magnitude without any perceptible change.
16. Bayer's star of the sixth magnitude 1° south of γ Virginis. "This star (fays 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 Ricciolus of the sixth magnitude, could not be seen by Maraldi in 1790; nor was it of the ninth magnitude, if at all visible in 1785.
18. The 91 and 92 Virginis. In 1685, 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 Dr Herlchel, that this star is variable. Bradley, Flamstead, &c. mark it of the second magnitude, but in 1786 it was only a bright fourth. It was frequently examined. 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. Marmaldi 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-glas in 1785.
25. A star marked by Bayer near Ursae majoris. This star could not be seen by Cassini; nor was Mr Pigot able to discover it with a night-glas in 1782.
26. The e, 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 southern 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. o Sagittarii. Dr 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. 8 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 Andromedae, and o Andromedae. 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 o Andromedae 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 Apparent Motions of the Heavenly Bodies. it in 1782; but is persuaded that it is the same with Motions of Flamsteed's 56th, marked f by Bayer, from which it is only a degree and a half distant. The 53d of Flamsteed, 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. Dr 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 particularised 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 to err a whole magnitude in the brightness of a star.
As these changes to which the fixed stars are liable Wollaston's do not seem to be subject to any certain rule, Mr Wollaston has given an easy method of observing whether variations they do take place in any part of the heavens or not, among the 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-glas, but of Dollond's construction, which magnifies about fix 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. A horizontal motion, except in the meridian, would be apt to mislead the judgement. 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 Part II.
Apparent known star, which is to be brought into the centre or Motions of common intersection of all the wires. The relative positions of such other stars as appear within the field are to be judged by the eye; whether at \( \frac{1}{2} \), \( \frac{1}{4} \), or \( \frac{1}{8} \) from the centre towards the circumference, or vice versa; 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 though 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. 80. shows part of the Corona Borealis delineated in this manner, and which was afterwards fully taken down by making the stars α, β, γ, δ, ε, ζ, θ, ι, κ, π, ξ, η, and ρ, 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 in fig. 81.
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, in figures 82 and 83.
A very remarkable appearance in the heavens is that called the galaxy, or milky-way. This is a broad circle, sometimes double, but for the most part single, surrounding the whole celestial concave. We perceive also in Apparent different parts of the heavens small white spots, which appear to be of the same nature with the milky-way. These spots are called nebulae.
We shall subjoin in this place, for the entertainment of the reader, the theories of Mr Michell and Dr Herschel, concerning the nature and position of the fixed stars.
"The very great number of stars (says Mr Mi. Mi. 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, of the fixed 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 other) may some time or other be discovered." Having then shewn 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 cases light 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 that the body emits light to be attracted by the same force in proportion it 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 these semidiameters inversely.
After proceeding in his calculations, in order to find Comparative the diameter and distance of any star, he proceeds thus: "According to Mr Bouguer the brightness of the fun and the fun exceeds that of a wax-candle in no less a pro-portion 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 still, I apprehend, appear as bright and luminous as the star Sirius, if removed to 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 app Apparent parent 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 suppose, 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 respect to the sun, we know that his whole 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 other times 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, Apparent I apprehend, as bright as Sirius, as I have observed Motions of above; whereas it is hardly to be expected, with any theHeavenly Bodies. 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."
Dr Herschel, improving on Mr Mitchell's idea of the fixed stars being collected into groups, and af- firmed by his own observations with the extraordinary cerning the telecopic powers already mentioned, has suggested a construction concerning the construction 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 Dr Herschel is of a very different opinion. "Hither-to (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 fitted, the idea of an expanded firmament of three dimensions; but the observations upon which I am now going to enter, will 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 flora 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, His obser- Dr Herschel first began to survey the Via Laetca, and found that it completely resolved the whitish appear- the Via. Lactea. ance into stars, which the telescopes he formerly used had not light enough to do. The portion he first ob- served 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
(a) Dr 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.
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 not 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 within 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 nebula (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 flary 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 latitude 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 nebula. One of these nebulous 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 nebulae, 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 nebulae variously arranged; large ones with small seeming attendants; narrow, but much extended lucid nebulae 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 nebulous atmosphere: a different fort again contain a nebulosity of the milky kind, like that wonderful inexplicable phenomenon about θ 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 the milky-way, is that in which the sun is placed, though perhaps not in the very centre of its thickness. We gather this from the appearance of the galaxy, which seems to compass the whole heavens, as it certainly must do if the sun is within the fame. 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 crowded according to the distance of the planes or number of stars contained in the thicknes or sides of the stratum.
Thus in fig. 83. an eye at S'within the stratum ab, will see the stars in the direction of its length ab, or Motions of theHeavenly Bodies. 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 apparent-fars 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 pg 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 inflamed 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 fame. 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 Of the sun's 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 method of knowledge of the sun's place in the sidereal stratum, gauging the heavens. Apparent one of which I have already begun to put in practice: Motions of I call it gauging the heavens; or the star-gauge. It consists in repeatedly taking the number of stars in ten 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 18.6 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 \( a \) fig. 84. be the stratum, and suppose the small circle \( g h l k \) to represent the space into which, by the light and power of a given telescope, we are enabled to penetrate, and let GHLK 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 revolving in several directions about an assumed point, and cut off by the bounds of the stratum. Thus, in fig. 85. let S be the place of an observer: \( S r r r, S r r r \), lines in the plane \( r S r, S r r \), 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, 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 nebulae 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 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. The direction of some of the capital strata or their branches. The well-known nebula of Cancer, visible Direction to the naked eye, is probably one belonging to a certain stratum, in which I suppose it to be so placed as to lie nearest to us. This stratum I shall call that of Cancer. It runs from \( \alpha \) 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 \( \gamma, \delta, \eta, \theta \). 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 rectangles 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 nebulae 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, Dr 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 the heavens, find it composed of shining stars, whole 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 Part II.
Apparent such a manner as to be almost equally distributed through Motions of 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 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. prop. 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 change 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 cluster. 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 the former cases, their will be great cavities or vacancies formed 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 for the mutual and combined attractions of the heavenly bodies to exert themselves in.
"From this theoretical view of the heavens, which 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 the 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 enclosed 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 above 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 nebulae 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 these 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 milkiness 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 nebulous 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 nebulae, 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 nebulae; 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, though in no less a space than what appeared before to 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 nebulae 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 form 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 enclosed in one of the nebulae, 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."
Dr 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 frustula 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, fix 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 \( na + \frac{n-1}{2} d + n \cdot \frac{n-1}{2} \frac{n-2}{3} d'' \), &c. 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 \( n^2 \). 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 \( B = \frac{r}{\tan \frac{1}{4} \text{ field}} \).
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; Apparent and the stars in each cone being in the ratio of the solidity, as being equally scattered, we have \( n = \sqrt{\frac{B^2 S}{B^2 S'}} \); 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 Dr Herschel prefers the former.
From the data now laid down, Dr Herschel next endeavours to prove that the earth is 'the planet of our fideal star belonging to a compound nebula of the third form,' saying a nebula.
"I shall now (says he) proceed to show, that the stupendous fideal 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 fideal 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 know 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 measures pased no less than 116,000 stars through the field of the 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
\[ n = \sqrt{\frac{B^2 S}{B^2 S'}} = 498; \]
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 cluster 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 Part II.
Apparent binations 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 nebulous appearance must have come on, I need not fear to have overrated 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 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 confident 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 authorize; 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 be no less than 4774; for when the visual ray \( r \) is given, the number of stars \( S \) will be \( = \frac{n^3}{E^3} \); where \( n = r + \frac{1}{r} \); 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 rectangular squares, approach nearer to each other than Motions of 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 arguments drawn from analogy. Among the great number of nebulae which I have now already seen, his doctrine amounts 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 interfaces 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 those 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 belong undoubtedly to our own system."
Having thus determined that the visible system of nature, by us called the universe, consisting of all the celestial 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. Dr 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 distances 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 seems 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) 88°. The section is one which makes an angle of 35° with our equator, crossing it in 124° and 324°. A celestial globe, adjusted to the latitude of 55° north, and having σ 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."
Dr 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 to be one that has fewer marks of antiquity than any of the rest. To explain this idea perhaps more clearly, we should recollect, that the condensation of clusters of stars 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 fidecal stratum. Motions of There are, moreover, many places in it in which, if we theHeavenly Bodies. may judge from appearances, there is the greatest rea- son to believe that the stars are drawing towards sec- 242 ondary centres, and will in time separate into clusters, fo of the de- as to occasion many subdivisions. Hence we may sur-mise, 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 sub- divisions which happened to it in length of time, occa- sioned 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, fo 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 fides 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 ne- bulae 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 80th Nebuleuse fans Etoiles of the Connaissance des Temps, which is one of the richest and most compressed clus- ters of small stars I remember to have seen, is situated just on the west border of it, and would almost autho- rize 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 more- over a small miniature cluster, or easily resolvable nebu- la, 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 fides; that is, towards Leo, Virgo, and Coma Berenices on one hand, and towards Cetus on the other; whereas the ground of the heavens becomes troubled as we ap- proach 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 Dr Herfchel, 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 Dr Herfchel's papers, the words condensations, 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. Dr Herfchel, 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, might 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, the stars of which I can see very distinctly; and on 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 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 underrated by supposing it only 600 times the distance of that star.
"Some of these round nebulae (says Dr Herfchel) 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 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."
Dr Herfchel 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 irrevolvable, 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-six or forty-eight thousand years in arriving from them to us. It would be worth while then to inquire, whether attraction is a virtue propagated in time or not; or whether it moves quicker or slower than light?
In the course of Dr Herfchel'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 parts of the great whole from approaching to each other. Apparent 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 admition 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 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 situation better than any one of the stars belonging to our system, on account of their being probably at a very great distance."
As the fixed stars constantly keep nearly the same situation relative to each other, astronomers have agreed to refer to them, as to so many fixed points, the different motions of the other heavenly bodies. Hence the reason of dividing them into constellations. But it was necessary besides, for the sake of perfect precision, to mark exactly the relative situation of every star in the celestial sphere. This is accomplished in the following manner.
"A great circle is supposed to pass through the two poles, and through the centre of every star. This circle is called a circle of declination. The arc of this circle included between the star and the equator measures the declination of the star. The declination of a star then is its perpendicular distance from the equator. It is north or south, according as the star is situated on the north or south side of the equator. All the stars situated in the same parallel of the equator have of course the same declination.
The declination then marks the situation of a star north or south from the equator. Precision requires still another circle from which their distance east or west may be marked, in order to give the real place. The circle of declination which passes through that point of the equator, called the vernal equinoctial point, has been chosen for that purpose. The distance of the circle of declination of a given star from that point measured on the equator, or the arc of the equator included between the vernal equinox and the circle of declination of the star, is called its right ascension. If we know the declination and the right ascension of a star, we know its precise situation in the heavens.
The declination of any star may be easily found by observing the following rule: Take the meridian altitude of the star, at any place where the latitude is known, the complement of this is the zenith distance, and is called north or south, as the star is north or south at the time of observation. Then, 1. When the latitude of the place and zenith distance of the star are of different kinds, namely, one north and the other south, their difference will be the declination; and it is of the same kind with the latitude, when that is the greatest of the two, otherwise it is of the contrary kind. 2. If the latitude and the zenith distance are of the same kind, i.e. both north or both south, their sum is the declination; and it is of the same kind with the latitude.
To prove the truth of this rule, turn to fig. 86. where Z is the zenith of the place, EQ the equinoctial, and EZ the latitude. 1. Let r represent the place of a star on the meridian, and Zr the zenith distance, the latitude being greater: then Er (the declination) will be equal to EZ—Zr (the zenith distance); again, let c be the place of a star in the meridian, when the zenith distance exceeds the latitude; then Ec (the declination) = Zc (the zenith distance)—EZ (the latitude). And it is manifest, that in the former instance Z and r are on the same side of the equinoctial; and that in the latter case Z and c are on contrary sides. 2dly, Let y be the place of a star on the meridian, having its zenith distance Zy of the same kind with EZ the latitude of the place: then Ey (the declination) = EZ + Zy; and the declination is of the same kind as the latitude, because Z and y are on the same side of the equinoctial. Q. E. D.
For an example, suppose that in north latitude 52° 15', the meridian altitude of a star is 51° 28' on the south; then 38° 32' the zenith distance, being taken from 52° 15' the latitude, leaves 13° 43' for declination of the star north.
Having, by means like the above, found the declination of a star, it becomes requisite, in the next place, to know the right ascension, as its situation with regard to the equator will then be known. Now the right ascension being estimated from the point where the equator and ecliptic intersect each other in the spring, a point which is marked out by nothing that comes under the cognizance of our senses; some phenomenon, therefore, must be chosen, whose right ascension is either given, or may be readily known at any time that the right ascensions of other objects may be discovered by comparison with it. For this purpose nothing appears Part II.
Apparent so proper as the sun; because its motion is the most simple, and its right ascension quickly found.
For if, in fig. 87, we have given QS the declination of the fun (which may be easily taken every day at noon by observation), and the angle SEQ the obliquity of the ecliptic—i.e. one leg of a right-angled spherical triangle, and its opposite angle, to find the adjacent leg EQ, the right ascension—it may be done by this proportion; as the tangent of the obliquity of the ecliptic : the tangent of the declination :: radius : the fine of the right ascension reckoned from the nearer equinoctial point.
For example: suppose on the 13th of February the sun's fourth declination is found to be 13° 24', and the obliquity of the ecliptic is 23° 28'; we shall thus find the sun's right ascension:
<table> <tr> <th>As tangt. 23° 28'</th> <th>9.6376106</th> </tr> <tr> <th>To tangt. 13° 24'</th> <th>9.3770930</th> </tr> <tr> <th>So is radius</th> <th>10.0000000</th> </tr> </table>
To fine 33° 16' 58" 9.7393924
Here 33° 16' 58" is the sun's distance from Y; but as the declination is at that time decreasing, and the sun approaching Y, this must be taken from 360°, and the remainder 326° 43' 2" is the right ascension.
In a similar manner may the sun's right ascension be calculated for every day at noon, and arranged in tables for use: for any intermediate time between one day at noon and the following, the right ascension may be determined by proportion.
The longitude ES of the sun, when required, may be readily found by the rules to ascertain the hypothenuse of the same triangle.
The apparent diurnal motion of the heavenly bodies being uniform, and performed in circles parallel to the equator, the interval of the times in which two stars pass over any meridian must bear the same proportion to the period of the diurnal motion, as that arc of the equator intercepted between the two secondaries passing through the stars, does to 360°, as is evident from the nature of the sphere: we may therefore find the right ascension of a star thus: Let an accurate pendulum clock be so regulated that the index may pass over the twenty-four hours during the time in which any fixed star after departing from the meridian will return to it again, which is rather less than twenty-four hours. Then let the index of a clock thus regulated be set to twelve o'clock when the sun is on the meridian; and observe the time the index points to, when the fixed star whose right ascension is sought comes to the meridian; which may be most accurately known by means of a transit telescope. Let these hours and parts, as marked by the clock, be converted into degrees, &c. of the equator, by allowing 15° to an hour; and the difference between the right ascensions of the fixed star and the sun will be known; this difference added to the sun's right ascension for that day at noon, gives the right ascension of the fixed star sought.
Or, if a clock whose dial plate is divided into 360°, instead of twelve hours, be ordered in such a manner, that the index may pass round the whole circle in the interval which a star requires to come to the same meridian again, and another index be so managed as to point out the sexagesimal parts: then, when the sun is on the meridian, let the indices of the clock be put to his right ascension at noon that day; and when the star comes to the meridian, its right ascension will be shown by the clock, without any kind of reduction.
The stars are referred likewise to the ecliptic as well as to the equator. In that case the terms longitude and latitude are used.
The longitude of any of the heavenly bodies is an arc of the ecliptic contained between the first point of the head Aries, and a secondary to the ecliptic or circle of latitude, passing through the body; it is always measured according to the order of the signs. If the body be supposed seen from the centre of the earth, it is called geocentric longitude; but if it be supposed seen from the centre of the sun, then is the longitude heliocentric.
The latitude of a heavenly body is its distance from the ecliptic, measured upon a secondary to the ecliptic drawn through the body. If the latitude be such as is seen from the earth's centre, it is called geocentric latitude; but if it be supposed seen from the centre of the sun, it is heliocentric.
The equator being the principal circle which respects the earth, the latitudes and longitudes of terrestrial objects are referred to it; and, for a similar reason (the sun's motion in the ecliptic rendering that the principal of the celestial circles), the situations of heavenly objects are generally ascertained by their latitudes and longitudes referred to the ecliptic: it has therefore become a useful problem to find the latitudes and longitudes of the stars, &c. having their declinations, and right ascensions, with the obliquity of the ecliptic, given. One of the best methods of performing this problem has been thus investigated: Let How Found. S be the place of the body (fig. 88.), EC the ecliptic, EQ the equator; and SL and SR being respectively perpendicular to EC and EQ, ER will represent the right ascension, SR the declination, EL the longitude, and SL the latitude; then, by spheres, rad.: fine ER :: co-tang. SR : co-tang. SER; and SER = CEQ = SEL. Also, co-fine SER : rad. :: tang. ER : tang. ES; and rad. : co-fine SEL :: tang. ES : tang. EL; therefore, co-fine SER : co-fine SEL :: tang. ER : tang. EL; whence we readily get
\[ \frac{\text{co-fine SEL} \times \text{tang. ER}}{\text{co-fine SER}} = \text{the tangent of EL, the longitude.} \]
Then, rad. : fine of EL :: tang. SEL : tang. SL, the latitude.
But the same thing may be performed very expeditiously by means of the following excellent rule, given by Dr Maskelyne, the present worthy astronomer royal:
1. The fine of the right ascension + co-tang. declination —10 = co-tang. of arc A, which call north, or south, according to the declination is north or south. 2. Call the obliquity of the ecliptic south in the fix first signs of right ascension, and north in the fix last. Let the sum of arc A, and obliquity of ecliptic, according to their titles, = arc B with its proper title. [If one be north and the other south, the proper title is that which belongs to the greater; and in this case, arc B is their difference.] 3. The arithmetical complement of co-fine arc A + co-fine arc B × tang. right ascension = tangent of the longitude: this is of the same kind as the right ascension, unless arc B be more than 90°, when the quantity found of the same kind as Apparent the right ascension must be subtracted from 12 signs, Motions of or 365°. 4. The sine of longitude + tang. arc the Heaven- ly Bodies. B = 10 = tang. of the required latitude, of the same title as arc B. Note, If the longitude be found near 0° or near 180°, for the fine of longitude, in the last operation, substitute tang. longitude + co-fine longitude — 10 ; and then the last operation will be tang. longitude + co-fine longitude + tang. arc B = 20 = tang. latitude. By fine tang. &c. are meant logarithm fine, log. tang. &c.
This rule may be exemplified by inquiring what are the latitude and longitude of a star whose declination is 12° 59' north, and right ascension 4° 29' 38", the obliquity of the ecliptic being 29° 28' 5".
Here, fine of right ascension 4° 29' 38" 9°7937486 Co-tang. of declination 12 59 10°6372126
Co-tang. of arc A, north 24 31 10°34069612 Obliquity of ecliptic south 23 28 Arc B, north - 1 3 cos. 9°9999271 Arith. comp. of co-fine arc A 0°0410347 Tangent of right ascension 9°7678344
Tangent of longitude 147° 13' 26" 9°8087962 Or 4° 27° 13' 26", answering to 27° 13' 26" of Leo. Then, fine of longitude - 9°7334843 Tangent of arc B - 8°2031153
Tang. of latitude, north, 34' 6" 7°9995996
Astronomers have observed that the stars vary in right ascension and in declination, but keep the same latitude : hence it was concluded that their variations in declination and right ascension were owing to the revolution of the celestial sphere round the poles of the ecliptic. Or they may be accounted for by supposing that the poles of the equator revolve slowly round those of the ecliptic. This revolution is called the precession of the equinoxes. A more particular account of it will be necessary.
By a long series of observations, the shepherds of Asia were able to mark out the sun's path in the heavens; he being always in the opposite point to that which comes to the meridian at midnight, with equal but opposite declination. Thus they could tell the stars among which the sun then was, although they could not see them. They discovered that this path was a great circle of the heavens, afterwards called the ECLIPITIC; which cuts the equator in two opposite points, dividing it, and being divided by it, into two equal parts. They farther observed, that when the sun was in either of those points of intersection, his circle of diurnal revolution coincided with the equator, and therefore the days and nights were equal. Hence the equator came to be called the EQUINOCTIAL LINE, and the points in which it cuts the ecliptic were called the EQUINOCTIAL POINTS, and the sun was then said to be in the equinoxes. One of these was called the VERNAL and the other the AUTUMNAL EQUINOX.
It was evidently an important problem in practical astronomy to determine the exact moment of the sun's occupying these stations; for it was natural to compute the course of the year from that moment. Accordingly this has been the leading problem in the astronomy of all nations. It is susceptible of considerable precision, without any apparatus of instruments. It is only ne- cessary to observe the sun's declination on the noon of the Heaven- two or three days before and after the equinoctial day. ly Bodies.
On two consecutive days of this number, his declination must have changed from north to south, or from south to north. If his declination on one day was observed to be 21' north, and on the next 5' south, it follows that his declination was nothing, or that he was in the equinoctial point about 23 minutes after 7 in the morning of the second day. Knowing the precise moments, and knowing the rate of the sun's motion in the ecliptic, it is easy to ascertain the precise point of the ecliptic in which the equator intersected it.
By a series of such observations made at Alexandria Hippar- chus, the father of our astronomy, found that the point of the autumnal equinox was about fix degrees to the eastward of the star called SPICA VIRGINIS. Eager to determine every thing by multiplied obervations, heran- facked all the Chaldæan, Egyptian, and other records, to which his travels could procure him access, for observa- tions of the same kind; but he does not mention his having found any. He found, however, some obser- vations of Arisillius and Timochares made about 150 years before. From these it appeared evident that the point of the autumnal equinox was then about eight degrees east of the fame star. He difficults these observations with great sagacity and rigour; and, on their authority, he affirms that the equinoctial points are not fixed in the heavens, but move to the westward about a degree in 75 years or somewhat less.
This motion is called the PRECESSION OF THE EQUINOXES, because by it the time and place of the sun's prece- quincofial station precedes the usual calculations : it is of the fully confirmed by all subsequent obervations. In 1750 the autumnal equinox was observed to be 20° 21' west- ward of Spica Virginis. Supposing the motion to have been uniform during this period of ages, it follows that the annual precession is about 50'' ; that is, if the celestial equator cuts the ecliptic in a particular point on any day of this year, it will on the same day of the follow- ing year cut it in a point 50'' to the west of it, and the sun will come to the equinox 20' 23'' before he has completed his round of the heavens. Thus the equinoctial or tropical year, or true year of seafons, is so much shorter than the revolution of the sun or the sidereal year.
It is this discovery that has chiefly immortalized the name of Hipparchus, though it must be acknowledg- ed that all his astronomical researches have been conducted with the same sagacity and intelligence. It was natural therefore for him to value himself highly for the discovery. It must be acknowledged to be one of the most singular that has been made, that the re- volution of the whole heavens should not be stable, but its axis continually changing. For it must be ob- served, that since the equator changes its position, and the equator is only an imaginary circle, equidistant from the two poles or extremities of the axis; these poles and this axis must equally change their positions. The equinoctial points make a complete revolution in about 25,745 years, the equator being all the while inclined to the ecliptic in nearly the same angle. Therefore the poles of this diurnal revolution must describe a circle Part II.
Apparent circle round the poles of the ecliptic at the distance of Motions of about 23° degrees in 25,745 years; and in the time of the Heavenly Bodies.
Timochares the north pole of the heavens must have been 30 degrees eastward of where it now is.
Hipparchus has been accused of plagiarism and infidelity in this matter. It is now very certain that the precession of the equinoxes was known to the astronomers of India many ages before the time of Hipparchus. It appears also that the Chaldeans had a pretty accurate knowledge of the year of seasons. From their faros we deduce their measure of this year to be 365 days 5 hours 49 minutes and 11 seconds, exceeding the truth only by 26", and much more exact than the year of Hipparchus. They had also a sidereal year of 365 days 6 hours 11 minutes. Now what could occasion an attention to two years, if they did not suppose the equinoxes moveable? The Egyptians also had a knowledge of something equivalent to this: for they had discovered that the dog-star was no longer the faithful fore-warner of the overflowing of the Nile; and they combined him with the star Fomalhaut* in their mythic calendar. This knowledge is also involved in the precepts of the Chinese astronomy, of much older date than the time of Hipparchus.
But all these acknowledged facts are not sufficient for depriving Hipparchus of the honour of the discovery, or fixing on him the charge of plagiarism. This motion was a thing unknown to the astronomers of the Alexandrian school, and it was pointed out to them by Hipparchus in the way in which he ascertained every other position in astronomy, namely, as the mathematical result of actual observations, and not as a thing deducible from any opinions on other subjects related to it. We see him, on all other occasions, eager to confirm his own observations, and his deductions from them, by every thing he could pick up from other astronomers; and he even adduced the above-mentioned practice of the Egyptians in corroboration of his doctrine. It is more than probable then that he did not know anything more. Had he known the Indian precession of 54" annually, he had no temptation whatever to withhold him from using it in preference to one which he acknowledges to be inaccurate, because deduced from the very short period of 150 years, and from the observations of Timochares, in which he had no great confidence.
Small periodical irregularities in the inclination of the equator to the ecliptic, and in the precession of the equinoxes, were discovered and examined by Bradley with great facility. He found that the pole described an epicycle, whose diameter was about 18", having for its centre that point of the circle round the pole of the ecliptic in which the pole would have been found independent of this new motion. He also observed, that the period of this epicyclical motion was 18 years and seven months. It struck him, that this was precisely the period of the revolution of the nodes of the moon's orbit. He gave a brief account of these results, to Lord Macclesfield, then president of the Royal Society, in 1747. Mr Maehin, to whom he also communicated the observations, gave him in return a very neat mathematical hypothesis, by which the motion might be calculated.
Let E (fig. 89.) be the pole of the ecliptic, and SPQ a circle distant from it 23° 28', representing the circle described by the pole of the equator during one revolution of the equinoctial points. Let P be the place of this last-mentioned pole at some given time. Round P describe a circle ABCD, whose diameter AC is 18". The real situation of the pole will be in the circumference of this circle; and its place, in this circumference, depends on the place of the moon's ascending node. Draw EPF and GPL perpendicular to it; let GL be the colure of the equinoxes, and EF the colure of the solstices. Dr Bradley's observations showed that the pole to be supposed was in A when the node was in L, the vernal equinox. If the node recede to H, the winter solstice, the pole is in B. When the node is in the autumnal equinox at G, the pole is at C; and when the node is in F, the summer solstice, the pole is in D. In intermediate situations of the moon's ascending node, the pole is in a point of the circumference ABCD, three signs or 90° more advanced.
Dr Bradley, by comparing together a great number of observations, found that the mathematical theory, if an ellipse and the calculation depending on it, would correspond much better with the observations, if an ellipse were substituted for the circle ABCD, making the longer axis AC 18", and the shorter, BD, 16". M. d'Alembert determined, by the physical theory of gravitation, the axis to be 18" and 13".4.
These observations, and this mathematical theory, must be considered as so many facts in astronomy, and we must deduce from them the methods of computing the places of all celestial phenomena, agreeable to the universal practice of determining every point of the heavens by its longitude, latitude, right ascension, and declination.
It is evident, in the first place, that the equation of Obliquity of the pole's motion makes a change in the obliquity of the ecliptic. The inclination of the equator to the ecliptic is measured by the arch of a great circle intercepted between their poles. Now, if the pole be in O instead of P, it is plain that the obliquity is measured by EO instead of EP. If EP be considered as the mean obliquity of the ecliptic, it is augmented by 9" when the moon's ascending node is in the vernal equinox, and consequently the pole in A. It is, on the contrary, diminished 9" when the node is in the autumnal equinox, and the pole in C; and it is equal to the mean when the node is in the colure of the solstices. This change of the inclination of the earth's axis to the plane of the ecliptic was called the nutation of the axis by Sir Isaac Newton.
Dr Bradley also discovered a general and periodical motion in all the stars, which alter a little their relative situations. To form an idea of this motion, let us suppose that each star describes annually a small circumference parallel to the ecliptic, whose centre is the mean position of the star, and whose diameter, as seen from the earth, subtends an angle of about 40"; and that it was in that circumference as the fun in its orbit, but so that the fun always precedes it by 90°. This circumference, projected upon the surface of the celestial sphere, appears under the form of an ellipse, more or less flattened according to the height of the star above the equator, the smaller axis of the ellipse being to the greater axis as the fine of that height to radius. These periodical movements of the stars have received the name of aberrations of the fixed stars. Besides these general motions, particular motions have been detected in several stars, excessively slow indeed, but which a long succession of ages has rendered sensible. These motions have been chiefly observed in Sirius and Arcturus. But astronomers suppose that all the stars have similar motions, which may become evident in process of time.
No method of ascertaining the distance of fixed stars 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 eyelids, 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 scarcely 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 they shone by borrowed light, they would be as invisible without telescopes as the satellites of Jupiter are; 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, Flamstead, 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 the 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 vol. lxxii. micrometers can furnish. The method and its theory p. 62. will be seen by the following investigation, extracted from his paper on the subject. Let O, E, (fig. 9c.) 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, first, 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 postulata, 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 OE (fig. 91.) 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 c O; 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 c of the third. Let us now suppose the angle O a E, or parallax of the whole orbit of the earth, to be \( \frac{1}{n} \) of a degree; then we have PE = O a E = \( \frac{1}{n} \); and because very small angles, having the same subtense OE, may be taken to be in the inverse ratio of the lines O a, O b, O c, &c. we shall have O b E = \( \frac{1}{2n} \), O c E = \( \frac{1}{3n} \), &c. Now when the earth is removed to E, we shall have PE = b E = \( \frac{1}{2n} \), and PE a - PE b = a E = \( \frac{1}{2n} \), i.e. the stars a, b, will appear to be \( \frac{1}{2n} \) distant. We also have PE c = E c = \( \frac{1}{3n} \), and PE a - PE c = a E c = \( \frac{1}{3n} \); i.e. the stars a, c, will appear to be \( \frac{1}{3n} \) distant when the earth is at E. Now, since we have b EP = \( \frac{1}{2n} \), and c EP = \( \frac{1}{3n} \), therefore b EP - c EP = b E c = \( \frac{1}{2n} - \frac{1}{3n} \); i.e. the stars b, c, will appear to be only \( \frac{1}{6n} \) 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 will \( p = \frac{m-M}{mM} P \); 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 \( \frac{12 \times 1}{12-1} P \), or \( \frac{1}{11} \) of the total parallax of a fixed star of the first magnitude; and if we should, by observation, find the partial parallax between two such stars to amount to \( r'' \), we shall have the total parallax \( P = \frac{1 \times 1 \times 12}{12-1} = r'' .0999 \). If the stars are of the third and twenty-fourth magnitude, the partial parallax will be \( \frac{24-3}{3 \times 24} = \frac{21}{72} P \); 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 O a b c 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° 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, whereof the two semiparallaxes are the sides; a general expression for which will be \( \sqrt{\frac{m-M}{2MP}} \times \frac{SS}{BR} + 1 \); for the stars will apparently describe two ellipses in the heavens, whose transverse axes will be to each other in the ratio of M to m (fig. 93.), and A a, B b, C c, D d, will be the cotemporary situations. Now, if b Q be drawn parallel to AC, and the parallelogram b q BQ be completed, we shall have \( b Q = \frac{1}{2} CA - \frac{1}{2} e a = \frac{1}{2} C c - \frac{1}{2} p \), or semiparallax 90° before or after the fun, and B b may be resolved into, or is compounded of, b Q and b q; but \( b q = \frac{1}{2} BD - \frac{1}{2} b d \) the semiparallax in the conjunction or opposition. We also have \( R : S :: b Q : b q = \frac{pS}{2R} \); therefore the distance
\( B b \) (or D d) \( = \sqrt{\frac{p^2}{2} \times \frac{pS}{2R}} \); and by substituting the value of \( p \) in this expression, we obtain
\[ \sqrt{\frac{m-M}{2MM}} \times \frac{qS}{RR} + 1, \] as above. When the stars are in the pole of the ecliptic, b q will become equal to b Q, and B b will be \( 7071 \frac{m-M}{MM} \). 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. 92.) 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 a Ex may be taken to be equal to a Ox; and as the foregoing formula gives us the angles x E b x E c, we are to add a Ex or 5" to x E b, and we shall have a Eb. 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 \( \frac{DMm-dMm}{m-M} = P \).
Suppose the two stars now to differ only in latitude, one being in the ecliptic, the other, e.g., 5 north when seen at O. This case may also be resolved by the former; for imagine the stars b, c, (fig. 91.) to be elevated at right angles above the plane of the figure, so that a O b, or a O c, may make an angle of 5" at O; then, instead of the line O a b c, E a, E b, E c, EP, 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 a b, a c, at E, and from thence the parallax B. Let the triangle a b g (fig. 94.) represent the situation of the stars; a b is the subtense of 5", the angle under which they are supposed to be seen at O. The quantity b, 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 moving 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 b, the hypothenuse of the triangle a b g; therefore, in general, putting a b = d, and a b = D, we have \( \sqrt{\frac{DD-dd \times MM}{m-M}} = P \). 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 a b g (fig. 160.) represent the situation of the stars, a b = D being their distance seen at O, a b = D their distance seen at E. That the change b g, 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 Apparent Motions of the Heavenly Bodies.
Now let the angle of position \( b \) \( a \) \( c \) be taken by a micrometer, or by any other method sufficiently exact; then, by solving the triangle \( a \) \( b \) \( a \), we shall have the longitudinal and latitudinal differences \( a \) \( a \) and \( b \) \( a \) of the two stars. Put \( a = x \), \( b = y \), and it will be \( x + b \beta \)
\[ = a q, \text{ whence } D = \sqrt{x + \frac{m - M P}{M m}} + y y; \]
and
\[ \frac{\sqrt{D^2 - y^2} \times M^2 m - M m}{m - M} = P. \]
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.
CHAP. VI. Of the Figure of the Earth.
Having now described the apparent motions of the heavenly bodies, let us return to the earth, in order to examine the information which has been collected concerning its figure.
We have seen already, that the earth is spherical. The force of gravity constantly directed towards its centre retains bodies on its surface, though situated on places diametrically opposite, or though antipodes to each other. The sun and stars appear always above the earth; for above and below are merely relative to the direction of gravity.
As soon as the spherical figure of the earth was discovered, curiosity naturally led men to endeavour to measure its dimensions. Hence it is probable, that attempts of that nature were made in very ancient times. The reference which several of the ancient measures have to the size of the globe is a confirmation of this. But among the moderns, Picard was the first who executed the task with any degree of success. He measured a degree of the meridian in France about the middle of the 17th century.
Since a meridian, or any other circle on a sphere, may be conceived to be divided into 360 equal parts, called degrees, and these into minutes and seconds, as explained by the writers on trigonometry, the circumference of the earth, and thence its diameter, may be determined by measuring the length of a degree on the meridian or any other great circle. To perform this important problem, there have been various methods invented by different philosophers of early and later times; one of these methods, which unites considerable accuracy with great facility, will be readily understood from fig. 95, where PB and ST represent two mountains or very high buildings, the distance PS between which must be very nicely determined by longimetry: then, by measuring the angles RBT and RTB with an accurate instrument, their sum taken from 180° leaves the angle BRT, which is measured also by the arc PS; whence PS is known in parts of the whole circle. Thus, if the angle BTR be 89° 45' 32", the angle TBR 89° 54' 28", and the distance PS 23\( \frac{1}{3} \) English miles; then the angle R or arc PS being equal to 180° - 89° 45' 32" + 89° 54' 28" = 20° it will be, as 22° : 60' or 1° :: 23\( \frac{1}{3} \) : 69\( \frac{1}{3} \) English miles, length of a degree. Hence the circumference of the earth is (according to this example) 24912 miles, and its diameter nearly 7930 miles.—A material advantage attending this method is, that there is no occasion to measure the altitudes of the mountains, an object which can seldom be attained without considerable difficulty.
The method which is given above is, it must be confessed, as well as all the other methods which aim at the measurement of a degree without having recourse to the heavenly bodies, liable to some inaccuracy; for, by reason of the changes in the state of the atmosphere, distant terrestrial objects never appear in their true places; they always seem more or less elevated or distant, according to the nature of the season, and the time of the day. On this account—and because it could not escape observation, that as persons changed their situation on the earth by moving towards the north or the south, the stars and other heavenly bodies either increased or decreased their apparent altitudes proportionally—the measurement of a degree was attempted even by the earliest philosophers, by means of known fixed stars. Every person who is acquainted with plane trigonometry will admit, that the distance of two places, north and south of each other, may be accurately measured by a series of triangles; for if we measure the distance of any two objects, and take the angles which each of them make with a third, the triangle formed by the three objects will become known; so that the other two sides may be as truly determined by calculation, as if they had been actually measured. And by making either of these sides the base of a new triangle, the distances of other objects may be found in the same manner; and thus by a series of triangles, properly connected at their bases, we might measure any part of the circumference of the earth. And if these distances were reduced to the north and south, or meridian line, and the altitude of some star was measured at the extremities of the distance, Apparent distance, the difference of the altitudes would be equal Motions of to the length of the grand lines in degrees, minutes, &c. whence the length of a degree would be known.
This method was, we believe, first practised by Eratosthenes in Egypt; and has been frequently used since with greater and greater accuracy, in proportion as the instruments for taking angles became, by gradual improvements, more exact and minute.
By this method, or some others not widely different, and which it is needless here to explain, the length of a degree has been measured in different parts of the earth; the results of the most noted of these measurements it may be proper to give.
Snell found the length of a degree by two different methods: by one method he made it 57664 Paris toises, or 342384 feet; and by the other 57957 toises, or 342342 feet.
M. Picard, in 1669, found by mensuration from Amiens to Malvoisin, the quantity of a degree to be 57660 toises, or 342360 feet; being nearly an arithmetical mean between the numbers of Snell.
Our countryman Norwood, about the year 1635, by measuring between London and York, determined a degree at 367196 English feet, or 57300 Paris toises, or 69 miles 288 yards.
Mulchenbroek, in 1700, with a view of correcting the errors of Snell, found by particular observations that the degree between Alkmaer and Bergen-op-zoom contained 57933 toises.
Messrs Maupertuis, Clairaut, Monnier, and others from France, were sent on a northern expedition, and began their operations in July 1735; they found the length of a degree in Sweden to be 57439 toises, when reduced to the level of the sea. About the same time Messrs Godin, Bouguer, and Condamine, from France, with some philosophers from Spain, were sent to South America, and measured a degree in the province of Quito in Peru; the medium of their results gives about 56750 toises for a degree.
M. de la Caille, being at the Cape of Good Hope in 1752, found the length of a degree on the meridian there to be 57037 toises. In 1755 Father Bofovich found the length of a degree between Rome and Rimini in Italy to be 56972 toises.
In 1764, F. Beccaria measured a degree near Turin; from his measurement he deduced the length of a degree there 57024 toises. At Vienna the length of a degree was found 57991 toises.
And in 1766 Messrs Mason and Dixon measured a degree in Maryland and Pennsylvania, North America, which they determined to be 363763 English feet, or 569944 Paris toises.
The difference of these measures leads us to conclude that the earth is not exactly spherical, but that its axis which passes through the poles, is shorter than that which passes through the equator. But the observations which have been made to determine the magnitude and figure of the earth, have not hitherto led to results completely satisfactory. They have indeed demonstrated the compression or oblateness of the terrestrial spheroid, but they have left an uncertainty as to the quantity of that compression, extending from about the 170th, to the 330th part of the radius of the equator. Between these two quantities, the former of which is nearly double of the latter, most of the results are placed, but in such a manner that those best entitled to credit are much nearer to the least extreme Motions of than to the greatest. Sir Isaac Newton, as is well known, supposing the earth to be of uniform density, assigned for the compression at the poles \( \frac{1}{230} \), nearly a mean between the two limits just mentioned; and it is probable, that, if the compression is less than this, it is owing to the increase of the density toward the centre. Bofovich, taking a mean from all the measures of degrees, so as to make the positive and negative errors equal, found the difference of the axis of the meridian \( = \frac{1}{248} \). By comparing the degrees measured by Father Leiganic in Germany, with eight others that have been measured in different latitudes, La Lande finds \( \frac{1}{311} \), and, suppressing the degree in Lapland, which appears to err in excess, \( \frac{1}{331} \) for the compression. La Place makes it \( \frac{1}{321} \); Sejour \( \frac{1}{307} \), and lastly, Carouge and La Lande \( \frac{1}{300} \).
These anomalies have induced some astronomers, especially M. de la Place, to give up the spheroidal figure of the earth altogether, to suppose that it is not the earth, a solid of revolution, and that its surface is a curve of double curvature. Mr Playfair, on the other hand, in an excellent dissertation on the subject, published in the fifth volume of the Edinburgh Transactions, supposes, that the anomalies may be owing to the different densities of the strata near the surface where the degrees were measured, occasioning errors in the measurement.
The position of the different places on the earth's surface is determined by their distance from the equator, and longitudes called their latitude, and from a first meridian called their longitude. The latitude is easily ascertained by observing the height of the pole: The longitude is calculated by observing some celestial phenomenon, as an eclipse of Jupiter's satellites at the same instant in two places situated in different meridians. The difference in point of apparent time in the two places, gives their distance east or west from each other, and consequently the difference of their longitude; for it is not noon at the same time in all the different parts of the earth's surface. When it is noon at London, it is only eleven o'clock in all the places 15° west from London, while it is one o'clock in all places 15° east from London. Every 15° east or west causes the difference of an hour. Hence the difference in time, when any celestial phenomenon is observed, gives us the distance east and west, or in longitude, between the places where it is observed.
The eclipses of Jupiter's satellites are of the greatest service in determining the longitude of places on this earth, astronomers therefore 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 tion 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 hour's 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 telescopic 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, 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.
From its impracticability, the seaman is obliged to have recourse to other celestial phenomena, and the most useful are the motions of the moon. On this subject we shall satisfy ourselves with inferring the following observations of Mr Lowe, who has pointed out a very simple method of ascertaining the longitude on land.
Although the method of determining the difference of longitude at sea from the lunar observations has been accurately laid down by Dr Maskelyne and other able nautical astronomers, it has, however, happened that several writers on longitude and astronomy have, in the course of the last twenty years, given rules for finding the difference of longitude at land from the moon's transits, either so erroneous or imperfect, that the adoption thereof might do a serious injury both to navigation and geography: they have given examples, but no demonstrations; or at least such obscure and imperfect ones, as prove that they had not a clear conception of the matter.
It is for these reasons that the following demonstration of a rule both easy and accurate for finding the difference of longitude is now proposed. The data are the observed increase of the moon's right ascension in passing from the first to the second meridian, and the increase of the sun's and moon's right ascension in twelve hours apparent time, which may be had from the Nautical Almanack.
Demonstration.—Let the circle ABC represent the Fig. 97- equator, P its pole, and APD the first meridian, as that of Greenwich. Suppose that the centres of the sun, the moon, and a fixed star, are on that meridian at the same moment of time as represented at A, and that they move from thence to the westward with their respective velocities, the earth being considered as at rest. Then, after twelve hours apparent time, the sun will be at D, the opposite point to A, or 180° distant from it; but the fixed star, moving in appearance over a greater space than 180° in twelve hours apparent time, will be at E; while the moon, with a motion apparently slower than the sun and the star, will appear after twelve hours at the point B, or on a meridian BP. But ED is the distance of the sun from the star after an interval of twelve hours apparent time, and EB the distance of the moon, or, in other words, the increase of their respective right ascensions: and since ED and EB are known from the Nautical Almanack, if we subtract the first from the last, we have DB, equal to the difference between the increase of the sun's and moon's right ascension in twelve hours apparent time. Now the difference of longitude between the two meridians AP and BP is the arc A S B, equal to A B D less the arc DB; that is, equal to 180° less the difference between the increase of the sun's and moon's right ascension in twelve hours; and, since the increase of the moon's right ascension from the time of its passing the meridian AP to the time of its passing BP is known from observation, and equal to EB, we can make the following proportion for finding the difference of longitude between any other two meridians, AP and S P, from the observed increase of the moon's right ascension: &c.
As EB : A S D—DB :: ; B : A S the difference of longitude; or, in more familiar language, as the increase of the moon's right ascension in twelve hours apparent time is to 180° or 12 h. less the difference between the increase of the sun's and moon's right ascension in that time :: fo is any other observed increase of the moon's right ascension between two meridians: to their difference of longitude.
If the increase of the moon's right ascension in 12 hours were uniform, or such that equal parts of it would be produced in equal times, the above rule would be strictly accurate; but as that increase arises from a motion continually accelerated or retarded, and seldom uniform but for a short space of time, it will therefore be necessary to find the mean increase of the moon's right ascension when it is at the intermediate point between A and S, in order to determine their difference of longitude with the greatest precision; and for that purpose, Taylor's Tables of Second Difference are very useful.
Example.—April the 8th, 1800, the transit of the moon's first limb was observed at the royal observato- Part II.
Apparent ry (A); and, allowance being made for the error of Motions of the clock, its right ascension was
Add the time that the moon's semi-diameter took to pass the meridian
Right ascension of the moon's centre 12 36 26.6 On a meridian (a) far to the westward, the transit of the moon's first limb was observed the same day, and being reduced to the centre, its right ascension was 12 47 56.7
Increase of right ascension between A and β. 0 11 30.1 The increase of the moon's right ascension in 12 hours apparent time per The Nautical Almanac was 0 26 3 The increase of the sun's in the same time 0 1 49.65
Difference 0 24 13.35
And 12 hours minus this difference is = 11 h. 35 m. 46.65 sec.; therefore, as 26 m. 3 sec. : 11 h. 35 m. 46.65 sec. :: 11 m. 30.1 sec. : to 5 h. 12 sec. the correct difference of longitude between A and β.
By reducing the three terms to seconds, and using logarithms, the operation is much shortened.
In a book published by Mr Mackay on longitude about 15 or 16 years ago, there is a rule given, and also an example, for finding the difference of longitude at land from the transits of the moon, but no demonstration. The rule, when divested of its high-founding enunciation, runs thus:
As the increase of the moon's right ascension in 12 hours apparent time : is to 180° :: fo is any other observed increase between two meridians : to their difference of longitude. It follows from this, that the moon as well as the sun would, in 12 hours apparent time, pass over an arc of 180°, although the apparent motion of the moon to the westward in 12 hours, or 180° of space, be less than that of the sun by fix or seven degrees; and so much error would this method produce, if the two places differed about 180° in longitude.
The above example, wrought according to MacKay's rule, would come out thus:
<table> <tr> <th></th> <th>H. M.</th> <th>Sec.</th> </tr> <tr> <td>As 26 m. 3 sec. : 12 h. :: 11 m. 30.1 sec. to 5</td> <td>17</td> <td>53.7</td> </tr> <tr> <td>But the correct difference as above is</td> <td>5</td> <td>7</td> </tr> <tr> <td>Error</td> <td>0</td> <td>10 41.7</td> </tr> </table>
which amounts to more than 2°, or 150 miles, in a difference of longitude little exceeding five hours.
Mr Edward Pigot adopts the very same rule for determining the difference of longitude between Greenwich and York, and states the result in the Philosophical Transactions for 1785, p. 417.
Mr Vince has inserted this rule and example in his Treatise of Practical Astronomy; but we have to regret that they were not accompanied with a strict demonstration.
The Rev. Mr Wollaston, in the appendix to his Vol. III. Part I.
Fasciculus Afronumicus, published two or three years ago, has given a rule, without demonstration or example, for finding the difference of longitude from the moon's transits, which produces the same error as Mackay's and Pigot's, although worded differently from theirs. Mr Wollaston makes the first term of his proportion apparent, and the third mean time; this renders the result erroneous. Since the motions of the sun, moon, and planets are computed for apparent time, and given so in the Nautical Almanack, mean time is not at all requisite for resolving the difference of longitude either at sea or at land. We shall therefore endeavour to apply Mr Wollaston's rule, according to its literal meaning, for finding the difference of longitude from the above observations.
The right ascension of the moon's centre on the meridian of Greenwich being known, we can easily deduce the mean and apparent time corresponding to it; and in like manner the mean and apparent time at the distant meridian β. The apparent and mean time of the transits of the moon's centre over the meridians of A and β, when trifly computed, were as follows:
<table> <tr> <th colspan="2">Apparent Time.</th> <th colspan="2">Mean Time.</th> </tr> <tr> <th>H. M</th> <th>Sec.</th> <th>H. M</th> <th>Sec.</th> </tr> <tr> <td>At A</td> <td>-</td> <td>11</td> <td>26 47.81</td> <td>11</td> <td>28 33.5</td> </tr> <tr> <td>At β</td> <td>-</td> <td>11</td> <td>37 29.5</td> <td>11</td> <td>39 11.4</td> </tr> </table>
Time later at β than at A 0 10 41.69 0 10 37.9
From the increase of the moon's right ascension in 12 hours 26 3 Subtract the increase of the sun's right ascension in that time 1 49.65
The moon's retardation in 12 hours 24 13.35
Then, "As twice the moon's retardation in 12 hours: is to 24 hours ::
"So is the mean time later at β than at A : to the difference of longitude well from A."
After doubling 24 m. 13.25 sec. and also 12, which is totally unnecessary, as the result would be the same if they stood single, we state the following proportion:
As 48 m. 26.7 sec. : 34 h. :: 10 m. 37.9 sec. to 5 h. 15 m. 1.3 sec. the difference of longitude between A and β.
But as the third term is improperly reduced to mean time, we shall take the apparent time above found, and then 48 m. 26.7 sec. : 24 h. :: 10 m. 41.69 sec. to 5 h. 17 m. 53.7 sec.; the same as results from MacKay's and Pigot's rules.
We shall only remark, that 5 h. 17 m. 53.7 sec. is the apparent time that the moon took in passing from the meridian of A to the meridian of β; but from what has been demonstrated, the apparent time at β will be equal to the difference between the increase of the sun's and moon's right ascension in that interval of apparent time; for DB, or 24 m. 13.35 sec. is the difference for 12 hours, and therefore by proportion δ β, or 10 m. 41.69 sec. will be the difference for 5 h. 17 m. 53.7 sec.; subtracting the former from the latter, we have 5 h. 7 m. 12 sec. the difference of longitude as before, and A clear proof that the authors above mentioned have omitted to deduct the apparent time at the distant place or Station S, from the apparent time at Greenwich.
A very important fact relative to the earth has been ascertained by astronomers, namely, that the weight of bodies does not continue the same when carried to different parts of it. It is impossible to ascertain this variation by the balance, because it affects equally the bodies weighed and the weight by which we estimate its gravity. But the pendulum affords a certain method of detecting every such change; because the number of oscillations made by a given pendulum in a given time depends upon the force of gravity. The smaller that force, the fewer vibrations will it make. Therefore, if the force of gravity diminish, the pendulum will move slower; if it decreases, it will oscillate with more celerity. In different pendulums the slowness of vibration is proportional to the length of the pendulum: If a pendulum be lengthened it moves slower, if it be shortened it moves swifter than before. Mr Richer in a voyage made to Cayenne, found that the pendulum of his clock did not vibrate so frequently there, as it did when at Paris; but that it was necessary to shorten it by about the eleventh part of an inch to make it vibrate in exact seconds. The nearer the equator a pendulum is placed it vibrates the slower, the nearer the pole it is placed it vibrates the faster. Hence it follows that the force of gravity is greatest at the poles, and that it gradually diminishes as we approach the equator, where it is smallest.
PART III. OF THE REAL MOTIONS OF THE HEAVENLY BODIES.
WE have now enumerated and explained the apparent motions of the heavenly bodies. Nothing can appear more intricate and perplexed, or more remote from what we are accustomed to consider as the simplicity of nature. Hence mankind have in all ages been tempted to consider them as merely apparent, and not real; and the object of astronomers has always been to detect the real motions of the heavenly bodies from those which they exhibit to the eye of a spectator on the earth. Neither industry nor address was spared to gain this desirable end. Hypothesis was formed after hypothesis; every new supposition was a step towards the truth; and at last the real motions have not only been ascertained but demonstrated in the most satisfactory manner. It shall be our object in this part of our treatise to lay before our readers the result of these discoveries.
CHAP. I. Of the Rotation of the Earth.
WE find that the sun, and those planets on which there are visible spots, turn round their axis: 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 feel 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 axis 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 axis 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 towards 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 centrifugal force is less, on account of their diurnal motion being slower. For both these reasons, bodies carried from the poles towards the equator gradually lose their weight. Experiments prove, that a pendulum which vibrates seconds near the poles vibrates slower near the equator, which shows 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 \( \frac{169}{276} \) 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 84 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
(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. Part III.
Real Mo. of a ship on smooth water can be sensible of the ship's motions of the 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 remotest 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 quietest 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 384 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 the 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 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 main-mast, 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 through on the floor, the drops will fall just as far forward on the floor when the ship fails 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 from 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. Men 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.
CHAP. II. Of the Revolution of the Planets round the Sun.
The apparent motions of the planets lead us to conclude that they all move in orbits nearly circular round the sun, while the sun moves round the earth: that the orbits of Venus and Mercury are nearer the sun than the earth; but the orbits of the other planets include the earth within them. All the apparent motions are reconcilable to this opinion, and lead us to form it. It removes all the inexplicable intricacy of their apparent motions.
But the earth itself is a planet, and bears a very exact resemblance to the rest. Shall we suppose all the other planets to revolve round the sun while it alone remains stationary? Or shall we suppose that the earth, like the other planets, revolves round the sun in the course of a year? The phenomena in both cases will be exactly the same, but the motion of the earth will reduce the whole system to the greatest simplicity, whereas the motion of the sun carrying with it the revolving planets would leave the whole complicated and involved. Various opinions on this subject have been maintained by astronomers.
Concerning the opinion 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 Real Motions of the Heavenly Bodies.
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 at 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 now 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 second century after Christ, the very name of the Pythagorean hypothesis was suppressed by a system erected by the famous geographer and astronomer Claudius Ptolemaeus. 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 the 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. 98.
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 falvo 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 his system hath ever afterwards been called the Copernican, and is represented fig. 99. Here the sun is supposed to be in the centre; next him revolves the 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 Tycho's 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 ascent to the earth's motion, which was the foundation of Copernicus's scheme. He therefore invented another system, where- Fig. 100. by he avoided the aforbing 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 fave 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 fun 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 fun 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 fun in a year, and not the fun 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 fun, and the satellites round the planets; the relative distances of all which are well known. But, if we suppose the fun to move round the earth, and compare its period with the moon's by the above rule, it will be found that the fun would take no less than 173,510 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; Part III.
Real Mo- 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 25,920 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 objections that can be made against the earth's motion round the sun is, that in opposite 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 side 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 the same, even to a mathematical demonstration. He considered this matter in the following manner: he imagined CA, fig. 101. 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 Real Mo- B. Joining the points BC, he supposed the line CB tions of the 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 fo 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 the 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 3½; 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 7' less at the autumnal than at the vernal equinox.
By calculating exactly the quantity of aberration Velocity 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 small an- 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 \( \frac{1}{8} \) of an inch diameter; and though he reduced it to \( \frac{1}{75} \) 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 disfregarded.
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.
The following experiment will give a plain idea of the diurnal or annual motions of the earth, together with the different length 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 twitted 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 of the thread, and therefore has a perpetual equinox. 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 equinox, namely, because the 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 enclosed by the wire.—But now define the person who holds the wire to hold it obliquely in the position A B C D, raising the side S just as much as he depresses the side V G, that the flame may be till 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 V G: and, if the circle be properly 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; while 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 from 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 excepted; 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 farther into the dark hemisphere, and the south pole advances more into the light, as the globe comes nearer to S: 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 Part III.
Real Mo- circles it is dark to the north frigid zone, and light to tions of the south.
Heavenly Bodies. Continue both motions; and as the globe moves through the quarter B, the north pole advances towards 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 a 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 course round the sun, considering its orbit as having no inclination; and its axis as inclining 23° 1/2 degrees from a line perpendicular to the plane of its orbit, and keeping the same oblique direction in all parts of its annual course; or, as commonly termed, keeping always parallel to itself.
Let a, b, c, d, e, f, g, h, be the earth in eight different 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 a b c d, &c., its axis N s keeps the same obliquity, and is still parallel to the line MN s. 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 evident by the figure which is taken from Dr Long's astronomy. 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 Fig. 103. 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 AE 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 place.
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 fix-o'clock hour-circle, and therefore the days and nights are equally long at all places: for every part of the meridian AETL a comes into the light at fix in the morning, and, revolving with the earth according to the order of the hour-letters, goes into the dark at fix in the evening. There are 24 meridians or hour-circles drawn on the earth in this figure, to show the time of sun-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 AE; 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 tract 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. 104.) 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. 103.; 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, Real Mo- night, supposing no inequalities in the horizon, and no tions of the 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 half-way round its orbit; or, from the 23d of September till the 20th of March. In the middle between these times, viz. on the 22d of December, the north pole is as far as it can be in the dark, which is 23 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 seasons are contrary, because, when the northern hemisphere inclines towards the sun, the southern declines from him.
Taking it for granted, then, that the earth revolves round the sun, let us see what effect that motion has upon the apparent motions of the other planets. 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 motion. Were the earth to stand still in any part of 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. 105, 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. 106, let ABCD be the orbit of the earth; e f g h that of Mercury; O the sun; GKI an arch 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 h. 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 longest elongations of the 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. 107. Here we have part of the extended ecliptic marked ω, θ, 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 into 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 from 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 periodical revolutions; and to denote this, two numeral 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. 108. 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 Part III.
Real Mo. 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 in its inferior semicircle, is said to be in perigee, and 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 at others. This difference in magnitude in Mercury is nearly as \( \frac{5}{7} \) to 1; and in Venus, no less than 3.2 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, 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. 109.
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 them 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 tions of the 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 F 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 F a or F c. 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. 111. 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. 112. 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 P g and P n, 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 Real Mo. swiftest when the earth is at d, and continues diminutions of the ing till it changes to retrograde, it must be insensible 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 insensible; for any motion gradually decreasing till it changes into a contrary one gradually increasing, must at the time of the change be altogether insensible.
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. 110,) 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. 113. 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 as he goes through in a month. Now, suppose the earth to be at x when Jupiter is at a, a line drawn through r and a shows Jupiter's place in the celestial ecliptic to be at A. In a month's time the earth will have moved from 1 to 2, Jupiter from a to b; and a line drawn from 2 to b will show his geocentric place to be in B. In another month, the earth will be in 3, 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 AB, BC, CD, DE; retrograde in EF, FG, GH, HI; and direct again in IK, KL, LM, MN. 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.
Chap. III. Of the Orbits of the Planets, and the Laws of their Motions.
It would be exceedingly easy to ascertain the position of the planets for any given time, if their orbits were circular and uniform. But they exhibit very sensible inequalities in this respect, the laws of which are exceedingly important in astronomy, as furnishing the only clue which can lead us to the theory of the celestial motions. To ascertain these irregularities, and detect their laws, it is necessary to abstract from their apparent motions the effects produced by the motion of the earth. In the first place then, we must determine the nature and dimensions of the earth's orbit.
We have seen formerly that the sun apparently moves round the earth in an ellipse, having the earth in the focus. We have only to reverse the position to obtain the orbit of the earth. It moves round the sun in an ellipse, having that luminary in the focus; so that its radius vector describes areas proportional to the times. In general, all the remarks made formerly on the supposed orbit of the sun relative to its eccentricity, &c. apply accurately to the real orbit of the earth.
The figure of the earth's orbit being thus ascertained, let us see how astronomers have been able to determine that of the other planets. Let us take the planet Mars as an example, which, from the great eccentricity of its orbit, and its nearness to the earth, furnishes an excellent medium for discovering the laws of the planetary motions.
The motion of Mars round the sun and his orbit would be known, if we had at any given time, the angle formed by its radius vector, and a fixed straight line passing through the centre of the sun, together with the length of that radius vector. To simplify the problem, a time is chosen when one of these quantities may be had separately from the other. This happens at the oppositions, when we see the planet in the same point of the ecliptic to which it would be referred by a spectator in the sun. The difference in the velocity and periodic times of the earth and Mars causes the planet to appear when in opposition in different points of the ecliptic successively. By comparing together a great number of such oppositions, the relation which subsists between the time and the angular motion of Mars round the sun, (called heliocentric), may be discovered. Different methods present themselves for that purpose. But in the present case the problem is simplified by considering that the principal inequalities of Mars returning in the same manner at every sidereal revolution, the whole of them may be expressed by a rapidly converging series of the sines of the angles multiplied by its mean motion. The relative changes in the length of the radius vector, may be determined by comparing together observations made about the quadrature when the planet being about 95° from the sun, that radius presents itself under the greatest angle possible. In the triangle formed by the straight lines which join the centres of the earth, the sun, and Mars, the angle at the earth is obtained by observation, that at the sun is ascertained by the law of Mars's heliocentric motion. Hence the radius vector is deduced in parts of the earth's radius vector. By comparing together a num- Part III.
Real Mo- ber of such radii vectors determined in this manner, tions of the the law of their variations, corresponding to the angles which they make with a straight line fixed in position, may be determined. In this manner Kepler determined the orbit of Mars, and found it to be an ellipse with the fun in the focus. He inferred that the other planets moved likewise in ellipses round the fun, and this inference has been confirmed by actual examination.
To a spectator placed in the fun, all the planets would appear to describe circles annually in the heavens; for though their motions are really elliptical, the eccentricity is so small, that the difference between them and true circles is not easily perceived even on earth; and at the fun, 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 fun, 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 fun 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 fun. Thus (fig. 114.), when the earth is at a, if we draw a line from a through the fun at S, the point G, in the sphere of the heavens where the line terminates, is the place where the fun 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 fun, 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 fun'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 geocentric latitude of a superior planet may be understood from fig. 115. 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 the south limit at C. If he were at that time viewed from S, the centre of the fun, 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 the earth. When the earth, for in- Real Mo- tance of the g, and his geocentric latitude will be FG. When the Heavenly Bodies 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 Nodes of a planet cuts the ecliptic, are called its nodes; and that planet, which the planet passes through as it goes into north latitude, is called the ascending node, and is marked thus Ω; and the opposite to this is called the descending node, and is marked γ. 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 deviation of the orbit 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 motion of the planets is swiftest at the perihelion when the radius vector is shortest: it diminishes as the radius vector increases, and is at its minimum at the aphelion. When Kepler compared these two quantities in the planet Mars, he observed that the velocity of the planet was always proportional to the square of the radius vector, so that the product of that velocity multiplied into the square of the radius vector is a constant quantity. This product is double the area described by the radius vector in the given time. Hence that area, supposing the radius vector to set out from a fixed line, increases as the time. This Kepler announced by saying, that the areas described by the radius vector are proportional to the times. These laws are precisely those followed by the earth in her motion round the fun. Hence Kepler established as the fundamental laws of the motions of the planets the two following:
1. The orbits of the planets are ellipses, having the fun in their focus.
2. The areas described by the radius vector of each planet are proportional to the times of describing them. These laws suffice for determining the motions of the planets round the fun: But it is necessary to know for each of the planets seven quantities, called the elements of their elliptical motion. Five of these elements relative to the motion of the ellipse are, 1. The duration of the sidereal revolution. 2. Half the greater axis or the mean distance of the planet from the fun. 3. The eccentricity of the orbit. 4. The mean longitude of the planet at a given time. 5. The longitude of its perihelion at the same epoch. The other two elements relate to the position of the orbits. They are, 6. The longitude of the nodes of the orbit at a given epoch, or the points where the orbit intersects the ecliptic. 7. The inclination of the orbit to the plane of the ecliptic. The following table exhibits a view of these elements. <table> <tr> <th>Sidereal revolutions.</th> <th>Mean distances.</th> <th>Eccentricity in 175c.</th> <th>Secular variation in the eccentricity.</th> <th>Mean longitude in 1750.</th> <th>Longitudes of the perihelion in 1750.</th> <th>Sidereal and secular motion of the perihelion.</th> <th>Inclination of the orbits to the ecliptic in 1750.</th> <th>Secular variation in the inclination to the ecliptic.</th> <th>Longitudes of the ascending nodes in 1750.</th> <th>Sidereal and secular motion of the nodes.</th> </tr> <tr> <td>Days.</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td>Mercury</td> <td>87.969255</td> <td>0.387100</td> <td>0.205513</td> <td>0.000003369</td> <td>281.3194</td> <td>81.7491</td> <td>"1735.50</td> <td>7.7778</td> <td>55.09</td> <td>50.3836</td> <td>-2332.90</td> </tr> <tr> <td>Venus</td> <td>224.700817</td> <td>0.723332</td> <td>0.006885</td> <td>0.000062905</td> <td>51.4963</td> <td>141.9759</td> <td>099.07</td> <td>3.7701</td> <td>13.80</td> <td>82.7993</td> <td>-5673.60</td> </tr> <tr> <td>Earth</td> <td>365.256384</td> <td>1.000000</td> <td>0.016814</td> <td>0.000045572</td> <td>311.1218</td> <td>309.5790</td> <td>3671.03</td> <td>0.0000</td> <td>0.00</td> <td>0.0000</td> <td>0.00</td> </tr> <tr> <td>Mars</td> <td>686.979579</td> <td>1.523693</td> <td>0.093088</td> <td>0.000009085</td> <td>24.4219</td> <td>368.3006</td> <td>4834.57</td> <td>2.0556</td> <td>-4.45</td> <td>52.9377</td> <td>-7027.41</td> </tr> <tr> <td>Jupiter</td> <td>4332.602208</td> <td>5.202792</td> <td>0.048077</td> <td>0.000134245</td> <td>4.1201</td> <td>11.5012</td> <td>2030.25</td> <td>1.4636</td> <td>-67.40</td> <td>108.8062</td> <td>-4509.50</td> </tr> <tr> <td>Saturn</td> <td>10759.077213</td> <td>9.540724</td> <td>0.056223</td> <td>0.000261553</td> <td>257.0438</td> <td>97.9466</td> <td>4967.64</td> <td>2.7762</td> <td>-47.87</td> <td>123.9327</td> <td>-5781.54</td> </tr> <tr> <td>Herschel</td> <td>30689.000000</td> <td>19.183620</td> <td>0.046683</td> <td>0.000026228</td> <td>353.9610</td> <td>185.1262</td> <td>759.85</td> <td>0.8599</td> <td>9.38</td> <td>80.7015</td> <td>-10608.00</td> </tr> </table>
The sign — denotes a retrograde motion.
In this table, drawn up by M. de La Place, the decimal notation is employed; the circle being divided into 400°, the degree into 100', the minute into 100", and so on: we did not alter it, in order to give the reader a specimen of this notation, and because the usual notation is employed in the following table.
We think it proper to subjoin here Dr Maskelyne's view of the planetary system for 1801, Dec. 1.
<table> <tr> <th></th> <th>I.</th> <th>II.</th> <th>III.</th> <th>IV.</th> <th>V.</th> <th>VI.</th> <th>VII.</th> <th>VIII.</th> <th>IX.</th> <th>X.</th> </tr> <tr> <th>Apparent mean diameters, as seen from the earth.</th> <th>Mean diameters as seen from the sun.</th> <th>Mean diameters in English miles.</th> <th>Mean distances from the sun round numbers of miles.</th> <th>More accurate proportional numbers of the preceding mean distances.</th> <th>Densities to that of water, which is 1.</th> <th>Proportions of the quantities of matter.</th> <th>Inclinations of orbits to the ecliptic in 1780.</th> <th>Inclinations of axes to orbits.</th> <th>Rotations diurnal or round their own axes.</th> </tr> <tr> <td>The Sun</td> <td>3' 2" 1",5</td> <td>883246</td> <td>1 1/3</td> <td>333928</td> <td>7° 0' 0"</td> <td>25d 14h 8m 0f</td> </tr> <tr> <td>Mercury</td> <td>10</td> <td>16"</td> <td>3224</td> <td>37000000</td> <td>38710</td> <td>0.1654</td> <td>3 23 35</td> <td>0 23 21</td> </tr> <tr> <td>Venus</td> <td>58</td> <td>30</td> <td>7087</td> <td>68000000</td> <td>72333</td> <td>5 1/2</td> <td>0.8899</td> <td>0 0 0</td> <td>66 32</td> <td>1</td> </tr> <tr> <td>The Earth</td> <td>17,2</td> <td>7911,73</td> <td>95000000</td> <td>100000</td> <td>4 1/2</td> <td>0.025</td> <td>5 9 3</td> <td>88 17</td> <td>29 17 44 3</td> </tr> <tr> <td>The Moon</td> <td>31 8</td> <td>4,6</td> <td>2180</td> <td>95000000</td> <td>100000</td> <td>5 1/2</td> <td>0.0875</td> <td>1 51 0</td> <td>59 22</td> <td>0 24 39 22</td> </tr> <tr> <td>Mars</td> <td>27</td> <td>10</td> <td>4189</td> <td>144000000</td> <td>152369</td> <td>3 7</td> <td>10 37 56,6 in 1801.</td> <td>34 50 40 in 1801.</td> <td>1 18 56 in 1780.</td> <td>90 nearly.</td> <td>0 9 55 37</td> </tr> <tr> <td>Ceres</td> <td>1</td> <td>160</td> <td>260000000</td> <td>273550</td> <td>3 1/2</td> <td>0.0875</td> <td>1 18 56 in 1780.</td> <td>2 29 50 in 1780.</td> <td>0 46 20 in 1780.</td> <td>60 probably.</td> <td>0 10 16 2</td> </tr> <tr> <td>Pallas</td> <td>0,5</td> <td>80</td> <td>266000000</td> <td>279100</td> <td>1 1/4</td> <td>312,1</td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td>Jupiter</td> <td>39</td> <td>37</td> <td>89170</td> <td>490000000</td> <td>520279</td> <td>1 1/4</td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td>Saturn</td> <td>18</td> <td>16</td> <td>79042</td> <td>900000000</td> <td>954072</td> <td>0 1/3</td> <td>97,76</td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td>Herschel</td> <td>3 54</td> <td>4</td> <td>35112</td> <td>180000000</td> <td>1908352</td> <td>0 88</td> <td>16,84</td> <td></td> <td></td> <td></td> <td></td> </tr> </table> <table> <tr> <th></th> <th>XI.</th> <th>XII.</th> <th>XIII.</th> <th>XIV.</th> <th>XV.</th> <th>XVI.</th> <th>XVII.</th> <th>XVIII.</th> </tr> <tr> <th></th> <th>Tropical revolutions.</th> <th>Sidereal revolutions.</th> <th>Places of Aphelia, January 1800.</th> <th>Secular motions of the Aphelia.</th> <th>Eccentricities; the mean distances being 100000.</th> <th>Greatest equations of the centres.</th> <th>Longitudes of \( \Omega \); or places of ascending nodes in 1750.</th> <th>Secular motions of nodes.</th> </tr> <tr> <td>The Sun</td> <td>87<sup>d</sup> 23<sup>h</sup> 14<sup>m</sup> 32,7<sup>s</sup></td> <td>87<sup>d</sup> 23<sup>h</sup> 15<sup>m</sup> 43,6<sup>s</sup></td> <td>8<sup>f</sup> 14<sup>o</sup> 20' 50''</td> <td>10 33' 45''</td> <td>7955,4</td> <td>23° 40' 00''</td> <td>1f 15° 20' 43''</td> <td>1° 12' 10''</td> </tr> <tr> <td>Mercury</td> <td>224 16 41 27,5</td> <td>224 46 49 10,6</td> <td>10 7 59 1 1 21 0</td> <td>408</td> <td>0 47 20</td> <td>2 14 26 18</td> <td>0 51 40</td> <td></td> </tr> <tr> <td>Venus</td> <td>365 5 48 49</td> <td>305 6 9 12</td> <td>9 8 40 12</td> <td>0 19 35</td> <td>1681,395</td> <td>1 55 30,9</td> <td></td> <td></td> </tr> <tr> <td>The Earth</td> <td>686 22 18 27,4</td> <td>686 23 30 35,6</td> <td>5 2 24 15</td> <td>1 51 40</td> <td>14183,7</td> <td>10 40 40</td> <td>1 17 38 38</td> <td>0 46 40</td> </tr> <tr> <td>The Moon</td> <td></td> <td></td> <td>10 25 57 14</td> <td></td> <td>8140,64</td> <td>9 20 8</td> <td>2 20 58 40 in 1802.</td> <td>5 22 28 57 in 1802.</td> </tr> <tr> <td>Mars</td> <td>1681 12 9 0</td> <td></td> <td></td> <td></td> <td>24630</td> <td></td> <td></td> <td></td> </tr> <tr> <td>Ceres</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td>Pallas</td> <td></td> <td>1703 16 48 0</td> <td></td> <td></td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td>Jupiter</td> <td>4330 14 39 2</td> <td>4332 14 27 10,8</td> <td>6 11 8 20 in 1800.</td> <td>1 34 33</td> <td>25013,3</td> <td>5 30 38</td> <td>3 7 55 32 in 1750.</td> <td>0 59 30</td> </tr> <tr> <td>Saturn</td> <td>10746 19 16 15,5</td> <td>10759 1 51 11,2</td> <td>8 29 4 11 in 1800.</td> <td>1 50 7</td> <td>53640,42</td> <td>6 26 42</td> <td>3 21 32 22 in 1750.</td> <td>0 55 30</td> </tr> <tr> <td>Herschel</td> <td>30637 4 0 0</td> <td>30737 18 0 0</td> <td>11 16 30 31 in 1800.</td> <td>1 29 2</td> <td>90804</td> <td>5 27 16</td> <td>2 12 47 in 1788.</td> <td>1 44 35</td> </tr> </table>
From the above tables it appears that this time of the revolution of the planets increases with their distance from the sun. This induced Kepler to suspect that some relation existed between them. After many attempts continued for 17 years, he at last discovered that the squares of the periodic times of the planets are proportional to the cubes of the greater axis of their orbits.
CHAP. IV. Of the Orbits of the Comets.
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 Italian and Pythagorean schools; for they held them to be so far off 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 Aritotle 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 spear; others Real Motions of the knowledge 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 Kepler and 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 agreeably 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 17th century. He maintained that comets "are spirits, which have 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 irreconcileable 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 Bernoulli, 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 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 the great comet of 1682. This descended 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 observations. 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. in the consulate of Lampadius and Orestes about the year 531, and in the year 44 B. C. before Julius Caesar was murdered; and hence 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 their planetary helion. Part III.
Real Mo-planetary regions since they have been looked upon as celestial bodies, and observed accordingly: besides, it may often happen, 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 of which will easily appear from fig. 116. wherein S represents the sun, E the earth, ABCD the sphere of 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 KLMN 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 BD perpendicular to SE, which will divide the sphere CLMN into two hemispheres, one of which, BCD, is toward the sun, the other, DAB, opposite. Now it is manifest, that the spherical portion LMN, which is in the hemisphere BCD towards the sun, is larger than the portion NKL in the hemisphere opposite to him: and consequently a greater number of comets will appear in the hemisphere BCD than in that marked DAB.
Though the orbits of all comets are very eccentric ellipses, there are vast differences among them; excepting Mercury, there are no great differences among the planets, either as to the eccentricity of their orbits, or the inclination of their planes; but the planes of some comets are almost perpendicular to others, and some of their ellipses are much wider than others. The narrowest ellipsis of any comet hitherto observed was that of 1682. 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, in a large work, wherein he gives the opinion of various 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 of 1577 to be about 210 semidiameters of the earth, 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 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 the 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 Cyfatus 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 interpolation of these bodies between the earth and sun, we might account for those caledon darknefes which cannot be derived from any interposition of the moon. Such are those mentioned by Herodotus, lib. vii. cap. 37. and lib. ix. cap. 70.; 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 would take place in any lunar eclipse.
Various conjectures have been formed respecting the tails of comets; though it is acknowledged by concerning 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, we cannot easily determine. Appian, 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. Descartes and his followers were of opinion, that the tail of a comet was owing to the refrac- Real Motion of its head: but if this were the case, the planets of the 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. Though 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 rarefied 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 familiarity 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, entitled, 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 above-mentioned opinion of Sir Isaac, "We find (says he) in this account, that Sir Isaac ascribes 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 exceedingly thin vapours of which comets tails are formed, move through 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 polar 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 of 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 Real Mo. possible it should move foremost, as it is that a torch of the 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 thickness of a comet's tail, does not suffer the least diminution. And yet, if the tail can reflect the light of the sun so 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 flender 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 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-luminous 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 starions of the to us, that ray must be turned far out of its way in palling 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-luminous substance."
But whatever probability the Doctor's conjecture concerning the materials whereof the tails are formed account 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 raresied part which is before will continually fly off opposite to the sun, being displaced by that which comes from behind; for though 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 less 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 raresied vapours and atmosphere ascending on the contrary side; just as in a common fire there is a constant stream of dense air ascending, which pushes up another of raresied air, flame, and smoke. The refraction of a 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 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 Rest Mr. cumbent on Dr Hamilton, therefore, to have explained the 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 in 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 freer (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 subtle 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 or air may be easily destroyed or broken by resistance, 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 Real Mother 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 Electric impression of solid bodies with such facility, that we not 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 untractable 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 this would always be imperceptible to us. The velocity of comets is indeed sometimes inconceivably great. velocity of Mr Brydone observed one at Palermo, in July 1793, a comet observed 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 tail, 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 sub-jects them to intense and inconceivable degrees of heat, heat of the Newton calculated that the heat of the comet of 1680 comet of must have been near 2000 times as great as that of red-hot iron. The calculation is founded upon this principle, 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-glas 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 so disunite the parts as to make them fly off every way in atoms. This comet, according to Halley, in passing through 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 Flamstead'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 through 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, through 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 greater part of the comet's atmosphere, would, after the subiding 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 a very great depth 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 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 intense 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 of Hevelius, they were composed of several masses compacted together, with a transparent fluid interspersed, but the nature of the apparent changes in the nucleus may be only on the comets' 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 their surfaces, 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 for ever 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 scarce 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 fun itself. Sir Isaac Newton hath farther concluded, that this comet must be considerably retarded in every succeeding revolution by the atmosphere of the fun within which it enters; and thus 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 whose condensed vapours 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 increase wholly from fluids; and again, as to their greatest part, turn by putrefaction into earth; an earthy slime being perpetually precipitated to the bottom of 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 recedes from the sun: that they are kindled up, and receive their alarming appearance, in their near approach to this glorious luminous: but that those without tails are seldom or ever seen but on their way to the sun; and he does not recollect 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 has 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 to change their places as the comet came near them.
A very strange opinion we find set forth in a book entitled "Observations and Conjectures on the Nature 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 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: Real Mo- As 1, the square of one year, the earth's periodical time, tions of the is to 5776 the square of 76, the comet's periodical time; to 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 3588 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 1682 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 these great men have determined: but it is the least 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 1682. Mr Betts, in attempting to compute the transverse axis of its orbit, found it come out fo 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 Dr Halley tables of comets, found such a similitude in the elements, calculated of those of 1531, 1607, and 1682, that he was induc'd 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 eccentric soever it might be, instead of the parabolic orbit which he had given given for the comet of 1682, he set about adapting the plan of that orbit to an ellipse 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 1697 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 ellipse 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 its perihelion August 24.; that of 1697, October 16; and that of 1682, September 4.; so that the first 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 these two planets: and observes, that in the summer of the year 1681, the comet in its descent was for some time so 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 contrarywise, 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 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 attractions 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 determined the period 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.
CHAP. V. Of the Motions of the Satellites.
The moon is the satellite which moves round the earth, and as her apparent and real motions are the same, we have already given an account of her elliptical orbit and irregularities.
Jupiter is attended by four satellites. If we represent the femidiameters of Jupiter's equator by unity, then the mean distances of the satellites from Jupiter, will be represented by the following numbers.
<table> <tr> <th>Satellite</th> <th>Femidiameters.</th> <th>Orbits and distances of Jupiter's satellites.</th> </tr> <tr> <td>First satellite</td> <td>5.697300</td> <td rowspan="4">Orbits and distances of Jupiter's satellites.</td> </tr> <tr> <td>Second satellite</td> <td>9.065898</td> </tr> <tr> <td>Third satellite</td> <td>14.461628</td> </tr> <tr> <td>Fourth satellite</td> <td>25.436000</td> </tr> </table>
The durations of their revolutions are respectively,
<table> <tr> <th>Satellite</th> <th>Days.</th> </tr> <tr> <td>First satellite</td> <td>1.769137787069931</td> </tr> <tr> <td>Second satellite</td> <td>3.551181016734599</td> </tr> <tr> <td>Third satellite</td> <td>7.14552807541524</td> </tr> <tr> <td>Fourth satellite</td> <td>16.68091930608634</td> </tr> </table>
If we compare the distances of these satellites with their periodic times, we observe the same relation pointed out by Kepler between the distances of the planets from the sun and the duration of their revolutions: for the squares of the periodic times of the satellites are proportional to the cubes of their distance from Jupiter's centre.
The frequent eclipses of these satellites have enabled astronomers to ascertain their motion, with much more precision than could have been attained merely by observing their distances from Jupiter. The following points have been ascertained.
The orbit of the first satellite is circular, at least its eccentricity is inensible; it coincides nearly with Jupiter's equator, which is inclined to the orbit of the planet at an angle of 3.9999°.
The ellipticity of the orbit of the second satellite is also inensible; its inclination to Jupiter's orbit varies, ties in their as does also the position of its nodes. These irregularities are represented pretty well, by supposing the inclination of the orbit to the equator of Jupiter 1750.968°, and that its nodes move retrograde in that plane in a period of 30 years.
A small eccentricity is observed in the orbit of the third satellite. The extremity of its longer axis next Jupiter, called the periode, has a direct motion. The eccentricity of the orbit has been observed to vary considerably. The equation of the centre was at its maximum about the end of the 17th century; it then a- mounted to about 862"; it gradually diminished, and in the year 1775 it was at its minimum, and amounted only to about 229.7". The inclination of the orbit of this satellite to that of Jupiter, and the position of its nodes, are variable. These different variations are represented pretty nearly, by supposing the orbit inclined to that of Jupiter, at an angle of about 726", and giving to the nodes a retrograde motion in the plane of the equator, completed in the period of 137 years.
The orbit of the fourth satellite is very sensibly elliptical. Its perijove has a direct motion, amounting to about 2112". This orbit is inclined to that of Jupiter, at an angle of about 147". It is in consequence of this inclination, that the fourth satellite often passes behind the planet relatively to the sun without being eclipsed. From the first discovery of this planet, till the year 1760, the inclination of its orbit appeared constant: but it has sensibly increased since that period.
Besides all these variations, the satellites of Jupiter are subjected to several irregularities, which disturb their elliptical motion, and render their theory very complicated. These irregularities are most conspicuous in the three first satellites.
Their mean motions are such, that the mean motion of the first satellite, together with twice the mean motion of the third, is nearly equal to thrice the mean motion of the second. The same relation holds in their synodical motions. The mean longitude both synodical and fidereal of the first three satellites, seen from the centre of Jupiter, is such, that the longitude of the first, minus thrice that of the second, plus twice that of the third, is nearly equal to the semicircumference. This relation is so very near the truth, that one is tempted to consider it as rigorous, and to ascribe the supposed errors to the imperfection of observations. It will hold at least for a long time to come, and shews us that the three satellites cannot be eclipsed at once.
The periods and laws of the principal irregularities of these satellites are the same in all. The irregularity of the first advances or retards its eclipses 26" of time at its maximum. If we compare the changes on this inequality, with the relative positions of the two first satellites, we find that it disappears when these two satellites, seen from the centre of Jupiter, are in opposition at the same time; that it increases gradually, and acquires its maximum when the first satellite, at the instant of opposition, is 45° more advanced than the second; that it vanishes when the first is 90° before the second. Beyond that point it becomes negative and retards the eclipses, and increases till the two satellites are 135 degrees from each other, when it acquires its negative maximum. Then it diminishes and disappears when they are 180° distant. In the second half of the circumference the very same laws are observed as in the first. From these phenomena it has been concluded, that there exists in the motion of the first satellite round Jupiter, an inequality amounting to 1733.6" at its maximum, and proportional to the fine of twice the excess of the mean longitude of the first satellite above that of the second; which excess is equal to the difference between the mean synodical longitudes of the two satellites. The period of this inequality does not amount to 4 days. How comes it then, it will be asked, to change into a period of 437-75 Real Days, with respect to the eclipses of the first satellite? Let us suppose, that the first and second satellites set out together from their mean opposition to the sun. During every revolution of the first satellite, in consequence of its mean synodical motion, it will be in mean opposition. Suppose a fictitious star, whose angular motion is owing to the excess of the mean synodical motion of the first satellite, over that of the second, then twice the difference of the mean synodical motions of the two satellites will in the eclipses of the first be equal to a multiple of the circumference together with the motion of the fictitious star. Of course the fine of this last motion will be proportional to the inequality of the first satellite in its eclipses, and may represent that inequality. Its period is equal to the duration of the revolution of the fictitious star, which according to the mean motion of the two satellites is 437-75 days. Thus it is determined with more precision than by direct observation.
The irregularity of the second satellite follows a law similar to that of the first; but its sign is always contrary. It accelerates or retards the eclipses 932" in time when at its maximum. When compared with the position of the two satellites, we perceive that it disappears when they are in opposition to the sun at the same time: that it retards the time of the eclipses more and more, till the two satellites are distant from each other 90° at the time when they take place; then the retardation diminishes and vanishes altogether, when the two satellites are 180° from each other at the time of the eclipses. It then accelerates the eclipses in the other half of the circumference precisely as it had retarded them before. From these observations it has been concluded that there exists in the motion of the second satellite an irregularity of 3647" at its maximum proportional, (but with a contrary sign) to the sign of the excess of the mean longitude of the first satellite above that of the second, which excess is equal to the difference of the mean synodical motions of the two satellites.
If the two satellites set out together from their mean opposition to the sun; the second satellite will be in mean opposition every time that it completes a synodical revolution. If we suppose, as before, a star whose angular motion is equal to the excess of the mean synodical movement of the first satellite, or twice that of the second, then the difference of the two synodical movements of the two satellites will, at the eclipses of the second, equal a multiple of the circumference together with the motion of the fictitious star. Of course the inequality of the second during its eclipse will be proportional to the fine of the angular motion of that fictitious star. Hence the reason that the period and law of that irregularity are the same as in the irregularity of the first satellite.
If the third satellite produces in the motion of the second an inequality resembling that which the second seems to produce in the motion of the first, that is to say, proportional to the fine of twice the difference of the mean longitudes of the second and third satellite; that new inequality will coincide with that which is due to the first satellite. For in consequence of the relation which the mean longitude of the three first satellites have to each other, the difference of the mean longitudes of the two first satellites is equal to the semicircumference cunference together with twice the difference of the mean longitudes of the second and third satellites, so that the fine of the first difference is the same as the fine of double the second difference, but with a contrary sign. The inequality produced by the third satellite in the motion of the second, will therefore have the same sign, and will follow the same law as the inequality observed in that motion. It is, therefore, very probable that this inequality is the result of two inequalities depending on the first and third satellite. If in the course of ages, the preceding relation between the mean longitudes of these three satellites should cease to exist, these two inequalities, at present compounded, would separate, and their respective values might be discovered.
The inequality relative to the third satellite in its eclipses, compared with the respective positions of the second and third, offers the same relations with the inequality of the second compared with the respective situations of the two first. There exists then in the motion of the third satellite, an inequality which at its maximum amounts to 268''. If we suppose a star whose angular motion is equal to the excess of the mean synodical motion of the second satellite, above twice the mean synodical motion of the third, the inequality of the third satellite will in its eclipses be proportional to the motion of this fictitious star. But in consequence of the relation which exists between the mean longitude of the three satellites, the fine of this motion is the same (except its sign), with that of the motion of the first fictitious star which we formerly considered. Therefore the inequality of the third satellite in its eclipses has the same period, and follows the same laws, with the inequalities of the two first satellites: such are the laws of the principal irregularities of the three first satellites of Jupiter.
Let us now consider the satellites of Saturn, which are seven in number. The satellites of Saturn have not as yet proved so useful to astronomy or geography as those of Jupiter; principally because they cannot be seen unless very powerful telescopes be used. Five of those satellites were discovered in the year 1685, by Cassini and Huygens, who used telescopes consisting of two simple lenses, but upwards of 100 feet in length; and those were called 1st, 2d, 3d, &c. reckoning from the planet. Two others were discovered by Dr Herschel in the years 1787 and 1788, and these are smaller and nearer to the planet, on which account they ought to have been called the first and second, at the same time that the other five ought to have been called 3d, 4th, 5th, 6th, and 7th; but, imagining that this might create some confusion in the reading of old astronomical books, the five old satellites have been suffered to retain their numerical names, and the two new satellites are now called the 6th and the 7th; so that the 7th is the nearest to the planet, then comes the 6th, then the 1st; and this is followed by the 2d, 3d, 4th, and 5th.
The inclinations of the orbits of the 1st, 2d, 3d, and 4th satellites, to the ecliptic, are from 30° to 31°.
That of the 5th is from 17° to 18°. Of all the satellites of the solar system, none, except the 5th of Saturn, has been observed to have any spots, from the motion of which the rotation of the satellite round its own axis might be determined. Then the 5th satellite of Saturn, as Dr Herschel has discovered, turns round its own axis; and it is remarkable, that, like our moon, it revolves round its axis exactly in the same time that it revolves round its primary.
The following table states the particulars which have been ascertained with respect to the satellites of Saturn.
<table> <tr> <th>Satellites.</th> <th>Periods.</th> <th>Dift. in semi-dia. of Saturn.</th> <th>Dift. in miles.</th> <th>App. diam. of orbits.</th> </tr> <tr> <td>Seventh</td> <td>0 22 40 46</td> <td>2 1/2</td> <td>107,000</td> <td>0 57</td> </tr> <tr> <td>Sixth</td> <td>1 8 53 9</td> <td>3 1/2</td> <td>135,000</td> <td>1 14</td> </tr> <tr> <td>First</td> <td>1 21 18 27</td> <td>4 1/2</td> <td>170,000</td> <td>1 27</td> </tr> <tr> <td>Second</td> <td>2 17 41 22</td> <td>5 1/2</td> <td>217,000</td> <td>1 52</td> </tr> <tr> <td>Third</td> <td>4 12 25 12</td> <td>8</td> <td>303,000</td> <td>2 36</td> </tr> <tr> <td>Fourth</td> <td>15 22 41 13</td> <td>18</td> <td>704,000</td> <td>6 18</td> </tr> <tr> <td>Fifth</td> <td>79 7 48 0</td> <td>54</td> <td>2,050,000</td> <td>17 4</td> </tr> </table>
The planet Herschel, with its six satellites, have been entirely discovered by Dr Herschel. The planet itself may be seen with almost any telescope; but its satellites cannot be perceived without the most powerful instruments, and the concurrence of all other favourable circumstances. One of these satellites Dr Herschel found to revolve round its primary in 8d. 17h. 1m. 19 sec.; the period of another he found to be 13d. 11h. 5m. 1.5 sec. The apparent distance of the former from the planet is 33''; that of the second 44'' 1/2. Their orbits are nearly perpendicular to the plane of the ecliptic.
The other four satellites were discovered a considerable time after, and of course Dr Herschel has had less time to make observations upon them. They are altogether very minute objects; so that the following particulars must be considered as being not accurate but probable. "Admitting the distance of the interior satellite to be 25'' 5, its periodical revolution will be 5d. 21h. 25m."
"If the intermediate satellite be placed at an equal distance between the two old satellites, or at 38'' 57', its period will be 10d. 23h. 4m. The nearest exterior satellite is about double the distance of the farthest old one; its periodical time will therefore be about 38d. 1h. 49m. The most distant satellite is full four times as far from the planet as the old second satellite; it will therefore take at least 107d. 16h. 40m. to complete one revolution. All these satellites perform their revolutions in their orbits contrary to the order of the signs; that is, their real motion is retrograde." PART IV. OF THE THEORY OF UNIVERSAL GRAVITATION.
HAVING in the last two parts of this treatise given an account of the apparent and real motions of the heavenly bodies, it only remains for us to compare these motions with the laws established by mathematicians, in order to ascertain the forces that animate the solar system, and to acquire notions of the general principle of gravitation on which they depend. To develop this part of the subject properly, three particulars claim our attention. We must in the first place lay down the laws of motion as established by mathematicians; in the second place, we must apply these laws to the heavenly bodies, which will furnish us with the theory of gravitation; and, in the third place, we must apply this theory to the planetary system, and demonstrate that the whole motions of the heavenly bodies are explicable by that theory, and merely cases of it. These particulars shall be the subject of the three following chapters.
CHAP. I. Of the Laws of Motion.
The laws of motion, by which all matter is regulated, and to which it is subject notwithstanding the variety of phenomena which it continually exhibits, constitute the first principles of mechanical philosophy. They will claim a separate place hereafter in this work, under the title of Dynamics; but some notions of them are requisite in order to understand the theory of gravitation. We shall satisfy ourselves in this place with the following short sketch.
A body appears to us to move when it changes its situation with respect to other bodies which we consider as at rest. Thus in a vessel falling down a river, bodies are said to be in motion when they correspond successively to different parts of the vessel. But this motion is merely relative. The vessel itself is moving along the surface of the river, which turns round the axis of the earth, while the centre of the earth itself is carried round the sun, and the sun with all its attendant planets is moving through space. This renders it necessary to refer the motion of a body to the parts of space, which is considered as boundless, immoveable, and penetrable. A body then is said to be in motion when it corresponds successively to different parts of space.
Matter, as far as we know, is equally indifferent to motion or rest. When in motion it moves for ever unless stopped by some cause, and when at rest it remains so, unless put in motion by some cause. The cause which puts matter in motion is called a force. The nature of moving forces is altogether unknown, but we can measure their effects.
Whenever a force acts upon matter it puts it in motion, if no other force prevent this effect; the straight line which the body describes, is called the direction of the force. Two forces may act upon matter at the same time. If their direction be the same, they increase the motion; if their direction be opposite they destroy each other; and the motion is nothing if the two forces be equal; it is merely the excess of the one force above the other if the motions be unequal. If the directions of the two forces make with each other any angle whatever, the resulting motion will be in a direction between the two. And it has been demonstrated, that if lines be taken to represent the direction and amount of the forces, if these lines be converted into a parallelogram by drawing parallels to them; the diagonal of that parallelogram will represent the direction and quantity of the resulting motion. This is called the composition of forces.
For two forces thus acting together, we may substitute their result, and vice versa. Hence we may decompose a force into two others, parallel to two axes situated in the same plane, and perpendicular to each other.
Thus finding that a body A, fig. 117, has moved from A to C, we may imagine either that the body has been impelled by a single force in the direction of AC, and proportionate to the length of AC, or that it has been impelled by two forces at once, viz. by one in the direction of AD, and proportionate to the length of AD; and by another force in the direction of AB or DC, and proportionate to AB or DC. Therefore, if two sides of any triangle (as AD and DC) represent both the quantities and the directions of two forces acting from a given point, then the third side (as AC) of the triangle will represent both the quantity and the direction of a third force, which acting from the same point, will be equivalent to the other two, and vice versa.
Thus also in fig. 118, finding that the body A has moved along the line AF from A to F in a certain time; we may imagine, 1st, that the body has been impelled by a single force in the direction and quantity represented by AF; or 2dly, that it has been impelled by two forces, viz. the one represented by AD, and the other represented by AE; or thirdly, that it has been impelled by three forces, viz. those represented by AD, AB, and AC; or lastly, that it has been impelled by any other number of forces in any directions; provided all these forces be equivalent to the single force which is represented by AF.
This supposition of a body having been impelled by two or more forces to perform a certain course; or, on the contrary, the supposition that a body has been impelled by a single force, when the body is actually known to have been impelled by several forces, which are, however, equivalent to that single force; has been called the composition and resolution of forces.
The knowledge of these principles gives mathematicians an easy method of obtaining the result of any of forces, number of forces whatever acting on a body. For every particular force may be resolved into three others, parallel to three axes given in position, and perpendicular to each other. It is obvious, that all the forces parallel to the same axis are equivalent to a single force, equal to the sum of all those which act in one direction, diminished by the sum of those which act in the opposite direction. Thus the body will be acted on by three forces perpendicular to each other; if the direction of these forces be represented by the sides of a parallelopiped, the resulting force will be represented by the diagonal of that parallelopiped.
The indifference of a material body to motion or rest, and its perseverance in either state when put into it, is called the vis inertiae of matter. This property is considered as the first law of motion. Hence, whenever the state of a body changes, we ascribe the change to the action of some cause; hence the motion of a body when not altered by the action of some new force, must be uniform and in a straight line.
In such uniform motions the space passed over is proportional to the time: but the time employed to describe a given space will be longer or shorter according to the greatness of the moving force. This difference in the time of traversing the same space gives us the notion of velocity, which in uniform motions is the ratio between the space and the time employed in traversing it. As space and time are heterogeneous quantities, they cannot indeed be compared together; it is the ratio between the numbers representing each that constitutes velocity. A unity of time, a second for instance, is chosen, and in like manner a unity of space, as a foot. Thus, if one body move over 20 feet in one second, and another only 10, then the velocity of the first is double that of the second; for the ratio between 20 and 1 is twice as great as the ratio of 10 to 1. When the space, time, and velocity, are represented by numbers, we have the space equal to the velocity multiplied by the time, and the time equal to the space divided by the time.
The force by which a body is moved is proportional to the velocity, and therefore is measured by the velocity. This has been disputed by some philosophers, but has been sufficiently established. We shall consider it, therefore, as a matter of fact, referring the reader for a discussion of the subject to the article DYNAMICS.
When a body is put in motion by forces which not only act at first, but which continue to act uniformly, it will describe a curve line, the nature of which depends upon the forces which occasion the motion. Gravitation is an instance of a force which acts in this manner. Let us consider it a little. It appears to act in the same manner in a body at rest and in motion. A body abandoned to its action acquires a very small velocity the first instant; the second instant it acquires a new velocity equal to what it had the first instant; and thus its velocity increases every instant in proportion to the time. Suppose a right-angled triangle, one of the sides of which represents the time, and the other the velocity. The fluxion of the surface of the triangle being equal to the fluxion of the time multiplied by that of the velocity, will represent the fluxion of the space. Hence the whole triangle will represent the space described in a given time. But the triangle increasing as the square of either of its sides, it is obvious, that in the accelerated motion produced by gravitation, the velocities increase with the times, and the heights from which a body falls from rest increase as the squares of the times or of the velocities. Hence, if we denote by 1 the space through which a body falls the first second, it will fall 4 in 2", 9 in 3", and so on; so that every second it will describe spaces increasing as the odd numbers 1, 3, 5, 7, &c. This important point will perhaps be rendered more intelligible by the following diagram.
Let AB, fig. 119, represent the time during which a body is descending, and let BC represent the velocity acquired at the end of that time. Complete the triangle ABC, and the parallelogram ABCD. Also suppose the time to be divided into innumerable particles, e i, im, mp, po, &c. and draw ef, ik, mn, &c. all parallel to the base BC. Then, since the velocity of the descending body has been gradually increasing from the commencement of the motion, and BC represents the ultimate velocity; therefore the parallel lines ef, ik, mn, &c. will represent the velocities at the ends of the respective times A e, A i, A m, &c. Moreover, since the velocity during an indefinitely small particle of time may be considered as uniform; therefore the right line ef will be as the velocity of the body in the indefinitely small particle of time ei; ik will be as the velocity in the particle of time im, and so forth. Now the space passed over in any time with any velocity is as the velocity multiplied by the time; viz. as the rectangle under that time and velocity; hence the space passed over in the time ei with the velocity ef, will be as the rectangle ef; the space passed over in the time im with the velocity ik, will be as the rectangle mk; the space passed over in the time mp with the velocity mn, will be as the rectangle pn, and so on. Therefore the space passed over in the sum of all those times, will be as the sum of all those rectangles. But since the particles of time are infinitely small, the sum of all the rectangles will be equal to the triangle ABC. Now since the space passed over by a moving body in the time AB with a uniform velocity BC, is as the rectangle ABCD, (viz. as the time multiplied by the velocity) and this rectangle is equal to twice the triangle ABC (Eucl. p. 31. B. I.) therefore the space passed over in a given time by a body falling from rest, is equal to half the space passed over in the same time with an uniform velocity, equal to that which is acquired by the descending body at the end of its fall.
Since the space run over by a falling body in the time represented by AB, fig. 120, with the velocity BC is as the triangle ABC, and the space run over in any other time AD, and velocity DE, is represented by the triangle ADE; those spaces must be as the squares of the times AB AD; for the similar triangles ABC, and ADE, are as the squares of their homologous sides, viz. ABC is to ADE as the square of AB is to the square of AD. (Eucl. p. 29. B. VI.)
When a body is placed upon an inclined plane, the force of gravity which urges that body downwards, acts with a power far much less, than if the body descended freely and perpendicularly downwards, as the elevation of the plane is less than its length.
The space which is described by a body descending freely from rest towards the earth, is to the space which it will describe upon the surface of an inclined plane in the same time as the length of the plane is to its elevation, or as radius is to the sine of the plane's inclination to the horizon.
If upon the elevation BC, fig. 121, of the plane BD, Theory of Universal Gravitation.
Theory of Universal Gravitation.
As a diameter, the semicircle BEGC be described, the Universal part BE of the inclined plane, which is cut off by the semicircle, is that part of the plane over which a body will descend, in the same time that another body will descend freely and perpendicularly along the diameter of the circle, viz. from B to C, which is the altitude of the plane, or fine of its inclination to the horizon.
The time of a body's descending along the whole length of an inclined plane, is to the time of its descending freely and perpendicularly along the altitude of the plane, as the length of the plane is to its altitude; or as the whole force of gravity is to that part of it which acts upon the plane.
A body by descending from a certain height to the same horizontal line, will acquire the same velocity whether the descent be made perpendicularly or obliquely, over an inclined plane, or over many successive inclined planes, or lastly over a curve surface.
From these propositions, which have been sufficiently established by mathematicians, it follows, that in the circle ABC (fig. 122.), a body will fall along the diameter from A to B, or along the chords CB, DB, in exactly the same line by the action of gravity.
When a body is projected in any line whatever not perpendicular to the earth's surface, it does not continue in that line, but continually deviates from it, describing a curve, of which the primary line of direction is a tangent. The motion of the body relative to this line is uniform. But if vertical lines be drawn from this tangent to the curve, it will be perceived that its velocity is uniformly accelerated in the direction of these verticals. They are proportional to the squares of the corresponding parts of the tangent. This property shows us that the curve in which the body projected moves is a parabola.
The oscillations of the pendulum are regulated likewise by the same law of gravitation. The fundamental proportions respecting pendulums are the following:
If a pendulum be moved to any distance from its natural and perpendicular direction, and there be let go, it will descend towards the perpendicular; then it will ascend on the opposite side nearly as far from the perpendicular, as the place whence it began to descend; after which it will again descend towards the perpendicular, and thus it will keep moving backwards and forwards for a considerable time; and it would continue to move in that manner for ever, were it not for the resistance of the air, and the friction at the point of suspension, which always prevent its ascending to the same height as that from which it lastly began to descend.
The velocity of a pendulum in its lowest point is as the chord of the arch which it has described in its descent.
The very small vibrations of the same pendulum are performed in times nearly equal; but the vibrations through longer and unequal arches are performed in times sensibly different.
As the diameter of a circle is to its circumference, so is the time of a heavy body's descent from rest through half the length of a pendulum to the time of one of the smallest vibrations of that pendulum.
It is from these propositions, and the experiments made with pendulums, that the space described by a body falling from rest by the action of gravity has been ascertained.
The late Mr John Whitehurst, an ingenious member of the Royal Society, seems to have contrived and performed the least exceptionable experiments relatively to this subject. The result of his experiments shews, that the length of the pendulum which vibrates seconds in London, at 113 feet above the level of the sea, in the temperature of 60° of Fahrenheit's thermometer, and when the barometer is at 30 inches, is 39,116 inches; whence it follows that the space which is passed over by bodies descending perpendicularly, in the first second of time, is 16,887 feet. This length of a second pendulum is certainly not mathematically exact, yet it may be considered as such for all common purposes; for it is not likely to differ from the truth by more than \( \frac{1}{750} \)th part of an inch.
By these propositions, also, the variations of gravity in different parts of the earth's surface and on the tops of mountains has been ascertained. Newton also has shewn, by means of the pendulum, that gravity does not depend upon the surface nor figure of a body.
The motion of bodies round a centre affords another well known instance of a constant force. As the motion of matter left to itself is uniform and rectilinear, it is obvious that a body moving in the circumference of a curve, must have a continual tendency to fly off at a tangent. This tendency is called a centrifugal force, while every force directed towards a centre is called a central or centripetal force. In circular motions the central force is equal, and directly contrary, to the centrifugal force. It bends constantly, to bring the body towards the centre, and in a very short interval of time, its effect is measured by the versed sine of the small arch described.
Let A (fig. 123.) 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 EFG 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 resume 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 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. 124.) being the centre of a centripetal force, let a body at B set out in the di- Theory of rection of the straight line BC, perpendicular to the Universal Gravitation. line AB. It will be easily conceived, that there is no other point in the line BC fo 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 some where 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 towards 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 CB, prolonged beyond B to G, it appears that AD is inclined to the line GC more obliquely than AB, 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 inclination at I and K will be less oblique than at B; and Sir Isaac Newton has particularly shewn, 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 further remark, that if the centripetal power, while the body increases its distance from the body centre, retain sufficient strength to make the lines round a drawn from the centre to the body to become at centre extended 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 defend, and describe a curve in all respects like that which it has described already, provided the centripetal power, everywhere 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, supposed 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 forward, and would arrive again at the point B in the same space of time as was taken up in its passage from B to I; the velocity of the body at its return from the point B being the same as that wherewith it first set out from that point.
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 to 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 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 turned 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 it 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 IEH and KEL, are altogether like and equal to each other. EK therefore being equal to EI, and EL equal to KH, and KL equal to HL, it is plain, that the body, after its return to E, being turned out of the line FE into ED by an impulse acting upon it in E after the manner above mentioned, it will receive such a velocity by this impulse as will carry it through EL in the same time it would have taken to go through EK, if it had passed through it undisturbed. It has already been observed, that the time in which the body would pass Universal over EK, with the velocity wherewith it returns, is equal to the time it took up in going forward from E to I; that is, to the time in which it would have gone through EH with the velocity wherewith it moved from D to E; therefore the time in which the body will pass from E to L, after its return into the line ED, is the same as would have been taken up by the body in passing through the line EH with the velocity wherewith it first moved in the line DE. Since, therefore, EL and EH are equal, the body returns into the line DE with the velocity which it had before in that line.—Again, we may affirm, that the second impulse in E is equal to the first; for, as the impulse in E, whereby the body was turned out of the line DE into the line EF, is of such strength, that if the body had been at rest when this impulse had acted upon it, it would have communicated as much motion to it, as would have been sufficient to carry it through a length equal to HI, in the time wherein the body would have passed from E to H, or in the time wherein it passed from E to I. In the same manner, on the return of the body, the impulse in E, whereby it is turned out of the line FE into ED, is of such strength, that if it had acted on the body at rest, it would have caused it move through a length equal to KL in the same time as the body would employ in passing through EK with the velocity wherewith it returns in the line FE: therefore the second impulse, had it acted on the body at rest, would have caused it to move through a length equal to KL, in the same space of time as would have been taken up by the body in passing through a length equal to HI were the first impulse to act on the body while at rest; that is, the effects of the first and second impulse on the body when at rest would be the same; for KL and HI are equal: consequently the second impulse is equal to the first. Thus, if the body be returned through FE with the velocity wherewith it moved forward, it has been shewn how, by the repetition of the impulse which acted on it in E, the body will return again into the line DE with the velocity which it had before in that line. By the same method of reasoning it may be proved, that when the body is returned back to D, the impulse which before acted on that point will throw the body into the line DC with the velocity which it first had in that line; and the other impulses being successively repeated, the body will at length be brought back again into the line BA with the velocity wherewith it set out in that line.—Thus these impulses, by acting over again in an inverted order all their operations on the body, bring it back again through the path in which it had proceeded forward; and this obtains equally whatever be the number of straight lines whereof this curve figure is composed. Now, by a method of reasoning of which Sir Isaac Newton made much use, and which he introduced into geometry, 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 num- ber 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 KIHB, 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. 127. 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. 127, 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 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 there-
to; 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. 126.) 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. 126. in which it will revolve from P to E, and from E to B, without end. If AE in fig. 126. 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. 128. in which the opposite points B and G are equally distant from A; and the opposite points E and L 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 the 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 propole the case of a body moving in the incurvated figure ABCDE (fig. 129.), which is composed of the straight lines, AB, BC, CD, DE, and AE; 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, Theory of Universal Gravitation.
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 strength as to turn it from the line CB into CD. At D, let the body, by another impulse, directed likewise towards the point F, be turned out of the line CD into DE. At E, let another impulse, directed likewise towards the point F, turn the body from the line DE into EA: and thus the body will, by means of these impulses, be carried through 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 wherewith 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 runs into the line AB, will pass through 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 as a curve figure which shall be concave towards it, as that marked ABC, fig. 130. and when it is returned 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 line may have some such figure as ABCDBE in fig. 131. In this curve line, if the body set out from B in the direction BF, and moved through the line Universal Gravitation. BCD till it returned to B; here the body would not enter again into the line BCD, because the two parts BD and BC of the curve line make an angle at the 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. 130, 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 of the force the figure has in any part of it. Sir Isaac Newton has requisite to laid down the following proposition as a foundation for dy in any 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. 132.), 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 set out from the point A, fig. 133. to move in the straight line AB; and after it had moved for some time in that line, it were to receive an 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 tri- angle 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 towards 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.
As the effect of a central force in a very small interval of time is measured by the verfed fine of the small arch described, we may easily compare the centrifugal force produced by the rotation of the earth with gravitation. At the equator, a body in consequence of the rotation of the earth describes an arch of 15" of the circumference of the earth, in 1" of time. The radius of the equator is about 16634778 French feet; the verfed fine of which is 0.0389704 feet. At the equator a body falls 11.23585 French feet in a second. The centrifugal force is to gravity as 0.0389704 to 11.23585, or nearly as 1 to 288.3. The centrifugal force diminishes gravity, and bodies only fall in consequence of the excess of the last above the first. If the whole force whose effect would be evident, were there no rotation, be called gravity; then at the equator the centrifugal force is about \( \frac{1}{288} \) of gravity. If the earth revolved 17 times faster than it does, the arch described in a second would be 17 times greater, and its verfed fine 289 times longer; the centrifugal force would then be equal to gravity, and at the equator, bodies would cease to have any weight.
In general the expression of a uniformly accelerating force, acting constantly towards the same point, is Universal equal to twice the space which it causes the body to describe, divided by the square of the time. Every accelerating force may be supposed constant for a very small interval of time, and acting in the same direction. The space described by a body moving in a circle in consequence of the central force, is the verfed fine of the small arch described; and this verfed fine is very nearly equal to the square of the arch divided by radius. The expression of the accelerating force is then the square of the arch described, divided by the square of the time, and by radius. The arch divided by the time gives the velocity. Hence the centripetal and centrifugal forces are equal to the square of the velocity divided by radius.
We have seen that gravity is equal to the square of the acquired velocity divided by twice the space gone through. Of course the centrifugal force is equal to gravity, if the velocity of the revolving body be that which it would acquire by falling from a height equal to half the radius of the circumference described. The velocities of different revolving bodies are as the circumferences which they describe divided by the time of their revolution. These circumferences are as their radii. The squares of the velocity of course are as the squares of the radii divided by the squares of the times. Hence centrifugal forces are to each other as the radii of the circumferences described divided by the squares of the times of the revolutions. Hence in different parallels of latitude, the centrifugal forces produced by the rotation of the earth are proportional to the radii of the parallels.
These remarks will give the reader an idea of the laws of motion. For a more particular investigation he must have recourse to those articles that treat particularly of Dynamics.
CHAP. II. Of Universal Gravitation.
The principles of dynamics being understood, let us make use of them to examine the motions of the heavenly bodies, in order to detect the general laws which produce and regulate these motions.
We have seen that the planets and comets move in ellipses round the sun, and that the areas described by their radii vectors are proportional to the time. The principles of dynamics laid down in the last chapter, inform us that this could not happen unless each of these bodies were constantly acted on by a force turning them from the straight line in the direction of the centre of these radii vectors. Hence it follows, that the planets are constantly acted upon by a force which urges them towards the sun as a centre.
Let us suppose that the planets revolve round the sun in circles, which is not very far from the truth. In that case, the squares of their velocities are proportional to the squares of the radii of their orbits, divided by the squares of the times of their revolution. But by the laws of Kepler, the squares of the times are as the cubes of the radii of the orbits of the planet, or of the distance. Therefore, the squares of the velocity are reciprocally as these radii. Perhaps this reasoning will be better understood by employing symbols. Let \( t = \) the Theory of the time, \( v \) = the velocity, and \( r \) = the radius, we have Universal Gravitation.
\[ v^2 \propto \frac{r^3}{r^2} = r. \] But \( r^2 \propto r^3 \), therefore, substituting \( r^3 \) in the first formula, we have \( v^2 \propto \frac{r^3}{r^3} \), but \( \frac{r^3}{r^3} = \frac{1}{r} \), therefore we have \( v^2 \propto \frac{1}{r} \), or \( v^2 \) always reciprocally proportional to \( r \). We have seen formerly that the central forces of different bodies revolving in a circle, are as the squares of the velocity divided by the radii of their orbits. Therefore, the tendency of the planets to the sun, then, are reciprocally as the squares of the radii of their orbits, or their distance from the sun. This will be better understood if we express it by symbols. We have
\[ v^2 \propto \frac{1}{r} \]
Let \( c \) denote the central force, \( c \propto \frac{v^2}{r} \); for \( v^2 \) substitutes its equivalent \( \frac{1}{r} \), and we have \( c \propto \frac{1}{r^2} \).
It is true that the orbits of the planets are not exactly circular; but as the law of the squares of the times, proportional to the cubes of the distances, is independent of the eccentricity of the planetary orbits, it is natural to suppose, that it would exist, even though the eccentricity were destroyed. The law, therefore, that the tendency to the sun is inversely as the square of the distance, is clearly indicated by this ratio.
An analogy leads us to suppose, that this law, which extends from one planet to another, holds also with respect to the same planet in all its different distances from the sun. That this is actually the case, follows with certainty from the elliptical orbits of the planets. When the planet is in its perihelion, its velocity is a maximum, and its tendency to separate from the sun in consequence of this velocity overcoming the tendency towards the sun, the radius vector increases in length, and forms obtuse angles with the direction of the planet. Hence it opposes, and of course, tends to diminish the velocity, till the planet reaches its aphelion. Then the radius vector becomes perpendicular to the curve, the velocity is at its minimum; and the tendency to separate from the sun being less than the tendency towards the sun, the planet approaches towards it, describing the second part of its elliptical orbit. In that part, the tendency to the sun increases the velocity of the planet, as in the former part it had diminished it: the planet accordingly comes to its perihelion with a maximum of velocity. Now the curvature of the ellipse being the same at the perihelion and aphelion, the radii of the equicurve circles will be the same, and, of course, the centrifugal forces in these two points will be to each other as the squares of the velocity. The sectors described in the same times being equal, the velocities at the aphelion and perihelion are reciprocally as the corresponding distances of the planet from the sun. Of course, the squares of the velocities are reciprocally as the squares of these distances, or at the perihelion and aphelion the centrifugal forces are equal to the tendency of the planet towards the sun. Therefore this tendency is inversely as the square of the distance of the planet from the sun.
We see then, in general, that all the planets tend towards the sun, with a force inversely as the square of their distance. Newton demonstrated, that this force would cause them, if projected with a given velocity, to describe ellipses round the sun as a centre. He demonstrated farther, that this tendency is the same in all the planets, varying only according to their distances. Hence it follows, that if they were all at rest, and placed at the same distance from the sun, they would all, in consequence of this tendency, fall into the sun at the same instant; the same result must be applied also to the comets, for in them also the squares of the times are undoubtedly proportional to the cubes of their distance from the sun.
The satellites tend equally to the sun with the planets around which they revolve. Were not the satellites under the influence of this tendency, instead of describing a circle round the earth, it would soon abandon it altogether. Unless the satellites of Jupiter and the moon tended towards the sun, irregularities would be perceptible in their orbits, which they do not exhibit. The planets, comets, and satellites, then, all tend to the sun in consequence of the action of the same force. While the satellites move round their planet, the entire system of planet and satellites is carried round the sun, and retained in their orbits by the same force. Of course, the motion of the satellites round the planet, is merely the same as if the planet were altogether at rest, and not acted upon by any foreign body.
Thus we have been led, without affuming any hypothesis, by the necessary consequence of the laws of the celestial movements, to consider the centre of the sun as the focus of a force, which extends itself indefinitely through space, diminishing inversely as the squares of the distance, and which attracts all bodies within the sphere of its activity. Each of Kepler's laws points out a property of this attractive force. The law of the areas proportional to the times, informs us, that the force is directed towards the sun; the elliptical figure of the planets proves to us, that its intensity diminishes as the square of the distance augments; and the law of the squares of the times proportional to the cubes of the distance, informs us, that the tendency, or gravitation of all the planets to the sun is the same, provided the distances were the same. We may call this force solar attraction, supposing, for the sake of a distinct conception, that it is a force residing in the sun.
The tendency or gravitation of the satellites towards their planets, is a necessary consequence of the areas described by their radii vectors being proportional to the times; that this gravitation is inversely as the square of their distance, is indicated by the ellipticity of their orbits. This ellipticity, indeed, being scarcely apparent in most of the satellites of Jupiter, Saturn, and Herschel, would leave some uncertainty, did not the third law, namely, the squares of the times being inversely as the cubes of their distance, demonstrate, that from one satellite to another, the tendency to the planet is inversely as the square of the distance.
This proof, indeed, is wanting with respect to our Moon's moon; but the defect may be supplied by the following considerations. Gravity, or the weight by which the same body tends towards the earth, extends itself to the top of the highest mountains, and the very trifling diminution which it experiences at that height, cannot permit us to doubt, that it would still be sensible at a considerably Part IV.
Theory of considerably greater distance from the earth's centre. Is it not natural to extend it as far as the moon, and to suppose that the force which retains that satellite in its orbit, is its gravitation towards the earth, just as it is the solar attraction which retains the planets in their orbits? The forces at least seem to be of the same nature; they both act upon every particle of bodies, and cause them to move at the same rate; for the solar attraction acts equally upon all bodies placed at the same distance from the sun, just as gravitation causes all bodies to fall from the same height with the same velocity. A body projected horizontally, falls upon the earth at some distance after describing a curve sensibly parabolic. It would fall at a greater distance, if the force of projection were more considerable; and, if projected with a certain velocity, it would not fall back at all, but revolve round the earth like a satellite. To make it move in the orbit of the moon, it would be necessary only to give it the same height and the same projecting force. But what demonstrates the identity of gravitation and of the force which retains the moon in its orbit is, that if we suppose gravity to diminish inversely as the square of the distance from the centre of the earth, at the distance of the moon it will be precisely equal to the moon's tendency to the earth.
Let A in fig. 134. 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 scarce 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\( \frac{1}{2} \) 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 the action of this power upon it, in one minute, would be equal to the number 16\( \frac{1}{2} \) multiplied twice into the number 60; that is, to 58050. But how fast bodies fall near the surface of the earth may be known by the pendulum; and by the exact experiments, they 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 be easily understood from what has been already laid 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.
We see then that the force which retains the moon in its orbit is gravitation, or that force which causes heavy bodies to fall to the ground. This comparison between gravity and the lunar tendency to the earth shows us, that, in our calculations, we ought to measure distance from the centre of gravity of the sun and of the planets; for this is obviously the case with the earth, and its tendency to the sun is precisely the same with that of the other planets.
The sun and the planets which have satellites, possessing, as we have seen, an attractive force inversely as the square of the distance, one is tempted to give the same property to the other planets also. The sphericity common to all these bodies, indicates clearly, that their particles are retained round their centre of gravity, by a force which at equal distances attracts them equally to that centre. But this important point is not left to analogical reasoning. We have seen, that if the planets and comets were placed at equal distances from the sun, their gravitation towards it would be proportional to their masses. But it may be considered as a general matter of fact, to which there is no exception, that action and reaction are equal and contrary. Of course all these bodies react upon the sun, and attract it in proportion to their masses, and consequently possess an attractive force proportional to their masses, and inversely as the square of their distance. The satellites also, in consequence of the same principle, attract the planets and the sun according to the same law. This attracting force is then common to all the heavenly bodies.
This force does not disturb the elliptical motion of the planets round the sun, when we consider only their mutual action. For the relative movement of a system of bodies does not change by giving them a common motion. Neither is the elliptical motion of the satellites disturbed by the revolution of the planets round the sun, for the very same reason.
The attractive force does not belong to these bodies only as wholes; but it belongs to every particle of matter of which each of them is composed. If the sun acted only upon the centre of the earth, without attracting every one of the particles of which it is composed individually, there would result tidal incomparably greater, and very different from those that we observe. observe. Besides, every body on the earth gravitates towards its centre, in proportion to its mass. It reacts of course upon the earth, and attracts it in the same ratio. Unless that were the case, or if any part of the earth, however small, did not attract the other part as it is attracted by it, the centre of gravity of the earth would be moved in space, in consequence of gravitation; which is impossible.
All these phenomena, compared with the laws of motion, lead us to this grand conclusion: All the particles of matter mutually attract each other, in proportion to their masses, and inversely as the squares of their distances. This is called universal gravitation, and was the discovery which crowned the happy industry, the consummate skill, and the unrivalled sagacity of Newton.
In universal gravitation, we readily perceive a cause of the irregularities and disturbances perceptible in the planetary motions. For as the planets and comets act upon each other, they ought to deviate a little from that exact ellipticity, which they would follow if they obeyed only the action of the sun. The satellites, disturbed equally by their mutual attraction, and by that of the sun, must deviate also from these laws. We see also, that the particles of which each heavenly body is composed, provided they be at liberty to move, ought to form themselves into a sphere, and that the result of their mutual action at the surface of this sphere ought to produce all the phenomena of gravity. We see also, that the rotation of the heavenly bodies round an axis ought to alter this sphericity somewhat by flattening them at the poles, and that the result of their mutual action not passing exactly through their centres of gravity, ought to produce in their axis of rotation motions similar to those which we perceive. We see also, that the particles of the ocean, unequally attracted by the sun and moon, ought to have an oscillation similar to the tides. But it will be necessary to consider the effects of gravitation more particularly; in order to show that it is established in the completest manner by all the phenomena. This shall be the subject of the next chapter.
CHAP. III. Of the Effects of Gravitation.
We shall in this chapter consider, in the first place, several points which could only be ascertained by the assistance of gravitation, and afterwards examine the several subjects hinted at towards the conclusion of the last chapter.
SECT. I. Of the Masses of the Planets.
It would appear, at first view, impossible to ascertain the respective masses of the sun and planets, and to calculate the velocity with which heavy bodies fall towards each when at a given distance from their centres; yet these points may be determined from the theory of gravitation without much difficulty.
It follows from the theorems relative to centrifugal forces, given in the first chapter of this part, that the force of the gravitation of a satellite towards its planet is to the gravitation of the earth towards the sun, as the mean distance of the satellite from its primary, divided by the square of the time of its sidereal revolution, or the mean distance of the earth from the sun divided by the square of a sidereal year. To bring these gravitations to the same distance from the bodies which produce them, we must multiply them respectively by the squares of the radii of the orbits which are described: and, as at equal distances the masses are proportional to the attractions, the mass of the earth is to that of the sun as the cube of the mean radius of the orbit of the satellite, divided by the square of the time of its sidereal motion, is to the cube of the mean distance of the earth from the sun, divided by the square of the sidereal year.
Let us apply this result to Jupiter. The mean distance of its 4th satellite subtends an angle of 1539".86 decimal seconds. Seen at the mean distance of the earth from the sun, it would appear under an angle of 7964".75 decimal seconds. The radius of the circle contains 636619".8 decimal seconds. Therefore the mean radii of the orbit of Jupiter's 4th satellite and of the earth's orbit are to each other as these two numbers. The time of the sidereal revolution of the 4th satellite is 16.6890 days; the sidereal year is 365.2564 days. These data give us \( \frac{1}{1066.08} \) for the mass of Jupiter, that of the sun being represented by 1. It is necessary to add unity to the denominator of this fraction, because the force which retains Jupiter in his orbit is the sum of the attractions of Jupiter and the sun. The mass of Jupiter is then \( \frac{1}{1067.08} \). The masses of Saturn and Herschel may be calculated in the same manner. That of the earth is best determined by the following method:
If we take the mean distance of the earth from the sun for unity, the arch described by the earth in a second of time will be the ratio of the circumference to the radius divided by the number of seconds in a sidereal year. If we divide the square of that arch by the diameter, we obtain \( \frac{1479565}{10^{10}} \) for its versed sine, which is the deflection of the earth towards the sun in a second. But on that parallel of the earth's surface the square of the sine of whose latitude is \( \frac{1}{4} \), a body falls in a second 16\( \frac{1}{2} \) feet. To reduce this attraction to the mean distance of the earth from the sun, we must divide the number by the feet contained in that distance; but the radius of the earth at the above-mentioned parallel is 19614648 French feet. If we divide this number by the tangent of the polar parallax, we obtain the mean radius of the earth's orbit expressed in feet. The effect of the attraction of the earth at a distance equal to the mean radius of its orbit, is equal to \( \frac{16\frac{1}{2}}{19614648} \) multiplied by the cube of the tangent of the solar parallax \( = \frac{1479560.5}{10^{10}} \). Hence the masses of the sun and earth are to each other as the numbers 1479560.5 and 4486113; therefore the mass of the earth is \( \frac{1}{329809} \), that of the sun being unity.
M. de la Place calculated the masses of Mars and Venus from the secular diminution of the obliquity of the ecliptic, and from the mean acceleration of the moon's motion. The mass of Mercury he obtained from its volume, supposing the densities of that planet and Part IV.
Theory of Universal Gravitation.
Theory of Universal Gravitation.
and of the earth reciprocally as their mean distance from the sun, a rule which holds, with respect to the earth, Jupiter, and Saturn. The following table exhibits the masses of the different planets, that of the sun being unity:
<table> <tr> <th></th> <th>Mercury</th> <th>Venus</th> <th>Earth</th> <th>Mars</th> <th>Jupiter</th> <th>Saturn</th> <th>Herschel</th> </tr> <tr> <td></td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> <td>1</td> </tr> <tr> <td></td> <td>2025810</td> <td>383137</td> <td>329809</td> <td>1846082</td> <td>1067.09</td> <td>3359.49</td> <td>19504</td> </tr> </table>
The densities of bodies are proportional to their masses divided by their bulks; and, when bodies are nearly spherical, their bulks are as the cubes of their semidiameters, of course the densities in that case are as the masses divided by the cubes of the semidiameters. For greater exactness, we must take that semidiameter of a planet which corresponds to the parallel, the square of the sine of which is equal to \( \frac{r}{R} \), and which is equal to the third of the sum of the radius of the pole, and twice the radius of the equator. This method gives us the densities of the principal planets as follows, that of the sun being unity:
<table> <tr> <th></th> <th>Earth</th> <th>Jupiter</th> <th>Saturn</th> <th>Herschel</th> </tr> <tr> <td></td> <td>3.93933</td> <td>0.86014</td> <td>0.49512</td> <td>1.13757</td> </tr> </table>
To have the intensity of gravitation at the surface of the sun and planets, let us consider, that, if Jupiter and the earth were exactly spherical, and destitute of their rotatory motion, gravitation at their equators would be proportional to the masses of these bodies divided by the squares of their diameters. But at the mean distance of the sun from the earth, the diameters of the equators of Jupiter and of the earth are to each other as the numbers 626.26 and 54.5. If then we represent the weight of a body at the earth's equator by 1, the same body, if transported to the equator of Jupiter, would weigh 2.599. But the difference of the centrifugal forces on the surface of the earth and Jupiter renders it necessary to diminish this last number by about \( \frac{1}{9} \). The same body at the surface of the sun would weigh 27.65.
SECT. II. Of the Perturbations in the Elliptical Orbit of the Planets.
If the planets were influenced only by the sun, they would describe ellipses round that luminary: but they act upon one another, and from these various attractions there result disturbances in their elliptical motions, discoverable by observation, and which it is necessary to determine, in order to be able to construct accurate tables of the planetary motions. The rigorous solution of this problem is above the reach of the mathematical analysis; mathematicians have been obliged to satisfy themselves with approximations.
The disturbances in the elliptical motions of the planets may be divided into two classes. The first class secular and affects the elements of the elliptical motion: they increase very slowly, and have been called secular inequalities. The other class depends upon the configuration of the planets, either with respect to each other, or with respect to their nodes and perihelions, and are renewed every time that the relative situation of the planets becomes the same. They are called periodical inequalities, to distinguish them from the secular, whose periods are much longer and altogether independent of the mutual configuration of the planets. Before proceeding farther, we beg leave to introduce the following quotation from Dr Pemberton, because it will convey some notion of these disturbances in a very familiar manner to our readers.
"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 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 forementioned 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 that power is lodged, remains to be discovered. Now it is proved, as shall afterwards be explained, that the three primary planets, Saturn, Jupiter, and the Earth, which have satellites revolving about them, are endowed with a power of caulking 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 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 pro- portion 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 continually, become sensible after a long series of years. Yet some of these other inequalities are discernible 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 ellipses, which should otherwise be 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 (c).
"Sir Isaac Newton after this proceeds to make an improvement in astronomy, by applying this theory to the farther correction of their motions. For as we have here observed the planets to possess a principle of 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 will describe about their common centre of gravity similar ellipses; and then, that the transverse axis of the ellipses, which would be described about the sun at rest in the same time, the same proportion as the quantity of solid matter in the sun and planet together bears to the first of two mean proportionals 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 assert, 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 leaft difference: and the present correction will not cause a 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 these small inequalities in the planets motions, which eternity of contains in it a very strong philosophical argument against the eternity of the world. It is this, that these inequalities must continually increase by flow 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 derived 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 afferter, that this does not detract from the divine wisdom. Certainly we cannot pretend to know all the omnificent 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
(c) 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. Theory of 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."
Sir Isaac Newton had no sooner discovered the universality and reciprocity of the deflections of the planets and the sun, than he also suspected that they were continually deflected towards each other. He immediately obtained a general notion of what should be the more general results of such a mutual action. They may be conceived in this way.
Let S (fig. 135.) represent the sun, E the earth, and I Jupiter, describing concentric orbits round the centre of the system. Make IS : EA = EI : SI. Then, if IS be taken to represent the deflection of the sun towards Jupiter, EA will represent the deflection of the Earth to Jupiter. Draw EB equal and parallel to SI, and complete the parallelogram EBAD. ED will represent the disturbing force of Jupiter. It may be resolved into EF perpendicular to ES, and EG in the direction of SE. By the first of these the earth's angular motion round the sun is affected, and by the second its deflection towards him is diminished or increased.
In consequence of this first part of the disturbing force, the angular motion is increased, while the earth approaches from quadrature to conjunction with Jupiter (which is the case represented in the figure), and is diminished from the time that Jupiter is in opposition till the earth is again in quadrature, westward of his opposition. The earth is then accelerated till Jupiter is in conjunction with the sun; after which it is retarded till the earth is again in quadrature.
The earth's tendency to the sun is diminished while Jupiter is in the neighbourhood of his opposition or conjunction, and increased while he is in the neighbourhood of his stationary positions. Jupiter being about 1000 times less than the sun, and 5 times more remote, IS must be considered as representing \( \frac{1}{500} \)th of the earth's deflection to the sun, and the forces ED and EG are to be measured on this scale.
In consequence of this change in the earth's tendency to the sun, the aphelion sometimes advances by the diminution, and sometimes retreats by the augmentation. It advances when Jupiter chances to be in opposition when the earth is in its aphelion; because this diminution of its deflection towards the sun makes it later before its path is brought from forming an obtuse angle with the radius vector, to form a right angle with it. Because the earth's tendency to the sun is, on the whole, more diminished by the disturbing force of Jupiter than it is increased, the aphelion of the earth's orbit advances on the whole.
In like manner the aphelia of the inferior planets advance by the disturbing forces of the superior: but the aphelion of a superior planet retreats; for these reasons, and because Jupiter and Saturn are larger and more powerful than the inferior planets, the aphelia of them all advance while that of Saturn retreats.
In consequence of the same disturbing forces, the node of the disturbed planet retreats on the orbit of the disturbing planet; therefore they all retreat on the ecliptic, except that of Jupiter, which advances by retreating on the orbit of Saturn, from which it suffers the greatest disturbance. This is owing to the particular position of the nodes and the inclinations of the orbits.
The inclination of a planetary orbit increases while the planet approaches the node, and diminishes while the planet retires from it.
M. de la Place has completed this deduction of the peculiar planetary inequalities, by explaining a peculiarity in the retrograde motions of Jupiter and Saturn, which has long employed the attention of astronomers. The accelerations and retardations of the planetary motions depend, as has been shown, on their configurations, or the relative quarters of the heavens in which they are. Those of Mercury, Venus, the Earth, and Mars, arising from their mutual deflections; and their more remarkable deflections to the great planets Jupiter and Saturn, nearly compensate each other, and no traces of them remain after a few revolutions: but the positions of the aphelia of Saturn and Jupiter are such, that the retardations of Saturn sensibly exceed the accelerations, and the anomalistic period of Saturn increases almost a day every century; on the contrary, that of Jupiter diminishes. M. de la Place shows, that this proceeds from the position of the aphelia, and the almost perfect commensurability of their revolutions; five revolutions of Jupiter making 21,075 days, while two revolutions of Saturn make 21,538, differing only 137 days.
Supposing the relation to be exact, the theory shows, that the mutual action of these planets must produce mutual accelerations and retardations of their mean motions, and affords the periods and limits of the secular equations thence arising. These periods include several centuries. Again, because this relation is not precise, but the odd days nearly divide the periods already found, there must arise an equation of this secular equation, of which the period is immensely longer, and the maximum very minute. He shews that this retardation of Saturn is now at its maximum, and is diminishing again, and will, in the course of years, change to an acceleration.
This investigation of the small inequalities is the most intricate problem in mechanical philosophy, and has been completed only by very slow degrees, by the arduous efforts of the greatest mathematicians, of whom M. de la Grange is the most eminent. Some of his general results are very remarkable.
He demonstrates, that since the planets move in one direction, in orbits nearly circular, no mutual disturbances make any permanent change in the mean distances and mean periods of the planets, and that the periodic changes are confined within very narrow limits. The orbits can never deviate sensibly from circles. None of them ever has been or will be a comet moving in a very eccentric orbit. The ecliptic will never coincide with the equator, nor change its inclination above two degrees. In short the solar planetary system oscillates, as it were, round a medium state, from which it never swerves very far.
This theory of the planetary inequalities, founded on the universal law of mutual deflection, has given to our tables a precision, and a coincidence with observation, that surpasses all expectation, and insures the legitimacy of the theory. The inequalities are most sensible in the motions of Jupiter and Saturn; and these present themselves in such a complicated state, and their periods are so long, that ages were necessary for discovering them. Theory of by mere observation. In this respect, therefore, the Universal Gravitation theory has outstripped the observations, on which it is founded. It is very remarkable, that the periods which the Indians assign to these two planets, and which appeared so inaccurate that they hurt the credit of the science of those ancient astronomers, are now found precisely such as must have obtained about three thousand years before the Christian era; and thus they give an authenticity to that ancient astronomy. The periods which any nation of astronomers assign to those two planets would afford no contemptible mean for determining the age in which it was observed.
The following circumstance pointed out by La Place is remarkable: Suppose Jupiter and Saturn in conjunction in the first degree of Aries; twenty years after, the conjunction will happen in Sagittarius; and after other twenty years, in Leo. It will continue in these three signs for 200 years. In the year 200 it will happen in Taurus, Capricornus, and Virgo; in the next 200 years, it will happen in Gemini, Aquarius, and Libra; and in the next 200 years, it will happen in Cancer, Pisces, and Scorpio: then all begins again in Aries. It is probable that these remarkable periods of the oppositions of Jupiter and Saturn, progressive for 40 years, and oscillating during 160 more, occasioned the astrological division of the heavens into the four trigons, of fire, air, earth, and water. These relations of the signs, which compose a trigon, point out the repetitions of the chief irregularities of the solar system.
M. de la Place observes (in 1796), that the planet Herschel gives evident marks of the action of the rest; and that when these are computed and taken into the account of its bygone motions, they put it beyond doubt that it was seen by Flamstead in 1690, by Mayer in 1756, and by Monnier in 1769.
Sect. III. Of the Disturbances in the Elliptical Motion of the Comets.
Before the time of Sir Isaac Newton 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 from astronomical observation, that the comets pass through the planetary regions, and are generally invisible at a greater distance than that of Jupiter. Hence, finding that they were evidently within the sphere of 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 ellipsis, 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 for ever 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. 136, where S represents the sun, C the comet, and ABDE Theory of its orbit; wherein the distance of S and D far exceeds Universal that of S and A. Hence those bodies are sometimes found at a moderate distance from the sun, and appear within the planetary regions; at other times they ascend to vast distances, far beyond the orbit of Saturn, and thus become invisible.
That the comets do move in this manner is proved They move by our author from computations built upon the ob-eccentric fervations 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 ellipsis DEAB; on which foundation Sir I- Isaac teaches a method of finding the parabola in which calculate any comet moves, by three observations made upon it the motion 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 ellipsis than a parabola, though the ellipsis 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 computation agrees 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. The comets may be considerably affected by the planets. The very important phenomenon of the return of the comet of 1682, which was to decide whether they were revolving planets describing ellipses, or bodies which come but once into the planetary regions, and then retire for ever, caused the astronomers to consider this matter with great care. Halley had shewn, in a rough way, that this comet must have been considerably affected by Jupiter. Their motion near the aphelion must be very slow, so that a very small change of velocity or direction, while in the planetary regions, must considerably affect their periods. Halley thought that the action of Jupiter might change it half a year. M. Clairaut, by considering the disturbing forces of Jupiter and Saturn through the whole revolution, showed that the period then running would exceed the former nearly two years (618 days), and assigned the middle of April 1759 for the time of its perihelion. It really passed its perihelion on the 12th of March. This was a wonderful precision, when we reflect that the comet had been seen but a very few days in its former apparitions.
A comet observed by Mr Proprerin and others in 1771 has greatly puzzled the astronomers. Its motions appear to have been extremely irregular, and it certainly came so near Jupiter, that his momentary influence was at least equal to the sun's. It has not been recognized since that time, although there is a great probability that it is continually among the planets.
It is by no means impossible, that, in the course of ages, a comet may actually meet one of the planets. The effect of such a concourse must be dreadful; a change of the axis of diurnal rotation must result from it, and the sea must desert its former bed and overflow the new equatorial regions. The shock and the deluge must destroy all the works of man, and most of the race. The remainder, reduced to misery, must long struggle for existence, and all remembrance of former arts and events must be lost, and every thing must be invented anew. There are not wanting traces of such devastations in this globe: firata and things are now found on mountain tops which were certainly at the bottom of the ocean in former times; remains of tropical animals and plants are now dug up in the circumpolar regions.
Sect. IV. Of the Irregularities in the Moon's Motion.
The moon is acted on at once by the sun and the earth: but her motion round the earth is only disturbed by the difference of the sun's action on these two bodies. If the sun were at an infinite distance it would act upon them both equally and in a parallel direction; of course, their relative motion would not be disturbed. But its distance, though very great, when compared with that of the moon, cannot be considered as infinite. The moon is alternately nearer and farther from the sun than the earth, and the straight line which joins the centre of the sun and moon forms angles more or less acute with the radius vector of the earth. Of course the sun acts unequally, and in different directions, upon the earth and moon; and from that diversity of action, there ought to result irregularities in the lunar motions, depending on the respective situation of the sun and moon.
Some of these inequalities, however, would take place, though the moon if undisturbed by the sun had moved in a circle concentrical to the earth, and in the plane 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 it 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. 137. 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 shewn 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 earth 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 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. 138. 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 towards 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 obli- Theory of quity 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 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 everywhere 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: For the action 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 orbits 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 is 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 fun, that is, are drawn by the fun 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, Theory of 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 when 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 too much the greater. This difference in the distance between the earth and the sun produces a further effect upon the moon's 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 recess from the universal 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 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. 139, 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 CEF 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. 140, 141, 142, 143, 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. 126. 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. 144. 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. 141. 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 AECD, 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 is in K, is less than when she was in A; or, that 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 spherical 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, do not amount to a semicircle. 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 AL 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 AKGHI 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. 142. 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 AKGHI 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 AKG 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 Theory of 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. 143.), and that the plane which touches the line AKGI 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 AKGHI being convex to all the planes which touch it, the part HI will wholly fall between the plane QVR and the 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. 140. 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 Theory of 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. 145. 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 BTD, and B the node falling between A, the place where the moon would be in the full, and G the place where the moon 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 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 BGD, and continue concave to the plane BGD till it crofts 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 fame. 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 allo 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 Theory 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 further 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 mount 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 Theory of earth, or the apparent motion of the sun himself. In Universal Gravitation. 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 nodes come 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 inclination of the orbit in the apulse 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. 146, represent one of the moon's nodes placed between the point of opposition 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 moon. 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 concave towards that plane. But the moon is longer in passing from A to G, and therefore the nodes go backward 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 consequently at H must be less than at A. Similar effects, 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 Theory of 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 duplicate 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 shewn, 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 remote 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 different 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, but diminished at the new and full; and in the intermediate places the influence of the earth is sometimes lessened, sometimes affixed, by the action of that luminary. 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 retardation of her motion so often mentioned. But besides 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 perfect ellipsis. One particular in which the lunar orbit will differ from a perfect elliptic figure, consists in the places where the motion of the moon is perpendicular to the line drawn from itself to the earth. In an ellipsis, 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 become perpendicular to the aforementioned line, and the moon return to the place when it set out, and have recovered again its greatest distance. But the moon in its real motion, after setting out as before, sometimes makes more than half a revolution before its motion 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 before she has made half a revolution, and recover again its greatest distance before she has made an entire revolution. The place where the moon is at its greatest distance is called the moon's apogeeon, and the place of her greatest di- stance her perigeon; and this change of place, where the moon comes successively to its greatest distance from the earth, is called the motion of the apogeeon. The manner in which this motion of the apogeeon is caused by the sun, comes now to be explained.
Sir Isaac Newton has shewn, that if the moon were attracted towards the earth by a composition of two powers, one of which was reciprocally in the duplicate proportion of the distance from the earth, and the other reciprocally in the triplicate proportion of the same 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 antecedence and themselves, that is, the same way as the moon itself 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 D 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. 147, 148.) denote the earth, and suppose the moon in the point A its apogeeon or greatest distance from the earth, moving in the direc- tion AF perpendicular to AB, and acted upon from ned. 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* See Newton's Principia, book 1, prop. 44. and the moon be acted upon by the difference of those powers; in both these cases the line AE, which shall be descried 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 towards 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 apogeeon; 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 apogeeon 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 apogeeon moves forward, the whole centripetal power increases faster, with the decrease of distance, that 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 to explain the moon's motion by an ellipsis, which may be done without any sensible error, we may collect in general, that when the power by which the moon is attracted to the earth, by varying the distance, increases in a greater than the duplicate proportion of the distance diminished, a motion in consequence must be ascribed to the apogeeon; but that when the attraction increases in a smaller proportion than that just mentioned, the apogeeon must have given to it a motion in antecedence. It is then observed by Sir Isaac Newton, that the former of these cases obtains when the moon is in the conjunction and opposition, and the latter when she is in the quarters; so that in the former the apogeeon moves according to the order of the signs; in the other, the contrary way. But, as has been already mentioned, the disturbance given to the action of the earth by the sun in the conjunction and opposition, being near twice as great as in the quarters, the apogeeon will advance with a greater velocity than recede, and in the compass of a whole revolution of the moon will be carried in consequence.
Sir Isaac shows, in the next place, that when the line AB coincides with the line that joins the sun and earth, the progressive motion of the apogeeon, 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 so great as to exceed the progressive; so that in this case the apogeeon, 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 apogeeon 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 apogeeon, arising from this last consideration, are much greater than what arise from the other.
This unsteady motion of the apogeeon gives rise to another inequality in the motion of the moon herself, so that it cannot at all times be explained by the same ellipsis. For whenever the apogeeon moves in consequence, the motion of the luminary must be referred to an orbit more eccentric than what the moon would strictly describe, if the whole power by which the moon was the moon's acted upon in its passing from the apogeeon changed orbit, according to the reciprocal duplicate proportion of its distance from the earth, and by that means the moon did describe an immovable ellipsis: and when the apogeeon 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 other is the ellipsis 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 defend nearer to it. On the other hand, when the apogeeon 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 apogeeon 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 apogeeon 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 farther, 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 defend 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 apogeeon 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 will become more and more eccentric. As soon, however, as the apogon 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 apogon becomes retrograde, the diminution of this proportion will be still farther continued, until the apogon 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 apogon is in conjunction with the sun, or in opposition to it, and least of all when the apogon is in the quarters. These changes in the nodes, the inclination of the orbit to the plane of the earth's motion, in the apogon 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 concentrical to the earth, his action will cause her approach nearer in the conjunction and opposition than in the 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 assigned to them the same period. The inclination of the moon's orbit, when least, is an angle about one-eighteenth 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 computation: which is also agreeable to the general observations of astronomers.
There is one empirical equation of the moon's motion which the comparison of ancient and modern eclipses obliges the astronomers to employ, without being able to deduce it, like the rest, a priori, from the theory of an universal force inversely proportional to the square of the distance. It has therefore been considered as a stumbling block in the Newtonian philosophy. This is what is called the secular equation of the moon's mean motion. The mean motion is deduced from a comparison of distant observations. The time between them, being divided by the number of intervening revolutions, gives the average time of one revolution, or the mean lunar period. When the ancient Chaldean observations are compared with those of Hipparchus, we obtain a certain period; when those of Hipparchus are compared with some in the 9th century, we obtain a period somewhat shorter; when the last are compared with those of Tycho Brahe, we obtain one still shorter; and when Brahe's are compared with those of our day, we obtain the shortest period of all—and thus the moon's mean motion appears to accelerate continually; and the accelerations appear to be in the duplicate ratio of the times. The acceleration for the century which ended in 1700 is about 9 seconds of a degree; that is to say, the whole motion of the moon during the 17th century must be increased 9 seconds, in order to obtain its motion during the 18th; and as much must be taken from it, or added to the computed longitude, to obtain its motion during the 16th: and the double of this must be taken from the motion during the 16th, to obtain its motion during the 15th, &c. Or it will be sufficient to calculate the moon's mean longitude for any time past or to come by the secular motion which obtains in the present century, and then to add to this longitude the product of 9 seconds, multiplied by the square of the number of centuries which intervene. Thus having found the mean longitude for the year 1200, add 9 seconds, multiplied by 36, for fix centuries. By this method we shall make our calculation agree with the most ancient and all intermediate observations. If we neglect this correction, we shall differ more than a degree from the Chaldean observation of the moon's place in the heavens.
The mathematicians having succeeded so completely in deducing all the observed inequalities of the planetary motions, from the single principle, that the deflecting forces diminished in the inverse duplicate ratio of the distances, were fretted by this exception, the reality of which they could not contest. Many opinions were formed about its cause. Some have attempted to deduce it from the action of the planets on the moon; others have deduced it from the oblate form of the earth, and the translation of the ocean by the tides; others have supposed it owing to the resistance of the ether in the celestial spaces; and others have imagined that the action of the deflecting force requires time for its propagation to a distance: But their deductions have been proved unsatisfactory, and have by no means the precision and evidence that have been attained in the other questions of physical astronomy. At last M. de la Place, of the Royal Academy of Sciences at Paris, has happily succeeded, and deduced the secular equation of the moon from the Newtonian law of planetary deflection. It is produced in the following manner.
Suppose the moon revolving round the earth, undisturbed by any deflection toward the sun, and that the time of her revolution is exactly ascertained. Now let Newtonian law of planetary deflection be added. This diminishes her tendency to the earth in opposition and conjunction, and increases it in the quadratures; but the diminutions exceed the augmentations both in quantity and duration; and the excess is equivalent to \( \frac{1}{79} \)th of her tendency to the earth. Therefore this diminished tendency cannot retain the moon in the same orbit: she must retire farther from the earth, and describe an orbit which is less incurvated by \( \frac{1}{79} \)th part; and she must employ a longer time in a revolution. The period therefore which we observe, is not that which would have obtained had the moon been influenced by the earth alone. We should not have known that her natural period was increased, had the disturbing influence of the sun remained unchanged; but this varies in the inverse triplicate ratio of the earth's distance from the sun, and is therefore greater in our winter, when the earth is nearer to the sun. This is the source of the annual equation, by which the lunar period in January is made to exceed that in July nearly 24 minutes. The angular velocity of the moon is diminished in general \( \frac{1}{79} \), and this numerical coefficient varies in the inverse ratio of the cube Theory of the earth's distance from the sun. If we expand this Universal inverse cube of the earth's distance into a series arranged according to the fines and coines of the earth's mean motion, making the earth's mean distance unity, we shall find that the series contains a term equal to \( \frac{1}{3} \) of the square of the eccentricity of the earth's orbit. Therefore the expression of the diminution of the moon's angular velocity contains a term equal to \( \frac{1}{4} \) of this velocity multiplied by \( \frac{2}{7} \) of the square of the earth's eccentricity; or equal to the product of the square of the eccentricity, multiplied by the moon's angular velocity, and divided by 119.33 (\( \frac{1}{7} \) of 179). Did this eccentricity remain constant, this product would also be constant, and would still be confounded with the general diminution, making a constant part of it: but the eccentricity of the earth's orbit is known to diminish, and its diminution is the result of the universality of the Newtonian law of the planetary deflections. Although this diminution is exceedingly small, its effect on the lunar motion becomes sensible by accumulation in the course of ages. The eccentricity diminishing, the diminution of the moon's angular motion must also diminish, that is, the angular motion must increase.
During the 18th century, the square of the earth's eccentricity has diminished 0,000015325, the mean distance from the sun being = 1. This has increased the angular motion of the moon in that time 0,0000001285. As this augmentation is gradual, we must multiply the angular motion during the century by the half of this quantity, in order to obtain its accumulated effect. This will be found to be 6' very nearly, which exceeds that deduced from a most careful comparison of the motion of the last two centuries, only by a fraction of a second.
As long as the diminution of the square of the eccentricity of the earth's orbit can be supposed proportional to the time, this effect will be as the squares of the times. When this theory is compared with observations, the coincidence is wonderful indeed. The effect on the moon's motion is periodical, as the change of the solar eccentricity is, and its period includes millions of years. Its effect on the moon's longitude will amount to several degrees before the secular acceleration change to a retardation.
Those who are not familiar with the disquisitions of modern analysis, may conceive this question in the following manner.
Let the length of a lunar period be computed for the earth's distance from the sun for every day of the year. Add them into one sum, and divide this by their number, the quotient will be the mean lunar period. This will be found to be greater than the arithmetical medium between the greatest and the least. Then suppose the eccentricity of the earth's orbit to be greater, and make the same computation. The average period will be found still greater, while the medium between the greatest and least periods will hardly differ from the former. Something very like this may be observed without any calculation, in a case very similar. The angular velocity of the sun is inversely as the square of his distance. Look into the solar tables, and the greatest diurnal motion will be found 3673", and the least 3433". The mean of these is 3553", but the medium of the whole is 3548". Now make a similar observation in tables of the motion of the planet Mars, whose eccentricity is much greater. We shall find that the medium between the greatest and least exceeds the true medium of all in a much greater proportion.
It has been supposed by some philosophers that the moon was originally a comet, which passing very near the earth, had been made to revolve round her by the force of attraction. But if we calculate ever so far backwards, we still find the moon revolving round the earth as the planets round the sun, which could not be the case if this opinion were true. Hence it follows, that neither the moon nor any of the satellites have ever been comets.
Sect. V. Of Irregularities in the Satellites of Jupiter.
The subferviency of the eclipses of Jupiter's satellites to geography and navigation had occasioned their motions to be very carefully observed, ever since these uses of them were first suggested by Galileo; and their theory is as far advanced as that of the primary planets. It has peculiar difficulties. Being very near to Jupiter, the great deviation of his figure from perfect sphericity makes the relation between their distances from his centre and their gravitations toward it vastly complicated. But this only excited the mathematicians so much the more to improve their analysis; and they saw, in this little system of Jupiter and his attendants, an epitome of the solar system, where the great rapidity of the motions must bring about in a short time every variety of configuration or relative position, and thus give us an example of those mutual disturbances of the primary planets, which require thousands of years for the discovery of their periods and limits. We have derived some very remarkable and useful pieces of information from this investigation; and have been led to the discovery of the eternal durability of the solar system, a thing which Newton greatly doubted of.
Mr Pound had observed long ago, that the irregularities of the three interior satellites were repeated in a period of 437 days; and this observation is found to be just to this day.
<table> <tr> <th></th> <th>Days.</th> <th>H.</th> <th>M.</th> </tr> <tr> <td>247 revolutions of the first occupy</td> <td>437</td> <td>3</td> <td>44</td> </tr> <tr> <td>123</td> <td>second</td> <td>437</td> <td>3</td> <td>42</td> </tr> <tr> <td>61</td> <td>third</td> <td>437</td> <td>3</td> <td>36</td> </tr> <tr> <td>26</td> <td>fourth</td> <td>435</td> <td>14</td> <td>16</td> </tr> </table>
This naturally led mathematicians to examine their motions, and see in what manner their relative positions or configurations, as they are called, corresponded to this period: and it is found, that the mean longitude of the first satellite, minus thrice the mean longitude of the second, plus twice the mean longitude of the third, always made 180 degrees. This requires that the mean motion of the first, added to twice that of the third, shall be equal to thrice the mean motion of the second. This correspondence of the mean motions is of itself a singular thing, and the odds against its probability seems infinitely great; and when we add to this the particular positions of the satellites in any one moment, which is necessary for the above constant relation of their longitudes, the improbability of the coincidence, as a thing quite fortuitous, becomes infinitely greater. Doubts were first entertained of the coincidence, Theory of coincidence, because it was not indeed accurate to a universal second. The result of the investigation is curious. When we follow out the consequences of mutual gravitation, we find, that although neither the primitive motions of projection, nor the points of the orbit from which the satellites were projected, were precisely such as suited these observed relations of their revolutions and their contemporaneous longitudes; yet if they differed from them only by very minute quantities, the mutual gravitations of the satellites would in time bring them into those positions, and those states of mean motion, that would induce the observed relations; and when they are once induced they will be continued for ever. There will indeed be a small equation, depending on the degree of unfavourableness of the first motions and positions; and this causes the whole system to oscillate, as it were, a little, and but a very little way on each side of this exact and permanent state. The permanency of these relations will not be destroyed by any secular equations arising from external causes; such as the action of the fourth satellite, or of the sun, or of a resisting medium; because their mutual actions will distribute this equation as it did the original error.
For a full discussion of this curious but difficult subject, we refer the reader to the dissertations of La Grange and La Place, and to the tables lately published by Delambre. These mathematicians have shown, that if the mass of Jupiter be represented by unity, that of his satellites will be represented by the following numbers.
<table> <tr> <th></th> <th></th> </tr> <tr> <td>First satellite</td> <td>0.0000172011</td> </tr> <tr> <td>Second satellite</td> <td>0.0000237103</td> </tr> <tr> <td>Third satellite</td> <td>0.0000872128</td> </tr> <tr> <td>Fourth satellite</td> <td>0.000544681</td> </tr> </table>
SECT. VI. Of Saturn's Ring.
The most important addition (in a philosophical view) which has been made to astronomical science since the discovery of the aberration of light and the nutation of the earth's axis, is that of the rotation of Saturn's ring. The ring itself is an object quite peculiar; and when it was discovered that all the bodies which had any immediate connexion with a planet gravitated toward that planet, it became an interesting question to ascertain what was the nature of this ring? What supports this immense arch of heavy matter without its resting on the planet? What maintains it in perpetual concentricity with the body of Saturn, and keeps its surface in one invariable position?
The theory of universal gravitation tells us what things are possible in the solar system; and our conjectures about the nature of this ring must always be regulated by the circumstance of its gravitation to the planet. Philosophers had at first supposed it to be a luminous atmosphere, thrown out into that form by the great centrifugal force arising from a rotation: but its well-defined edge, and, in particular, its being two very narrow rings, extremely near each other, yet perfectly separate, rendered this opinion of its constitution more improbable.
Dr. Herschel's discovery of brighter spots on its surface, and that those spots were permanent during the whole time of his observation, seem to make it more probable that the parts of the ring have a solid connexion. Mr. Herschel has discovered, by the help of those spots, that the ring turns round its axis, and that this axis is also the axis of Saturn's rotation. The time of rotation is 10h. 32\frac{1}{2}''. But the other circumstances are not narrated with the precision sufficient for an accurate comparison with the theory of gravity. He informs us, that the radii of the four edges of the ring are 599', 741', 774', 830', of a certain scale, and that the angle subtended by the ring at the mean distance from the earth is 46\frac{1}{2}'''. Therefore its elongation is 23\frac{1}{2}''. The elongation of the second Cassilian satellite is 56'', and its revolution is 2d. 17h. 44''. This should give, by the third law of Kepler, 17h. 10' for the revolution of the outer edge of the ring, or rather of an atom of that edge, in order that it may maintain itself in equilibrium. The same calculation applied to the outer edge of the inner ring gives about 13h. 36\frac{1}{2}'', and we obtain 11h. 16'' for the inner edge of this ring. Such varieties are inconsistent with the permanent appearance of a spot. We may suppose the ring to be a luminous fluid or vapour, each particle of which maintains its situation by the law of planetary revolution. In such a state, it would consist of concentric strata, revolving more slowly as they were more remote from the planet, like the concentric strata of a vortex, and therefore having a relative motion incompatible with the permanency of any spot. Besides, the rotation observed by Herschel is too rapid even for the innermost part of the ring. We think therefore that it consists of cohering matter, and of considerable tenacity, at least equal to that of a very clammy fluid, such as melted glass.
We can tell the figure which a fluid ring must have, so that it may maintain its form by the mutual gravitation of its particles to each other, and their gravitation to the planet. Suppose it cut by a meridian. It may be in equilibrium if the section is an ellipse, of which the longer axis is directed to the centre of the planet, and very small in comparison with its distance from the centre of the planet, and having the revolution of its middle round Saturn, such as agree with the Keplerian law. These circumstances are not very consistent with the dimensions of Saturn's inner ring. The distance between the middle of its breadth and the centre of Saturn is 670', and its breadth is 161', nearly one-fourth of the distance from the centre of Saturn. De la Place says, that the revolution of the inner ring observed by Herschel is very nearly that required by Kepler's law: but we cannot see the grounds of this assertion. The above comparison with the second Cassilian satellite shows the contrary. The elongation of that satellite is taken from Bradley's observations, as is also its periodic time. A ring of detached particles revolving in 10h. 32\frac{1}{2}' must be of much smaller diameter than even the inner edge of Saturn's ring. Indeed the quantity of matter in it might be such as to increase the gravitation considerably; but this would be seen by its disturbing the seventh and fifth satellites, which are exceedingly near it. We cannot help thinking, therefore, that it consists of matter which has very considerable probable tenacity. An equatorial zone of matter, tenacious enough, like melted glass, and whirled briskly round, might be thrown off, and, retaining its great velocity, would stretch out while whirling, enlarging in diameter and diminishing in thickness or breadth, or both, till the centrifugal force was balanced by the united force of gravity Theory of gravity and tenacity. We find the the equilibrium Universal will not be sensibly disturbed by considerable deviations, such as equal breadth, or even want of flatness. Such inequalities appear on the ring at that time of its disparition, when its edge is turned to the sun or to us. The appearances of its different sides are then considerably different.
Such a ring or rings must have an oscillatory motion round the centre of Saturn, in consequence of their mutual action, and the action of the sun, and their own irregularities: but there will be a certain position which they have a tendency to maintain, and to which they will be brought back, after deviating from it, by the ellipticity of Saturn, which is very great. The sun will occasion a nutation of Saturn's axis and a precession of his equinoxes, and this will drag along with it both the rings and the neighbouring satellites.
The atmosphere which surrounds a whirling planet cannot have all its parts circulating according to the third law of Kepler. The mutual attrition of the planet, and of the different strata, arising from their different velocities, must accelerate the flowly moving strata, and retard the rapid, till all acquire a velocity proportional to their distance from the axis of rotation; and this will be such that the momentum of rotation of the planet and its atmosphere remains always the same. It will swell out at the equator, and sink at the poles, till the centrifugal force at the equator balances the height of a superficial particle. The greatest ratio which the equatorial diameter can acquire to the polar axis is that of four to three, unless a cohesive force keeps the particles united, so that it constitutes a liquid, and not an elastic fluid like air; and an elastic fluid cannot form an atmosphere bounded in its dimensions, unless there be a certain rarity which takes away all elasticity. If the equator swells beyond the dimension which makes the gravitation balance the centrifugal force, it must immediately dissipate.
If we suppose that the atmosphere has extended to this limit, and then condenses by cold, or any chemical or other cause different from gravity, its rotation necessarily augments, preferring its former momentum, and the limit will approach the axis; because a greater velocity produces a greater centrifugal force, and requires a greater gravitation to balance it. Such an atmosphere may therefore desert, in succession, zones of its own matter in the plane of its equator, and leave them revolving in the form of rings. It is not unlikely that the rings of Saturn may have been furnished in this very way; and the zones, having acquired a common velocity in their different strata, will preserve it; and they are susceptible of irregularities arising from local causes at the time of their separation, which may afford permanent spots.
SECT. VII. Of the Atmospheres of the Planets.
By atmosphere is meant a rare, transparent, compressible, and elastic fluid surrounding a body. It is supposed that all the heavenly bodies possess atmospheres. The atmosphere of the earth is familiar to all its inhabitants. Observation points out the atmospheres of the sun and of Jupiter; but that of the other planets is scarcely perceptible.
The atmosphere becomes rarer in proportion to its distance from the body to which it belongs, in consequence of its elasticity, which causes it to dilate the Universal more the less it is compressed. If its most remote parts were still possessed of elasticity, they would separate indefinitely, and the whole would be scattered through space. To prevent this effect, it is necessary that the elasticity should diminish at a greater rate than the compressing force, and that when it reaches a certain degree of rarity its elasticity should vanish altogether.
All the atmospheric strata must gradually acquire the fame rotatory motion with the bodies to which they belong in consequence of the continual friction to which their different parts must be subjected, which will gradually accelerate or retard the different parts till a common motion is produced. In all these changes, and indeed in all those which the atmosphere undergoes, the sum of the products of the particles of the body and of its atmosphere multiplied by the areas described round their common centres of gravity by their radii vectors projected in the plane of the equator continue always the same, the times being the fame. If we suppose then, by any cause whatever, the height of the atmosphere is diminished, and a portion of it condenses on the surface of the planet; the consequence will be, that the rotatory motion of the planet and of its atmosphere will be accelerated. For, the radii vectors of the areas described by the particles of the primitive atmosphere becoming shorter, the sum of the products of all these particles by the corresponding areas cannot remain the same unless the rotatory motion augment.
At the upper surface of the atmosphere the fluid is retained only by its weight. Its figure is such that the direction resulting from the combination of the centrifugal forces and the attracting forces is perpendicular to it. It is flattened at the poles, and more convex at the equator. But this flattening has its limits. When a maximum the axis of the poles is to that at the equator as 2 to 3.
At the equator the atmosphere can only extend to the place where the centrifugal force and gravitation exactly balance each other; for if it pass that limit, it will be dissipated altogether. Hence it follows that the solar atmosphere does not extend as far as Mercury; consequently it is not the cause of the zodiacal light which appears to extend beyond even the earth's orbit.
The place where the centrifugal force and gravitation balance each other is so much the nearer a body the more rapid its rotatory motion is. If we suppose the atmosphere to extend to that limit, and then to condense by cooling, &c. at the surface of the planet the rotatory motion will increase in rapidity in proportion to this condensation, and the limit of the height of the atmosphere will constantly approach the planet. The atmosphere would of course abandon successively zones of fluid in the plane of the equator, which would continue to circulate round the body. We have shown in the last section that Saturn's ring may owe its origin to this cause.
We may add also, that the action of another body may considerably change the constitution of this atmosphere. Thus, supposing that the moon had we no originally an atmosphere, the limit will be that distance from the moon where the centrifugal force, arising from the moon's rotation, added to the gravitation Part IV.
Theory of Universal Gravitation.
The moon balances the gravitation to the earth. If the moon be \( \frac{1}{3} \)th of the earth, this limit will be about \( \frac{4}{5} \)th of the moon's distance from the earth. If at this distance the elasticity of the atmosphere is not annihilated by its rarefaction, it will be all taken off by the earth, and accumulate round it. This may be the reason why we see no atmosphere about the moon.
SECT. VII. Of the Tides.
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, 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 towards 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 ZIBL n KFNZ, whose longer axis n OZ 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 towards 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 moon to the moon again in 24\( \frac{1}{2} \) 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 at first 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 (E) 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 Fig. 150. 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
(e) 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. Theory of Universal Gravitation
Influence of the sun in raising tides.
Why they are not highest when the moon is in the meridian.
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 M is generally past the north and south meridian, as at p, when it is high water at Z and at n. The reason is obvious: for though the moon's attraction was to cease altogether when the 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 have done; and as experience shows that the day is hotter about three in the afternoon, than when the sun is on the meridian, because of the increase made to the heat already imparted.
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 several times, even though the sun and moon were annihilated, or their influence should cease; as, if a bason of water were agitated, the water would continue to move for some time after the bason was left to hand fill; 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. 157, 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 go 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 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 the tropic of Cancer, C the arctic circle, P the north pole, and the curves, 1, 2, 3, &c. 24 meridians or hour circles, intersecting each other in the poles: 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 ETP 6 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 the 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 I (under D) revolving as formerly, goes sooner from I to II (under F) than from II to I, because the parallel it describes is cut into unequal segments by the high-water circle HCO: but the points I and II 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 1/2 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. 153, when the moon is vertical to the equator EQ, 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 height 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, fo 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 return, 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 channels 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 likewise 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 likewise 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 likewise 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 likewise 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 cumulated 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 through 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 so small, that when the moon is vertical she attracts every part of them alike, and therefore by rendering all the water equally light, no part of it can be raised higher than another. The Mediterranean and Baltic seas have very small elevations, because the inlets by which they communicate with the ocean are so narrow, that they cannot, in so short a time, receive or discharge enough to raise or sink their surfaces sensibly.
For a more complete discussion of this important subject, we refer the reader to the article TIDE.
SECT. IX. Of the Precession of the Equinoxes, and the Nutation of the Earth's Axis.
It now remains to consider the precession of the equinoctial points, with its equations, arising from the nutation of the earth's axis as a physical phenomenon, and to endeavour to account for it upon those mechanical principles which have so happily explained all the other phenomena of the celestial motions.
This did not escape the penetrating eye of Sir Isaac Newton; and he quickly found it to be a consequence, and the most beautiful proof, of the universal gravitation of all matter to all matter; and there is no part of his immortal work where his sagacity and fertility of resource shine more conspicuously than in this investigation. It must be acknowledged, however, that Newton's investigation is only a shrewd guess, founded on assumptions, of which it would be extremely difficult to demonstrate either the truth or falsity, and which required the genius of a Newton to pick out in such a complication of abstract circumstances. The subject has occupied the attention of the first mathematicians of Europe since his time; and is still considered as the most curious and difficult of all mechanical problems. The most elaborate and accurate dissertations on the precession of the equinoxes are those of Sylvabella and Walnelly, in the Philosophical Transactions, published about the year 1754; that of Thomas Simpson, published in his Miscellaneous Tracts; that of Father Frisius, in the Memoirs of the Berlin Academy, and afterwards, with great improvements, in his Cosmographia; that of Universal Euler in the Memoirs of Berlin; that of D'Alembert in a separate dissertation; and that of De la Grange on the Libration of the Moon, which obtained the prize in the Academy of Paris in 1769. We think the dissertation of Father Frisius the most perspicuous of them all, being conducted in the method of geometrical analysis; whereas most of the others proceed in the fluxionary and symbolic method, which is frequently deficient in distinct notions of the quantities under consideration, and therefore does not give us the same perspicuous conviction of the truth of the results. In a work like ours, it is impossible to do justice to the problem, without entering into a detail which would be thought extremely disproportionate to the subject by the generality of our readers. Yet those who have the necessary preparation of mathematical knowledge, and wish to understand the subject fully, will find enough here to give them a very distinct notion of it; and in the article ROTATION, they will find the fundamental theorems, which will enable them to carry on the investigation. We shall first give a short sketch of Newton's investigation, which is of the most palpable and popular kind, and is highly valuable, not only for its ingenuity, but also because it will give our unlearned readers distinct and satisfactory conceptions of the chief circumstances of the whole phenomena.
Let S (fig. 134.) be the sun, E the earth, and M the moon, moving in the orbit NMCD n, which cuts the plane of the ecliptic in the line of the nodes N n, and investigates has one half raised above it, as represented in the figure, the other half being hid below the ecliptic. Suppose this orbit folded down; it will coincide with the ecliptic in the circle N m c d n. Let EX represent the axis of this orbit, perpendicular to its plane, and therefore inclined to the ecliptic. Since the moon gravitates to the sun in the direction MS which is all above the ecliptic, it is plain that this gravitation has a tendency to draw the moon towards the ecliptic. Suppose this force to be such that it would draw the moon down from M to i in the time that she would have moved from M to r, in the tangent to her orbit. By the combination of these motions, the moon will desert her orbit, and describe the line MR, which makes the diagonal of the parallelogram; and if no farther action of the sun be supposed, she will describe another orbit M d n', lying between the orbit M C D n and the ecliptic, and she will come to the ecliptic, and pass through it in a point n', nearer to M than n is, which was the former place of her descending node. By this change of orbit, the line EX will no longer be perpendicular to it; but there will be another line E x, which will now be perpendicular to the new orbit. Also the moon, moving from M to r, does not move as if she had come from the ascending node N, but from a point N lying beyond it; and the line of the nodes of the orbit in this new position is N' n'. Also the angle MN' m is less than the angle MN m.
Thus the nodes shift their places in a direction opposite to that of her motion, or move to the westward; the axis of the orbit changes its position, and the orbit itself changes its inclination to the ecliptic. These momentary changes are different in different parts of the orbit, according to the position of the line of the nodes. nodes. Sometimes the inclination of the orbit is increased, and sometimes the nodes move to the eastward. But, in general, the inclination increases from the time that the nodes are in the line of fyzigee, till they get into quadrature, after which it diminishes till the nodes are again in fyzigee. The nodes advance only while they are in the octants after the quadratures, and while the moon passes from quadrature to the node, and they recede in all other situations. Therefore the recess exceeds the advance in every revolution of the moon round the earth, and, on the whole, they recede.
What has been said of one moon, would be true of each of a continued ring of moons surrounding the earth, and they would thus compose a flexible ring, which would never be flat, but waved, according to the difference (both in kind and degree) of the disturbing forces acting on its different parts. But suppose these moons to cohere, and to form a rigid and flat ring, nothing would remain in this ring but the excess of the contrary tendencies of its different parts. Its axis would be perpendicular to its plane, and its position in any moment will be the mean position of all the axes of the orbits of each part of the flexible ring; therefore the nodes of this rigid ring will continually recede, except when the plane of the ring passes through the sun, that is, when the nodes are in fyzigee; and (says Newton) the motion of these nodes will be the same with the mean motion of the nodes of the orbit of one moon. The inclination of this ring to the ecliptic will be equal to the mean inclination of the moon's orbit during any one revolution which has the same situation of the nodes. It will therefore be least of all when the nodes are in quadrature, and will increase till they are in fyzigee, and then diminish till they are again in quadrature.
Suppose this ring to contract in dimensions, the disturbing forces will diminish in the same proportion, and in this proportion will all their effects diminish. Suppose its motion of revolution to accelerate, or the time of a revolution to diminish; the linear effects of the disturbing forces being as the squares of the times of their action, and their angular effects as the times, those errors must diminish also on this account; and we can compute what those errors will be for any diameter of the ring, and for any period of its revolution. We can tell, therefore, what would be the motion of the nodes, the change of inclination, and deviation of the axis, of a ring which would touch the surface of the earth, and revolved in 24 hours; nay, we can tell what these motions would be, should this ring adhere to the earth. They must be much less than if the ring were detached; for the disturbing forces of the ring must drag along with it the whole globe of the earth. The quantity of motion which the disturbing forces would have produced in the ring alone, will now (says Newton) be produced in the whole mass; and therefore the velocity must be as much less as the quantity of matter is greater: But still this can be computed.
Now there is such a ring on the earth: for the earth is not a sphere, but an elliptical spheroid. Sir Isaac Newton therefore engaged in a computation of the effects of the disturbing force, and has exhibited a most beautiful example of mathematical investigation. He first asserts, that the earth must be an elliptical spheroid, whose polar axis is to its equatorial diameter as 229 to 230.
Then he demonstrates, that if the sine of the inclination of the equator be called \( \pi \), and if \( t \) be the number of days (sidereal) in a year, the annual motion of a detached ring will be \( 360^\circ \times \frac{3\sqrt{1-\pi^2}}{4t} \). He then shows that the effect of the disturbing force on this ring is to its effect on the matter of the same ring, distributed in the form of an elliptical stratum (but still detached) as 5 to 2; therefore the motion of the nodes will be \( 360^\circ \times \frac{3\sqrt{1-\pi^2}}{10t} \), or \( 16' 16'' 24''' \) annually. He then proceeds to show, that the quantity of motion in the sphere is to that in an equatorial ring revolving in the same time, as the matter in the sphere to the matter in the ring, and as three times the square of a quadrantal arch to two squares of a diameter, jointly: Then he shows, that the quantity of matter in the terrestrial sphere is to that in the protuberant matter of the spheroid, as 52900 to 461 (supposing all homogeneous). From these premises it follows, that the motion of \( 16' 16'' 24''' \), must be diminished in the ratio of 10717 to 100, which reduces it to \( 9'' 07''' \) annually. And this (he says) is the precession of the equinoxes, occasioned by the action of the sun; and the rest of the \( 501'' \), which is the observed precession, is owing to the action of the moon, nearly five times greater than that of the sun. This appeared a great difficulty: for the phenomena of the tides show that it cannot much exceed twice the sun's force.
Nothing can exceed the ingenuity of this process. His determination is so celebrated and candid commentator, Daniel Bernoulli, say (in his Dissertation on the Tides, of the form which shared the prize of the French Academy with those of M'Laurin and Euler), that Newton saw through a veil which no one else could discover with a microscope magnified in the light of the meridian fun. His determination of the form and dimensions of the earth, which is the foundation of the whole process, is not offered as any thing better than a probable guess, in re difficillima; and it has been since demonstrated with geometrical rigour by M'Laurin.
His next principle, that the motion of the nodes of the rigid ring is equal to the mean motion of the nodes of the moon, has been most critically discussed by the first mathematicians, as a thing which could neither be proved nor refuted. Frisius has at least shewn it to be a mistake, and that the motion of the nodes of the ring is double the mean motion of the nodes of a single moon; and that Newton's own principles should have produced a precession of 18\( \frac{1}{4} \) seconds annually, which removes the difficulty formerly mentioned.
His third assumption, that the quantity of motion of the ring must be shared with the included sphere, was acquiesced in by all his commentators, till D'Alembert and Euler, in 1749, showed that it was not the quantity of motion round an axis of rotation which remained the same, but the quantity of momentum or rotary effort. The quantity of motion is the product of every particle by its velocity; that is, by its distance from the axis; while its momentum, or power of producing rotation, is as the square of that distance, and is to be had by taking the sum of each particle multiplied by the square of its distance from the axis. Since the Theory of Universal Gravitation.
We proceed now to the examination of this phenomenon upon the fundamental principles of mechanics. Because the mutual gravitation of the particles of matter in the solar system is in the inverse ratio of the squares of the distance, it follows, that the gravitations of the different parts of the earth to the sun or to the moon are unequal. The nearer particles gravitate more than those that are more remote.
Let PQ p E (fig. 155.), be a meridional section of the terrestrial sphere, and PO p g the section of the inscribed sphere. Let CS be a line in the plane of the ecliptic passing through the sun, so that the angle ECS is the sun's declination. Let NCM be a plane passing through the centre of the earth at right angles to the plane of the meridian PQ p E; NCM will therefore be the plane of illumination.
In consequence of the unequal gravitation of the matter of the earth to the sun, every particle, such as B, is acted on by a disturbing force parallel to CS, and proportional to BD, the distance of the particle from the plane of illumination; and this force is to the gravitation of the central particle to the sun, as three times BD is to CS, the distance of the earth from the sun.
Let AB a be a plane passing through the particle B, parallel to the plane EQ of the equator. This section of the earth will be a circle, of which A a is a diameter, and Q q will be the diameter of its section with the inscribed sphere. These will be two concentric circles, and the ring by which the section of the spheroid exceeds the section of the sphere will have AQ for its breadth; F p is the axis of figure.
<table> <tr> <th>EC</th> <th>their difference,</th> <th>CL</th> <th>QL</th> <th>The periphery of a circle to radius r</th> <th>The disturbing force at the distance r from the plane NCM</th> <th>The sine of declination ECS</th> <th>The cosine of ECS</th> </tr> <tr> <td>OC or PC</td> <td>\( \frac{a^2 - b^2}{a + b} \)</td> <td></td> <td></td> <td>\( \pi d^2 - x^2 \)</td> <td>f</td> <td>m</td> <td>n</td> </tr> </table>
It is evident, that with respect to the inscribed sphere, the disturbing forces are completely compensated, for every particle has a corresponding particle in the adjoining quadrant, which is acted on by an equal and opposite force. But this is not the case with the protuberant matter which makes up the spheroid. The segments NS s n and MT t m are more acted on than the segments NT t n and MS s m; and thus there is produced a tendency to a conversion of the whole earth, round an axis passing through the centre C, perpendicular to the plane PQ p E. We shall distinguish this motion from all others to which the spheroid may be subject, by the name Libration. The axis of this libration is always perpendicular to that diameter of the equator over which the sun is, or to that meridian in which he is.
Prob. I. To determine the momentum of libration corresponding to any position of the earth respecting the sun, that is, to determine the accumulated energy of the disturbing forces on all the protuberant matter of the spheroid.
Let B and b be two particles in the ring formed by the revolution of AQ, and so situated, that they are at equal distances from the plane NM; but on opposite sides of it. Draw BD, b d, perpendicular to NM, and FLG perpendicular to LT.
Then, because the momentum, or power of producing rotation, is as the force and as the distance of its line of direction from the axis of rotation, jointly, the combined momentum of the particles B and b, will be f.BD.DC-f.bddc, (for the particles B and b are urged in contrary directions). But the momentum of B is f.BF.DC+f.FD.DC, and that of b is f.b.Gd.C-f.d.Gd.C; and the combined momentum is f.BF.Dd-f.FD.DC+dC, = 2f.BF.LF-2f.LT.TC.
Because m and n are the sine and cosine of the angle ECS or LTC, we have LT=m.CL, and CT=n.CL, and LF=m.BL, and BF=n.BL. This gives the momentum \(=2f\ m\ n\ BL^2-CL^2\).
The breadth AQ of the protuberant ring being very small, we may suppose, without any sensible error, that all the matter of the line AQ is collected in the point Q; and, in like manner, that the matter of the whole ring is collected in the circumference of its inner circle, and that B and b now represent, not single particles, but the collected matter of lines such as AQ, which terminate at B and b. The combined momentum of two such lines will therefore be \(2m\ n f.AQ.BL^2-CL^2\).
Let the circumference of each parallel of latitude be divided into a great number of indefinitely small and equal parts. The number of such parts in the circumference, of which Q q is the diameter, will be \(n.QL\). To each pair of these there belongs a momentum \(2m\ n f.AQ.BL^2-CL^2\). The sum of all the squares of BL, which can be taken round the circle, is one half of as many squares of the radius CL: for BL is the fine of an arch, and the sum of its square and the square of its corresponding cosine is equal to the square of the radius. Therefore the sum of all the squares of the fines, together with the sum of all the squares of the cosines, is equal to the sum of the same number of squares of the radius; and the sum of the squares of the fines is equal to the sum of the squares of the corresponding cosines; therefore the sum of the squares of the radius is double of either sum. Therefore \(f n.QL.BL^2=\frac{1}{8} n.QL^3.QL^2\). In like manner the sum of the number \(n.QL\) of CL's will be \(n.QL.CL^2\). These sums, taken for the semicircle, are \(\frac{1}{4} n.QL^3.QL^2\), and \(\frac{1}{4} n.QL.CL^2\), or \(\frac{1}{4} n.QL^2+QL^2\), and \(n.QL^2.CL^2\): therefore the momentum of the whole ring will be \(2m\ n f.AQ.QL.n(\frac{1}{4} QL^2-\frac{1}{4} CL^2)\): for the momentum of the ring is the combined momenta of a number of pairs, and this number is \(\frac{1}{4} n.QL\).
By the ellipse we have OC : QL = EO : AQ, and AQ=QL \(\frac{EO}{OC}\), =QL \(\frac{d}{b}\); therefore the momentum of the ring is \(2m\ n f\frac{d}{b}QL^2.n(\frac{1}{4} QL^2-\frac{1}{4} CL^2)\), =\(m n f\frac{a}{b}\) \(QL^2.n(\frac{1}{4} QL^2-CL^2)\): but \(QL^2=b^2-x^2\); therefore \(\frac{1}{4} QL^2\) therefore the momentum of the ring is \( m n f \frac{d}{b} \Pi \left( b^4 - x^2 \right) \)
\[ \left( \frac{b^4 - 3x^2}{2} \right) = m n f \frac{d}{b} \Pi \left( \frac{b^4 - 4b^2 x^2 + 3x^4}{2} \right), = m n f \frac{d}{2b} \Pi (b^4 - 4b^2 x^2 + 3x^4). \]
If we now suppose another parallel extremely near to A a, as represented by the dotted line, the distance Ll between them being x, we shall have the fluxion of the momentum of the spheroid
\[ m n f \frac{d}{2b} \Pi (b^4 x - 4b^2 x^3 + 3x^4), \]
of which the fluent is
\[ m n f \frac{d}{2b} \Pi \left( \frac{3}{5} b^4 x - 4b^2 x^3 + 3x^4 \right). \]
This expresses the momentum of the zone E.A a Q, contained between the equator and the parallel of latitude A a. Now let x become = b, and we shall obtain the momentum of the hemispheroid
\[ m n f \frac{d}{2b} \Pi \left( b^5 - \frac{4}{5} b^5 + \frac{1}{3} b^5 \right), \]
and that of the spheroid
\[ m n f \frac{d}{b} \Pi \left( b^5 - \frac{4}{5} b^5 + \frac{1}{3} b^5 \right) = \frac{4}{15} m n f d \Pi b^4. \]
This formula does not express any motion, but only a pressure tending to produce motion, and particularly tending to produce a libration by its action on the cohering matter of the earth, which is affected as a number of levers. It is similar to the common mechanical formula w. d, where w means a weight, and d its distance from the fulcrum of the lever.
It is worthy of remark, that the momentum of this protuberant matter is just \( \frac{4}{15} \) of what it would be if it were all collected at the point O of the equator: for the matter in the spheroid is so that in the inscribed sphere as \( a^2 \) to \( b^2 \), and the contents of the inscribed sphere is \( \frac{4}{7} \Pi b^3 \). Therefore \( a^2 : a^2 - b^2 = \frac{4}{7} \Pi b^2 : \frac{4}{7} \Pi b^3 \), or \( \frac{a^2 - b^2}{a^2} \), which is the quantity of protuberant matter. We may, without sensible error, suppose \( \frac{a^2 - b^2}{a^2} = 2d \); then the protuberant matter will be \( \frac{4}{7} \Pi b^3 d \). If all this were placed at O, the momentum would be \( \frac{4}{7} \Pi db^3 f OH \cdot HC = \frac{4}{7} m n f d b^4 \), because OH·HC = mnbd; now \( \frac{4}{7} \) is 5 times \( \frac{4}{35} \).
Also, because the sum of all the rectangles OH·HC round the equator is half of as many squares of OC, it follows that the momentum of the protuberant matter placed in a ring round the equator of the sphere, or spheroid, is one half of what it would be if collected in the point G or E; whence it follows that the momentum of the protuberant matter in its natural place is two-fifths of what it would be if it were diffused in an equatorial ring. It was in this manner, that Sir Isaac Newton was enabled to compare the effect of the sun's action on the protuberant matter of the earth, with his effect on a rigid ring of moons. The preceding investigation of the momentum is nearly the same with his, and appears to us greatly preferable in point of periplicity to the fluxionary solutions given by later authors. These indeed have the appearance of greater accuracy, because they do not suppose all the protuberant matter to be condensed on the surface of the inscribed sphere: nor were we under the necessity of doing this; only it would have led to very complicated expressions had we supposed the matter in each line AQ collected in its centre of oscillation or gyration. We made a compensation for the error introduced by this which may amount to \( \frac{1}{11} \) of the whole, and should not be neglected, by taking d as equal to \( \frac{a^2 - b^2}{2a} \) instead of \( \frac{a^2 - b^2}{a + b} \). The consequence is, that our formula is the same with that of the later authors.
Thus far Sir Isaac Newton proceeded with mathematical rigour; but in the application he made two assumptions, or, as he calls them hypothesises, which have been found to be unwarranted. The first was, that when the ring of protuberant matter is connected with the inscribed sphere, and subjected to the action of the disturbing force, the same quantity of motion is produced in the whole mass as in the ring alone. The second was, that the motion of the nodes of a rigid ring of moons is the same with the mean motion of the nodes of a solitary moon. But we are now able to demonstrate, that it is not the quantity of motion, but of momentum, which remains the same, and that the nodes of a rigid ring move twice as fast as those of a single particle. We proceed therefore to,
PROB. II. To determine the deviation of the axis, Effects of and the retrograde motion of the nodes which result the libratory momentum of the earth's protuberant matter.
But here we must refer our readers to some fundamental propositions of rotary motions which are demonstrated in the article ROTATION.
If a rigid body is turning round an axis A, passing through its centre of gravity with the angular velocity a, and receives an impulse which alone would cause it to turn round an axis B, also passing through its centre of gravity, with the angular velocity b, the body will now turn round a third axis C, passing through its centre of gravity, and lying in the plane of the axis A and B, and the fine of the inclination of this third axis to the axis A will be to the fine of the inclination to the axis B as the velocity b to the velocity a.
When a rigid body is made to turn round any axis by the action of an external force, the quantity of momentum produced (that is, the sum of the products of every particle by its velocity and by its distance from the axis) is equal to the momentum or similar product of the moving force or forces.
If an oblate spheroid, whose equatorial diameter is a, and polar diameter b, be made to librate round an equatorial diameter, and the velocity of that point of the equator which is farthest from the axis of libration be v, the momentum of the spheroid is \( \frac{4}{15} \Pi a b^3 v \).
The two last are to be found in every elementary book of mechanics.
Let AN an (fig. 156.) be the plane of the earth's equator, cutting the ecliptic CNK in the line of the nodes or equinoctial points Na. Let OAS be the section of the earth by a meridian passing through the fun, so that the line OCS is in the ecliptic, and CA is an arch of an hour-circle or meridian, measuring the fun's declination. The fun not being in the plane of the equator, there is, by prop. 1, a force tending to produce a libration round an axis ZO α at right angles to the diameter AA of that meridian in which the fun is situated, Theory of and the momentum of all the disturbing forces is Universal Gravitation.
The product of any force by the moment of its action expresses the momentary increment of velocity; therefore the momentary velocity, or the velocity of libration granted in the time \( t \) is \( \frac{4}{3} m n f d^i \Pi b^i t \). This is the absolute velocity of a point at the distance 1 from the axis, or it is the space which would be uniformly described in the moment \( t \), with the velocity which the point has acquired at the end of that moment. It is double the space actually described by the libration during that moment; because this has been an uniformly accelerated motion, in consequence of the continued and uniform action of the momentum during this time. This must be carefully attended to, and the neglect of it has occasioned very faulty solutions of this problem.
Let \( v \) be the velocity produced in the point A, the most remote from the axis of libration. The momentum excited or produced in the spheroid is \( \frac{4}{3} \Pi a^b b^i v \) (as above), and this must be equal to the momentum of the moving force, or to \( \frac{4}{3} m n f d \Pi b^i t \); therefore we obtain \( v = \frac{\frac{4}{3} m n f d \Pi b^i t}{\frac{4}{3} \Pi a^b b^i} \), that is, \( v = m n f d i^i \frac{b^i}{a^2} \) or nearly \( m n f d i^i \), because \( \frac{b^i}{a^2} = 1 \) very nearly. Also, because the product of the velocity and time gives the space uniformly described in that time, the space described by A in its libration round Z z is \( m n f d i^i \), and the angular velocity is \( \frac{m n f d i^i}{a} \).
Let \( r \) be the momentary angle of diurnal rotation. The arch A r, described by the point A of the equator in this moment \( t \) will therefore be \( a r \), that is, \( a \times r \), and the velocity of the point A is \( \frac{a r}{t} \), and the angular velocity of rotation is \( \frac{r}{t} \).
Here then is a body (fig. 157.) turning round an axis OP, perpendicular to the plane of the equator z o z, and therefore situated in the plane ZPz; and it turns round this axis with the angular velocity \( \frac{r}{t} \). It has received an impulse, by which alone it would librate round the axis Zz, with the angular velocity \( \frac{m n f d i^i}{a} \). It will therefore turn round neither axis, but round a third axis OP', passing through O, and lying in the plane ZPz, in which the other two are situated, and the fine P'P of its inclination to the axis of libration Z z will be to the fine P'p of its inclination to the axis OP of rotation as \( \frac{r}{t} \) to \( \frac{m n f d i^i}{a} \).
Now A, in fig. 156. is the summit of the equator both of libration and rotation: \( m n f d i^i \) is the space described by its libration in the time \( t \); and \( a r \) is the space or arch A r (fig. 156.) described in the same time by its rotation: therefore, taking A r to A c (perpendicular to the plane of the equator of rotation, and lying in the equator of libration), as \( a r \) to \( m n f d i^i \), and Theory of completing the parallelogram A r m c, A m will be the compound motion of A, and \( a r : m n f d i^i = 1 : \frac{m n f d i^i}{a r} \), which will be the tangent of the angle m A r, or of the change of position of the equator. But the axes of rotation are perpendicular to their equator; and therefore the angle of deviation \( w \) is equal to this angle r A m. This appears from fig. 5.; for \( \Pi P' : P'p = O'p : P'p = OP : \tan. POP \); and it is evident that \( a r : m n f d i^i = \frac{r}{t} : m n f d i^i \), as is required by the composition of rotations.
In consequence of this change of position, the plane of the equator no longer cuts the plane of the ecliptic in the line N n. The plane of the new equator cuts the former equator in the line AO, and the part AN of the former equator lies between the ecliptic and the new equator AN'; while the part A n of the former equator is above the new one A n'; therefore the new node N', from which the point A was moving, is removed to the westward, or farther from A; and the new node n', to which A is approaching, is also moved westward, or nearer to A; and this happens in every position of A. The nodes therefore, or equinoctial points, continually shift to the westward, or in a contrary direction to the rotation of the earth; and the axis of rotation always deviates to the east side of the meridian which passes through the fun.
This account of the motions is extremely different from what a person should naturally expect. If the earth were placed in the summer solstice, with respect to us who inhabit its northern hemisphere, and had no rotation round its axis, the equator would begin to approach the ecliptic, and the axis would become more upright; and this would go on with a motion continually accelerating, till the equator coincided with the ecliptic. It would not stop here, but go as far on the other side, till its motion were extinguished by the opposing forces; and it would return to its former position, and again begin to approach the ecliptic, playing up and down like the arm of a balance. On this account this motion is very properly termed libration: but this very slow libration, compounded with the incomparably swifter motion of diurnal rotation, produces a third motion extremely different from both. At first the north pole of the earth inclines forward toward the sun; after a long course of years it will incline to the left hand, as viewed from the sun, and be much more inclined to the ecliptic, and the plane of the equator will pass through the fun. The south pole will come into view, and the north pole will begin to decline from the fun; and this will go on (the inclination of the equator diminishing all the while) till, after a course of years, the north pole will be turned quite away from the sun, and the inclination of the equator will be restored to its original quantity. After this the phenomena will have another period similar to the former, but the axis will now deviate to the right hand. And thus, although both the earth and fun should not move from their places, the inhabitants of the earth would have a complete succession of the seasons accomplished in a period of many centuries. This would be prettily illustrated by an iron ring poised very nicely on a cap like the card Theory of Universal Gravitation.
Part IV. ASTRONOMY.
Theory of a mariner's compass, having its centre of gravity coinciding with the point of the cap, so that it may whirl round in any position. As this is extremely difficult to execute, the cap may be pierced a little deeper, which will cause the ring to maintain a horizontal position with a very small force. When the ring is whirling very steadily, and pretty briskly, in the direction of the hours of a watch-dial, hold a strong magnet above the middle of the nearer semicircle (above the 6 hour point) at the distance of three or four inches. We shall immediately observe the ring rise from the 9 hour point, and sink at the 3 hour point, and gradually acquire a motion of precession and nutation, such as has been described.
If the earth be now put in motion round the sun, or the fun round the earth, motions of libration and deviation will still obtain, and the succession of their different phases, if we may so call them, will be perfectly analogous to the above statement. But the quantity of deviation, and change of inclination, will now be prodigiously diminished, because the rapid change of the sun's position quickly diminishes the disturbing forces, annihilates them by bringing the fun into the plane of the equator, and brings opposite forces into action.
We see in general that the deviation of the axis is always at right angles to the plane passing through the fun, and that the axis, instead of being raised from the ecliptic, or brought nearer to it, as the libration would occasion, deviates likewise; and the equator, instead of being raised or depressed round its east and west points, is twisted likewise round the north and south points; or at least things have this appearance: but we must now attend to this circumstance more minutely.
The composition of rotation shows us that this change of the axis of diurnal rotation is by no means a translation of the former axis (which we may suppose to be the axis of figure) into a new position, in which it again becomes the axis of diurnal motion; nor does the equator of figure, that is, the most prominent section of the terrestrial spheroid, change its position, and in this new position continue to be the equator of rotation. This was indeed supposed by Sir Isaac Newton; and this supposition naturally resulted from the train of reasoning which he adopted. It was strictly true of a single moon, or of the imaginary orbit attached to it; and therefore Newton supposed that the whole earth did in this manner deviate from its former position, till, however, turning round its axis of figure. In this he has been followed by Walmelly, Simpson, and most of his commentators. D'Alembert was the first who entertained any suspicion that this might not be certain; and both he and Euler at last showed that the new axis of rotation was really a new line in the body of the earth, and that its axis and equator of figure did not remain the axis and equator of rotation. They ascertained the position of the real axis by means of a most intricate analysis, which obscured the connexion of the different positions of the axis with each other, and gave us only a kind of momentary information. Father Frisius turned his thoughts to this problem, and fortunately discovered the composition of rotations as a general principle of mechanical philosophy. Few things of this kind have escaped the penetrating eyes of Sir Isaac Newton. Even this principle had been glanced at by him. He affirms it in express terms with respect to a body that is perfectly spherical (cor. 22. prop. 66. Theor. of book i.). But it was referred for Frisius to demonstrate it to be true of bodies of any figure, and thus to enrich mechanical science with a principle which gives simple and elegant solutions of the most difficult problems.
But here a very formidable objection naturally offers itself. If the axis of the diurnal motion of the heavens is not the axis of the earth's spheroidal figure, but an imaginary line in it, round which even the axis of figure must revolve; and if this axis of diurnal rotation has so greatly changed its position, that it now points at a star at least 12 degrees distant from the pole observed by Timocharis, how comes it that the equator has the very same situation on the surface of the earth that it had in ancient times? No sensible change has been observed in the latitudes of places.
The answer is very simple and satisfactory: Suppose that in 12 hours the axis of rotation has changed from the position PR (fig. 158.) to pr, so that the north pole, instead of being at P, which we may suppose to be a particular mountain, is now at p. In this 12 hours the mountain P, by its rotation round pr, has acquired the position π. At the end of the next 12 hours, the axis of rotation has got the position πg, and the axis of figure has got the position pr, and the mountain P is now at p. Thus, on the noon of the following day, the axis of figure PR is in the situation which the real axis of rotation occupied at the intervening midnight. This goes on continually, and the axis of figure follows the position of the axis of rotation, and is never further removed from it than the deviation of 12 hours, which does not exceed \( \frac{1}{72} \)th part of one second, a quantity altogether imperceptible. Therefore the axis of figure will always sensibly coincide with the axis of rotation, and no change can be produced in the latitudes of places on the surface of the earth.
We have hitherto considered this problem in the most general manner; let us now apply the knowledge we have gotten of the deviation of the axis or of the momentary action of the disturbing force to the explanation of the phenomena; that is, let us see what precession and what nutation will be accumulated after any given time of action.
For this purpose we must ascertain the precise deviation which the disturbing forces are competent to produce. This we can do by comparing the momentum of libration with the gravitation of the earth to the fun, and this with the force which would retain a body on the equator while the earth turns round its axis.
The gravitation of the earth to the fun is in the proportion of the fun's quantity of matter M directly, and to the square of the distance A inversely, and may therefore be expressed by the symbol \( \frac{M}{A^2} \). The disturbing force at the distance r from the place of illumination is to the gravitation of the earth's centre to the fun as 3 to A, (A being measured on the same scale which measures the distance from the plane of illumination).
Therefore \( \frac{3M}{A^2} \) will be the disturbing force f of our formula.
Let p be the centrifugal force of a particle at the distance r from the axis of rotation; and let t and T be the time of rotation and of annual revolution, viz. fidecal day and year. Then \( \rho : \frac{M}{A^2} = \frac{1}{T^2} : \frac{A}{T^2} \). Hence we derive \( \frac{3M}{A^2} = 3\rho \frac{t^2}{T^2} \). But since \( r \) was the angular velocity of rotation, and consequently \( 1 \times r \) the space described, and \( \frac{1 \times r}{t} \) the velocity; and since the centrifugal force is as the square of the velocity divided by the radius, (this being the measure of the generated velocity, which is the proper measure of any accelerating force), we have \( p = \frac{1^2 \times r^2}{1^2 \times t^2} = \frac{r^2}{t^2} \), and \( f = \frac{3 \cdot \frac{r^2}{t^2}}{\frac{t^2}{T^2}} \).
Now the formula \( f m n d \frac{d}{a} \) expressed the sine of the angle. This being extremely small, the sine may be considered as equal to the arc which measures the angle. Now, substitute for it the value now found, viz.
\[ \frac{3 \cdot r^2}{T^2} \times \frac{t^2}{T^2} \]
and we obtain the angle of deviation \( w = r \)
\[ \frac{3t^2}{T^2} mn \frac{d}{a}, \]
and this is the simplest form in which it can appear. But it is convenient, for other reasons, to express it a little differently: \( d \) is nearly equal to \( \frac{a^2 - b^2}{2a^2} \), therefore \( w = r \times \frac{3t^2}{2T^2} mn \frac{a^2 - b^2}{a^2} \), and this is the form in which we shall now employ it.
The small angle \( r \frac{3t^2}{2T^2} mn \frac{a^2 - b^2}{a^2} \) is the angle in which the new equator cuts the former one. It is different at different times, as appears from the variable part \( mn \), the product of the sine and cosine of the sun's declination. It will be a maximum when the declination is in the solstice, for \( mn \) increases all the way to 45°, and the declination never exceeds 23½°. It increases, therefore, from the equinox to the solstice, and then diminishes.
Let ESL (fig. 159.) be the ecliptic, EAC the equator, BAD the new position which it acquires by the momentary action of the sun, cutting the former in the angle BAE\( = r \frac{3t^2}{2T^2} mn \frac{a^2 - b^2}{a^2} \). Let S be the sun's place in the ecliptic, and AS the sun's declination, the meridian AS being perpendicular to the equator. Let \( \frac{a^2 - b^2}{a^2} \) be k. The angle BAE is then \( = r \frac{3t^2}{2T^2} kmn \). In the spherical triangle BAE we have fin. B : fin. AE\( = \) fin. A : fin. BE, or \( = A : BE \), because very small angles and arches are as their sines. Therefore BE, which is the momentary precession of the equinoctial point E, is equal to \( A \frac{\text{fin. AE}}{\text{fin. B}} = r \times \frac{3t^2}{2T^2} kmn \).
fin. R. ascenf.
fin. obli. ecl.
The equator EAC, by taking the position BAD, recedes from the ecliptic in the colour of the solstices CL, and CD is the change of obliquity or the nutation. For let CL be the solstitial colour of BAD, and c' the solstitial colour of EAC. Then we have fin. B : fin. E \( = \) fin. LD : fin. lc'; and therefore the difference of the arches LD and lc' will be the measure of the difference of the angles B and E. But when BE is indefinitely small, CD may be taken for the difference of LD and lc', they being ultimately in the ratio of equality. Therefore CD measures the change of the obliquity of the ecliptic, or the nutation of the axis with respect to the ecliptic.
The real deviation of the axis is the same with the change in the position of the equator, Pp being the measure of the angle EAB. But this not being always made in a plane perpendicular to the ecliptic, the change of obliquity generally differs from the change in the position of the axis. Thus, when the sun is in the solstice, the momentary change of the position of the equator is the greatest possible; but being made at right angles to the plane in which the obliquity of the ecliptic is computed, it makes no change whatever in the obliquity, but the greatest possible change in the precession.
In order to find CD the change of obliquity, observe that in the triangle CAD, R : fin. AC, or R : cof. AE\( = \)sin. A : fin. CD, \( = A : CD \) (because A and CD are exceedingly small). Therefore the change of obliquity (which is the thing commonly meant by nutation) CD\( = A \times \)cof. AE, \( = r \frac{3t^2}{2T^2} kmn \), cof. AE\( = r \frac{3t^2}{2T^2} \)
\( k \times \)fin. declin. \( \times \)cof. declin. \( \times \)cof. R. ascenf.
But it is more convenient for the purposes of astronomical computation to make use of the sun's longitude SE. Therefore make
<table> <tr> <th>The sun's longitude ES</th> <th></th> <th>= z</th> </tr> <tr> <th>Sine of the sun's long.</th> <th></th> <th>= x</th> </tr> <tr> <th>Cofine</th> <th></th> <th>\( \sqrt{1-x^2} = y \)</th> </tr> <tr> <th>Sine obliq. eclipt.</th> <th></th> <th>23½ = p</th> </tr> <tr> <th>Cofine obliq.</th> <th></th> <th>= q</th> </tr> </table>
In the spherical triangle EAS, right-angled at A (because AS is the sun's declination perpendicular to the equator), we have R : fin. ES\( = \)sin. E : fin. AS, and fin. AS\( = \)px. Also R : cof. AS\( = \)cof. AE : cof. ES, and cof. ES or \( y = \)cof. AS\( \times \)cof. AE. Therefore \( p \times y = \)fin. AS\( \times \)cof. AE\( = mn \times \)cof. AE.
Therefore the momentary nutation CD\( = r \frac{3t^2}{2T^2} kpxy \).
We must recollect that this angle is a certain fraction of the momentary diurnal rotation. It is more convenient to consider it as a fraction of the sun's annual motion, that so we may directly compare his motion on the ecliptic with the precession and nutation corresponding to his situation in the heavens. This change is easily made, by augmenting the fraction in the ratio of the sun's angular motion to the motion of rotation, or multiplying the fraction by \( \frac{T}{t} \); therefore the momentary nutation will be \( r \frac{3t^2}{2T^2} kpxy \). In this value \( \frac{3t^2kp}{2T} \) is a constant quantity, and the momentary nutation is proportional to xy, or to the product of the sine and cosine of the sun's longitude, or to the sine of twice the sun's longitude; for xy is equal to half the sine of twice z.
If therefore we multiply this fraction by the sun's momentary angular motion, which we may suppose, with abundant accuracy, proportional to z, we obtain the fluxion of the nutation, the fluent of which will ex- Part IV.
Theory of Universal Gravitation.
The precession of the whole nutation while the sun describes the arch \( x \) of the ecliptic, beginning at the vernal equinox. Therefore, in place of \( y \) put \( \sqrt{1-x^2} \), and in place of \( \dot{x} \) put \( \frac{\dot{x}}{\sqrt{1-x^2}} \), and we have the fluxion of the nutation for the moment when the sun's longitude is \( x \), and the fluent will be the whole nutation. The fluxion resulting from this process is \( \frac{3t kp}{2T} x \dot{x} \), of which the fluent is \( \frac{3t kp}{4T} x^2 \). This is the whole change produced on the obliquity of the ecliptic while the sun moves along the arch \( x \) ecliptic, reckoned from the vernal equinox. When this arch is 90°, \( x \) is 1, and therefore \( \frac{3t kp}{4T} \) is the nutation produced while the sun moves from the equinox to the solstice.
The momentary change of the axis and plane of the equator (which is the measure of the changing force) is \( \frac{3t k}{2T} m n \).
The momentary change of the obliquity of the ecliptic is \( \frac{3t kp}{2T} x \dot{x} \).
The whole change of obliquity is \( \frac{3t kp}{4T} x^2 \).
Hence we see that the force and the real momentary change of position are greatest at the solstices, and diminish to nothing at the equinoxes.
The momentary change of obliquity is greatest at the equants, being proportional to \( x \dot{x} \) or to \( x \dot{y} \).
The whole accumulated change of obliquity is greatest at the solstices, the obliquity itself being then smallest.
We must in like manner find the accumulated quantity of the precession after a given time, that is, the arch BE for a finite time.
We have ER : CD = fin. EA : fin. CA (or cof. EA) = tan. EA : 1, and EB : ER = 1 : fin. B. Therefore EB : CD = tan. EA : fin. B. But tan. EA = cof. E × tan. ES, = cof. E × \( \frac{\sin. long.}{\cos. long.} = \frac{q x}{\sqrt{1-x^2}} \).
Therefore EB : CD = \( \frac{q x}{\sqrt{1-x^2}} p \), and CD = EB : fin. obliq. eclip. ⊙. If we now substitute for CD its value found in No 40, viz. \( \frac{3t kp}{2T} x \dot{x} \), we obtain EB = \( \frac{3t}{2T} \times \frac{k q x^2 \dot{x}}{\sqrt{1-x^2}} \), the fluxion of the precession of the equinoxes occasioned by the action of the sun. The fluent of the variable part \( \frac{x \dot{x}}{\sqrt{1-x^2}} = x \dot{y} \), of which the fluent is evidently a segment of a circle whose arch is \( x \) and sine \( x \), that is, \( \frac{x - x \sqrt{1-x^2}}{2} \), and the whole precession, while the sun describes the arch \( x \), is \( \frac{3t}{2T} \times \frac{k q}{2} \left( 2 - x \sqrt{1-x^2} \right) \). This is the precession of the equinoxes while the sun moves from the vernal equinox along the arch \( x \) of the ecliptic.
In this expression, which consists of two parts, \( \frac{3t kp}{4T} x^2 \), and \( \frac{3t kp}{4T} \left( -x \sqrt{1-x^2} \right) \), the first is incomparably greater than the second, which never exceeds 1", and is always compensated in the succeeding quadrant. The precession occasioned by the sun will be \( \frac{3t kp}{4T} x \), and from this expression we see that the precession increases uniformly, or at least increases at the same rate with the sun's longitude \( x \), because the quantity \( \frac{3t kp}{4T} \) is constant.
In order to make use of these formulae, which are now reduced to very great simplicity, it is necessary to determine the values of the two constant quantities \( \frac{3t kp}{4T}, \frac{3t kp}{4T} \), which we shall call N and P, as factors of the nutation and precession. Now t is one sidereal day, and T is 366\( \frac{1}{4} \). k is \( \frac{a^2-b^2}{a^2} \), which according to Sir Isaac Newton is \( \frac{231^2-230^2}{231^2} = \frac{1}{115} \); p and q are the sine and cosine of 23° 28', viz. 0.39822 and 0.91729.
These data give \( N = \frac{1}{141030} \) and \( P = \frac{1}{61224} \) of which the logarithms are 4.85069 and 5.21308, viz. the arithmetical complements of 5.14931 and 4.78692.
Let us, for an example of the use of this investigation, compute the precession of the equinoxes when the utility fun has moved from the vernal equinox to the summer of the infolice, so that \( x \) is 90°, or 324000".
Log. 324000" = 5.51055 Log. P = 5.21308 Log. 5°29' = 0.72363
The precession therefore in a quarter of a year is 5,292 seconds; and, since it increases uniformly, it is 21",168 annually.
We must now recollect the assumptions on which this computation proceeds. The earth is supposed to be homogeneous, and the ratio of its equatorial diameter to its polar axis is supposed to be that of 231 to 230. If the earth be more or less protuberant at the equator, the precession will be greater or less in the ratio of this protuberance. The measures which have been taken of the degrees of the meridian are very inconsistent among themselves; and although a comparison of them all indicates a smaller protuberance, nearly \( \frac{1}{115} \) instead of \( \frac{1}{114} \), their differences are too great to leave much confidence in this method. But if this figure be thought more probable, the precession will be reduced to about 17" annually. But even though the figure of the earth were accurately determined, we have no authority to say that it is homogeneous. If it be denser towards the centre, the momentum of the protuberant matter will not be so great as if it were equally dense with the inferior parts, and the precession will be diminished on this account. Did we know the proportion of the matter in the moon to that in the sun, we could Theory of could easily determine the proportion of the whole observed annual precession of \(53^{\frac{1}{2}}\) which is produced by the sun's action. But we have no unexceptionable data for determining this; and we are rather obliged to infer it from the effect which the produces in disturbing the regularity of the precession, as will be considered immediately. So far, therefore, as we have yet proceeded in this investigation, the result is very uncertain. We have only ascertained unquestionably the law which is observed in the solar precession. It is probable, however, that this precession is not very different from \(2\alpha''\) annually; for the phenomena of the tides show the disturbing force of the sun to be very nearly \(\frac{1}{7}\) of the disturbing force of the moon. Now \(2\alpha''\) is \(\frac{2}{3}\) of \(50''\).
But let us now proceed to consider the effect of the moon's action on the protuberant matter of the earth; and as we are ignorant of her quantity of matter, and consequently of her influence in similar circumstances with the sun, we shall suppose that the disturbing force of the moon is to that of the sun as \(m\) to \(1\). Then (exenteris partibus) the precession will be to the solar precession \(\pi\) in the ratio of the force and of the time of its action jointly. Let \(t\) and \(T\) therefore represent a periodical month and year, and the lunar precession will be \(= \frac{m\pi t}{T}\). This precession must be reckoned on the plane of the lunar orbit, in the same manner as the solar precession is reckoned on the ecliptic. We must also observe, that \(\frac{m\pi t}{T}\) represents the lunar precession only on the supposition that the earth's equator is inclined to the lunar orbit in an angle of \(23^{\frac{1}{2}}\) degrees. This is indeed the mean inclination; but it is sometimes increased to above \(28^{\circ}\), and sometimes reduced to \(18^{\circ}\). Now in the value of the solar precession the cosine of the obliquity was employed. Therefore whatever is the angle \(E\) contained between the equator and the lunar orbit, the precession will be \(= \frac{m\pi t}{T} \cdot \frac{\text{Cof. E}}{\text{Cof. } 23^{\frac{1}{2}}}\), and it must be reckoned on the lunar orbit.
Now let \(\varphi B\) (fig. 160.) be the immovable plane of the ecliptic, \(\varphi E D \perp F\) the equator in its first situation, before it has been deranged by the action of the moon, AGRDBH the equator in its new position, after the momentary action of the moon. Let EGNFH be the moon's orbit, of which N is the ascending node, and the angle \(N = 5^{\circ} 8' 46''\).
Let \(N \varphi\) the long. of the node be
<table> <tr><th>Sine \(N \varphi\)</th><td>-</td></tr> <tr><th>Cofine \(N \varphi\)</th><td>-</td></tr> <tr><th>Sine \(\varphi = 23^{\frac{1}{2}}\)</th><td>-</td></tr> <tr><th>Cofine \(\varphi\)</th><td>-</td></tr> <tr><th>Sine \(N = 5.846\)</th><td>-</td></tr> <tr><th>Cofine \(N\)</th><td>-</td></tr> <tr><th>Circumference to radius \(1 = 6,28\)</th><td>-</td></tr> <tr><th>Force of the moon</th><td>-</td></tr> <tr><th>Solar precession (supposed \(= 14^{\frac{1}{2}}\) by observation)</th><td>-</td></tr> <tr><th>Revolution of \(C = 274^{\frac{1}{2}}\)</th><td>-</td></tr> <tr><th>Revolution of \(O = 366^{\frac{1}{2}}\)</th><td>-</td></tr> <tr><th>Revolution of \(N = 18\) years 7 months</th><td>-</td></tr> </table>
In order to reduce the lunar precession to the ecliptic, we must recollect that the equator will have the same inclination at the end of every half revolution of the sun or of the moon, that is, when they pass through the equator, because the sum of all the momentary changes of its position begins again each revolution. Therefore if we neglect the motion of the node during one month, which is only \(1^{\frac{1}{2}}\) degree, and can produce but an inensible change, it is plain that the moon produces, in one half revolution, that is, while he moves from H to G, the greatest difference that the can in the position of the equator. The point D, therefore, half way from G to H, is that in which the moveable equator cuts the primitive equator, and DE and DF are each \(90^{\circ}\). But S being the solstitial point, \(\varphi S\) is also \(90^{\circ}\). Therefore DS\(=\varphi E\). Therefore, in the triangle DGE, we have fin. ED : fin. G = fin. EG : fin. D, = EG : D. Therefore D = EG \times fin. G, = EG \times fin. E nearly. Again, in the triangle \(\varphi DA\) we have fin. A : fin. \(\varphi D\) (or cof. \(\varphi E\)) = fin. D : fin. \(\varphi A\), = D : \(\varphi A\). Therefore \(\frac{D \cdot \text{Cof. } \varphi E}{\text{Sin. A}} = \frac{EG \cdot \text{Sin. E} \cdot \text{Cof. } \varphi E}{\text{Sin. } 23^{\frac{1}{2}}}\), \(\frac{m\pi t}{T} \cdot \frac{\text{Sin. E} \cdot \text{Cof. E} \cdot \text{Cof. } \varphi E}{\text{Sin. } \varphi \cdot \text{Cof. } \varphi E}\).
This is the lunar precession produced in the course of one month, estimated on the ecliptic, not constant like the solar precession, but varying with the inclination of the angle E or F, which varies both by a change in the angle N, and also by a change in the position of N on the ecliptic.
We must find in like manner the nutation SR produced in the same time, reckoned on the colure of the same solstices RL. We have R : fin. DS = D : RS, and RS = D \times fin. DS, = D \times fin. \varphi E. But D = EG \times fin. E Therefore RS = ED \times fin. E \times fin. \varphi E, = \frac{m\pi t}{T} \cdot \text{Cof. } \varphi E \times \text{fin. E} \times \text{fin. } \varphi E. In this expression we must substitute the angle N, which may be considered as constant during the month, and the longitude \(\varphi N\), which is also nearly constant, by observing that fin. E : fin. \varphi N = fin. N : fin. \varphi E. Therefore RS = \frac{m\pi t}{T} \times \frac{\text{Sin. N} \cdot \text{Sin. } \varphi N \cdot \text{Cof. E}}{\text{Cof. } \varphi E}.
But we must exterminate the angle E, because it changes by the change of the position of N. Now, in the triangle \(\varphi EN\) we have cof. E = cof. \varphi N \times fin. N \times \text{fin. } \varphi - \text{cof. N} \cdot \text{cof. } \varphi, = y c - d b. And because the angle E is necessarily obtuse, the perpendicular will fall without the triangle, the cofine of E will be negative, and we shall have cof. E = b d - a c y. Therefore the nutation for one month will be \(= \frac{m\pi t}{T} \times \frac{c x (b d - a c y)}{b}\), the node being supposed all the while in N.
These two expressions of the monthly precession and may be nutation may be considered as momentary parts of the considered moon's action, corresponding to a certain position of the node and inclination of the equator, or as the fluxions of the whole variable precession and nutation, while the moon's action continually changes its place, and in the space of 18 years makes a complete tour of the heavens.
We must, therefore, take the motion of the node as the fluent of comparison, or we must compare the fluxions and nutations of the node's motion with the fluxions of the precession and nutation; therefore, let the longitude of the node be \(x\), and its monthly change \(= \dot{x}\); we shall then have Theory of Universal Gravitation.
Let T be = 1, in order that n may be 186, and substitute for t its value in the fluxion of the nutation, by putting \( \sqrt{1-x^2} \) in place of y. By this substitution we obtain \( m \pi n \frac{c}{e b} \left( \frac{d b x \dot{x}}{\sqrt{1-x^2}} - a c x \dot{x}^2 \right) \). The fluent of this is \( m \pi n \frac{c}{e b} \left( -d b \ln \frac{1}{\sqrt{1-x^2}} - a c x^2 \right) \). (Vide Simpson's Fluxions, § 77.) But when x is = 0, the nutation must be = 0, because it is from the position in the equinoctial points that all our deviations are reckoned, and it is from this point that the periods of the lunar action recommence. But if we make x = 0 in this expression, the term \( \frac{a c x^3}{2} \) vanishes, and the term \( -d b \ln \frac{1}{\sqrt{1-x^2}} \) becomes \( = -d b \); therefore our fluent has a constant part \( +d b \); and the complete fluent is \( m \pi n \frac{c}{e b} \left( d b - d b \ln \frac{1}{\sqrt{1-x^2}} - a c x^2 \right) \). Now this is equal to \( m \pi n \frac{c}{e b} (d b \times \text{versed fine } z) \).
fine, \( z = \frac{1}{2} a c \times \text{versed fine } 2z \): For the versed fine of z is equal to \( (1-\cos z) \); and the square of the fine of an arch is \( \frac{1}{4} \) the versed fine of twice that arch.
This, then, is the whole nutation while the moon's ascending node moves from the vernal equinox to the longitude \( \varphi N = z \). It is the expression of a certain number of seconds, because \( \pi \), one of its factors, is the solar precession in seconds; and all the other factors are numbers, or fractions of the radius r; even e is expressed in terms of the radius r.
The fluxion of the precession, or the monthly precession, is to that of the nutation as the cotangent of \( \varphi E \) is to the fine of \( \varphi \). This also appears by considering fig. 159. Pp measures the angle A, or change of position of the equator; but the precession itself, reckoned on the ecliptic, is measured by Po, and the nutation by po; and the fluxion of the precession is equal to the fluxion of nutation \( \times \frac{\cot. \varphi E}{\text{fine } \varphi} \), but cot. \( \varphi E = \frac{a d + b c y}{c x} \); therefore
\[ \frac{\cot. \varphi E}{\text{fine } \varphi} = \frac{a d + b c \sqrt{1-x^2}}{c x} : \]
This, multiplied into the fluxion of the nutation, gives
\[ \frac{m \pi n}{a b c} \left( \frac{a b d^2}{\sqrt{1-x^2}} + (b^2-a^2) d c - a b c^2 \sqrt{1-x^2} \right) \dot{x} \]
for the monthly precession. The fluent of this is
\[ \frac{m \pi n}{a b c} \left( a d^2 b x + (b^2-a^2) d c x - \frac{1}{2} a b c^2 \text{versed fine } 2z \right), \]
or it is equal to
\[ \frac{m \pi n}{a b c} \left( (d^2 - \frac{1}{2} c^2) a b x + (b^2-a^2) d c x - \frac{1}{2} a b c^2 \text{versed fine } 2z \right). \]
Let us now express this in numbers: When the node has made a half revolution, we have \( z = 180^\circ \), whose versed fine is 2, and the versed fine of \( 2z \), or \( 360^\circ \), is 0; therefore, after half a revolution of the node, the nutation becomes \( \frac{m \pi n c}{e b} 2 b d \). If, in this expression, we supposed \( m = 2 \frac{1}{2} \), and \( \pi = 14 \frac{1}{2}'' \), we shall find the nutation to be \( 19 \frac{1}{2}'' \).
Now the observed nutation is about \( 18'' \). This requires m to be \( 2 \frac{1}{2} \), and \( \pi = 16 \frac{1}{2}'' \). But it is evident, that no astronomer can pretend to warrant the accuracy of his observations of the nutation within \( 1'' \).
To find the lunar precession during half a revolution of the node, observe, that then z becomes \( = e \), and the fine of z and of \( 2z \) vanish, \( d^2 \) becomes \( 1 - \frac{1}{2} c^2 \), and the precession becomes \( \frac{m \pi n}{2} (d^2 - \frac{1}{2} c^2) = \frac{m \pi n}{2} (1 - \frac{1}{2} c^2) \), and the precession in 18 years is \( m \pi n \frac{1}{4} c^2 \).
We see, by comparing the nutation and precession for nine years, that they are as \( \frac{4 c d}{e} \) to \( 1 - \frac{1}{2} c^2 \) nearly as 1 to \( 17 \frac{1}{2} \). This gives 313'' of precession, corresponding to 18'', the observed nutation, which is about 35'' of precession annually produced by the moon.
And thus we see that the inequality produced by the moon in the precession of the equinoxes, and, more disturbing particularly, the nutation occasioned by the variable obliquity of her orbit, enables us to judge of her share in the whole phenomenon; and therefore informs us of her disturbing force, and therefore of her quantity of matter. This phenomenon, and those of the tides, are the only facts which enable us to judge of this matter; and this is one of the circumstances which has caused this problem to occupy so much attention. Dr Bradley, by a nice comparison of his observations with the mathematical theory, as it is called, furnished him by Mr Machin, found that the equation of precession computed by that theory was too great, and that the theory would agree better with the observations, if an ellipse were substituted for Mr Machin's little circle. He thought that the shorter axis of this ellipse, lying in the colure of the solstices, should not exceed 16''. Nothing can more clearly show the astonishing accuracy of Bradley's observations than this remark: for it results from the theory, that the pole must really describe an ellipse, having its shorter axis in the solstitial colure, and the ratio of the axes must be that of 18 to 16.8; for the mean precession during half a revolution of the node is \( \frac{m \pi n}{2} \left( d^2 - \frac{c^2}{2} \right) \); and therefore for the longitude \( z \), it will be \( \frac{m \pi n}{e} \left( d^2 - \frac{c^2}{2} \right) \); when this is taken from the true precession for that longitude, it leaves the equation of precession \( \frac{m \pi n}{a b c} \left( (b^2 - a^2) d c - \frac{1}{2} a b c^2 \text{versed fine } 2z \right) \); therefore when the node is in the solstice, and the equation greatest, we have it \( = \frac{m \pi n c d}{a b e} (b^2 - a^2) \). We here neglect the second term as insignificant. This greatest equation of precession is to \( \frac{2 m \pi n c d}{c} \), the nutation of 18", as \( b^2 - a^2 \) to \( 2 a b \); that is, as radius to the tangent of twice the obliquity of the ecliptic. This gives the greatest equation of precession 16", 8, not differing half a second from Bradley's observations.
Thus have we attempted to give some account of this curious and important phenomenon. It is curious, because it affects the whole celestial motions in a very intricate manner, and received no explanation from the more obvious application of mechanical principles, which so happily accounted for all the other appearances. It is one of the most illustrious proofs of Sir Isaac Newton's sagacity and penetration, which caught at a very remote analogy between this phenomenon and the libration of the moon's orbit.—It is highly important to the progress of practical and useful astronomy, because it has enabled us to compute tables of such accuracy, that they can be used with confidence for determining the longitude of a ship at sea. This alone fixes its importance; but it is still more important to the philosopher, affording the most incontrovertible proof of the universal and mutual gravitation of all matter to all matter. It left nothing in the solar system unexplained from the theory of gravity but the acceleration of the moon's mean motion; and this has at last been added to the list of our acquisitions by M. de la Place.
Que toties animos veterum torfere Sophorum, Queque scholas frustra rauco certamine vexant, Obvia conspicimus, nube pellente Mathef, Jam dubios nulla caligine praegavat error Quis superum penetrare domos, atque ardua caeli Scandere sublimis geniti concepsit acumen. Nec fas est proprius mortali attingere divos.
HALLEY.
SECT. X. Of the Libration of the Moon.
The only phenomena which still remain to be explained are the libration of the moon and the motion of the nodes of her equator. The moon, in consequence of her rotation round her axis, is a little flattened towards the poles; but the attraction of the earth must have lengthened the axis of the moon directed towards that planet. If the moon were homogeneous and fluid, she would assume the form of an ellipsoid, whose shorter axis would pass through her poles of rotation; the longer axis would be directed towards the earth, and in the plane of the moon's equator; and the mean axis, situated in the same plane, would be perpendicular to the two others. The excess of the longer over the shorter would be quadruple the excess of the mean axis over the shorter, and would amount to about \( \frac{1}{29711} \), the shorter axis being represented by unity.
It is easy to see, that if the longer axis of the moon deviate a little from the direction of the radius vector, which joins together the centres of the earth and moon, the attraction of the earth will tend to bring it towards that radius just as gravity tends to bring a pendulum towards the vertical position. If the rotation of the satellite had been at first sufficiently rapid to overcome this tendency, the time of a rotation would not have been equal to that of a revolution round the earth, and their difference would have discovered to us successively all the points of the moon's surface. But the angular motions of rotation and revolution having been at first but very little different, the force with which the longer axis separated from the radius vector was not sufficient to overcome the tendency toward the radius vector occasioned by the attraction of the earth. This last tendency, therefore, has rendered the two motions rigidly equal. And, as a pendulum driven from the vertical direction by a very small force constantly returns to it, making small oscillations on each side, in like manner the longer axis of the moon ought to oscillate on each side of the radius vector of her orbit. The libration of the moon then depends upon the small difference which originally subsisted between the angular motions of the moon's rotation and revolution.
Thus we see, that the theory of gravitation explains the equality which subsists between the mean rotation and revolution of the moon. It is only necessary to suppose, that the original difference between them was small. In that case the attraction of the earth would soon reduce them to a state of equality.
The singular coincidence of the nodes of the moon's equator, with those of its orbit, is also owing to the attraction of the earth. This was first demonstrated by La Grange. The planes of the equator and of the orbit of the moon, and the plane which passes through its centre, parallel to the ecliptic, have always nearly the same intersection. The secular movements of the ecliptic neither alter the coincidence of the nodes of these three planes, nor their mean inclination, which the attraction of the earth keeps always the same.
We have now examined all the phenomena of the heavenly bodies, and have found that they are all explicable on the theory of gravitation, and indeed necessary consequences of that theory. The exact coincidence of all the phenomena must be considered as a complete demonstration of the truth of the theory; and indeed places it beyond the reach of every possible objection. With respect to the nature of this force called gravitation, nothing whatever is known, nor is it likely that any thing ever will be known. The diffusion being evidently above the reach of the human faculties, all the different theories which have been published, explaining it by others, &c. have only served to show the weakness of human reason, when it attempts to leave the plain path of experience, and indulge in fancy and conjecture.
APPENDIX.
In the preceding article we have endeavoured to give as full a view as possible of astronomy; avoiding, at the same time, the introduction of minute details upon those subjects which are not essential, that the readers attention might not be distracted and diverted from objects of primary importance. But for the sake of those persons who may wish to indulge their taste for practical astronomy, we have thought proper to subjoin an appendix; in which we shall give, in the first place, the rules for calculating eclipses, and in the second, a description of the most important astronomical instruments.
I. Method of Calculating Eclipses.
The method of constructing tables for the calculation of eclipses will be understood from the following observations.
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 fix 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 unequal 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 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 fix 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 vernal 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 appears, 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 37 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 minutes 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 opposite points, which are called her nodes: and it is observed, 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 motions of the sun and moon were always equable. Hence 12 mean lunations contain 354 days 8 hours 48 minutes 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, consisting of 365 days 6 hours; and 13 mean lunations contain 383 days 21 hours 32 minutes 39 seconds 38 thirds 38 fourths, which exceeds the length of a common 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 later 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 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 minutes, 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 366 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 therefrom Of Calcula. from in January and February in those years; to which ting Eclip- all tables of this kind are subject, which begin the year es, &c. with January, in calculating the times of new or full moons.
The mean anomalies of the sun and moon, and the fun'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, being added to the radical anomalies of the sun and moon, and to the fun'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 fun's mean distance 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 number 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 42 fourths, with respect to the time of mean new moon. These being added together 65 times (always taking care to throw off a whole lunation when the days exceed 29 1/2) 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 fun's distance from the node found, for these centurial 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 fun'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 equally in all parts of their orbits. But we have already shewn, 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 fix signs: and that their mean places are always forwarder than their true places, whilst the anomaly is less than fix signs; and their two places are forwarder than the mean, whilst the anomaly is more.
Hence it is evident, that whilst the sun's anomaly is less than fix signs, the moon will overtake him, or be opposite to him, sooner than he could if his motion were equable; and later whilst his anomaly is more than fix 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 48 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 fix signs.
The sun is in his apogee on the 30th of June, and in his perigee on the 35th 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 fix signs, and added when the anomaly is more.—At the greatest it is 4 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 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 20 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 fun is in the same sign and degree either with the moon's apogee or perigee; shortest of all, or least eccentric, when the fun'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 fun'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 6 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 se- Appendix.
of Calcula. cond equation of the mean to the true syzygy. (See Taking Ecliptic IX.) : and the moon's anomaly, when equated by les, &c. 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 Fergufon 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, and 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 °, degree thus ′, 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 number 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 Of Calcula-Table IV. under the given month, where you are sure ting Eclipses, &c. 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, shew 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 h. 30 m. 25 sec. of tabular time is April 1st (in common reckoning) at 30 m. 25 sec. after 10 o'clock in the morning.
EXAMPLE I. Required the true time of New Moon in April 1764, New Style?
<table> <tr> <th>By the Precepts.</th> <th>New Moon</th> <th>Sun's Anomaly</th> <th>Moon's Anomaly.</th> <th>Sun from Node.</th> </tr> <tr> <td></td> <td>D. H. M. S.</td> <td>s o ' "</td> <td>s o ' "</td> <td>s o ' "</td> </tr> <tr> <td>March 1764,<br>Add 1 Lunation,</td> <td>2 8 55 36<br>29 12 44 3</td> <td>8 2 20 0<br>0 29 6 19</td> <td>10 13 35 21<br>0 25 49 0</td> <td>11 4 54 48<br>1 0 40 14</td> </tr> <tr> <td>Mean New Moon,<br>First Equation,</td> <td>31 21 39 39<br>+ 4 10 40</td> <td>9 1 26 19<br>11 10 59 18</td> <td>11 9 24 21<br>+ 1 34 57</td> <td>0 5 35 2</td> </tr> <tr> <td>Time once equated,<br>Second Equation,</td> <td>32 1 50 19<br>— 3 24 49</td> <td>9 20 27 1</td> <td>11 10 59 18</td> <td rowspan="2">Sun from Node,<br>and Arg. 4th equation.</td> </tr> <tr> <td>Arg. 3d equation.</td> <td></td> <td></td> <td></td> </tr> <tr> <td>Time twice equated,<br>Third Equation,</td> <td>31 22 25 39<br>+ 4 37</td> <td></td> <td></td> <td></td> </tr> <tr> <td>Time thrice equated,<br>Fourth Equation,</td> <td>31 22 30 7<br>+ 18</td> <td></td> <td></td> <td></td> </tr> <tr> <td>True New Moon,<br>Equation of days,</td> <td>31 22 30 25<br>— 3 48</td> <td></td> <td></td> <td></td> </tr> <tr> <td>Apparent time,</td> <td>31 22 26 37</td> <td></td> <td></td> <td></td> </tr> </table>
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. Qu. The true time of Full Moon in May 1762, New Style?
<table> <tr> <th>By the Precepts.</th> <th>New Moon</th> <th>Sun's anomaly.</th> <th>Moon's Anomaly.</th> <th>Sun from Node.</th> </tr> <tr> <td></td> <td>D. H. M. S.</td> <td>s o ' "</td> <td>s o ' "</td> <td>s o ' "</td> </tr> <tr> <td>March 1762,<br>Add 2 lunations,</td> <td>24 15 18 24<br>59 1 28 6</td> <td>8 23 48 16<br>1 28 12 39</td> <td>1 23 59 11<br>1 21 38 1</td> <td>10 18 49 14<br>2 1 20 28</td> </tr> <tr> <td>New Moon, May,<br>Subt. \( \frac{1}{2} \) Lunation,</td> <td>22 16 46 30<br>14 18 22 2</td> <td>10 22 0 55<br>0 14 33 10</td> <td>3 15 37 12<br>6 12 54 30</td> <td>0 20 9 42<br>0 15 20 7</td> </tr> <tr> <td>Full Moon, May,<br>First Equation,</td> <td>7 22 24 28<br>+ 3 16 36</td> <td>10 7 27 45<br>9 3 57 18</td> <td>9 2 42 42<br>+ 1 14 36</td> <td rowspan="2">Sun from Node,<br>and Arg. 4th equation.</td> </tr> <tr> <td>Arg. 3d equation.</td> <td></td> <td></td> <td></td> </tr> <tr> <td>Time once equated,<br>Second Equation,</td> <td>8 1 41 4<br>— 9 47 53</td> <td>1 3 30 27</td> <td>9 3 57 18</td> <td></td> </tr> <tr> <td>Time twice equated,<br>Third Equation,</td> <td>7 15 53 11<br>— 2 36</td> <td></td> <td></td> <td></td> </tr> <tr> <td>Time thrice equated,<br>Fourth Equation,</td> <td>7 15 50 35<br>+ 15</td> <td></td> <td></td> <td></td> </tr> <tr> <td>The Full Moon,</td> <td>7 15 50 50</td> <td></td> <td></td> <td></td> </tr> </table>
Anf. May 7th at 15 h. 50 min. 50 sec. past noon, viz. May 8th at 3 h. 50 sec. in the morning.
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.
EXAMPLE. Appendix.
EXAMPLE III. Required the true time of Full Moon in April, Old Style, A. D. 302 From 1730 subtract 1700 (or 17 centuries) and there remains 30.
<table> <tr> <th>By the Precepts.</th> <th>New Moon.</th> <th>Sun's Anomaly.</th> <th>Moon's Anomaly.</th> <th>Sun from Node.</th> </tr> <tr> <td></td> <td>D. H. M. S.</td> <td>s. o. ' "</td> <td>s. o. ' "</td> <td>s. o. ' "</td> </tr> <tr> <td>March 1730,<br>Add \( \frac{1}{2} \) Lunation.</td> <td>7 12 34 16<br>14 18 22 2</td> <td>8 18 4 31<br>0 14 33 10</td> <td>9 0 32 17<br>6 12 54 30</td> <td>1 23 17 16<br>0 15 20 7</td> </tr> <tr> <td>Full Moon,<br>1700 years subtr.</td> <td>22 6 56 18<br>14 17 36 42</td> <td>9 2 37 41<br>11 28 46 0</td> <td>3 13 26 47<br>10 29 36 0</td> <td>2 8 37 23<br>4 29 23 0</td> </tr> <tr> <td>Full d March A. D. 30.<br>Add 1 Lunation,</td> <td>7 13 19 36<br>29 12 44 3</td> <td>9 3 51 41<br>0 29 6 19</td> <td>4 13 50 47<br>0 25 49 0</td> <td>9 9 14 23<br>1 0 40 14</td> </tr> <tr> <td>Full Moon, April,<br>First Equation,</td> <td>6 2 3 39<br>+ 3 28 4</td> <td>10 2 58 0<br>5 10 58 40</td> <td>5 9 39 47<br>+ 1 18 53</td> <td>10 9 54 37<br>Sun from Node,<br>and Arg. fourth equation.</td> </tr> <tr> <td>Time once equated,<br>Second Equation,</td> <td>6 5 31 43<br>+ 2 37 48</td> <td>4 21 59 20<br>Arg. 3d equation.</td> <td>5 10 58 40<br>Arg. 2d equation.</td> <td></td> </tr> <tr> <td>Time twice equated,<br>Third Equation,</td> <td>6 8 29 31<br>— 2 54</td> <td></td> <td></td> <td></td> </tr> <tr> <td>Time thrice equated,<br>Fourth Equation,</td> <td>6 8 26 37<br>— 1 33</td> <td></td> <td></td> <td></td> </tr> <tr> <td>True Full Moon, April,</td> <td>6 8 25 4</td> <td></td> <td></td> <td></td> </tr> </table>
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 subtract the time and anomalies belonging to it from those of the mean new moon in March, the above found year of the 18th century; and the remainder will denote the time and anomalies, &c. of mean new moon in March, the given year before Christ.—Then, for the true time thereof in any month of that year, proceed as above taught.
Hence it appears, that the true time of Full Moon in April, A. D. 30, old style, was on the 6th day, at 25 m. 4 f. past eight in the evening.
EXAMPLE IV. Required the true time of New Moon in May, Old Style, the year before Christ 5852
The years 384 added to 1716, make 2300, or 23 centuries.
<table> <tr> <th>By the Precepts</th> <th>New Moon.</th> <th>Sun's Anomaly.</th> <th>Moon's Anomaly.</th> <th>Sun from Node.</th> </tr> <tr> <td></td> <td>D. H. M. S.</td> <td>s. o. ' "</td> <td>s. o. ' "</td> <td>s. o. ' "</td> </tr> <tr> <td>March 1716,<br>2300 years subtract,</td> <td>11 17 33 29<br>11 5 57 53</td> <td>8 22 50 39<br>11 19 47 0</td> <td>4 4 14 2<br>1 5 39 0</td> <td>4 27 17 5<br>7 25 27 0</td> </tr> <tr> <td>March before Christ 585.<br>Add 3 Lunations,</td> <td>0 11 35 36<br>88 14 12 9</td> <td>9 3 3 39<br>2 27 18 58</td> <td>2 28 15 2<br>2 17 27 1</td> <td>9 1 50 5<br>3 2 0 42</td> </tr> <tr> <td>May before Christ 585,<br>First Equation,</td> <td>28 1 47 45<br>— 1 37</td> <td>0 0 22 37<br>5 15 41 17</td> <td>5 15 42 3<br>— 46</td> <td>2 3 50 47<br>Sun from Node,<br>and Arg. fourth equation.</td> </tr> <tr> <td>Time once equated,<br>Second Equation,</td> <td>28 1 46 8<br>+ 2 15 1</td> <td>6 14 41 20<br>Arg. 3d equation.</td> <td>5 19 41 17<br>Arg. 2d equation.</td> <td></td> </tr> <tr> <td>Time twice equated,<br>Third Equation,</td> <td>28 4 1 9<br>+ 1 9</td> <td></td> <td></td> <td></td> </tr> <tr> <td>Time thrice equated,<br>Fourth equation,</td> <td>28 4 2 18<br>+ 12</td> <td></td> <td></td> <td></td> </tr> <tr> <td>True New Moon,</td> <td>28 4 2 30</td> <td></td> <td></td> <td></td> </tr> </table>
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 EXAMPLE.
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.
<table> <tr> <th rowspan="2">By the Precepts.</th> <th colspan="3">New Moon.</th> <th colspan="3">Sun's Anomaly.</th> <th colspan="3">Moon's Anomaly.</th> <th colspan="3">Sun from Node.</th> </tr> <tr> <th>D.</th><th>H.</th><th>M.</th> <th>S</th><th>°</th><th>"</th> <th>S</th><th>°</th><th>"</th> <th>S</th><th>°</th><th>"</th> </tr> <tr> <td>March 1800,<br>Add 1 Lunation,</td> <td>13</td><td>0</td><td>22</td> <td>17</td><td>8</td><td>23</td><td>19</td><td>55</td> <td>10</td><td>7</td><td>32</td><td>36</td> <td>11</td><td>3</td><td>58</td><td>24</td> </tr> <tr> <td></td> <td>29</td><td>12</td><td>44</td> <td>3</td><td>0</td><td>29</td><td>6</td><td>19</td> <td>0</td><td>25</td><td>48</td><td>0</td> <td>1</td><td>0</td><td>40</td><td>14</td> </tr> <tr> <td>From the sum,<br>Subtract 2000 years,</td> <td>42</td><td>13</td><td>6</td> <td>20</td><td>9</td><td>22</td><td>26</td><td>14</td> <td>11</td><td>3</td><td>41</td><td>36</td> <td>0</td><td>4</td><td>38</td><td>38</td> </tr> <tr> <td>27</td><td>18</td><td>9</td> <td>19</td><td>0</td><td>8</td><td>50</td><td>0</td> <td>0</td><td>15</td><td>42</td><td>0</td> <td>6</td><td>27</td><td>45</td><td>0</td> </tr> <tr> <td>N. M. bef. Chr. 201,<br>Add 6 Lunations,<br>half Lunations,</td> <td>14</td><td>18</td><td>57</td> <td>1</td><td>13</td><td>36</td><td>14</td><td>10</td><td>17</td><td>59</td><td>36</td> <td>5</td><td>6</td><td>53</td><td>38</td> </tr> <tr> <td></td> <td>177</td><td>4</td><td>24</td> <td>18</td><td>5</td><td>24</td><td>37</td><td>56</td> <td>5</td><td>4</td><td>54</td><td>3</td> <td>6</td><td>4</td><td>1</td><td>24</td> </tr> <tr> <td>14</td><td>18</td><td>22</td> <td>2</td><td>0</td><td>14</td><td>33</td><td>10</td> <td>6</td><td>12</td><td>54</td><td>30</td> <td>0</td><td>15</td><td>20</td><td>7</td> </tr> <tr> <td>Full moon, September,<br>First Equation,</td> <td>22</td><td>17</td><td>43</td> <td>21</td><td>3</td><td>22</td><td>47</td><td>20</td> <td>10</td><td>5</td><td>48</td><td>9</td> <td>11</td><td>26</td><td>15</td><td>9</td> </tr> <tr> <td></td> <td>—</td><td>3</td><td>52</td> <td>6</td><td>10</td><td>4</td><td>19</td><td>55</td> <td>—</td><td>1</td><td>28</td><td>14</td> <td colspan="4">Sun from Node, and Argument 4th equation.</td> </tr> <tr> <td>Time once equated,<br>Second Equation,</td> <td>22</td><td>13</td><td>51</td> <td>15</td><td>5</td><td>18</td><td>27</td><td>25</td> <td>10</td><td>4</td><td>19</td><td>55</td> <td colspan="4"></td> </tr> <tr> <td></td> <td>—</td><td>8</td><td>25</td> <td>4</td><td>Arg. 3d equation.</td><td>Arg. 2d equation.</td><td colspan="4"></td> </tr> <tr> <td>Time twice equated,<br>Third Equation,</td> <td>22</td><td>5</td><td>26</td> <td>11</td><td>—</td><td>—</td><td>58</td><td colspan="4"></td> </tr> <tr> <td>Time thrice equated,<br>Fourth Equation,</td> <td>22</td><td>5</td><td>25</td> <td>13</td><td>—</td><td>—</td><td>12</td><td colspan="4"></td> </tr> <tr> <td>True time at London,<br>Add for Alexandria,</td> <td>22</td><td>5</td><td>25</td> <td>1</td><td>2</td><td>1</td><td>27</td><td colspan="4"></td> </tr> <tr> <td>True time there,</td> <td>22</td><td>7</td><td>26</td> <td>28</td><td colspan="4"></td> </tr> </table>
Thus it appears, that the true time of Full Moon, at Alexandria, in September, old style, the year before Christ 201, was the 22d 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 year 4008 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.
<table> <tr> <th rowspan="2">By the Precepts.</th> <th colspan="3">New Moon.</th> <th colspan="3">Sun's Anomaly.</th> <th colspan="3">Moon's Anomaly.</th> <th colspan="3">Sun from Node.</th> </tr> <tr> <th>D.</th><th>H.</th><th>M.</th> <th>S</th><th>°</th><th>"</th> <th>S</th><th>°</th><th>"</th> <th>S</th><th>°</th><th>"</th> </tr> <tr> <td>March 1793,<br>Subtract 5800 years,</td> <td>30</td><td>9</td><td>13</td> <td>55</td><td>9</td><td>10</td><td>16</td><td>11</td> <td>8</td><td>7</td><td>37</td><td>58</td> <td>7</td><td>6</td><td>18</td><td>26</td> </tr> <tr> <td></td> <td>15</td><td>12</td><td>38</td> <td>7</td><td>10</td><td>21</td><td>35</td><td>0</td> <td>6</td><td>24</td><td>43</td><td>0</td> <td>9</td><td>13</td><td>1</td><td>0</td> </tr> <tr> <td>N. M. bef. Chr. 4007,<br>Add 7 Lunations,<br>half Lunations,</td> <td>14</td><td>20</td><td>35</td> <td>48</td><td>10</td><td>18</td><td>41</td><td>11</td> <td>1</td><td>12</td><td>54</td><td>58</td> <td>9</td><td>23</td><td>17</td><td>26</td> </tr> <tr> <td></td> <td>206</td><td>17</td><td>8</td> <td>21</td><td>6</td><td>23</td><td>44</td><td>15</td> <td>6</td><td>0</td><td>43</td><td>3</td> <td>7</td><td>4</td><td>41</td><td>38</td> </tr> <tr> <td>14</td><td>18</td><td>22</td> <td>2</td><td>0</td><td>14</td><td>33</td><td>10</td> <td>6</td><td>12</td><td>54</td><td>30</td> <td>0</td><td>15</td><td>20</td><td>7</td> </tr> <tr> <td>Full Moon, October,<br>First Equation,</td> <td>22</td><td>8</td><td>6</td> <td>11</td><td>5</td><td>26</td><td>58</td><td>36</td> <td>1</td><td>26</td><td>32</td><td>31</td> <td>5</td><td>13</td><td>19</td><td>11</td> </tr> <tr> <td></td> <td>—</td><td>13</td><td>26</td><td>1</td><td>26</td><td>27</td><td>26</td><td>—</td><td>5</td><td>5</td><td colspan="4">Sun from Node, and Argument 4th equation.</td> </tr> <tr> <td>Time once equated,<br>Second Equation,</td> <td>22</td><td>7</td><td>52</td> <td>45</td><td>4</td><td>0</td><td>31</td><td>10</td> <td>1</td><td>26</td><td>27</td><td>26</td> <td colspan="4"></td> </tr> <tr> <td></td> <td>+</td><td>8</td><td>29</td> <td>21</td><td>Arg. 3d equation.</td><td>Arg. 2d equation.</td><td colspan="4"></td> </tr> <tr> <td>Time twice equated,<br>Third Equation,</td> <td>22</td><td>16</td><td>22</td> <td>6</td><td>—</td><td>4</td><td>10</td><td colspan="4"></td> </tr> <tr> <td>Time thrice equated,<br>Fourth Equation,</td> <td>22</td><td>16</td><td>17</td> <td>56</td><td>—</td><td>51</td><td colspan="4"></td> </tr> <tr> <td>Full Moon at London,<br>Add for Babylon,</td> <td>22</td><td>16</td><td>17</td> <td>5</td><td>2</td><td>25</td><td>41</td><td colspan="4"></td> </tr> <tr> <td>True time there.</td> <td>22</td><td>18</td><td>42</td> <td>46</td><td colspan="4"></td> </tr> </table>
So that, on the meridian of London, the true time was October 23d, at 17 minutes 5 seconds past four in the morning; but at Babylon, the true time was October 23d, at 42 minutes 46 seconds past fix in the morning.—This is supposed by some to have been the year of the creation. Appendix.
Of Calcula.- 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 above-mentioned year in the 18th century will answer to the given year in which the new or full Of Calcula-moon is required; and take out the first new moon, ting Eclipses, &c.
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.
<table> <tr> <th>By the Precepts.</th> <th>New Moo.</th> <th>Sun's Anomaly</th> <th>Moon's Anomaly</th> <th>Sun from Node</th> </tr> <tr> <td>D. H. M. S.</td> <td>s o i "</td> <td>s o i "</td> <td>s o i "</td> </tr> <tr> <td>March 1780,<br>Add 400 years,</td> <td>23 23 1 34<br>17 8 43 29</td> <td>9 4 18 13<br>0 13 24 0</td> <td>1 21 7 47<br>1 28 0 6</td> <td>10 18 21 1<br>17 49 0</td> </tr> <tr> <td>From the Sum<br>Subtract 1 Lunation</td> <td>41 7 45 13<br>29 12 44 3</td> <td>9 17 42 13<br>0 29 6 19</td> <td>11 22 35 47<br>0 25 49 0</td> <td>6 10 1<br>40 14</td> </tr> <tr> <td>New Moon March 2180,<br>Add 4 Lunations,</td> <td>11 19 1 10<br>11 8 56 12</td> <td>8 18 35 54<br>3 26 25 17</td> <td>10 26 46 47<br>3 13 16 2</td> <td>4 5 29 47<br>2 40 56</td> </tr> <tr> <td>New Moon July 2180,<br>First Equation,</td> <td>7 21 57 22<br>— 1 3 39</td> <td>0 15 1 11<br>3 9 38 37</td> <td>2 10 2 49<br>— 24 12</td> <td>8 8 10 43<br>Sun from Node and Argument fourth equation.</td> </tr> <tr> <td>Time once equated,<br>Second Equation,</td> <td>7 20 53 43<br>+ 9 24 8</td> <td>10 5 22 34<br>Arg. 3d equation.</td> <td>2 9 38 37<br>Arg. 2d equation.</td> <td></td> </tr> <tr> <td>Time twice equated,<br>Third Equation,</td> <td>8 6 17 51<br>+ 3 56</td> <td></td> <td></td> <td></td> </tr> <tr> <td>Time thrice equated,<br>Fourth Equation,</td> <td>8 6 21 47<br>+ 1 8</td> <td></td> <td></td> <td></td> </tr> <tr> <td>True time, July,</td> <td>8 6 22 55</td> <td></td> <td></td> <td></td> </tr> </table>
True time, July 8th, at 22 minutes 55 seconds past fix 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 fix 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, taken 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. ASTRONOMY.
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 in the Forenoon.
<table> <tr> <th></th> <th>Sun's Longitude.</th> <th>Sun's Anomaly.</th> </tr> <tr> <td>To the radical year after Christ</td> <td>1701</td> <td>9 20 43 50</td> <td>6 13 1 0</td> </tr> <tr> <td>Add complete years</td> <td>60</td> <td>0 0 27 12</td> <td>11 29 26 0</td> </tr> <tr> <td></td> <td>3</td> <td>11 29 17 0</td> <td>11 29 14 0</td> </tr> <tr> <td>March</td> <td></td> <td>1 28 9 11</td> <td>1 28 9 0</td> </tr> <tr> <td>Bissextile Days</td> <td>20</td> <td>20 41 55</td> <td>20 41 55</td> </tr> <tr> <td>Hours</td> <td>22</td> <td>54 13</td> <td>54 13</td> </tr> <tr> <td>Minutes</td> <td>30</td> <td>1 14</td> <td>1 14</td> </tr> <tr> <td>Seconds</td> <td>25</td> <td>1</td> <td>1</td> </tr> <tr> <td>Sun's mean place at the given time</td> <td></td> <td>0 10 14 36</td> <td>9 1 27 23</td> </tr> <tr> <td>Equation of the Sun's centre, add</td> <td></td> <td>1 55 36</td> <td>Mean Anomaly.</td> </tr> <tr> <td>Sun's true place at the same time</td> <td></td> <td>0 12 10 12 or 12 10 12</td> <td></td> </tr> </table>
EXAMPLE II. Required the Sun's true place, October 23d, Old Style, at 16 hours 57 minutes past Noon, in the 4008th year before the year of Christ 1; which was the 4007th before the year of his birth, and the year of the Julian period 766.
<table> <tr> <th>By the Precepts.</th> <th>Sun's Longitude.</th> <th>Sun's Anomaly.</th> </tr> <tr> <td>From the radical numbers after Christ</td> <td>1</td> <td>9 7 53 10</td> <td>6 28 48 0</td> </tr> <tr> <td>Subtract those for 5000 complete years</td> <td></td> <td>1 7 46 40</td> <td>10 13 25 0</td> </tr> <tr> <td>Remains for a new radix</td> <td>900</td> <td>8 0 6 30</td> <td>8 15 23 0</td> </tr> <tr> <td>complete years</td> <td>80</td> <td>0 6 48 0</td> <td>11 21 37 0</td> </tr> <tr> <td>To which add,</td> <td>12</td> <td>0 0 36 16</td> <td>11 29 15 0</td> </tr> <tr> <td>to bring it to the given time</td> <td>October</td> <td>8 29 4 54</td> <td>8 29 4 0</td> </tr> <tr> <td></td> <td>Days 23</td> <td>22 40 12</td> <td>22 40 12</td> </tr> <tr> <td></td> <td>Hours 16</td> <td>39 26</td> <td>39 26</td> </tr> <tr> <td></td> <td>Minutes 57</td> <td>2 20</td> <td>2 20</td> </tr> <tr> <td>Sun's mean place at the given time</td> <td></td> <td>6 0 3 4</td> <td>5 28 33 58</td> </tr> <tr> <td>Equation of the Sun's centre subtract</td> <td></td> <td>3 4</td> <td>Sun's Anomaly.</td> </tr> <tr> <td>Sun's true place at the same time</td> <td></td> <td>6 0 0 0 or 6 0 0 0</td> <td></td> </tr> </table>
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 h. 25 m. 41 sec. for the longitude of Babylon, we shall have for the time of the same equinox, at that place, October 23d, at 19 h. 22 m. 41 sec.; which, in the common way of reckoning, is October 24th, at 22 m. 41 sec. 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 23d, at 42 m. 46 sec. after fix 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 fyzigies; 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 Appendix.
Of Calcula. mean distance from the ascending node is 0° 5' 35" 2".
See Example I.
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° 5' 35" 2", which motion is found in Table XII. for 50 minutes 46 seconds, to be 2' 12". 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° 10' 26" 19"; and then we shall have the sun's true distance from the node, at the true time of new moon, as follows:
<table> <tr> <th>Sun from Node.</th> <th>°</th> <th>'</th> <th>"</th> </tr> <tr> <td>At the mean time of new moon in April 1764</td> <td>0</td> <td>5</td> <td>35</td> <td>2</td> </tr> <tr> <td>Sun's motion from the node for 50 minutes</td> <td>2</td> <td>10</td> <td></td> <td></td> </tr> <tr> <td>Sun's motion from the node for 46 seconds</td> <td>2</td> <td></td> <td></td> <td></td> </tr> <tr> <td>Sun's mean distance from node at true new moon</td> <td>0</td> <td>5</td> <td>37</td> <td>14</td> </tr> <tr> <td>Equation of mean distance from node, add</td> <td>2</td> <td>5</td> <td>0</td> <td></td> </tr> <tr> <td>Sun's true distance from the ascending node</td> <td>0</td> <td>7</td> <td>42</td> <td>14</td> </tr> </table>
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 10 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. 10. The 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. 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 Of Calcula in the table only to every 6th degree), and thereby ting Eclipses, &c. take out the moon's horizontal parallax; which for the above time, answering to the anomaly 11° 9' 24" 21", is 54' 43".
3. To find the sun's distance from the nearest solstice, viz. the beginning of Cancer, which is 3° or 90° from the beginning of Aries. It appears by Example I. (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° 12° 10' 12"; that is, the sun's place at that time is γ Aries, 12° 10' 12".
Therefore from <table> <tr> <th></th> <th>s</th> <th>°</th> <th>'</th> <th>"</th> </tr> <tr> <td></td> <td>3</td> <td>0</td> <td>0</td> <td>0</td> </tr> </table> Subtract the sun's longitude or place <table> <tr> <th></th> <th>s</th> <th>°</th> <th>'</th> <th>"</th> </tr> <tr> <td></td> <td>0</td> <td>12</td> <td>10</td> <td>12</td> </tr> </table> Remains the sun's distance from the solstice <table> <tr> <th></th> <th>s</th> <th>°</th> <th>'</th> <th>"</th> </tr> <tr> <td></td> <td>2</td> <td>17</td> <td>49</td> <td>48</td> </tr> </table> Or 77° 49' 48"; 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° 12°, and making proportions for the 10' 12", take out the sun's declination answering to his true place, and it will be found to be 4° 49' 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° 7' 42" 14". See above.
Therefore, enter Table XVI. with 0 signs at the top, and 7 and 8 degrees at the left hand, and take out 36' and 39", the latitude for 7°; and 41' 51", the latitude for 8°: and by making proportions between these latitudes for the 42' 14", by which the moon's distance from the node exceeds 7 degrees, her true latitude will be found to be 40' 18" north ascending.
6. To find the moon's true horary motion from the sun. With the moon's anomaly, viz. 11° 9' 24" 21", enter Table XVII. and take out the moon's horary motion; which, by making proportions in that Table, will be found to be 30' 22". Then, with the sun's anomaly, 9° 10' 26" 19", take out his horary motion 2' 28" from the same table; and subtracting the latter from the former, there will remain 27' 54" 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° 35', without any sensible error.
8, 9. 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 semidiameter 16' 6", and the moon's anomaly gives her semidiameter 14' 57".
10. 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' 3".
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
<table> <tr> <th></th> <th>1</th> <th>10</th> <th>30</th> <th>25</th> </tr> <tr> <td>o</td> <td></td> <td></td> <td></td> <td></td> </tr> <tr> <td>"</td> <td></td> <td></td> <td></td> <td></td> </tr> </table>
2. Semidiameter of the earth's disk 54 53 3. Sun's distance from the nearest folt. 77 49 48 4. Sun's declination, north 4 49 0 5. Moon's latitude, north ascending 45 18 6. Moon's horary motion from the sun 27 54 7. Angle of the moon's visible path with the ecliptic 5 35 0 8. Sun's semidiameter 16 6 9. Moon's semidiameter 14 57 10. Semidiameter of the penumbra 31 3
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 semi-disk 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 h, in the periphery of the semidisk; and draw the straight line gVh, 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 fix 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 C XII 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 fix signs.
Open the sector till the radius (or distance of the two 90's) of the fines be equal to the length of Vh, and take the fine of the sun's distance from the fullice (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 gVh, 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° 59' will be the colatitude, which take in your compasses from the line of chords, making CA or CB the radius, and set it from h (where the earth's axis meets the periphery of the disk) to VI and ting help 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 F XII G and DLE.
Bisect LK XII in K, and through the point K draw the black line VI K VI. Then make CB the radius of a line of fines on the sector, take the colatitude of London 38° 59' 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, 8o, 9n, &c. parallel to the earth's axis C XII P.
With the small extent K XII as a radius, describe the quadrantal arc XIIf, and divide it into fix equal parts, as XII, a, ab, bc, cd, de, and ef; and through the division points a, b, c, d, e, draw the occult lines VII e V, VIII 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, 8o, &c. in the points VII VIII IX X XI, 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 VII, &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 take 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 r to x in the bisecting line Cz, making ix parallel to Cy; and through x, at right angles to the axis of the moon's orbit CM, draw the straight line N wxy 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 in the middle of the general eclipses; the point x is that where the conjunction Appendix.
Of Calcula. of the sun and moon falls, according to equal time by tig Eclipses, &c. and the point y is the ecliptical conjunction of the sun and moon.
Take the moon's true horary motion from the sun, 27' 34", 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 60 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 on 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 37 1/2 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 on the path of London, at m, namely at 47 1/2 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 on the path of the penumbra's centre at m, in the 47 1/2 minutes 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 centre 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; 47 1/2 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 answers to apparent or 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 were 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 Flamstead'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 II. 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 follows:
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, 4. To find the semidiameters of the sun and moon. Enter Table XVIII. with their respective anomalies, the sun's being 10° 7' 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 37' 23", the sun 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 at the true time of full moon); and this distance increased by fix signs will be the moon's true distance from the same node; and consequently the argument for finding her true latitude.
Thus, in Example II. the sun's mean distance from the ascending node was 0° 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 of 0° 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° 27' 45"; and lastly add fix signs: so shall the moon's true distance from the ascending node be found as follows:
<table> <tr> <th>Sun from node at mean full moon</th> <td>0 4 49 35</td> </tr> <tr> <th>His motion from it in 6 hours</th> <td>1 5 35</td> </tr> <tr> <th>33 minutes</th> <td>1 26</td> </tr> <tr> <th>38 seconds</th> <td>2</td> </tr> <tr> <th>Sum, subtract from the uppermost line</th> <td>17 3</td> </tr> <tr> <th>Remains his mean distance at true full moon</th> <td>0 4 32 32</td> </tr> <tr> <th>Equation of his mean distance, add</th> <td>1 38 0</td> </tr> <tr> <th>Sun's true distance from the node</th> <td>0 6 10 32</td> </tr> <tr> <th>To which add</th> <td>0 0 0 0</td> </tr> <tr> <th>And the sum will be</th> <td>0 6 10 32</td> </tr> </table>
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 the 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° 32".
Now collect these elements together for use.
<table> <tr> <th>1. True time of full moon in May 1762</th> <td>D. H. M. S.</td> <td>8 3 59 50</td> </tr> <tr> <th>2. Moon's horizontal parallax</th> <td></td> <td>0 57 23</td> </tr> <tr> <th>3. Sun's semidiameter</th> <td></td> <td>0 15 56</td> </tr> <tr> <th>4. Moon's semidiameter</th> <td></td> <td>0 15 38</td> </tr> <tr> <th>5. Semidiameter of the earth's shadow at the moon</th> <td></td> <td>0 41 37</td> </tr> <tr> <th>6. Moon's true latitude, south descending</th> <td></td> <td>0 32 21</td> </tr> <tr> <th>7. Angle of her visible path with the ecliptic</th> <td></td> <td>5 35 0</td> </tr> <tr> <th>8. Her true horary motion from the sun</th> <td></td> <td>30 52</td> </tr> </table>
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. of Calculating Eclipses, &c.), and divide it into 60 equal parts, each part ting Eclipses, &c.
Draw the right line ACB (fig. 160. a.) for part of the ecliptic, and CD perpendicular thereto for the southern part of its axis; the moon having fourth 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 on 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 femicircle 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.
Bisect 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 table.
Take the moon's latitude 32' 21", from the scale with your compasses, and set it from C to G in the line CG g; 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 signifying the instant of full moon (viz. 50 minutes 50 seconds after III in the morning) may be in the point G, where the line of the moon's path cuts the line that bisects 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.
<table> <tr> <th rowspan="2">A.D.</th> <th colspan="3">Mean New Moon in March.</th> <th colspan="3">Sun's Mean Anomaly.</th> <th colspan="3">Moon's mean Anomaly.</th> <th colspan="3">Sun's mean Dilt. from the Node</th> </tr> <tr> <th>D.</th><th>H.</th><th>M.</th><th>S.</th> <th>D.</th><th>H.</th><th>M.</th><th>S.</th> <th>D.</th><th>H.</th><th>M.</th><th>S.</th> </tr> <tr><td>1700</td><td>8</td><td>16</td><td>11</td><td>25</td><td>8</td><td>19</td><td>58</td><td>48</td><td>1</td><td>22</td><td>39</td><td>37</td><td>6</td><td>14</td><td>41</td><td>7</td></tr> <tr><td>1701</td><td>7</td><td>13</td><td>44</td><td>59</td><td>8</td><td>20</td><td>59</td><td></td><td>28</td><td>7</td><td>43</td><td>27</td><td>1</td><td>23</td><td>14</td><td>8</td></tr> <tr><td>1702</td><td>16</td><td>22</td><td>34</td><td>18</td><td>27</td><td>36</td><td>51</td><td>11</td><td>7</td><td>53</td><td>47</td><td>8</td><td>1</td><td>16</td><td>53</td><td></td></tr> <tr><td>1703</td><td>6</td><td>7</td><td>21</td><td>58</td><td>16</td><td>55</td><td>43</td><td></td><td>9</td><td>17</td><td>43</td><td>52</td><td>8</td><td>9</td><td>19</td><td>42</td></tr> <tr><td>1704</td><td>24</td><td>4</td><td>53</td><td>57</td><td>9</td><td>5</td><td>44</td><td>54</td><td>8</td><td>23</td><td>20</td><td>57</td><td>19</td><td>8</td><td>2</td><td>43</td></tr> <!-- Remainder of the table follows the same structure --> </table>
<table> <tr> <th rowspan="2">A.D.</th> <th colspan="3">Mean New Moon in March.</th> <th colspan="3">Sun's mean Anomaly.</th> <th colspan="3">Moon's mean Anomaly.</th> <th colspan="3">Sun's mean Dilt. from the Node</th> </tr> <tr> <th>D.</th><th>H.</th><th>M.</th><th>S.</th> <th>D.</th><th>H.</th><th>M.</th><th>S.</th> <th>D.</th><th>H.</th><th>M.</th><th>S.</th> </tr> <tr><td>1753</td><td>22</td><td>17</td><td>48</td><td>49</td><td>3</td><td>6</td><td>28</td><td>2</td><td>8</td><td>19</td><td>21</td><td>5</td><td>4</td><td>23</td><td>28</td><td></td></tr> <tr><td>1754</td><td>12</td><td>22</td><td>37</td><td>25</td><td>22</td><td>25</td><td>2</td><td>0</td><td>8</td><td>7</td><td>26</td><td>5</td><td>12</td><td>20</td><td>15</td></tr> <tr><td>1755</td><td>11</td><td>21</td><td>59</td><td>58</td><td>11</td><td>38</td><td>12</td><td>1</td><td>10</td><td>27</td><td>55</td><td>1</td><td>20</td><td>29</td><td>2</td></tr> <!-- Remainder of the table follows the same structure --> </table> <table> <tr> <th colspan="9">TABLE II. Mean New Moon, &c. in March, New Style, from A. D. 1752 to A. D. 1800.</th> <th colspan="10">TABLE III. Mean Anomalies, and Sun's mean Distance from the Node, for 3 mean Lunations.</th> </tr> <tr> <th rowspan="2">Y. of Chr.</th> <th rowspan="2">Mean New Moon in March.</th> <th rowspan="2">Sun's mean Anomaly.</th> <th rowspan="2">Moon's mean Anomaly.</th> <th rowspan="2">Sun's mean Diff. from the Node.</th> <th colspan="4">Mean Lunations.</th> <th colspan="4">Sun's mean Anomaly.</th> <th colspan="4">Moon's mean Anomaly.</th> <th colspan="4">Sun's mean Dist. from the Node.</th> </tr> <tr> <th>D</th><th>H</th><th>M</th><th>S</th> <th>s</th><th>o</th><th>1'</th><th>1"</th> <th>s</th><th>o</th><th>1'</th><th>1"</th> <th>s</th><th>o</th><th>1'</th><th>1"'</th> </tr> <tr><td>1752</td><td>14 20 16 6</td><td>8 14 44 16</td><td>3 2 42 15</td><td>3 25 40 27</td><td>1</td><td>29</td><td>12</td><td>44</td><td>3</td><td>20</td><td>6</td><td>19</td><td>4</td><td>49</td><td>1</td><td>0</td><td>40</td><td>14</td></tr> <tr><td>1753</td><td>5 4 4 2</td><td>8 4 0 8</td><td>1 12 30 20</td><td>4 3 3 14</td><td>2</td><td>39</td><td>1</td><td>28</td><td>12</td><td>3</td><td>28</td><td>1</td><td>21</td><td>38</td><td>1</td><td>2</td><td>1</td><td>20</td></tr> <tr><td>1754</td><td>2 37 22</td><td>8 22 22 20</td><td>18 7 26</td><td>5 12 26 15</td><td>3</td><td>38</td><td>14</td><td>12</td><td>2</td><td>27</td><td>18 28</td><td>2</td><td>17</td><td>27</td><td>2</td><td>3</td><td>0</td><td>42</td></tr> <tr><td>1755</td><td>12 11 35 50</td><td>8 11 38 12</td><td>10 27 55 31</td><td>5 20 29 2</td><td>4</td><td>41</td><td>8</td><td>56 12</td><td>3</td><td>3 6</td><td>25 17</td><td>3</td><td>13</td><td>16 2</td><td>4</td><td>2</td><td>40</td><td>50</td></tr> <tr><td>1756</td><td>8 8 36 9</td><td>0 4 24</td><td>10 32 37</td><td>6 29 12 3</td><td>5</td><td>47</td><td>15</td><td>40 15</td><td>4</td><td>5 24 31</td><td>37</td><td>5</td><td>9 5 2</td><td>5</td><td>3</td><td>21</td><td>2</td></tr> <!-- Table continues... --> </table>
<table> <tr> <th colspan="10">TABLE IV. The Days of the Year, reckoned from the beginning of March.</th> </tr> <tr> <th>Days</th> <th>Jan.</th> <th>Feb.</th> <th>Mar.</th> <th>Apr.</th> <th>May</th> <th>June</th> <th>July</th> <th>Aug.</th> <th>Sep.</th> <th>Oct.</th> <th>Nov.</th> <th>Dec.</th> </tr> <tr><td>1</td><td>32</td><td>62</td><td>93</td><td>123</td><td>154</td><td>185</td><td>215</td><td>246</td><td>276</td><td>307</td><td>338</td><td>368</td></tr> <tr><td>2</td><td>33</td><td>63</td><td>94</td><td>124</td><td>155</td><td>186</td><td>216</td><td>247</td><td>277</td><td>308</td><td>339</td><td>369</td></tr> <!-- Table continues... --> </table> <table> <tr> <th rowspan="2">Lunar.</th> <th colspan="2">Days. Decimal Parts.</th> <th colspan="4">Days. H. M. S. Th. Fo.</th> </tr> <tr> <th>Days</th> <th>Dec. Parts.</th> <th>29</th> <th>12</th> <th>44</th> <th>3</th> <th>2</th> <th>38</th> </tr> <tr><td>1</td><td>29.53059851080</td><td></td><td>29</td><td>12</td><td>44</td><td>3</td><td>2</td><td>38</td></tr> <tr><td>2</td><td>59.06181702166</td><td></td><td>59</td><td>1</td><td>28</td><td>6</td><td>5</td><td>57</td></tr> <!-- Table continues for all 100 columns --> </table>
<table> <tr> <th>Lunations.</th> <th>Years</th> <th colspan="2">First New Moon.</th> <th colspan="2">Sun's mean Anomaly.</th> <th colspan="2">M's mean Anomaly.</th> <th colspan="2">Sun from Node.</th> </tr> <tr> <th></th> <th></th> <th>D.</th> <th>H. M. S.</th> <th>s</th> <th>°</th> <th>s</th> <th>°</th> <th>s</th> <th>°</th> </tr> <tr><td>11132</td><td>900</td><td>9</td><td>12</td><td>53</td><td>47</td><td>1</td><td>4</td><td>3</td><td>22</td><td>29</td><td>4</td><td>24</td><td>25</td></tr> <!-- Table continues for all 100 columns --> </table>
<table> <tr> <th rowspan="2">Lunations.</th> <th rowspan="2">Years</th> <th colspan="5">The first mean New Moon, with the mean Anomalies of the Sun and Moon, and the Sun's mean Difflance from the Ascending Node, next after complete Centuries of Julian Years.</th> </tr> <tr> <th>Fist New Moon.</th> <th>Sun's mean Anomaly.</th> <th>M's mean Anomaly.</th> <th>Sun from Node.</th> </tr> <tr> <th>D.</th> <th>H. M. S.</th> <th>s</th> <th>°</th> <th>s</th> <th>°</th> <th>s</th> <th>°</th> </tr> <tr><td>1237</td><td>100</td><td>4</td><td>8</td><td>10</td><td>52</td><td>6</td><td>21</td><td>8</td><td>15</td><td>22</td><td>4</td><td>19</td><td>27</td></tr> <!-- Table continues for all 100 columns --> </table> TABLE VII. The annual or first Equation of the mean to the true Sisygy.
Argument. Sun's mean Anomaly.
<table> <tr> <th rowspan="2">Degrees</th> <th colspan="6">Subtract.</th> <th rowspan="2">Degrees</th> </tr> <tr> <th>Signs</th><th>Sign</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> </tr> <tr><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td></td></tr> <tr><td>0° 0' 0"</td><td>2 3 12</td><td>3 35</td><td>4 10 53</td><td>3 39 39</td><td>2 7 45 3°</td><td></td></tr> <tr><td>1° 4 18</td><td>2 6 53</td><td>3 37 10</td><td>4 10 57</td><td>3 37 19</td><td>2 3 55 29</td><td></td></tr> <tr><td>2° 8 35</td><td>2 10 36</td><td>3 39 18</td><td>4 10 55 3</td><td>3 35 6</td><td>2 0 1 28</td><td></td></tr> <tr><td>3° 12 51</td><td>2 14 15</td><td>3 31 23</td><td>4 10 49 3</td><td>3 32 50</td><td>1 56 5 27</td><td></td></tr> <tr><td>4° 17 8</td><td>2 17 52</td><td>3 43 26</td><td>4 10 39 3</td><td>3 30 1</td><td>1 52 6 26</td><td></td></tr> <tr><td>5° 21 24</td><td>2 21 27</td><td>3 45 25</td><td>4 10 24</td><td>3 28 5</td><td>1 48 4 25</td><td></td></tr> <tr><td>6° 25 39</td><td>2 25 9</td><td>3 47 19</td><td>4 10 4</td><td>3 25 35</td><td>1 41 1 24</td><td></td></tr> <tr><td>7° 28 55</td><td>2 28 29</td><td>3 49 7</td><td>4 9 3 23</td><td>1 39 36 25</td><td></td><td></td></tr> <tr><td>8° 34 11</td><td>2 31 57</td><td>3 50 4</td><td>4 10 3 20</td><td>1 35 49 22</td><td></td><td></td></tr> <tr><td>9° 38 26</td><td>2 35 22</td><td>3 52 29</td><td>4 8 37 17</td><td>1 31 41 21</td><td></td><td></td></tr> <tr><td>10° 42 39</td><td>2 38 44</td><td>3 54 4</td><td>4 7 59 3</td><td>1 27 31 20</td><td></td><td></td></tr> <tr><td>11° 46 52</td><td>2 42 3</td><td>3 55 35</td><td>4 7 16 3</td><td>1 23 19 10</td><td></td><td></td></tr> <tr><td>12° 51 4</td><td>2 45 18</td><td>3 57 2</td><td>4 6 29 3</td><td>1 19 5 16</td><td></td><td></td></tr> <tr><td>13° 55 17</td><td>2 48 39</td><td>3 58 27</td><td>4 5 37 3</td><td>1 14 49 17</td><td></td><td></td></tr> <tr><td>14° 59 27</td><td>2 51 40</td><td>3 59 49</td><td>4 4 41 3</td><td>1 10 32 16</td><td></td><td></td></tr> <tr><td>15° 1 3 36</td><td>2 54 48</td><td>3 1 7</td><td>4 3 49 3</td><td>0 7</td><td>1 6 15 15</td><td></td></tr> <tr><td>16° 7 45</td><td>2 57 53</td><td>4 2 18</td><td>4 2 35</td><td>2 57</td><td>1 5 61 14</td><td></td></tr> <tr><td>17° 11 53</td><td>3 54 4</td><td>4 3 23</td><td>4 1 20</td><td>2 33 49</td><td>1 57 36 13</td><td></td></tr> <tr><td>18° 16 3</td><td>3 51 4</td><td>4 2 42</td><td>4 0 12</td><td>2 30 50</td><td>1 53 15 12</td><td></td></tr> <tr><td>19° 20 6</td><td>3 6 45</td><td>4 1 58</td><td>3 58 52</td><td>2 47 18</td><td>0 48 52 11</td><td></td></tr> <tr><td>20° 24 10</td><td>3 9 36</td><td>4 6 10</td><td>3 57 27</td><td>2 43 57</td><td>0 44 28 10</td><td></td></tr> <tr><td>21° 28 12</td><td>3 12 24</td><td>4 6 58</td><td>3 55 59</td><td>2 40 33</td><td>0 40 2 9</td><td></td></tr> <tr><td>22° 32 12</td><td>3 15 9</td><td>4 7 41</td><td>3 54 26</td><td>2 37 6</td><td>0 35 36 8</td><td></td></tr> <tr><td>23° 36 10</td><td>3 17 51</td><td>4 8 21</td><td>3 52 49</td><td>2 33 35</td><td>0 31 17 7</td><td></td></tr> <tr><td>24° 40 6</td><td>3 20 30</td><td>4 8 57</td><td>3 51 9</td><td>2 30 26</td><td>0 26 44 6</td><td></td></tr> <tr><td>25° 44 1</td><td>3 23 5</td><td>4 9 29</td><td>3 49 26</td><td>2 26 26</td><td>0 22 17 5</td><td></td></tr> <tr><td>26° 47 54</td><td>3 25 36</td><td>4 9 55</td><td>3 47 38</td><td>2 22 47</td><td>0 17 50 4</td><td></td></tr> <tr><td>27° 51 46</td><td>3 28 4</td><td>4 10 16</td><td>3 45 44</td><td>2 19 5</td><td>0 13 23 3</td><td></td></tr> <tr><td>28° 55 37</td><td>3 30 26</td><td>4 10 33</td><td>3 43 45</td><td>2 15 20</td><td>0 8 56 2</td><td></td></tr> <tr><td>29° 59 26</td><td>3 32 45</td><td>4 10 45</td><td>3 41 40</td><td>2 11 35</td><td>0 4 29 1</td><td></td></tr> <tr><td>30° 0 12</td><td>3 35</td><td>4 10 53</td><td>3 39 30</td><td>2 7 45</td><td>0 0</td><td></td></tr> </table>
<table> <tr> <th>Deg.</th><th>11 Signs</th><th>10 Signs</th><th>9 Signs</th><th>8 Signs</th><th>7 Signs</th><th>6 Signs</th> </tr> <tr><td>Add</td><td></td><td></td><td></td><td></td><td></td><td></td></tr> </table>
TABLE VIII. Equation of the Moon's mean Anomaly.
Argument. Sun's mean Anomaly.
<table> <tr> <th rowspan="2">Degrees</th> <th colspan="6">Subtract.</th> <th rowspan="2">Degrees</th> </tr> <tr> <th>Signs</th><th>Sign</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> </tr> <tr><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td></td></tr> <tr><td>0° 0' 0"</td><td>0 46 45</td><td>1 21 32</td><td>1 35</td><td>1 23</td><td>4 48 19</td><td>3 0</td></tr> <tr><td>1° 37</td><td>48 10</td><td>1 22 21</td><td>1 35</td><td>1 22 14</td><td>4 6 51 29</td><td></td></tr> <tr><td>2° 13</td><td>49 34</td><td>1 23 10</td><td>1 35</td><td>1 21 24</td><td>4 5 23 28</td><td></td></tr> <tr><td>3° 45 2</td><td>50 53</td><td>1 23 57</td><td>1 35</td><td>1 20 32</td><td>4 3 54 27</td><td></td></tr> <tr><td>4° 6 28</td><td>52 19</td><td>1 24 41</td><td>1 34 57</td><td>1 19 38</td><td>4 2 24 26</td><td></td></tr> <tr><td>5° 8 6</td><td>53 40</td><td>1 25 24</td><td>1 34 50</td><td>1 18 42</td><td>4 0 53 25</td><td></td></tr> </table>
<table> <tr> <th>Signs</th><th>Sign</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Degrees</th> </tr> <tr><td>0</td><td>9 42</td><td>55</td><td>26</td><td>6</td><td>34 43</td><td>1 17 4 39 21 24</td></tr> <tr><td>1</td><td>11 20</td><td>56</td><td>21</td><td>26</td><td>34 33</td><td>1 16 4 37 49 23</td></tr> <tr><td>2</td><td>12 56</td><td>57</td><td>38</td><td>27</td><td>34 22</td><td>1 15 4 36 15 22</td></tr> <tr><td>3</td><td>14 33</td><td>58</td><td>36</td><td>28</td><td>34 9</td><td>1 14 4 34 40 21</td></tr> <tr><td>4</td><td>16 10</td><td>1</td><td>13</td><td>28</td><td>33 53</td><td>1 13 4 33 5 20</td></tr> <tr><td>5</td><td>17 47</td><td>1</td><td>29</td><td>20</td><td>33 37</td><td>1 12 3 31 11 16</td></tr> <tr><td>6</td><td>19 23</td><td>1</td><td>24</td><td>29</td><td>33 22</td><td>1 11 3 29 54 18</td></tr> <tr><td>7</td><td>20 59</td><td>1</td><td>30</td><td>30</td><td>32 38</td><td>1 10 2 28 18 17</td></tr> <tr><td>8</td><td>22 35</td><td>1</td><td>58</td><td>30</td><td>32 19</td><td>1 26 40 16 11</td></tr> <tr><td>9</td><td>24 10</td><td>1</td><td>6 18</td><td>31</td><td>32 14</td><td>1 8 8 25 3 13</td></tr> <tr><td>10</td><td>25 45</td><td>1</td><td>7 27</td><td>31</td><td>31 5</td><td>1 6 58 23 23 14</td></tr> <tr><td>11</td><td>27 19</td><td>1</td><td>8 36</td><td>32</td><td>31 23</td><td>1 5 40 21 41 13</td></tr> <tr><td>12</td><td>28 52</td><td>1</td><td>9 42</td><td>32</td><td>30 53</td><td>1 4 32 20 7 12</td></tr> <tr><td>13</td><td>30 23</td><td>1</td><td>10 49</td><td>32</td><td>30 25</td><td>1 3 19 18 28 11</td></tr> <tr><td>14</td><td>31 57</td><td>1</td><td>11 54</td><td>33</td><td>29 54</td><td>1 2 1 16 48 10</td></tr> <tr><td>15</td><td>33 29</td><td>1</td><td>12 58</td><td>33</td><td>29 26</td><td>1 0 43 15 9</td></tr> <tr><td>16</td><td>35 2</td><td>1</td><td>14 1</td><td>33</td><td>28 45</td><td>0 59 26 13 28 8</td></tr> <tr><td>17</td><td>36 32</td><td>1</td><td>15 1</td><td>34 1</td><td>28 9</td><td>0 58 7 11 48 7</td></tr> <tr><td>18</td><td>38 1</td><td>1</td><td>16 1</td><td>34 18</td><td>27 30</td><td>0 56 4 10 7 6</td></tr> <tr><td>19</td><td>39 29</td><td>1</td><td>16 59</td><td>34 36</td><td>26 50</td><td>0 55 23 8 2 5</td></tr> <tr><td>20</td><td>40 59</td><td>1</td><td>17 57</td><td>34 45</td><td>26 27</td><td>0 54 1 6 44 4</td></tr> <tr><td>21</td><td>42 26</td><td>1</td><td>18 52</td><td>34 48</td><td>25 5</td><td>0 52 37 5 3 3</td></tr> <tr><td>22</td><td>43 54</td><td>1</td><td>19 47</td><td>34 54</td><td>24 39</td><td>0 51 12 3 21 2</td></tr> <tr><td>23</td><td>45 19</td><td>1</td><td>20 40</td><td>34 58</td><td>23 52</td><td>0 49 45 1 40 0</td></tr> <tr><td>24</td><td>46 45</td><td>1</td><td>21 32</td><td>35 1</td><td>23 4</td><td>0 48 19 0 0 0</td></tr> </table>
<table> <tr> <th>Deg.</th><th>11 Signs</th><th>10 Signs</th><th>9 Signs</th><th>8 Signs</th><th>7 Signs</th><th>6 Signs</th> </tr> <tr><td>Add</td><td></td><td></td><td></td><td></td><td></td><td></td></tr> </table>
TABLE IX. The second Equation of the mean to the true Sisygy.
Argument. Moon's equated Anomaly.
<table> <tr> <th rowspan="2">Degrees</th> <th colspan="6">Add</th> <th rowspan="2">Degrees</th> </tr> <tr> <th>Signs</th><th>Sign</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> </tr> <tr><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td>H. M. S.</td><td></td></tr> <tr><td>0° 0' 0"</td><td>5 12 48</td><td>8 47 8</td><td>9 46 44</td><td>8 8 59</td><td>4 34 33 3</td><td></td></tr> <tr><td>1° 10 58</td><td>5 21 56</td><td>8 51 45</td><td>9 45 4</td><td>8 3 12</td><td>4 26 1 26</td><td></td></tr> <tr><td>2° 21 56</td><td>5 30 57</td><td>8 56 10</td><td>9 45 12</td><td>7 57 23</td><td>3 17 25 8</td><td></td></tr> <tr><td>3° 32 54</td><td>5 39 51</td><td>9 0 25</td><td>9 44 11</td><td>7 51 3 4 8 7</td><td>2 42 7 27</td><td></td></tr> <tr><td>4° 43 52</td><td>5 48 37</td><td>9 4 31</td><td>9 42 59</td><td>7 45 4 0 7</td><td>2 37 4 3 23 23</td><td></td></tr> <tr><td>5° 54 50</td><td>5 57 17</td><td>9 8 25</td><td>9 41 36</td><td>7 39 46</td><td>2 31 23 23 23 23</td><td></td></tr> <tr><td>6° 1 54 8</td><td>6 5 51</td><td>9 12</td><td>9 40 3</td><td>7 33 36</td><td>3 42 32 24</td><td></td></tr> <tr><td>7° 16 46</td><td>6 14 19</td><td>9 15 43</td><td>9 38 19</td><td>7 27 22</td><td>3 33 38 23</td><td></td></tr> <tr><td>8° 27 44</td><td>6 22 41</td><td>9 19 5</td><td>9 36 24</td><td>7 21 2</td><td>3 24 42 22</td><td></td></tr> <tr><td>9° 38 40</td><td>6 30 57</td><td>9 22 14</td><td>9 34 18</td><td>7 14 3 15 41 21</td><td>3 15 41 21 15 41 21</td><td></td></tr> <tr><td>10° 49 33</td><td>6 39 4</td><td>9 25 12</td><td>9 32 1</td><td>7 7 50</td><td>3 6 5 4 20</td><td></td></tr> <tr><td>11° 2 0 33</td><td>6 47 9</td><td>9 27 54</td><td>9 29 33</td><td>7 1 2</td><td>2 57 43 1</td><td></td></tr> <tr><td>12° 11 10</td><td>6 54 46</td><td>9 30 29</td><td>9 26 54</td><td>6 54 8</td><td>2 48 39 17</td><td></td></tr> <tr><td>13° 21 54</td><td>7 2 24</td><td>9 32 58</td><td>9 24 4</td><td>6 47 9</td><td>2 39 34 17</td><td></td></tr> <tr><td>14° 32 34</td><td>7 9 52</td><td>9 35 14</td><td>9 21</td><td>6 40 6</td><td>2 30 28 16</td><td></td></tr> <tr><td>15° 43 29</td><td>7 17 9</td><td>9 37 12</td><td>9 17 51</td><td>6 32 56</td><td>2 21 19 15</td><td></td></tr> </table> <table> <tr> <th colspan="9">TABLE IX. Concluded.</th> <th colspan="6">TABLE XII. The Sun's mean Long'tude, Motion, and Anomaly, Old Style.</th> </tr> <tr> <th>Signs</th><th>H. M. S.</th><th>Signs</th><th>H. M. S.</th><th>Signs</th><th>H. M. S.</th><th>Signs</th><th>H. M. S.</th><th>Signs</th><th>H. M. S.</th><th>Sun's mean Longitude.</th><th>Sun's mean Anomaly.</th><th>Sun's mean Motion.</th><th>Sun's mean Anomaly.</th> </tr> <tr><td></td><td></td><td>5</td><td>12</td><td>48</td><td>3</td><td>47</td><td>8</td><td>46</td><td>44</td><td>8</td><td>59</td><td>+34</td><td>33</td><td>30</td></tr> <!--Rest omitted for brevity.--> </table>
<table> <tr> <th colspan="2">TABLE X. The third equation of the mean to the true Syzygy.</th> <th colspan="2">TAB.XI. The fourth equation of the mean to the true Syzygy.</th> <th colspan="2">TABLE XII. The Sun's mean Long'tude, Motion, and Anomaly, Old Style.</th> </tr> <tr> <th colspan="3">Argument. Sun's Anomaly—Moons' Anomaly.</th> <th colspan="3">Argument. Sun's mean distance from the Node.</th> <th colspan="3">Sun's mean Motion and Anomaly.</th> <th colspan="3">Sun's mean dist. from the Node.</th> </tr> <tr> <th>Signs</th><th>Sub.</th><th>Add.</th> <th>Signs</th><th>Add.</th><th>Add.</th> <th>Signs</th><th>M.</th><th>S.</th><th>M.</th><th>S.</th><th>M.</th><th>S.</th> </tr> <tr><td></td><td>0</td><td>0</td><td>0</td><td>22</td><td>4</td><td>130</td><td>0</td><td>0</td><td>1</td><td>1</td><td>72</td><td>39</td></tr> <!--Rest omitted for brevity.--> </table>
<table> <tr> <th>Sigs.</th><th>Sigs.</th><th>Sigs.</th><th>Degres</th> <th>Sub.</th><th>Sub.</th><th>Add.</th><th>Add.</th><th>Degres</th> <th>Ho</th><th>I</th><th>I</th><th>Subtract.</th> <th>M.</th><th>S.</th><th>M.</th><th>S.</th><th>Deges</th> <th>H</th><th>O</th><th>I</th><th>Subtract.</th> <th>M.</th><th>S.</th><th>M.</th><th>S.</th> </tr> <tr><td>0</td><td>0</td><td>2</td><td>22</td><td>4</td><td>130</td><td>0</td><td>0</td><td>1</td><td>1</td><td>72</td><td>39</td><td>Miss 114</td><td>Add 8</td><td>0</td><td>0</td><td>1</td><td>1</td><td>0</td><td>0</td><td>25</td><td>43</td><td>50</td><td>48</td></tr> <!--Rest omitted for brevity.--> </table>
In Leap years, after February, each day and one day's motion. TABLE XIII. Equation of the Sun's centre, or the difference between his mean and true place.
<table> <tr> <th colspan="2">Argument.</th> <th colspan="5">Sun's mean Anomaly.</th> <th colspan="6">Subtract</th> </tr> <tr> <th>Signs</th><th>Degrees</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> </tr> <tr><td colspan="13"></td></tr> <tr><td colspan="13">Add</td></tr> </table>
TABLE XIV. The Sun's Declination.
<table> <tr> <th colspan="2">Argument.</th> <th colspan="5">Sun's true place.</th> </tr> <tr> <th>Signs</th><th>Degrees</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> </tr> <tr><td colspan="7"></td></tr> <tr><td colspan="7">Add</td></tr> </table>
TABLE XV. Equation of the Sun's mean Distance from the Node.
<table> <tr> <th colspan="2">Argument.</th> <th colspan="5">Sun's mean Anomaly.</th> <th colspan="6">Subtract</th> </tr> <tr> <th>Signs</th><th>Degrees</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> </tr> <tr><td colspan="13"></td></tr> <tr><td colspan="13">Add</td></tr> </table>
TABLE XVI. The Moon's Latitude in Eclipses.
<table> <tr> <th colspan="2">Arg. Moon's equated Distance from the Node.</th> <th colspan="5">Moon's Horizontal Parallax.</th> <th colspan="5">Moon's Semidiameter.</th> <th colspan="5">Moon's Horizon. Motion.</th> <th colspan="5">Moon's Sun and Moon's Aomaly.</th> </tr> <tr> <th>Signs</th><th>Degrees</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> </tr> <tr><td colspan="24"></td></tr> <tr><td colspan="24">Add</td></tr> </table>
TABLE XVII. The Moon's horizontal Parallax, with the Semidiameters and true Horary Motions of the Sun and Moon, to every sixth degree of their mean Anomalies, the quantities for the intermediate degrees being easily proportioned by sight.
<table> <tr> <th colspan="2">Moon's Sun and Moon's Aomaly.</th> <th colspan="5">Moon's Horizontal Parallax.</th> <th colspan="5">Moon's Semidiameter.</th> <th colspan="5">Moon's Horizon. Motion.</th> <th colspan="5">Moon's Sun and Moon's Aomaly.</th> </tr> <tr> <th>Signs</th><th>Degrees</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> <th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th><th>Signs</th> </tr> <tr><td colspan="24"></td></tr> <tr><td colspan="24">Add</td></tr> </table>
This Table shows the Moon's Latitude a little beyond the utmost Limits of Eclipses. II. Description of Astronomical Instruments serving to illustrate the Motions of the Heavenly Bodies.
The machine represented by fig. 161. 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 pillars. 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 principal 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 the equinoctial circle, making an angle of 23 1/2 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 femicircle, moveable upon two points fixed in θ and ω. This femicircle 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 to be 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 1/2 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 cafe, by which its motion represents the phases of the moon according to her age. Without the orbits of 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 of Astronomy 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 glass 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 glass 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 θ, 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 shewn 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 rotation 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. 162.) 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 parallellism 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 23 1/2 days on its axis, of which the north pole inclines toward the eighth degree of Pices 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 earth, 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 87 1/3 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 24 1/3 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 Part 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 fun again, in 24 hours of the same time. No. 6. is a fidereal dial-plate under the earth, and No. 7. a solar dial-plate on the cover of the machine. The index of the former shows fidereal, 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 fidereal days, or apparent revolutions of the stars. In the time that the earth makes 365 1/4 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 shows at what time the sun rises and sets to all places throughout the year. The earth's axis inclines 23 1/2 degrees from the axis of the ecliptic; the north pole inclines towards 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 compass. 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 sunrising, 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 above-mentioned horizon is rectified to the latitude of any given place, the times of the moon's rising and setting, together with her amplitude, are shewn 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 fines and degrees in 18 1/3 years. The degrees of the moon's latitude to the highest at NL (north latitude) and lowest at SL (south latitude), are engraven 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 shewn 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 shewn 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 moons and node, she will be eclipsed, and the annual index shows the day of that eclipse. There is a circle of 29 1/2 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. 163.) 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 A a E b B, 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. 162.) 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. Appendix.
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. 164.) 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 wheel 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 theron. 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 1 1/2 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 confirmation 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 magnitudes; 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 see by this planetarium that the motions of the planets should always be regular and uniformly the of Astronomical Instruments; 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 presently shewn. (5.) By the machine you see the planets move all the same way, viz. from west to east continually: but in the heavens we seem 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 brafs ball for the sun, and restoring the earth to its proper situation among the planets, then every thing 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. 165. 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 mean time 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. 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 seasons 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 23 1/2 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 betwixt 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 orreries 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. 166.) 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 aforesaid case) through the shadow of the earth, and is by that means eclipsed. And the orbit A (fig. 164.) 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 candle light only, you will see (the machine being in motion) the immersions and emersions 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 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 Situations.
Having dwelt 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 assistance of an ephemeris or almanack, which among other almanacks is published annually by the Stationers Company.
"The ephemeris contains a diary or daily account of the planets 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 fix 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.
<table> <tr> <th>Days.</th> <th>Day increaf.</th> <th>Helioc. long. h</th> <th>Helioc. long. v</th> <th>Helioc. long. d</th> <th>Helioc. long. Θ</th> <th>Helioc. long. Ψ</th> <th>Helioc. long. χ</th> </tr> <tr> <td>1</td> <td>3</td> <td>11 16 19 56 17</td> <td>11</td> <td>1 25 30 11 17 37</td> <td>0 4 35 7 58</td> <td></td> <td></td> </tr> <tr> <td>7</td> <td>3</td> <td>35 16 56 17</td> <td>43 4</td> <td>23 17 37 10 7 25</td> <td>23</td> <td>23</td> <td>23</td> </tr> <tr> <td>31</td> <td>3</td> <td>59 17 7 18</td> <td>15 7</td> <td>15 23 36 19 38 11 7 9</td> <td></td> <td></td> <td></td> </tr> <tr> <td>91</td> <td>4</td> <td>23 17 17 18</td> <td>47 10</td> <td>6 29 33 29 8 28 33</td> <td></td> <td></td> <td></td> </tr> <tr> <td>52</td> <td>4</td> <td>47 17 28 19</td> <td>19 12</td> <td>5 5 30 8 38 15 49</td> <td></td> <td></td> <td></td> </tr> </table> " 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 (b), is in 17° 17', or 17 degrees (rejecting the minutes, being in this case useless) of Capricornus (♑); of Jupiter (♃) in 18° of Aquarius (♒); Mars (♂) in 10° of Cancer (♋); the earth (⊕) in 29° of Virgo (♍); Venus (♀) in 26° of Sagittarius, (♐); Mercury (☿) in 28 degrees 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 assigning 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 (♑); 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 the 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 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 sort 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 apogee point of her orbit, and the months in which the sun and moon must be eclipsed.
"On the great immovable plate A (see fig. 167.) are the months and days of the year, and the signs and degrees of the zodiac so placed, that when the annual index h 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, and is carried round the immovable plate, with it, by means of the knob n. The carrying this frame and index round the immovable 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 fixed on the axis a, which is fixed in the centre of the immovable 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 with 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 23½ degrees from a light 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, fixed 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 Ω₁, to which the descending node is opposite below e, but hid from view by the globe e. The half Ω₁ T e of this orbit is on the north side of the ecliptic OP, and the other half e S Ω₁ 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 U U, 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, fixed into the upper plate of the frame at X. A straight wire W proceeds from the fun Z, and points always towards the centre of the earth 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 h shows the sun's place in the ecliptic for every day of the year, by turning the frame round the axis of the immovable 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 still keep parallel to those on the immovable 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 19½ degrees every Julian year; and the moon's apogal wire UU is moved forward, or 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 h goes round the circles of months and signs on the immovable 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 h to any given day of the year, on the immovable 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 description of Africa.
"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 24½ degrees south from the equator.
"If you bring the annual-index h to the beginning of January, and turn the moon's orbit ST by its supporting wires Q and R till the ascending node (marked Ω) 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 apogal 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 apogal 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 months and signs upon it, you will see that the earth's axis and beginning of Cancer will still keep towards 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 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 parallelism 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 18 years and a half."
The Cometarium, (fig. 168.). 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 degrees of eccentricity. It was invented by the late Dr Defagullier.
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 h, 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 ecliptic orbit is divided into 12 equal parts or signs, with their respective degrees, and so is the circle n o p q r 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, Description as is shewn by the small knob on the end of the wire of Astronomical Instruments; 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 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.
ABCD EFGHIKLM is a circular orbit for showing the equable motion of a body round the fun 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 a b, b c, &c.
Now suppose the two bodies Y and I 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 I all the way from a to g, and from A to G: but I 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, 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 fun, 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. 169. 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 focuses, 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 h. On H, the axis of the handle or winch N in fig. 216, is an endless screw in fig. 217, working in the wheels D and E, whose numbers of teeth being equal, and should be equal to the number of lines, a S, b S, c S, &c. in fig. 168, 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 h of the elliptical 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 h moves so much faster than the end I, although it goes no sooner round the centre E. But then the quick-moving end h of the plate FF leads about the short end h K of the plate GG with the same velocity; Description and the slow-moving end I of the plate FF coming of Astrono- half round as to B, must then lead the long end k of the plate GG as slowly about : so that the elliptical 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 GC from its axis at K : or in other words, to infintce in two points, if the distance K k be four, five, or fix times as great as the distance K h, the point h will move in that position, four, five, or fix 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 K k and K h. 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. 169, is the sun S in fig 168 : which sun, by the wire fixed to it, carries the ball r 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. 169, is the sun S in fig. 168, 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 above 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. 170. On the north pole of the axis, above the hour-circle, is fixed an arch MKH of 23 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 E and H, to which are fixed the quadrantal wires N and O, having two little balls on their ends for the sun and moon, as in the figure. The collet D is fixed to the circular plate F, whereon the 29 1/2 days of the moon's age are engraven, beginning just under the sun'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 sun appears to do) but has a declination of 5 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 when they go below the horizon in an oblique sphere.
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 the 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 point of the horizon the sun and moon balls rise and set, for these agree with the points of the compass on which the sun and moon rise and set in the heavens on the given day; and the hour index shows the time 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 forementioned 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 appluses 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. 171. 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 1000,000 of miles. M is the moon, whose centre is 1/45 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 1/4 days is to 29 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 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 l, which points out the days of the months answering the days of the moon's age, shewn by the index F, in the circle of 29 1/2 equal parts at the other end of the bar. On the Appendix.
Description the axis of the wheel L is put the piece D, below of Astrono- the cock C, in which this axis turns round; and in mical In- D are put the pencils e and m directly under the earth stru- 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 caufe its spiked feet (of which two appear at P and P, the third being supposed to be hid from light 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, crosswise to the bar; which done, move the bar slowly round the axis g of the wheel Y; and as the earth D 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 1/2 equal parts. And as this last index points to the different days in its circle, the like numeral figures may be fet 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 7/8 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.
III. 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 Description northward as southward.
This place, called an Observatory, should contain some, if not all, of the following instruments:
I. 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 Brahe 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 Siffon, Graham, Bird, and other eminent mathematical instrument makers in London. The construction of them being generally the same in all the sizes, we shall here describe one made by the late Joh. Siffon, under the direction of the late Mr. Graham. Fig. 172. represents the instrument as already fixed to the wall. It is of copper, and of about five feet radius. The frame is formed of flat bars, and strengthened by edge bars affixed underneath perpendicularly to them. The radii HB, HA, being divided each into four equal parts, serve 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 EF or ef, which goes into a hol- Description low screw in order to restore the instruments 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 observation. In order to prevent the unsteadiness of so great a machine, 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 are 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 FO, which is perpendicular to the plane of the instrument, and which would pass through the centre if it was continued. The 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 near 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 takes 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 so 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 quarters of minutes, that the angular height of the object observed is equal to. The remainder is easily estimated 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 brads, CDAB (fig. 173.), which is a small portion of X (fig. 172.) 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 division difference, or 30"; and so on unto the 20th and last of all. All the differences being accumulated each of the 20th 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 15", 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 30", 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 30", 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 15" of the limb of a quadrant five feet radius, and simply divided into 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 15" 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 5' 15" 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 8" 47 1/2"'. 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. 174. 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 brads, 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 Appendix.
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 again, 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 distance 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. 175.) 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 description length of the telescope, or the radius of the sector, is of astronomical instruments 24 feet; the breadth of the radius, near the end C, is 1 1/2 inch; and at the end D two inches. The breadth of the limb AB is 1 1/2 inch; and its length fix 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 thickness 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-glas, 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 by 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 hfo 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 hfo, 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 mn; if the cross hairs cover the same star supposed at b, the elevation of the axis hfo is exactly right; but if it be necessary to move the telescope into the position uv, in order to point to this star at c, the arch mu, which measures the angle mfu or bfc, will be known; and then the axis hfo 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 fix-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 ofpf, 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 afp and afp. Unless the star be 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 a 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 Roemer 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. 176.), to which the middle of the telescope is fixed, is about 2 1/2 feet long, tapering gradually towards its ends, which terminate in cylinders well turned and smoothed. The telescope CD, which is about four feet and 1 1/2 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 stock 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 8 1/2 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 a 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 PV, 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 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 sextantal 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.
3. Compound Transit Instrument. Some instruments have been contrived to answer both kinds of observations, viz. either a transit or equal altitudes. Fig. 178. 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 Graham, mounted and fixed 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 a b, 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. 177.), 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. 178.) 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 dove-tail slider; which slider is Appendix.
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 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 times of its arrival at that circle; it follows, that having once found the difference of right ascension of two stars about 60 degrees asunder, 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 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. 179.) 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 <table> <tr> <th>Measures of the several circles and divisions on them.</th> <th>Radius indec.</th> <th>Limb divided to</th> <th>Nomus of 30 gives seconds</th> <th>Divided on limb into parts of inc.</th> <th>Divided by Nomus into parts of inc.</th> </tr> <tr> <td>Azimuth or horizontal circle</td> <td>5</td> <td>1</td> <td>15'</td> <td>30''</td> <td>45th</td> <td>1350th</td> </tr> <tr> <td>Equatorial or hour circle</td> <td>5</td> <td>1</td> <td>15' in time</td> <td>30''</td> <td>45th</td> <td>1350th</td> </tr> <tr> <td>Vertical semicircle for declination or latitude.</td> <td>5</td> <td>5</td> <td>15'</td> <td>30''</td> <td>42d</td> <td>1260th</td> </tr> </table>
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 shewn 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 eye-tubes 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 14.4 inches; and when all the parts are horizontal is about 29 inches high: the weight of the equatorial and apparatus is only 59 lb. avoidupos, 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, or some corresponding line represented by the small brafs 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 brafs 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 brafs 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-semicircle; 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-semicircle; repeat this till the bubble be right in both positions. In order to make the brafs 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-semicircle 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 aforesaid 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 brafs rod. The polar axis remaining nearly horizontal as before, and the declination-semicircle at 0°, adjust the bubble by the hour circle; then turn the declination-semicircle 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 captain screws at the feet of the two supports on one side of the axis of mo-
PLATE LIX.
Fig. 1. Fig. 2. Fig. 3. Fig. 4. Fig. 5. Fig. 6. Fig. 7. Fig. 8. Fig. 9. Fig. 10. Fig. 11.
PLATE LX.
Fig. 12.
Fig. 13.
Fig. 14.
Fig. 16.
Fig. 15. Fig. 18. Fig. 19. Fig. 20.
Fig. 17. The Moon in her mean libration with the Spots according to Roccoti Cassini &c.
Fig. 21. Fig. 22. Fig. 23. Fig. 24. Fig. 25. Fig. 26.
South
E. Mitchell fecip.
PLATE LXII.
The Motion of Venus and Mercury in respect of the Earth.
Venus
The Earth
Mercury
Fig. 27. Fig. 28. Fig. 29. Fig. 30. Fig. 31. Fig. 32. Fig. 33. Fig. 34.
W. Archibald Sculp.
PLATE LXIII.
Fig. 36. Fig. 37. Fig. 38. Fig. 39. Fig. 40. Fig. 41. Fig. 42. Fig. 43. Fig. 44. Fig. 45. Fig. 46. Fig. 47. Fig. 48. Fig. 49. Fig. 50. Fig. 51. Fig. 52. Fig. 53. Fig. 54. Fig. 55. Fig. 56. Fig. 57. Fig. 58. Fig. 59. Fig. 60. Fig. 61. Fig. 62. Fig. 63. Fig. 64. Fig. 65.
PLATE LXIV.
Fig. 66. P M C E O N
Fig. 68.
Fig. 69.
Fig. 70.
Fig. 71.
Fig. 72.
Fig. 73. Distance of the fourth Satellite from the third.
Fig. 74. SATURN
JUPITER
MARS ♂ EARTH ♀ & MOON ♀ VENUS ♀ MERCURY ♀
Fig. 75.
Fig. 76.
Fig. 77. Distance of the Moon from the Earth
SATURN & his Satellites
EARTH MOON Distance of the Moon from the Earth
Fifth Satellite thrice the distance of the Fourth
PLATE LXV.
Fig. 77. Mar. 5. Feb. 5. Jan. 25. Jan. 5. Dec. 29. Dec. 21. Dec. 12. the Sun
Orbit of the Earth
Fig. 78.
Fig. 79.
Fig. 80.
North
Fig. 81.
*λ *μ *ε *κ *υ *ν *μμ *ζ *δ *γ *α *β *η *ο *κ *λλ *ι *ρ *σ *τ *υ *θ *φ *χ *ψ *ω *ε' *γ'
Bootes Bootes Bootes Bootes
Regis Sculp.
PLATE LXVI.
The Principal fixed Stars in the North Hemisphere Delineated on the Plane of the Equator.
PLATE LXVII.
Fig. 82.a
The Principal fixed Stars in the South Hemisphere Delineated on the Plane of the Equator.
W. Train Sculpt.
PLATE LXVIII.
Fig. 82. b
Northern Hemisphere with the Figures of the Constellations.
W. Train Sculp.
PLATE LXIX.
Fig. 82. c
Southern Hemisphere with the Figures of the Constellations.
W. Train Sculpt.
PLATE LXX.
Fig. 83. Fig. 84. Fig. 86. Fig. 85. Fig. 87. Fig. 88. Fig. 89. Fig. 91. Fig. 92. Fig. 93. Fig. 94.
W. Arnold & Son, London.
PLATE LXXI.
Fig. 96 & 103.
Fig. 98.
Ptolemaic System.
Fig. 95.
Fig. 97.
PLATE LXXII.
Fig. 101. C A B D
Fig. 102. A a B b C c D d E I F H K I II E I II
Fig. 100. Tycho's System.
Fig. 104.
Fig. 105.
Fig. 106.
E. Mitchell sculp.
PLATE LXXIII.
Fig. 107.
Fig. 108.
Fig. III.
Fig. III2.
Fig. III4.
Fig. III6.
Fig. III5.
Fig. III3.
W. Archibald sculp. ASTRONOMY
PLATE LXXIV.
Fig. III7. Fig. III8. Fig. III9. Fig. 120. Fig. 121. Fig. 122.
The motion of Saturn Jupiter and Mars in respect of the Earth.
Saturn Jupiter Mars The Earth
Fig. 123. Fig. 124. Fig. 125. Fig. 126. Fig. 127. Fig. 128. Fig. 129. Fig. 130. Fig. 131.
View of the proportional Magnitudes of the Planetary Orbits. Orbit of the Georgium Sidus.
Proportional Magnitude of the Primary Planets. Saturn 5 Jupiter 2 Georgium Sidus Mars 6 Earth 8 Venus 9 Mercury 3
Apparent Magnitude of the Sun seen from each Planet. From Mercury From Venus From Earth From Mars From Jupiter From Saturn From Georgium Sidus
N.B. 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 'e'
PLATE LXXVI.
Fig. 132. Fig. 137. Fig. 138. Fig. 139. Fig. 99 & 133. Fig. 136. Fig. 140. Fig. 146. Fig. 142. Fig. 141. Fig. 145. Fig. 143. Fig. 144. Fig. 147. Fig. 148.
Reile Sculp. Edin."
PLATE LXXVII.
Fig 149.
Fig 150.
THE SOLAR SYSTEM
Reffo Sculp. Edin?
PLATE LXXVIII.
Fig. 151. Fig. 152. Fig. 153. Fig. 154. Fig. 157. Fig. 155. Fig. 156. Fig. 158. Fig. 159. Fig. 160. Fig. 159.a.
W. Archibald Smulp.
PLATE LXXIX.
Fig. 158. a.
Path of the Renumerous Comet Earth's Sondial Moon's Latitude Sundialometer of the Pyramids Moon's Hourly motion, for the Sun State Sundialometer
Fig. 160. a.
Fig. 163.
Fig. 170.
Fig. 175.
Reff's Sculp. Fig.161. The GRAND ORRERY by Rowley
Fig.162. FERGUSONS ORRERY
PLATE LXXXI.
Fig. III. TRAJECTORIUM LUNARE.
Fig. 164. PLANETARIUM by Jones.
Fig. 165.
Fig. 166.
M. Anselmehold sculp't ASTRONOMY
Fig. 167. Mechanical Paradox
Fig. 172. Mural Quadrant
Fig. 168. Cometarium
Fig. 169.
Fig. 173.
E. Mitchell sculp. Fig. 174. Portable Astronomical Quadrant.
Fig. 176. Transit Instrument.
Fig. 179. Universal Equatorial.
Fig. 177
Fig. 178. Compound Transit Instrument.
E. Mitchell sculp't Appendix.
Description 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 cross 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 unscrewing the nearest 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 cross hairs; then invert the telescope, which is done by turning the hour-circle half round; and if the centre of the cross 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 cross 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 \( \varnothing \) on the declination semicircle 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 twelve 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 colatitude of the place, and set the declination semicircle 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 cross 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 daylight. Having elevated the equatorial circle to the colatitude 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.
<table> <tr> <th>ABERRATION of the fixed stars, No 265</th> <th>Apogee of the moon's orbit, of the planets,</th> <th>No 81</th> <th>Axis of the world,</th> <th>No 37</th> </tr> <tr> <td>Academy, Royal, of sciences founded, 30</td> <td>Aphides, moon's,</td> <td>293</td> <td>Astrological division of the heavens,</td> <td>371</td> </tr> <tr> <td>Adam supposed to have understood astronomy.</td> <td>Arabians cultivate astronomy in the dark ages,</td> <td>ib.</td> <td>Atmosphere, why not seen about the moon,</td> <td>399</td> </tr> <tr> <td>Alphonso de Caffile patronises astronomers, by whom the tables are constructed,</td> <td>Archimedes determines the relative distances of the planets,</td> <td>17</td> <td>B</td> <td></td> </tr> <tr> <td>American knowledge of astronomy,</td> <td>Ascension, right,</td> <td>13</td> <td>Bailly investigates the Indian astronomy,</td> <td>4</td> </tr> <tr> <td>Anomaly, mean,</td> <td>Attraction, laws of, act among the stars,</td> <td>250</td> <td>Bayer, John, forms a celestial atlas, improves the nomenclature of the stars,</td> <td>ib.</td> </tr> <tr> <td>Antediluvians, their knowledge of astronomy,</td> <td>Benares, observatory,</td> <td>230</td> <td>A a</td> <td>Bianchini,</td> </tr> </table>
Bianchini, observations of Venus, No 146 Bissextile, or Leap year, 63 Bodier, falling, velocity calculated, 352 weight increases towards the poles, 276 Bodin's opinion of comets, 301 Bradley, Dr, succeeds Dr Halley, 34 discovers the aberration of light, ib. nutation of the earth's axis, ib. Briggs, Henry, improves logarithms, 28 Brydone observes the prodigious velocity of a comet, 319 conjectures about comets without tails, 322 Caesar, Julius, reforms the year, 63 Caille, de la, constructs solar tables, 36 Calendar, Gregorian, 64 Cassini appointed to the observatory at Paris, 30 his observations of Venus, 141, 144, 149, Ceres, planet, elements not precisely known, 183 Chaldeans cultivated astronomy early, 6 Chinese, their knowledge of astronomy, 2 give names to the zodiac, 3 Clouds, solar, two regions, 77 Cole's hypothesis of comets 323 Cometarium, 428 Comets, account of, 183 atmospheres and phaæs, 184 appearance of one in 1618, 1680, 1744, 1759, 186, 191 tails, 187 observations on, by Hevelius and Hooke, 188 supposed by the ancients to be planets, 298 Aristotle's opinion of, 299 one species only exists, 300 opinion of Kepler and Bodin, 301 Bernoulli's opinion of, 302 true doctrine revived by Tycho Brahe, 303 motion determined by Newton, 304 return of one predicted by Dr Halley, 305 periodical times of different ones determined, 306 why sometimes invisible in perihelion, 397 more seen in hemisphere towards the sun, 308 differences in eccentricities of orbits, 309 opinions of their substance, 310 distances and diameters of some computed, 311 occasion eclipses, 312 conjectures concerning their tails, 313
Comets, conjectures by Appian and Tycho Brahe, No 313 by Descartes, 314 by others, ascribed to electricity, ib. by Dr Hamilton, 315 velocity observed by Brydone, 319 in 1680 subjected to great heat, 320 conjectures of their nature by Hevelius, 321 about those without tails, 322 Cole's hypothesis, 323 periodical times, 324 Hally calculates their return, 325 return at unequal intervals, why, 326 at what distance visible, 372 move in eccentric ellipses, 373 motion, how to calculate, 374 are affected by the planets, 375 consequence of meeting a planet, 376 Constellations, names of, 199 Copernicus restores the system of Pythagoras, 22 his diffidence and fear of giving offence, retard the publication of his system, ib.
Day, astronomical, 55 sidereal, 56 varies in length, 57 mean astronomical, 58 Declination, rules for finding, 249 Densities of planets calculated, 356 Dollond improves the telescope, 31 Dunn, Mr, his account of the solar spots, 69
Earth, true figure discovered, 34 figure spherical, 269 dimensions ascertained, 270 different measurements, 271 anomalies in the figure, accounted for by Mr Playfair, 272 latitude and longitude found, 273 motion round the sun, proved from its figure, 275 from celestial appearances, 277 objections to, answered, 278 demonstrated from the aberration of light, 286 diurnal motion and changes of seasons illustrated, 290 effects of motion on the appearances of the planets, 292 Earth and moon move about a common centre, 333 protuberant matter, effects of, 411 libratory momentum, applied to nutation and precession, 412 various modes of application, 413 real and momentary change greatest at the solstices, 414 moon's action, 419 Eclipses, method of calculating, 355 tables for calculating, p. 169, 170. lunar, 99 solar, 103 duration of, 110 Ecliptic, obliquity of, 43 Egyptians early cultivators of astronomy, 6 Elements of the planets, tables of, 297 Equation of the centre explained, 31 of time computed, 60 Equator, 37 Equinoctial line, 255 points, ib. Equinoxes, precession of, discovered by Hipparchus, 257 importance of the discovery, 259 small irregularities, discovered by Dr Bradley, 261
Falling bodies, velocity of, calculated, 352 Flamsteed, Mr, appointed astronomer royal, 30 makes a catalogue of the stars, 33 Fontana improves telescopes, 31 Forces, moving, cause of motion, 332 composition of, 333 revolution, 334 accelerating, 336 central, 338 centripetal, effect of, 340 centrifugal compared with gravitation, 342 French philosophers cultivate astronomy, 35
G Galaxy, or milky-way, seems to surround the heavens, 211 Globe, celestial, 429 Golden number, 61 Gravitation, general law of, 355 Grecian astronomy improved by Thales, 9 by Anaximander, ib. Greeks, unknown when astronomy was first cultivated among them, 8 Index.
Greenwich observatory built, H 30
Valley, Dr., appointed astronomer royal, 33 discovers the acceleration of the moon, ib. recommends the method of finding the longitude now followed, ib. account of new stars, 200 predicts the return of a comet, 305 calculates their return, 325
Hamilton's, Dr., opinion of comets, 315 insufficient, 317
Heavens, division of, 197 method of gauging, by Herschel, 224 interior construction, 229
Herschel, Dr., improves telescopes, 36 discovers a planet, ib. fix satellites, ib. observations on the sun, 79 adopts new terms to express the appearances, 71 opinion of the construction of the universe, 217 of the via lactea, 218 method of gauging the heavens, 225 hypothesis of celestial appearances, 223 method of finding the parallax of fixed stars, 268 planet discovered by, 36 its satellites, 183 fix in number, 330
Tavelius, a zealous astronomer, 30 compiles his Selenographia, ib. his observatory and instruments burnt, ib. conjectures of the nature of comets, 321 Hicetas taught the diurnal motion of the earth, 11 Tipparchus discovers the eccentricity of the planetary orbits, 14 makes a catalogue of the fixed stars, 15 discovers the precession of the equinoxes, 257 charged with plagiarism, 260
Hooke, Dr., improves telescopes, 30 Horizon explained, 37 Horrox, a young astronomer of great talents, 29 predicted and observed the transit of Venus for the first time, ib. formed a theory of the moon, ib.
Indians, their knowledge of astronomy, 4
Indians, authenticity of their astronomy, N° 379
Instruments, astronomical, first improved in England, 31 description of, p. 171—184.
Josephus mentions the grand year known to Seth, 1 Jupiter's belts first discovered, 166 spots in them, 167 account of one, 168 no difference of seasons, 169 moons, four in number, 170 distance and periodic times, 171 eclipses, 172 appear sometimes as dark spots, 173 vary in light and magnitude, 174 shadows sometimes visible on Jupiter's disk, 175 three eclipsed every revolution, 176 eclipses, when visible, 177 orbits and distances, 327 irregularities in their motions, 328
K
Kepler discovers the famous laws in astronomy, 26 law explained, 49 opinion of comets, 301 discovers the cause of the tides, 400
L
Latitudes of heavenly bodies, 252 how found, 253 Libration of the moon, 133 theory, 134 Line, meridian, method of drawing, 40 Logarithms invented by Baron Napier, improved by Urfinus and Briggs, 28 Longitude of the heavenly bodies, 251 of places on the earth, method of finding, 274 Long, Dr., his account of the solar spots, 66 Louville's observations on the moon's ring, 226 Lowe's, Mr., method of finding the longitude, 274 Lunation, or month, 61
M
Mackay, Mr., method of finding the longitude, 274 Mars, spots first seen, 155 bright about the poles, 156 Dr Herschel's account of, 157 appear and disappear, 158 white about the poles, 162 position of the poles, 159 seasons, 160 resembles the earth, 161 Mars, his form spheroidal, N° 163 difference of diameter, 164 atmosphere, 165
Maskelyne, Dr., improves the lunar method of finding the longitude, 36 Mercury's apparent motions, 135 diameter, 136 nature, 137 Meridian explained, 37 line, method of drawing, 40 Milky-way, 211 Moon's motion in her orbit, 79 orbit elliptical, 80 eccentricity, 81 evolution, 83 variation, 84 annual equation, 85 revolution of her nodes, 86 parallax, method of determining, 87 distance, 89 phases, 90 is opaque, 91 mode of measuring the year, 93 the earth appears a moon to it, 92 longitude found, 94 nonagefimal degree, 98 eclipses, 99 period, 100 why visible when eclipsed, 101 eclipses observed with difficulty, 102 number in a year, 104 total and annual, 106 extent of shadow and penumbra, 107 size, 111 light, 112 spots, 113 names of, ib. inequalities of surface, 114 method of measuring mountains, 115 mountains height of, overrated, 116 volcanoes, 119 substance, conjectures of, 120 atmosphere, existence of, disputed, 121 ring observed in eclipses, 123 lightning, 127 height of atmosphere accounted for, 128 has no sensible atmosphere, 131 libration, 133 theory, 134 tendency the fame as gravitation, 350 motion explained, 351 inequalities, 377 nearest the earth when least attracted, 379 orbit, cause of dilatation, 380 Moon's orbit changed by the action of the sun, 381 nodes, 382 motion explained, 384 inclination, 383 motion, irregularities from being elliptical, 385 orbit, inequality in the eccentricity, 391 inequalities computed, 392 mean distance, secular equation, 393 has no atmosphere, why, 399 Motion, definition of; 331 Motions, of the, 51
N Nadir, 37 Napier, Baron, invents logarithms, 26 Nebula, our sidereal system, one, 236 extent, 239 how to be delineated, 241 Nebulae, in the milky-way, 219 arranged in strata, 220 assume various shapes, 221 how formed, 231 vacancies, how occasioned, 232 decay and recomposition, 242 universe composed of, 243 size and distance, 244 time of forming, 245 planetary, 247 Newton, Sir Isaac, his discoveries, determines the motions of the comets, 304 his opinion of comets defended, 316 observations on the precession of equinoxes, 407 sketch of his investigation, 408 determination of the form and dimensions of the earth, 409 examination of phenomena of precession on mechanical principles, 410 Node, ascending, 86 Nodules on the luminous clouds of the sun, 75 Number, golden, 61 Nutation, lunar, 421 compared with precession, 423
O Observatory, portable, 434 Openings formed by the sun's luminous clouds being removed, 72 Oscillation of the planetary system, 369
P Pallas, planet, elements not precisely known, 183 Perigee of the moon's orbit, 81 of the planets, 293 Phoenicians taught astronomy by the Egyptians, 7 apply it to navigation, ib. Philolaus affirms the annual motion of the earth round the sun, 11
Planets, No 37 apogee and perigee, 293 difference of apparent diameters, 294 appearances of superior, explained, 295 orbits and laws of their motions, ib. heliocentric circles, 296 nodes, 297 tables of elements, ib. revolve round the sun, 343 in consequence of a force in the sun, 344 the same tendency in all, 346 and fame in their satellites, 347 react on the sun, 354 densities calculated, 356 masses, table of, 357 gravity at their surfaces, 359 secular and periodical inequalities, 360 motion of the aphelion, 361 motions, method of correcting, 363 Jupiter and Saturn influence each others motions, 362 deflection of, towards each other, 366 Pendulum regulated by gravitation, 337 Precession of equinoxes, 257—259 observations by Newton and others, 407 Newton's investigation, 408 lunar, 420 greatest equation, 425 Ptolemy, his system erroneous, 16 Ptolemy Philadelphus encourages the sciences in Egypt, 12 Purbuck improves astronomy, 19 Pythagoras improves astronomy, 11 correct notions of the solar system, ib. of the moon's light, ib. of the milky-way, ib.
Q Quadrants, 431 Quadratures of the moon, 90
R Revolution of a body round a centre explained, 339 Regiomontanus constructs astronomical apparatus, 20 calculates lunations and eclipses, ib. writes a theory of planets and comets, ib. Ridges of the sun's luminous clouds, 74 Ring of Saturn, 395 discovery concerning, by Dr Herschel, 396 probably consistent, 397 origin, 398
Roemer discovers the progressive motion of light, 33 Rothman, an astronomer, S Satellites, tend to the sun, 347 to their primaries, 349 irregularities in Jupiter's, 394 Saturn, telescopic appearance of, 178 ring discovered by Huygens, 179 supposed to revolve round its axis, 180 diameter, ib. satellites, two discovered by Herschel, 182 number, 329 Seasons, explained, 44 changes illustrated, 290 different, explained, 291 Sector, equatorial, 432 Selenographia compiled by Hevelius, 30 Shallows of the sun's luminous clouds, 93 Shepherds, Asiatic, observe the heavens, 255 Signs of the zodiac, 52 Society, Royal, founded in London, 80 Stars, fixed, occultations by the moon, 130 Style, old, new, ib. Sun, annual motion, altitude, 39 motion, method of ascertaining, not uniform, 42 diameter varies, 46 distance varies, 47 motion varies, 48 orbit, elliptical, varies, 50 distance determined, 54 spots, first discovered, 65 Long, Dr, his account of them, 66 move from west to east, 67 observed by different astronomers, 68 Dunn, Mr, his account of them, 69 appearances of the luminous clouds, 72—76 two regions of clouds, 77 theory of phenomena, 78 eclipses, 103 beginning and ending account of one by Dr Halley, 124 his place in the universe, 224 his centre attracts all bodies, moves round the common centre of gravity, 364 Syzygies of the moon, 90 Stars, fixed, number increased by telescopes, 192 difference in magnitude, 193 telescopic, 194 unformed, 195 division in constellations, and utes, 196
Tides, why high at full moon, N° 401 influence of the sun, 402 not highest when the moon is in the meridian, 403 turn on the axis of the moon's orbit, 404 irregularities accounted for, 405 Trajectorium lunare, 430 Transit instruments, 433 Tycho Brahe observes the connexion of Saturn and Jupiter, 24 makes a more accurate quadrant, ib. superintends the building of Uraniburg, his observatory, 25 revives the true doctrine of comets, 301
U Ulug Beg cultivates astronomy, 17 Uraniburg, built by Tycho Brahe, and furnished with instruments, 25
V Velocity, motion, 335 Venus, apparent motions, 138 revolution round her axis, 139 doubts of the time, 147 spots first discovered, 140 seem to move from south to north, and why, 142 appearances at different times, 143 observations by Cassini, 141 by Bianchini, 146 satellites discovered by Cassini, 149 and by Mr Short, 150 Mr Montaigne, 151 difficult to be seen, 152
Venus, atmosphere of, observed by Mr Hirlt, N° 153 Volcanoes in the moon, W 119 Walther cultivates astronomy, and constructs instruments, 20 Weight of bodies increases towards the poles, 276 Werner, John early attachment to astronomy, 21 observes the motion of a comet, ib. proposes a method of finding the longitude at sea, ib. discovers the precession of the equinoxes, ib. constructs a planetarium, ib.
William IV. landgrave of Hesse-Cassel, an astronomer, 23 World, argument against its eternity, 36 Wright, Edward, makes observations on the sun's altitude, 26 improves the theory of his motion, ib. computes tables of his declination, ib. Year, tropical, 59 federal, ib. Roman, 62 reformed by Julius Caesar, 63 leap, ib. lunar, 93
Z Zenith, 37 Zodiac, Chinese names of the signs, 3 signs of, 52 division of, 198