It was more probably suggested to different nations by the phases of the moon. The Egyptians had likewise been attentive to the courses of the planets. Diodorus Siculus affirms that they could explain the phenomena of the stations and retrogradations; and Macrobius ascribes to them the knowledge of the real motions of Mercury and Venus, and says that they regarded these planets as satellites of the sun. This notion would do credit to their philosophy; but it is unfortunately not mentioned by any other author, and for this reason the testimony of Macrobius is suspected. The state of their practical as- tronomy may in some measure be inferred from the means they employed to determine the magnitude of the sun's apparent diameter. By comparing the time, observed by means of a clepsydra, which the sun takes to mount above the horizon at the equinox, with that in which he makes a complete revolution of the sky, they estimated his dia- meter at 28° 48'. An observation of this kind is liable to great uncertainty; and as there is no evidence that the Egyptians possessed the slightest knowledge of spherical trigonometry, they would probably make no allowance for the obliquity of the equator to their horizon; and if this

as left out of the calculation, as it probably was, their diameter, already too small, ought to have been still farther reduced, and to have amounted only to 24° 42'. It has been conjectured by Goguet, that the obelisks of

ere intended to serve the purpose of gnomons; and this conjecture acquires some probability from their needle-shaped form, and the narrowness of their bases re- latively to their heights. It has however been proved by MM. Jollois and Devilliers, in their description of Thebes, that the obelisks were connected with the walls of temples and palaces; a disposition which rendered them entire- ly unfit for the purposes of astronomical observation. Their summits were also of so unfavourable a form, that the Romans were obliged to surmount them with a ball in order to obtain a distinct and well-defined shadow. The pyramids have also been adduced as evidences of the early progress of astronomy among the Egyptians; for History, the faces of these stupendous masses are turned directly towards the four cardinal points, from which it is evident that the people by whom they were constructed were at least acquainted with the method of tracing a meridional line.

From this brief account it appears, that the only cir- cumstances with which we are acquainted that imply the knowledge of astronomical methods among the Egyptians, are the length of the year, the doubtful discovery of the true motions of Mercury and Venus, and the position of the pyramids. The Chaldean observations were of use to Hipparchus and Ptolemy in the determination of some important elements; but those of the Egyptians exercised no influence whatever on the future progress of the science.

The Phenicians are also generally enumerated among Phoeni- cian nations who cultivated astronomy at a very early clas- period, though it does not appear, from any facts mention- ed by ancient authors, that they addicted themselves to the observation of the heavens, or made any discoveries relative to the motions of the planets. That they excelled in the art of navigation is certain, from the commercial intercourse which they carried on with many places on the coasts of Africa and Spain, and in the principal islands of the Mediterranean; and it may readily be allowed that in their long voyages they would direct their course during the night by the circumpolar stars. If they had any speculative notions of astronomy, these were probably derived from the Chaldeans or Egyptians.

In China, astronomy has been cultivated from the re- Chinese motest ages, and always been considered as a science in- dispensably necessary to the civil government of the state. The Chinese boast of a series of eclipses, recorded in the annals of the nation, extending over a period of 3858 years, all of which, they pretend, were not only carefully observed, but calculated and figured previous to their oc- currence. The same motives which led the Chaldeans and Egyptians to attend to the celestial phenomena, namely, the regulation and division of time, had equal in- fluence among the Chinese; and we accordingly find the care of the calendar occupying the attention of their earliest princes. The emperor Fou-Hi, whose reign com- menced about 2857 years before our era, is said to have assiduously studied the motions of the celestial bodies, and laboured to instruct his ignorant subjects in the mys- teries of astronomy. But as they were yet in too rude a condition to be able to comprehend his theories, he was obliged to content himself with giving them a rule for the computation of time by means of the numbers 10 and 12, the combination of which produces the cycle of 60 years, which is the standard or unit from which they deduce their hours, days, and months. Tradition is silent with respect to the sources from which Fou-Hi derived his own know- ledge. The Chinese attribute to him also the invention of arithmetic and music. In the year 2608 B.C., Hoang- Ti caused an observatory to be built, for the purpose of correcting the calendar, which had already fallen into great

and appointed one set of astronomers to observe the course of the sun, another that of the moon, and a third that of the stars. It was then discovered that the twelve lunar months do not exactly correspond with a solar year; and that, in order to restore the coinci- dence, it was necessary to intercalate seven lunations in the space of nineteen years. If this fact rested on undoubted evidence, it would follow that the Chinese had anticipated the Greeks by 2000 years in the dis- covery of the Metonic cycle. The reign of Hoang-Ti is also rendered memorable by the institution of the Mathematical Tribunal, for promoting the science of History, astronomy, and regularly predicting eclipses, to which an extraordinary importance has always been attached in China. The members of this celebrated tribunal were made responsible with their lives for the accuracy of their predictions, by a law of the empire, which ordained that, "whether the instant of the occurrence of any celestial phenomenon was erroneously assigned, or the phenomenon itself not foreseen and predicted, either negligence should be punished with death." In the reign of Tchong-Kang, the two mathematicians of the empire, Ho and Hi, were the victims of this absurd and sanguinary law; an eclipse having taken place which their skill had not enabled them to foresee.

The emperor Yao, who mounted the throne, according to the Chinese annals, about the year 2317 B.C., gave a new impulse to the study of astronomy, which had begun already to decline. He ordered his astronomers to observe with the utmost care the motions of the sun and moon, of the planets and the stars, and to determine the exact length of each of the four seasons. He sent Hi-Tchong to the east to observe the star situated at the point of the vernal equinox, Hi-Tchong to the south to examine that at the summer solstice, Ho-Tchong to the west, and Ho-Tchou to the north, to observe those situated respectively at the autumnal equinox and winter solstice. These docile observers found stars in the positions assigned by the emperor; but the extraordinary resemblance of their names imparts a fabulous air to the whole relation, and excites a very excusable incredulity even with regard to those statements which involve no improbability. To this emperor are attributed the Chinese division of the zodiac into 28 constellations, called the houses of the moon, and the severe laws already noticed in regard to the erroneous prediction of the celestial phenomena.

From the time of Yao the Chinese year consisted of 365½ days. They also divided the circle into 365¼ degrees, so that the sun daily described in his orbit an arc of one Chinese degree. Their common lunar year consisted of 364¾ days; and by combining this number with 365¼, they formed the period of 4617 years, after which the sun and moon again occupy the same relative positions.

The earliest Chinese observations we are acquainted with, sufficiently precise to afford any result useful to astronomy, were made by Tcheou-Kong, whose reign commenced about the year 1100 before our era. Two of these observations are meridional altitudes of the sun, observed with great care at the village of Loyang, at the time of the summer and winter solstices. The obliquity of the ecliptic thus determined at that remote epoch is $23°\ 54'\ 3\frac{1}{2}''$; a result which perfectly agrees with the theory of universal gravitation. Another observation, made about the same time, relates to the position of the winter solstice in the heavens; and it also corresponds to within a minute of a degree with the calculations of Laplace. Laplace considers this extraordinary conformity as an indubitable proof of the authenticity of those ancient observations.

The golden age of Chinese astronomy extended from the reign of Fou-Hi to the year 480 B.C.; that is, over a space of 2500 years. It is only, however, towards the latter part of this long period that the history of China becomes in any degree authentic; and the true date which must be assigned for the commencement of observations on which any reliance can be placed, is the year 722 B.C.; that is, 25 years posterior to the era of Nabonassar. From that period to the year 400 B.C., Confucius reckons a series of 36 eclipses, and of these 31 have been verified by modern astronomers. After this the science fell into great neglect, notwithstanding the inveterate tenacity with which the Chinese in general adhere to their ancient customs. The decline of their astronomy is ascribed, whether justly or not, to the barbarous policy of the emperor Tsin-Chi-Hong-Ti, who, in the year 291 B.C., ordered all the books to be destroyed, excepting those only which related to agriculture, medicine, and astrology, the only sciences which he considered as being of any use to mankind. His fury, it is true, was principally directed against those of Confucius, the stern morality of which he felt to be a censure on his own profligacy; but those of science and astronomy were included in the general destruction. In this manner, it is said, the precious mass of astronomical observations and precepts which had been accumulating for ages was irretrievably lost.

Liou-Pang, the successor of Tsin-Chi-Hong, endeavoured to repair the disaster, by re-establishing the tribunal of the mathematics, and ordering a new series of observations to be undertaken. About the year 104 B.C., the astronomer Sse-Ma-Tsien gave some precepts for the calculation of eclipses, the motions of the planets, and the syzygies. He employed instruments of copper, the nature and construction of which are however not very well understood, for measuring the extent of the 28 zodiacal constellations; and he observed the meridional altitudes of the sun, by means of a gnomon 8 feet high. The differences of right ascensions, and the intervals between the risings, settings, and culminations of the stars, were measured by clepsydrae. It would appear that after this period astronomical observations continued for some time to be made in China with considerable regularity. In the 164th year of our era the astronomer Tchang-Heng constructed armillary spheres and a celestial globe. He also formed a catalogue of stars, which is said to have contained 2500, but without either latitudes or longitudes; a circumstance which gives us a very unfavourable idea of the state of practical astronomy at that time. About the eighth century of our era, all knowledge of the science seems to have been again lost. The predictions were erroneous; and the Chinese witnessed, with superstitious terror, eclipses of which their astronomers had given them no intimation. This induced the emperor Hieng-Tsong to call to his court the astronomer Y-Hang, by whose indefatigable activity a reform was speedily effected. With a view to determine the situations of the principal places of the empire, this astronomer constructed gnomons, spheres, astrolabes, quadrants, and other instruments; and sent one company of mathematicians to the south, and another to the north, with directions to observe daily the altitudes of the sun and the polar star. The latitudes of the cities were determined by observing the shadow of the gnomon, and the longitudes by eclipses of the moon. Y-Hang had the mortification of announcing two eclipses which did not take place. On these occasions he alleged the usual excuse, namely, that his calculus was not in error, but that the celestial bodies had deviated from their ordinary courses out of respect to the virtues of the emperor. The fate of Ho and Hi had probably suggested to the Chinese astronomers this ingenious mode of disarming the emperor's resentment by flattering his vanity.

On considering attentively the accounts which have been given of the Chinese astronomy, we find that it consisted only in the practice of observations which led to nothing more than the knowledge of a few isolated facts. The missionaries who were sent out by the Jesuits about the end of the seventeenth century, to whom we are indebted for what is known of the early history of China, either seduced by some appearances of truth, or thinking it prudent to conciliate the people whom they were attempting to convert, adopted their marvellous relations regarding the antiquity of their science, and spread them over Europe. As the history of the nation begins to become more authentic, their astronomy shrinks into its real but insignificant dimensions. Superstitiously attached to their ancient usages, and blindly adopting the habits of their ancestors, the Chinese continued to observe the heavens from century to century without making the slightest advances in theoretical knowledge. In later times they have adopted many improvements, for which they are entirely indebted to foreigners. During the time of the caliphs many Mahometans passed into China, carrying with them the astronomical methods and knowledge of the Arabians. The missionaries introduced the science of Europe; and the most that can be said in praise of the Chinese is, that their government sometimes relaxed so far its spirit of jealousy and exclusion, as to afford protection to these strangers, adopt their arts, and place them at the head of the mathematical tribunal.

The astronomy of the Indians forms one of the most curious problems which the history of science presents to the consideration of the learned, and one which, notwithstanding the numerous dissertations to which it has given rise, still continues involved in great uncertainty. Of the science of the ancient nations, of which we have already spoken, the accounts which have come down to our times are founded on conjecture and tradition; for few monuments remain to confirm or confute the glowing descriptions which authors have given of its high antiquity and great perfection. But the claims of the Indians rest on more solid foundations. We are in possession of the tables from which they compute the eclipses and places of the planets, and of the methods by which they effect the computation: we have, in short, an Indian astronomy committed to writing, which represents the celestial phenomena with considerable exactness, and which, therefore, could only be produced by a people far advanced in science. But the difficulty of the problem consists in determining the sources from which this science originated, and the epoch of its existence; whether it was created by the people who now blindly follow its precepts without understanding its principles, or was communicated to them by another race of a bolder and more original genius, through channels with which we are unacquainted. Some authors regard India as the cradle of all the sciences, particularly of astronomy, which they suppose to have been cultivated there from the remotest ages; others date the origin of the Indian astronomy from the period when Pythagoras travelled into that country, and carried thither the arts and sciences of the Greeks; a third opinion is, that astronomy was conveyed to India by the Arabians in the ninth century of our era, and that the Brahmins are only entitled to the humble merit of adapting the rules and practices of that people to their own peculiar methods of calculation. We shall endeavour to describe very briefly the existing monuments of the Indian astronomy, which furnish the only data from which a rational conjecture can be formed relative to its antiquity and precision.

We possess four different sets of tables of Indian astronomy. The first which were known in Europe were brought from Siam by La Loubère, who had resided in that country as ambassador from Louis XIV. They were communicated by him to the celebrated Cassini, who, notwithstanding the difficulties arising from the complicated and useless operations which they directed, succeeded in detecting the principles on which they were constructed, and in explaining their use and signification. The date of these tables corresponds to the 21st of March in the year 638 of our era. They suppose two species of years, the solar tropical year, which they make to consist of 365 days 5 hours 50 min. and 4 sec., and the solar anomalistic year, that is, the period in which the sun returns to its apogee, which they estimate at 365 days 6 hours 12 min. 36 sec. This determination of the length of the solar year is too great only by 1 min. 15 sec. By means of the same tables the longitudes of the sun and moon are determined with considerable accuracy. They contain a correction for the sun's mean place, which corresponds to the equation of the centre. At 90° from the apogee, where the inequality of the sun's motion is greatest, they estimate the requisite correction at 2° 12', which is about 16' too great. This determination deserves to be particularly remarked, because, on account of a secular inequality of the eccentricity of the sun's orbit, there was once a time when the greatest value of the equation of the centre was nearly 2° 12'; and this fact is adduced as a proof of the remote antiquity of the observations from which the tables in question have been constructed. These tables suppose the apogee to retain always the same position relatively to the fixed stars; in reality it advances or gains on the stars about 10° annually; but the supposition is still much nearer the truth than in the system of Ptolemy, where the apogee is supposed to be absolutely at rest with regard to the plane of the sun's orbit, and consequently to fall back among the stars by the whole quantity of the precession of the equinoxes, or about 50° annually. With regard to the motions of the moon, they are deduced from a period of 19 years, in which are comprehended nearly 235 lunations; so that the cycle of Meton appears to have been known in Siam as well as in China. The moon's apogee is supposed to have been in the beginning of the movable zodiac 621 days after the epoch of the 21st of March 638, and to make an entire revolution in the heavens in the space of 3232 days. The first of these suppositions agrees with Mayer's tables to within a degree, and the second differs from them only by 11 hours 14 min. 31 sec. They contain only one correction for the two principal inequalities of the moon's motion, the equation of the centre and the evocation.

A second set of Indian tables was sent from Chrismabouran, a town in the Carnatic, by Father Du Champ, to De Lisle, about the year 1750. They are fifteen in number. They give the mean motions of the sun, moon, and planets; equations of the centre for the sun and moon; and two corrections for each of the planets, one of which corresponds to the apparent, the other to the real inequality. The epoch of these tables is not so ancient as that of the former. It corresponds to the 10th of March, at sunrise, in the year 1491 of our era, when the sun and moon were in conjunction.

A third set of astronomical tables was sent from India by Father Patomillet, and received by De Lisle about the same time with those of Chrismabouran. These have not the name of any particular place affixed to them; but being calculated for the latitude of 16° 16', Bailly thinks it probable that they came from Naraspour. Their epoch is midnight, between the 17th and 18th of March 1569.

The fourth and last set of Indian tables which we possess have been published in the Memoirs of the Academy of Sciences. They were communicated by a learned Brahmin of Tirvalore, a small town on the Coromandel coast, to the French astronomer Legentil, who had gone to India to observe the transit of Venus in 1769. The tables History of Tirvalore, though somewhat different in form, present many points of resemblance with those formerly known in Europe. They suppose the same length of the year, the same inequalities of the sun and moon, and they are adapted nearly to the same meridian. But while they correspond with the other tables in these elements, they differ from them greatly in the antiquity of their epoch, which goes back to the famous era of the Calyougham, that is, the beginning of the year 3102 before Christ.

Now, the only question to be determined with regard to the antiquity of the Indian astronomy is, whether this epoch is real or fictitious; that is, whether the state of the heavens at the commencement of the Calyougham, as assumed in these tables, was actually determined by observation, or computed backwards from observations of more modern date. The solution to this question can only be obtained from the internal evidence afforded by the tables themselves; by examining whether the elements and precepts which they furnish are of sufficient accuracy to enable the places of the sun, moon, and planets to be calculated through a period of 44 centuries, without involving errors which the refined accuracy of the modern tables furnishes the means of detecting. A comparison of the Indian with the modern tables has been made at great length by Bailly, who imagines that he finds ample evidence of the reality of the era in question, and of the existence of an astronomy prior to that period, hardly yielding in accuracy to that which modern science has built on the theory of universal gravitation. The theory of Bailly has been adopted, and put forth with additional clearness and evidence, by the late Professor Playfair.

One of the principal arguments which these illustrious authors bring forward in support of it is founded on the longitudes of the sun and moon. The mean place of the moon at the commencement of the Calyougham, that is, at midnight, between the 17th and 18th of February 3102 B.C., is stated by the Indian tables to be 30°6'. Her mean place, computed from Mayer's tables, without taking into account the acceleration, with which the Indians in the 15th century were of course unacquainted, is 30°51'16". Hence there would be a discrepancy of 5°8'44". But, according to the theory and last tables of Laplace, the moon, in virtue of the acceleration of her mean motion, has passed over an arc of very nearly 6° more than she would have done had her mean motion continued uniform from the period of the Calyougham to the date of Mayer's tables. This added to 30°51'16" gives 30°51'16" for the mean longitude of the moon at the epoch of the Calyougham, differing from the Indian determination by only 5°16". Now, it is argued that this is a degree of accuracy which could have been reached only by actual observation, especially since, if the tables had been computed backwards, the error arising from the acceleration alone would have amounted to more than 5°. Bailly computes the place of the moon at the same epoch, from all the tables, Greek and Arabian, to which the Indians can be supposed to have had access, and the discrepancies are so great as to render his conclusion almost inevitable, that the Indian tables could not possibly have been drawn from such sources. The tables of Ptolemy make the moon's longitude at that time 11°52'7" greater than the Indian tables; and those of Ulugh-Beigh, constructed at Samarcan in 1437, give a difference of 6° also in excess.

Similar results are obtained from the consideration of other elements. According to the tables of Tirvalore, the tropical year consists of 365 days 5 hours 50 min. 35 sec. Laccaille makes it 365 days 5 hours 48 min. 49 sec. The difference is 1 min. 46 sec. Now the tropical year, being affected by the precession of the equinoxes, is subject to a secular inequality, which, according to the theory of Lagrange, renders it actually shorter by 40.5 sec. at the present time than it was at the commencement of the Calyougham. The error of the Indian tables is thus reduced to 1 min. 5.5 sec. In like manner, the obliquity of the ecliptic, which has been gradually diminishing during a great number of centuries, is supposed in the Indian tables to be greater than it is now found to be by observation. The Brahmins estimate it at 24°. The formula of Lagrange makes the variation, in 4800 years, amount to 23°32". This therefore must be added to its obliquity in 1700, that is, to 23°28'41", in order to have the true obliquity at the commencement of the Calyougham. The sum is 23°51'13" and falls short of the Indian determination by 8'47". We shall mention only another element, the equation of the centre of the sun. Bailly calculates that, according to the theory of Lagrange, the equation of the sun's centre, at the epoch of the tables, was 2°6'28". The Indians make it 2°10'32". The difference is only about 4', and incomparably less than could have resulted from calculation by any methods which we can suppose the Indians to have possessed.

These arguments, it must be admitted, are exceedingly specious, but they are not by any means convincing. Even with the best modern tables we could not, as Bailly himself acknowledges, answer for the accuracy of the places of the sun and moon computed for so remote an epoch. The corrections for the secular inequalities amount in that long period to considerable quantities; and these corrections are deduced by theory from elements with respect to which there exists great uncertainty. And if we cannot be sure of the true places by computing backwards from our own tables, with what degree of confidence can we pronounce upon the accuracy of the places assigned in the tables of the Indians? It may be said that comparisons of this kind can never be supposed to give results perfectly alike. Granted; but if the discrepancies are such that the lapse of a thousand years more or less is required to establish a rigorous conformity, what becomes of the famous epoch of the Calyougham? Some of the elements of the Indian tables could not have the values assigned to them but at a long period before that epoch. In order to find their equation of the sun's centre, for example, it is necessary, according to the results of modern theory, to go back to 6000 years before our era. The argument, therefore, proves too much, and is consequently inconclusive. The different sets of tables of which we have spoken are closely allied with each other, and the most probable supposition is, that they are all derived from those of Chrismabouram, of which the epoch is 1491. At that era the Indians were acquainted with the instruments, the geometry, and the researches of the Arabians and Greeks. Through this channel the tables seem to have come into their possession. The Brahmins adapted them to their own particular methods of computation, and threw back their epoch to the period when, according to these tables, all the planets were in conjunction with the sun. Every circumstance connected with the science of the Indians conspires to give us the humblest ideas of its value. Their methods of computation are encumbered with the unnecessary multiplications and divisions of enormous numbers, endless additions, subtractions, and reductions, for the purpose of obtaining numbers which could be put into technical verses, and even adapted to songs; so that the astronomer might be enabled to effect his calculations from memory alone, without its being necessary to have Astronomy.

The origin of astronomy in Greece, as in other early nations, ascends beyond the period of authentic history, and is concealed amidst the fables and traditions of the remotest times. During the darkness of the heroic ages some gleams of an acquaintance with the motions of the stars occasionally burst forth; and some traces appear of astronomical observations, probably derived from Egypt, the country which also furnished Greece with its gods and its arts. The Greeks seem to have divided the heavens into constellations about 13 or 14 centuries before the Christian era; for the sphere of Eudoxus, which is probably one of the fruits of the famous voyage of the Argonauts, must be referred to that period. Their early attention to the appearances of the heavens is sufficiently attested by their mythological fables, the greater part of which are only allegories of the celestial motions, and of the operations of nature. The lively fancy and brilliant imagination of this ingenious people strewed flowers in the most rugged paths, and spread agreeable images over the driest and most uninviting subjects; hence the sky was quickly covered with legends of the loves and exploits of gods and heroes. It would be foreign to our present purpose to enter into an enumeration of these fables, or attempt to trace their connection with the first dawns of astronomy: we shall content ourselves with barely alluding to Uranus, to Atlas and his son Hesperus, who gave his name to the planet Venus; also to his daughters the Atlantides, from whom the Pleiades received their appellation; to Endymion, who, on the summit of Mount Latmos, held nocturnal converse with the chaste Diana; to Hercules; and Chiron the centaur, who taught men the use of the constellations; Museus, who imagined the figures of men and animals which cover the celestial sphere; Orpheus and Linus, who explained the theogonies; Atreus, from whose banquet the sun fled back with horror; and Tiresias, who was struck blind for having witnessed some secret of the gods.

The true foundations of Grecian science were laid by Thales, who was born at Miletus 640 years before our era. He was descended from an illustrious family, which had formerly reigned in Phoenicia, and inherited an ample fortune, which he expended in collecting the expiring embers of oriental science. Instigated by the love of knowledge, he travelled first into Crete, and afterwards into Egypt, where he was initiated into the mysteries of the priests, to whom, in return, he is said to have taught the method of measuring the height of the pyramids by comparing their shadows with those of known objects. Returned to his own country, he publicly taught the truths he had collected during his travels, and formed a sect which has been distinguished by the title of the History, Ionian School. His doctrines regarding astronomy contain a few truths which do honour to his sagacity and observation, though they are mixed with much error and absurdity. He taught that the stars are formed of fire; that the moon receives her light from the sun, and is invisible at her conjunctions, because she is hid in the sun's rays. He also taught the sphericity of the earth, which he placed at the centre of the world. He divided the sphere into five zones, by the arctic and antarctic circles, and the two tropics; and held that the equator is cut obliquely by the ecliptic, and perpendicularly by the meridian. He is also said to have observed eclipses; and Herodotus relates that he predicted the famous one which put a stop to the war between the Medes and the Lydians. It does not appear, however, that he ventured to assign either the day or the month of the eclipse, so that his prediction must have been confined to the year. According to Callimachus, he determined the positions of the stars which form the Lesser Bear, by which the Phoenicians guided themselves in their voyages. It is difficult, however, to conceive how Thales, unacquainted with instruments, could determine the positions of stars with so much accuracy as to render any essential assistance to the navigator. It is probable that he only pointed out the configuration, and some of the more brilliant stars of that constellation, among which he might remark that which is nearest the pole of the world.

Thales was succeeded by Anaximander, to whom is also attributed the invention of the sphere, and the knowledge of the zodiac. According to Diogenes Laertius, 610 B.C., he supposed, like his master Thales, the earth to be spherical, and placed at the centre of the universe; but Plutarch ascribes to him the less philosophical opinion of its resemblance to a column. He supposed the sun to be of equal magnitude with the earth. He invented the gnomon, and placed one at Lacedaemon to observe the solstices and equinoxes. But the circumstance which does most honour to Anaximander, and which entitles him to the gratitude of posterity, is the invention of geographical charts. He is said also to have believed in the plurality of worlds,—a sublime idea, which was adopted by almost every succeeding philosopher of Greece.

Anaximenes succeeded Anaximander in the Ionian school, and maintained nearly the same doctrines. Pliny says he was the first who taught the art of constructing dials,—an invention which, as we have just seen, has also been ascribed to Anaximander. These two philosophers probably revived the knowledge of an instrument the use of which had been forgotten amidst the general rudeness and ignorance of their countrymen. Before their time the Greeks only marked the divisions of the day by the different lengths of the sun's shadow.

Anaxagoras was the disciple and successor of Anaximenes. If this philosopher really entertained the ridiculous opinions ascribed to him by Plutarch, the Ionian school must rather have retrograded than advanced in sound philosophy from the time of Thales. He is said to have believed that the sun is a mass of red-hot iron, or of heated stone, somewhat bigger than the Peloponnesus,—that the heaven is a vault of stones, which is prevented from tumbling only by the rapidity of its circular motion,—and that the sun is prevented from advancing beyond the tropics by a thick and dense atmosphere, which forces

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1 For an account of the Indian astronomy, see Bailly, Astronomie Indienne; also a Memoir by Professor Playfair, in the Edinburgh Transactions, vol ii.; or in the 3d volume of his Works; and the Papers of Jones, Bentley, and Davis, in the Calcutta Memoirs. The theory of Bailly is most satisfactorily refuted by Delambre. See his Histoire de l'Astronomie Ancienne, tom. i. History; him to retrace his course. These absurd notions are probably greatly exaggerated; but it does not appear that Anaxagoras contributed much to extend the knowledge of the heavens. A melancholy interest is, however, excited in his behalf, on account of the persecution which he suffered in consequence of his liberal opinions and his disregard for the superstitious notions of his age. Having shown the reason of the eclipses of the moon, he was accused of ascribing to natural causes the attributes and power of the gods. Having taught the existence of only one God, he was accused of impiety and treason towards his country. Sentence of death was pronounced on the philosopher and all his family; and it required the powerful interest of his friend and disciple Pericles to obtain a commutation of this iniquitous sentence into one of perpetual banishment.

Pythagoras, born 530 B.C.

While the Ionian sect was so successfully employed in cultivating and propagating a knowledge of nature in Greece, another, still more celebrated, was founded in Italy by Pythagoras. This renowned philosopher was in early youth a disciple of Thales. In quest of knowledge, which in those days could only be obtained by visiting the sages of foreign lands, he travelled into Egypt, Phoenicia, Chaldea, and India, where his memory is said still to subsist. Through the favour of Amasis, king of Egypt, to whom he was recommended by Polycrates, the tyrant of Samos, he was admitted into the sacred college at Memphis, though with great reluctance on the part of the priests. The severe ordeal through which these charlatans compelled him to pass, before they would consent to initiate him into their mysteries, was sufficient to have deterred the most courageous votary of knowledge; and Pythagoras was probably the only stranger who ever succeeded in fully exploring their secrets. After an absence of thirty years he returned to Greece, and began to give instructions in his native island of Samos. Soon after, he passed over to the Grecian colony established at Taranto in Italy, and settled at Crotona, where he speedily acquired a splendid reputation. He was the first who assumed the modest title of philosopher, or lover of wisdom; formerly those who devoted themselves to the acquisition of learning were called sophists or sages.

Pythagoras is said to have acquired in Egypt the knowledge of the obliquity of the ecliptic, and that of the identity of the morning and evening stars. What he chiefly deserves to be commemorated for in the history of astronomy, is his philosophical doctrine regarding the motion of the earth. He taught publicly that the earth is placed at the centre of the universe; but among his chosen disciples he propagated the doctrine that the sun occupies the centre of the planetary world, and that the earth is a planet circulating about the sun. This system, which still retains his name, being called the old or Pythagorean system of the universe, is that which was revived by Copernicus. It is, however, only just to the memory of this last mentioned great man to observe, that there is a vast difference between the bare statement of the possibility of a fact, and the demonstration of its existence by irrefragable arguments. Pythagoras having remarked the relation which subsists between the tone of a musical chord and the rapidity of its vibration, was led by analogy to extend the same relation to the planets, and to suppose that they emit sounds proportional to their respective distances, and form a celestial concert too melodious to affect the gross organs of mankind. Another fancy into which he was led by his passion for analogies, was the application of the five geometrical solids to the elements of the world. The cube symbolically represented the earth; the pyramid, fire; the octaedron, air; the icosaedron, or twenty-sided figure, water; and the dodecaedron, or figure with twelve faces, the exterior sphere of the universe. Pythagoras left no writings; and it is doubtful whether he really entertained many of the opinions and reveries which have usually been ascribed to him.

Philolaus of Crotona, a disciple of Pythagoras, embraced the doctrine of his master with regard to the revolution of the earth about the sun. He supposed the sun to be a disk of glass which reflects the light of the world. He made the lunar month consist of 29½ days, the lunar year of 354 days, and the solar year of 365½ days.

Nicetas of Syracuse seems to have been the first who openly taught the Pythagorean system of the universe. Cicero, on the authority of Theophrastus, the ancient historian of astronomy, gives him the credit of maintaining that the apparent motion of the stars arises from the diurnal motion of the earth about its axis; but this rational doctrine seems to have been first broached by Heraclides of Pontus, and Eudoxus, a disciple of Pythagoras.

The introduction of the Metonic cycle forms an era in the history of the early astronomy of Greece. The Chaldeans, as we have already stated, established several lunar periods; and the difficulty of reconciling the motions of the sun and moon, or of assigning a period at the end of which these two luminaries again occupy the same positions relatively to the stars, had long embarrassed those who had the care of regulating the festivals. Meton and Euctemon had the honour of first obviating this difficulty, at least for a time; for the motions of the sun and moon being incommensurable, no period can be assigned which will bring them back to precisely the same situations. These two astronomers formed a cycle of nineteen lunar years, twelve of which contained each 12 lunations, and the seven others each 13, which they intercalated among the former. It had long been known that the synodic month consisted of 29½ days nearly; and in order to avoid the fraction, it had been usual to make the twelve synodic months, which compose the solar year, to consist of 29 and 30 days alternately; the former being called deficient and the latter full months. Meton made his period to consist of 125 full and 110 deficient months, which gives 6940 days for the 235 lunations, and is nearly equal to 19 solar years. This cycle commenced on the 16th of July in the year 438 B.C. It was received with acclamation by the people assembled at the Olympic games, and adopted in all the cities and colonies of Greece. It was also engraved in golden letters on tables of brass, whence it received the appellation of the golden number, and has been the basis of the calendars of all the nations of modern Europe. It is still in ecclesiastical use, with such modifications as time has rendered necessary.

Eudoxus of Cnidus, about the year 370 B.C., obtained great reputation as an astronomer. According to Pliny, he introduced the year of 365½ days into Greece. Archimedes says that he supposed the diameter of the sun to be nine times greater than that of the moon, which shows that he had in some degree overcome the illusions

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1 "Nicetas Syracusius, ut ait Theophrastus, coelum, solem, lunam, stellas, supera denique omnia, stare censet; neque, praeter terram, rem ullam in mundo moveri; quae cum circum axem se summa celeritate convertat et torquet, cadem effici omnia, quasi, statio terra, coelum moveretur." (Cicero, Acad. Quest. i Opera, tom. iv. p. 38, edit. Bipont.) Copernicus himself could not have stated the doctrine with greater precision. The titles of three of his works have been preserved—the *Period or Circumference of the Earth*, the *Phenomena*, and the *Mirror*. His observatory was still standing at Cnidus in the time of Strabo. His memory deserves to be honoured for the contempt which he evinced for the Chaldean predictions, and for having contributed to separate true astronomy from the reveries of judicial astrology. Eudoxus seems to have been the first who attempted to give a mechanical explanation of the apparent motions of the planets. He supposed each planet to occupy a particular part of the heavens, and that the path which it describes is determined by the combined motion of several spheres performed in different directions. The sun and moon had each three spheres; one revolving round an axis which passes through the poles of the world, and which occasions the diurnal motion; a second revolving round the poles of the ecliptic, in a contrary direction, and causing the annual and monthly revolutions; the third revolving in a direction perpendicular to the first, and causing the changes of declination. Each of the planets had a fourth sphere to explain the stations and retrogradations. As new inequalities and motions were discovered, new spheres were added, till the machinery became so complicated as to be altogether unintelligible.

Although Plato can hardly be cited as an astronomer, yet the progress of the science was accelerated by means of the lights struck out by his sublime and penetrating genius. He seems to have had just notions of the causes of eclipses; and he imagined that the celestial bodies originally moved in straight lines, but that gravity altered their directions, and compelled them to move in curves. He proposed to astronomers the problem of representing the courses of the stars and planets by circular and regular motions. Geometry was assiduously cultivated in the school of Plato; and on this account he claims a distinguished place among the promoters of true astronomy.

Astronomy is also under some obligations to Aristotle. In a treatise which he composed on this science, he recorded a number of observations which he had made; and, among others, mentions an eclipse of Mars by the moon, and the occultation of a star in the constellation Gemini by the planet Jupiter. As such phenomena are of rare occurrence, their observation proves that he had paid considerable attention to the planetary motions.

A great number of astronomers about this time appear on the stage, whose labours and observations prepared the way for the reformation of the science which was shortly after effected by Hipparchus. Helicon of Cizyene is renowned for the prediction of an eclipse, which took place, as Plutarch affirms, at the time announced. History records the names of only three individuals in ancient Greece who predicted eclipses, Thales, Helicon, and Eudemus. Eudemus composed a history of astronomy, a fragment of which, consisting of only a few lines, is preserved by Fabricius in the *Bibliotheca Graeca*. In this it is mentioned that the axes of the ecliptic and equator are separated from each other by the side of a pentadecagon, which is equivalent to saying that they contain an angle of $24^\circ$. This is the first value which we find assigned by the Greeks to the obliquity of the ecliptic. It is given in round numbers, and may easily be supposed to contain an error of a quarter of a degree.

Calippus is celebrated for the period which he formed of four Metonic cycles. Having observed, by means of an eclipse of the moon which took place about six years before the death of Alexander, that the Metonic cycle contained an error of a fourth of a day, he introduced the period of 940 lunations, containing four Metonic cy-

In the history of the various sects which have hitherto come under our review, we meet only with some useful remarks, with numerous hypotheses and conjectures, but with scarcely any appearance of regular and connected science. Up to this date the astronomical knowledge of the Greeks was confined to a few facts, the discovery of which implies no theory, and scarcely the aid even of the simplest instruments. The order and arrangement of the planets, the causes of eclipses, the identity of the morning and evening stars, the approximate length of the year, that of the lunar month, the obliquity of the ecliptic, and the cycles of Meton and Calippus, were almost the sole results of their astronomical speculations. In the Alexandrian school we meet for the first time with regular and systematic observations. We there find angular distances measured with appropriate instruments, and calculations made according to the rules of trigonometry.

After the premature death of Alexander, his principal generals shared among themselves his magnificent conquests, and Egypt fell to the lot of Ptolemy Soter. This prince was distinguished by an ardent love of science, and a desire to promote every species of liberal knowledge. He accordingly invited to his court, which he had established at Alexandria, the most eminent philosophers of Greece, and fixed them there by his liberality and munificent protection. His son, Ptolemy Philadelphus, who inherited his throne, also inherited his genius and love of science and learning. A superb edifice, styled the Museum, was assigned to the use of the men of science whom he had attracted to his capital, to which he also added an observatory, and the famous library, which had been collected with great care and at a vast expense by Demetrius Phalerius. The prince took great delight in the Museum; he visited it frequently, entered into familiar conversation with its inmates on the subject of their various pursuits, and by his own example stimulated their History-zeal and encouraged their inquiries. This noble institution, which survived all the vicissitudes of nine centuries, was the means of conferring incalculable benefits on the human race; and the name of its founder, Ptolemy Philadelphus, will be gratefully remembered while science and learning occupy a place in the estimation of mankind.

The first astronomers of the Alexandrian school were Aristillus and Timocharis, who flourished under the first Ptolemy, about 300 years before Christ. The chief object of their labours was the determination of the relative positions of the principal stars of the zodiac, instead of merely announcing their risings and settings, as had been the practice of the orientals and the ancient Greeks. The observations of these two astronomers conducted Hipparchus to the important discovery of the precession of the equinoxes, and served as the basis of the theory which Ptolemy, some centuries afterwards, gave of that phenomenon.

Aristarchus of Samos, the next in order of the Alexandrian astronomers, composed a treatise on the Magnitudes and Distances of the sun and moon, which has been preserved to our times. In this treatise he describes an ingenious method which he employed to obtain the relative distances of the two luminaries. At the instant when the moon is dichotomized, that is, when the exact half of her disk appears to a spectator on the earth to be illuminated by the sun's light, the visual ray passing from the centre of the moon to the eye of the observer is perpendicular to the line which joins the centre of the moon and sun. At that instant, therefore, he measured the angular distance of the two bodies, and finding it to be 87 degrees, he concluded, by the resolution of a right-angled triangle, that the distance of the sun is between eighteen and nineteen times greater than that of the moon. This method is perfectly correct in theory, but it is difficult to be assured of the exact instant of the moon's dichotomy, and in an angle of such magnitude a very small error greatly affects the result. The error of Aristarchus is very considerable, the true angle being about $87^\circ 50'$. The estimated distance of the sun is by consequence far too small; yet the determination, faulty as it was, contributed to expand greatly the existing notions relative to the boundaries of the universe, for the Pythagoreans had taught that the sun is only three, or at most three and a half times more distant than the moon. Another delicate observation made by Aristarchus was that of the magnitude of the sun's diameter, which, as we learn from Archimedes, he determined to be the 720th part of the circumference of the circle which the sun describes in his diurnal revolution. This estimate is not very far from the truth, and the observation is by no means an easy one. He embraced the doctrine of Pythagoras respecting the earth's motion, and appears to have entertained juster notions than any of the astronomers who preceded him, on the magnitude and extent of the universe. The treatise on the Magnitudes and Distances is published in the third volume of the works of Dr Wallis, with a Latin translation by Commandine, and some notes.

successor of Aristarchus, was a native of Cyrene, and invited to Alexandria by Ptolemy Evergetes, who appointed him keeper of the royal library. He is supposed to have been the inventor of armillary spheres, a species of instrument extensively used by the ancient astronomers. By means of an instrument of this kind he observed the distance between the tropics to be the whole circumference of a great circle as 11 to 83; a ratio equivalent to $47^\circ 42' 39''$, half of which gives $23^\circ 51' 19.5''$ for the obliquity of the ecliptic. This is a very important observation, and confirms the gradual diminution of the obliquity as indicated by theory. Eratosthenes is celebrated for being the first who attempted, on correct principles, to determine the magnitude of the earth. Having remarked, by some means with which we are unacquainted, that Syene, the most southern of the cities of ancient Egypt, is situated nearly on the same meridian with Alexandria, he conceived the idea of determining the amplitude of the celestial arch intercepted between the zeniths of the two places, and of measuring at the same time their distance on the ground; operations which would afford data for the determination of the whole length of the terrestrial meridian. Syene was known to be situated exactly under the tropic; for at the summer solstice the gnomon had no shadow, and the sun's rays illumined the bottom of a deep well in that city. On the day of the solstice he found the meridional distance of the sun from the zenith of Alexandria to be $7^\circ 12'$, or a fiftieth part of the circumference. It had also been ascertained by the hematists or surveyors of Alexander and the Ptolemies, that the itinerary distance between Alexandria and Syene was 5000 stadia; therefore $5000 \times 50 = 250,000$ stadia form the circumference of a great circle of the earth, or the length of the terrestrial meridian. Unfortunately, on account of the uncertainty respecting the length of the stadium here employed, we possess no means of estimating the degree of approximation afforded by this rude though ingenious attempt; but the idea does immortal honour to Eratosthenes, and the moderns have added nothing to his method: their better success is owing solely to the progress of the arts and the perfection of astronomical instruments.

About this time the science of astronomy was enriched by the discoveries of some of the distinguished geometers whose labours have so greatly extended the glory of the Alexandrian school. Euclid, the celebrated author of the Elements, lived in the reign of the first Ptolemy. He composed a book on the sphere, which probably served as a model for future works of the same kind, and was the first who treated in a geometrical manner of the phenomena of the different inclinations of the sphere. Conon of Samos, the friend of Archimedes, collected the records of eclipses, which had been observed by the ancient Egyptians; and Callinicus ascribes to him the constellation of Berenice's hair. Archimedes, whose profound genius and deep knowledge of geometry and mechanics entitle him to the appellation of the Newton of the ancients, also claims a high rank among the cultivators of astronomy. His celebrated planetarium, which represented the motions of the sun, moon, planets, and starry sphere, has been a frequent theme of the admiration and praises of the poets:

oli, rerumque fidem, legesque deorum, Ecce Syracusae transtulit arte senex.

Apollonius of Perga solved the important problem of the Apollonians and retrogradations of the planets by means of epicycles and deferents; and he is entitled to the glory of having formed the alliance between the two sciences of geometry and astronomy, which has been productive of the greatest advantages to both.

Astronomy, which had as yet only consisted of a know-Hipparchus, one of the most astonishing men of antiquity, and perhaps the greatest of all in the sciences which are not purely speculative. This illustrious founder of astronomical science was born at Nice in Bythinia, and ob- observations of eclipses recorded by the Chaldeans, he was enabled to determine the period of the moon's revolution relatively to the stars, to the sun, to her nodes, and to her apogee. These determinations are among the most precious relics of ancient astronomy, inasmuch as they corroborate the results of theory in one of its finest deductions—the acceleration of the mean lunar motion—and thus furnish one of the most delicate tests of the truth of Newton's law of gravitation. It was, indeed, by a comparison of the observations of Hipparchus with those of the Arabian and modern astronomers, that Dr Halley was led to the discovery of that curious and important phenomenon. Hipparchus also determined the eccentricity of the lunar orbit, and its inclination to the plane of the ecliptic; and the values which he assigned to these elements, making allowance for the evaporation and the inequalities of the moon's motion in latitude, are to a few minutes the same as those which are now observed. He had also an idea of the second inequality of the moon's motion, namely, the evaporation, and made all the necessary preparations for a discovery which was reserved for Ptolemy. He likewise approximated to the parallax of the moon, which he attempted to deduce from that of the sun, by determining the length of the frustum cut off from the cone of the terrestrial shadow by the moon when she traverses it in her eclipses. From the parallax he concluded that the greatest and least distances of the moon are respectively equal to 78 and 67 semi-diameters of the earth, and that the distance of the sun is equal to 1300 of the same semi-diameters. The first of these determinations exceeds the truth; the second falls greatly short of it; the distance of the sun being nearly equal to 24,000 terrestrial semi-diameters. It may, however, be remarked that Ptolemy, who undertook to correct Hipparchus with regard to the parallax, deviated still farther from the truth.

The apparition of a new star in the time of Hipparchus induced him to undertake the formation of a catalogue of all the stars visible above his horizon, to fix their relative positions, and mark their configurations, in order that posterity might have the means of observing any changes which might in future take place in the state of the heavens. This arduous undertaking was rewarded by the important discovery of the precession of the equinoxes, one of the fundamental elements of astronomy. By comparing his own observations with those of Aristillus and Timocharis, he found that the first point of Aries, which, in the time of these astronomers, or 150 years before, corresponded with the vernal equinox, had advanced two degrees, according to the order of the signs, or at the rate of 48 seconds a year. This determination is not very far from the truth; for, according to modern observations, the rate of the precession is about 50-1 seconds annually. His catalogue contained 1080 stars: it is generally, but erroneously, stated to have contained only 1022, after that of Ptolemy, in which the nebulous and some obscure stars are omitted. He also commenced a series of observations to furnish his successors with the means of forming a theory of the planets. Hipparchus likewise invented the planisphere, or method of representing the starry firmament on a plane surface, which afforded the means of solving the problems of spherical trigonometry in a manner often more exact and more commodious than the globe itself. He was the first who demonstrated the methods of calculating triangles, whether rectilineal or spherical; and he constructed a table of chords, from which he drew nearly the same advantages as we derive at present from the tables of sines. Geography is also indebted to him for the happy idea of fixing the position of places on

the earth by means of their latitudes and longitudes; and he was the first who determined the longitude by the eclipses of the moon.

These various labours and brilliant discoveries give a high idea of the industry and genius of Hipparchus. His writings have unfortunately all perished, excepting a commentary on the poem of Aratus; but the principal elements of his theories, together with a few observations, have been preserved in the Almagest of Ptolemy.

After the death of Hipparchus, nearly three centuries elapsed before any successor arose worthy of the name. During this long period astronomy gained no essential advancement. Some rude observations, scarcely superior to those of the Chaldeans, and a few meagre treatises, are the only monuments which exist to testify that science had not fallen into utter oblivion in an age so fertile of poets and orators. Geminos and Cleomedes wrote treatises, which have been preserved to our times; Agrippa and Menelaus are said to have observed; the Roman calendar was reformed by Julius Caesar and the Egyptian astronomer Sosigenes; and Posidonius measured a degree, and remarked that the laws of the tides depend on the motions of the sun and moon.

Ptolemy was born at Ptolemais in Egypt, and flourished at Alexandria about the 130th year of our era, under the reigns of Adrian and Antoninus. This illustrious ornament of the Alexandrian school is entitled by his own discoveries to the high rank among astronomers which has universally been assigned to him; but the most signal service which he conferred on science was the collection and arrangement of the ancient observations. Out of these materials he formed the Μεγάλη Σύνταξις, or Great Composition, a collection which exhibits a complete view of the state of astronomy in the time of Ptolemy, and which contains the germ of most of the methods in use at the present day.

The hypothesis which Ptolemy adopted for the purpose of explaining the apparent motions, was that which had been followed by Hipparchus. We have already seen that the genius of Pythagoras, soaring above the illusions of sense, had conceived the sun to be situated at the centre of the universe, and the earth to circulate, like the other planets, about the sun; and that the same opinion was entertained and supported by Aristarchus and a few other astronomers. It would seem, however, that this philosophical idea never gained much ground in antiquity, even among the learned. The vulgar prejudice respecting the immobility of the earth continued to prevail; and it had become an inveterate axiom, that all the celestial motions must be circular and uniform. Ptolemy himself, who felt in its full force the difficulty of reconciling the appearances with the notion of a uniform circular motion, adopted the common opinion without scruple as a primordial law of the universe; for, says he, this perfection belongs to the essence of celestial things, which neither admit of disorder nor irregularity. To save this chimera—the uniform circular motion—Apollonius imagined the ingenious apparatus of epicycles and deferents; and Hipparchus advanced a step farther, by placing the centre of the sun's circle at a small distance from the earth. Ptolemy adopted both hypotheses, and supposed the planet to describe an epicycle by a uniform revolution in a circle, the centre of which was carried forward uniformly in an eccentric round the earth. By means of these suppositions, and by assigning proper relations between the radii of the epicycle and deferent circle, and also between the velocity of the planet and the centre of its epicycle, he was enabled to represent with tolerable accuracy the apparent motions of the planets, and particularly the phenomena of the stations and retrogradations, which formed the principal object of the researches of the ancient astronomers. The notions of Apollonius and Hipparchus were thus reduced to a systematic form, and the proportions of the eccentrics and epicycles of all the planets assigned, by Ptolemy; on which account, the system has been generally ascribed to him, and obtained the name of the Ptolemaic System of the universe. As a first attempt to bring the celestial motions within the grasp of geometry, it does infinite honour to the genius of its inventors. It is, however, totally irreconcilable with the precision of modern observations; for it is impossible to represent on this hypothesis the variations of the distances of the planets at the same time with their apparent motions. But this difficulty could scarcely be felt by Ptolemy, inasmuch as it was impossible, before the invention of the telescope and micrometer, to form any accurate estimate of the variations of the apparent diameter of a planet, and consequently of its distance. It must be admitted, however, that the Ptolemaic hypothesis might be sufficient for the wants of practical astronomy, that is, for calculating the places of the planets and forming tables of their motions, were it not for its extreme complication. The discovery of every new irregularity in the planetary motions exacts the addition of a new epicycle; and such was the confusion resulting from this circumstance, that Alphonso X., despairing of being able to comprehend the complicated machinery, was tempted to exclaim, that if the Deity had called him to his counsels at the creation of the world, he could have given good advice. Yet, notwithstanding all its defects, the system of Ptolemy gained a complete ascendency over the minds of mankind, and, so difficult is it to leave the beaten path, continued to be implicitly followed by every astronomer during fourteen centuries, having been only finally exploded by Kepler's discovery of the elliptic orbit of Mars.

The most important discovery which astronomy owes to Ptolemy is that of the Evection of the moon. Hipparchus had discovered the first lunar inequality, or the equation of the centre, which serves to correct the mean motion at the syzygies, and had also remarked the necessity of another correction for the quadratures. He even undertook a set of observations, with a view to ascertain its amount and its law; but death put a stop to his labours before he had brought them to a successful issue. Ptolemy completed the investigation, and discovered that the eccentricity of the lunar orbit is itself subject to an annual variation, depending on the motion of the line of the apsides. The variation of the position of the apsides produces an inequality of the moon's motion in her quarters, which has been technically denominated the evction. The equation given by Ptolemy, though of course empirical, is remarkably exact.

Ptolemy employed a very simple process for determining the moon's parallax, which was probably suggested to him by the situation of Alexandria, where he observed. He determined the latitude of a place a little to the south of that city, over the zenith of which the moon was observed to pass when her northern declination was the greatest possible. But when the moon is in the zenith, or in the same straight line with the observer and the centre of the earth, she has no parallax; consequently the obliquity of the ecliptic and the latitude of the station being known, the moon's greatest northern latitude was also determined. The next step was to observe the moon's meridian altitude fifteen days after the first observation, when her southern latitude was necessarily the greatest possible. This observation gave the apparent altitude of the moon, but her greatest northern and southern decli- nations being supposed equal, her true altitude, as seen from the centre of the earth, was easily computed from the previous observation, and the difference between the true and apparent altitudes gave the amount of the parallax.

The observations of Hipparchus relative to the motion of the stars in longitude, or the regression of the equinoctial points, were confirmed by Ptolemy, although he mistook its amount, and diminished a quantity which Hipparchus had already estimated too low. According to Hipparchus, the regression is at the rate of two degrees in 150 years. Ptolemy reduced it to one degree in 90 years. This disagreement would seem to indicate an error of more than a degree in the observations, which can with difficulty be admitted, considering the accordance which subsists among the different observations cited by Ptolemy in support of his own determination. For this and some other reasons Ptolemy has been accused of altering the observations of Hipparchus, and accommodating them to his own theory; but there does not appear to be any just ground for the imputation. The error with regard to the regression probably arose from the circumstance, that Hipparchus had assigned too great a value to the length of the year, whence the motion of the sun with regard to the equinoxes would be made too slow, and the longitudes employed by Ptolemy consequently diminished.

Ptolemy has been called the Prince of astronomers—a title which may perhaps be justified by the universal and long-continued prevalence of his system, but to which he has no claim from the number or value of his own observations. After a laborious and minute examination of the Almagest, Delambre doubts whether anything, saving the author's declarations, is contained in that great work, from which it can be decisively inferred that Ptolemy ever observed at all. He indeed frequently makes mention of observations made by himself; but his solar tables, rate of the precession, eclipses, determination of the moon's motion and parallax, and, above all, his catalogue of stars, render it impossible to doubt that the greater part of the results which he has given as observations are merely computed from the tables of Hipparchus. It is therefore difficult to allow to Ptolemy that good faith and "astronomical probity which forms one of the most indispensable qualities of an observer." He never in any instance cites a single observation more than is just necessary for the object he has immediately in view, and consequently, by precluding all comparison of one observation with another, has deprived us of the means even of guessing at the probable amount of the errors of his solar, lunar, and planetary tables. If an astronomer, as Delambre justly remarks, were to adopt the same course at the present day, he would be certain of forfeiting all claim to confidence; but Ptolemy stood alone; he had neither judges nor rivals; he claimed admiration, and received it; and now no one condescends to calculate the few observations he has left us. (Delambre, Astronomie Ancienne, tom. i. Discours Préliminaire.) His catalogue contains only 1029 stars, and is therefore less extensive than that of Hipparchus, but it is exceedingly valuable on account of its details.

The name Almagest (Μαγεστης, with the Arabic prefix) was bestowed on the Syntaxis by the Arabians, into whose language it was translated in the ninth century. The first Latin translation was from the Arabic, and published at Venice in 1515. It abounds in Arabic words and idioms, and is very inaccurate and barbarous. The second Latin translation was made from the original Greek by George of Trebizond, and is greatly superior to the first. It was published at Basle in 1541, and in 1551. The History. Greek text was published at the same time in 1538. Ptolemy was the author of numerous other works connected with astronomy, of which his Geography, in eight books, is the best known. It contains a list of all the places of which the latitudes and longitudes had at that time been determined. His treatise on Optics was supposed to be lost, till an imperfect Latin translation, from an Arabic version, was lately discovered in the king's library at Paris. The last book of this work contains a theory of astronomical refraction, more complete than any which existed before that of Cassini. It would seem that Ptolemy had not discovered the refraction at the time he composed the Almagest, no mention being made of the subject in that work. The explanation which he gives of the phenomenon is natural and satisfactory, indeed entirely conformable with that which is now universally adopted. The idea and explanation remained buried in the Optics till reproduced by Alhazen; but neither Ptolemy nor Alhazen attempted to estimate the amount of the refraction. His Planisphere and Analemma, in which he treats of the stereographic and orthographic projections of the sphere, show a perfect acquaintance with spherical trigonometry. In the last-mentioned work he makes use of the sines, and his constructions comprehend three of the four general theorems in modern use. Divers treatises also on music, dialling, chronology, and mechanics, attest the universality of Ptolemy's genius, and his unremitting application to the pursuits of science. Like Archimedes, he had a desire to transmit to posterity the history of his labours by a public monument. In the temple of Serapis, at Canopus, he is said to have consecrated a marble pillar, with an inscription containing the principal elements of his astronomy, such as the length of the year, the eccentricity of the solar and lunar orbits, the dimensions and forms of the epicycles of the planets, &c.

On the death of Ptolemy astronomy ceased to be cultivated among the Greeks. The Alexandrian school subsisted indeed for some centuries after; but genuine science had fled, and its place been usurped by the vain wranglings of theologians and grammarians. During the long period of six or seven centuries, the labours of those who assumed the name of astronomers were confined to needless or trifling commentaries on the works of Hipparchus and Ptolemy, and were productive of no observations, or even remarks, having a tendency to enlarge the boundaries of the science. The genius of the Roman dominion was unfavourable to the development or exercise of the higher faculties of the human intellect; and the natural sciences, with the liberal arts, faded away under the withering influence of military despotism.

From the brief account which has now been given, it will be easily inferred that the Greeks cultivated astronomy rather as a speculative than a practical science. None of their numerous sects ever evinced any taste for observation or experiment; and hence, while geometry made great and rapid advances in their hands, physics and experimental philosophy were entirely neglected. The prevailing passion for speculation pervaded even their astronomy. They explained the doctrine of the sphere, and the apparent motions of the planets; and framed ingenious theories to account for such phenomena as came immediately under the cognizance of their senses; but if we except the observations of Hipparchus and Ptolemy, and perhaps two solstitial distances of the sun from the zenith, observed by Eratosthenes, we remark among them no observations made with instruments capable of measuring angular distances. Before Hipparchus, no mention is made of the astrolabe; and the recorded determinations do not History give us a very favourable idea of the accuracy of that instrument. On casting our eyes over the catalogue of Ptolemy, we scarcely ever meet with a fraction of a degree smaller than one-twelfth, that is to say, less than five minutes; whence we may infer that the astrolabe only measured twelfth parts of a degree. Occasionally, indeed, the fractions one-fourth and three-fourths occur; but these were most probably inserted by estimation. The Greeks of Alexandria committed an error of no less than 15' with regard to the altitude of the pole, one of the most essential elements to an observer; and it does not appear that they were ever able to determine the time to within a quarter of an hour. Yet notwithstanding these circumstances, which indicate that the art of observation was still in its infancy, the science of astronomy is vastly indebted to the labours and speculations of the Greeks. The complicated but ingenious hypotheses of Ptolemy prepared the way for the elliptic orbits and laws of Kepler, which, in their turn, conducted Newton to the great discovery of the law of gravitation.

Astronomy of the Arabians.

While the nations of western Europe were involved in the thickest shades of ignorance and barbarism, the torch of science was rekindled, and blazed forth with extraordinary splendour, among the Saracens. The burst of fanaticism which enabled the followers of Mahomet to carry their religion and their arms over the fairest portion of the ancient world subsided, in a great measure, after a century and a half of uninterrupted conquest, and was succeeded by a period of repose, during which they cultivated the arts of peace and civilisation with the same ardour which had characterized their achievements in arms. Under the enlightened and munificent protection of the caliphs, Bagdad became what Alexandria had been under the Ptolemies, the centre of politeness and knowledge.

The accounts which we possess of the Saracen literature are imperfect and scanty; but the first of the caliphs who appears to have encouraged the study of astronomy was Aboujafar, surnamed Almansur, or the Victorious, who reigned in the eighth century. His grandson Almamon, the seventh of the Abbasides, and second son of the famous Haroun Al Raschid, who reigned at Bagdad from 813 to 833, is celebrated for the protection which he gave to learning, and the zeal with which he laboured to propagate the sciences of the Greeks among his subjects. In granting peace to the emperor Michael III., he stipulated for liberty to collect in Greece all the writings of the philosophers. These he transported into his own country, and caused to be translated into Arabic. Finding it mentioned in the geography of Ptolemy that a degree of the earth was equivalent to 500 stadia, he resolved to have this fact verified by a new measurement; and in obedience to the commands of the caliph, a company of mathematicians assembled in the spacious plain of Sinnar, where, having observed the altitude of the pole, they separated themselves into two parties, and proceeded in opposite directions along the meridian, measuring the distance they passed over till the altitude of the pole varied one degree. Being unacquainted with the nature of the instruments made use of in these geodetic operations, we cannot estimate the probable accuracy of the result; but as it agreed perfectly with the statement of Ptolemy, we have a right to infer that the measurement was executed in a very inadequate manner, and that the mathematicians of the caliph adopted the ancient determination from want of confidence in their own.

The Syntax of Ptolemy was translated into Arabic History, under the reign of Almamon, by Isaac Ben Honain. The translation was afterwards revised by Thabet or Thebeth Ben Korah, and it was about this time that it received the appellation of Almagest. Astronomical observations, which, as we have had occasion to remark, had been greatly neglected by the successors of Hipparchus, formed a principal object of the attention of the Arabians. By the orders of Almamon, the obliquity of the ecliptic was observed, and found to be $23^\circ 33'$. According to the modern tables, the obliquity at that time was $23^\circ 36' 34''$, so that the error was less than that of Hipparchus and Ptolemy, in their determination of the same element. This observation supposes instruments of some accuracy. Among the astronomers whom Almamon drew to his court, we find the names of Habash of Bagdad, who composed three books of astronomical tables; Ahmed, or Mohammed Ben Cothair, better known by the name of Alfragan, or Alfranius, who, from his great expertise in computing, was styled the calculator. He composed an elementary treatise on astronomy, which was only an abridged extract of the works of Ptolemy; and likewise wrote on sun-dials, and gave a description of the astrolabe. The Jew Meshala, whose treatise on the elements was published at Nuremberg in 1549, also lived in the time of Almansor or Almamon.

The most celebrated of the Arabian astronomers was Albategnius, or Muhammed Ben Geber Albattani, so called from Batan, a city of Mesopotamia, where he was born. He was a prince of Syria, and resided at Amete or Racha, in Mesopotamia; but many of his observations were made at Antioch. Having studied the Syntax of Ptolemy, and made himself acquainted with the methods practised by the Greek astronomers, he began to observe, and soon found that the places assigned to many of the stars in Ptolemy's tables were considerably different from their actual situations, in consequence of the error which that great astronomer had committed with regard to the precession of the equinoxes. Albategnius measured the rate of the precession with greater accuracy than had been done by Ptolemy; and he had still better success in his attempt to determine the eccentricity of the solar orbit; his value of which differs extremely little from that which results from modern observations. In assigning the length of the year, however, he fell into an error of more than two minutes; but this proceeded, as has been shown by Dr Halley, from too great confidence in the observations of Ptolemy. Albategnius also remarked that the place of the sun's apogee is not immovable, as former astronomers had supposed, but that it advances at a slow rate, according to the order of the signs—a discovery which has been confirmed by the theory of gravitation. A new set of astronomical tables, more accurate than those of Ptolemy, likewise resulted from the indefatigable labours of Albategnius; and his observations, important in themselves, are doubly interesting on account that they form a link of connection between those of the astronomers of Alexandria and of modern Europe. The works of Albategnius were published in 1537, under the title of De Scientia Stellarum.

Thebeth Ben Korah, another Arabian, acquired celebrity by proposing an explanation of the motions of the stars, which, under the name of the "System of Trepidation," was eagerly received by the astronomers of the middle ages, and disfigured the tables of Alphonso, and even those of Copernicus. He ascribed to the eighth sphere, or that of the fixed stars, two motions; one the diurnal motion, the other that of trepidation, performed in small circles round the first points of Aries and Libra, and of Astronomy.

He therefore supposed two ecliptics, one fixed in the ninth sphere, the other movable in the eighth. According to this construction, the motion of the stars is sometimes direct and sometimes retrograde.

The Arabs have been said to be not only the cultivators but the apostles of the sciences, on account of the activity with which they propagated them among all the nations subjected to their dominion. The Fatimite caliphs, who reigned in Egypt during two centuries, rivalled their predecessors the Ptolemies in the encouragement which they gave to astronomy. Under the caliph Hakem, who reigned from 996 to 1021 of our era, Ebn Jounis acquired a splendid reputation. He constructed a set of tables, and composed a sort of celestial history, in which he has recorded numerous observations of his own and of other astronomers belonging to the same country. This work, imperfectly known through some extracts, long excited the curiosity of astronomers, as it was supposed to contain observations tending to establish the acceleration of the mean motion of the moon. A manuscript copy of it, belonging to the university of Leyden, was, in 1804, transmitted to the French Institute, and translated by Professor Caussin. It contains 28 observations of eclipses from the year 829 to 1004; seven observations of the equinoxes; one of the summer solstice; one of the obliquity of the ecliptic made at Damascus, by which the value of that element is found to be $23^\circ 35'$; and likewise a portion of tables of the sun and moon, with some other matter illustrative of the state of astronomy among the Arabsians. The observations which regard the acceleration of the mean lunar motion are two eclipses of the sun and one of the moon, observed by Ebn Jounis, near Cairo, in the years 977, 978, and 979, and they agree with theory in confirming the existence of that phenomenon.

The Saracen conquests in Spain were attended with the same happy results as in Egypt, and science flourished in that country while the rest of Europe was involved in the darkest shades of ignorance. Arzachel is supposed to be the author of the Toledo Tables, constructed about the year 1150, but which, on account of the established reputation of those of Albategnius, were never in great estimation. He made some changes in the dimensions which had been assigned by Hipparchus and Ptolemy to the solar orbit, and deserves the praise of having been an exact and attentive observer. Alhazen, who flourished in the same country about the same period, contributed to the progress of astronomy by a treatise on Optics, in which he clearly indicated the necessity of making an allowance for the celestial refraction in astronomical observations. His treatise contained a theory of reflection and refraction, an explanation of the cause of the twilight, and of the magnitude of the horizontal moon. Averroes, a physician of Cordova, made an abridgement of the Almagest in the twelfth century, and Almansor found the obliquity of the ecliptic to be $23^\circ 38' 30''$, which proves that practical astronomy had now attained to a tolerable degree of exactness.

e inquire what effect the labours of the Arabsians and their disciples had on the progress of astronomy, we shall find that their services were confined entirely to the practical part. In point of theory they did absolutely nothing. They admitted all the hypotheses of Ptolemy without the slightest alteration, even with timid and superstitious respect, and did not advance a single step towards the discovery of the solar system. But with regard to instruments and methods of calculation, their improvements were numerous and important. They constructed instruments on a larger scale, and divided them with greater care; and, even from the time of Almamon, we remark among them History, new and more exact determinations of the obliquity of the ecliptic, of the positions of some stars, of the precession, of the length of the year, and of the eccentricity of the sun's orbit. To these fundamental points they added numerous observations of eclipses and conjunctions; they industriously sought out and corrected the errors of Ptolemy's tables; they perceived the necessity of marking the instant of each phenomenon with greater care; and their determinations of the commencement and end of eclipses are in general accompanied with the altitude of a star, which afforded them the means of calculating the hour, angle, and the true time. In cases where less precision was wanted, they made use of their clepsydrae and solar dials, to the construction of which they paid particular attention. Trigonometry derived signal advantages from their constant care to facilitate the calculations of spherical astronomy. Albategnius substituted the sines for the chords,—a most important improvement, the idea of which was probably suggested to him by the Analemma of Ptolemy. By this happy substitution the solution of all rectangular spherical triangles was reduced to four general formulae, of which the Greeks had the equivalent in a much less commodious form. The same astronomer also appears to have invented a very remarkable rule for the oblique-angled triangles, perfectly identical with one of the four general formulæ now in use. Ebn Jounis, and his contemporary Aboul Wefa, were acquainted with the tangents and secants, and employed them very dexterously in reducing complicated binomial expressions to a single and simpler term. They also employed subsidiary arcs and other artifices in the calculus of the sines, in order to facilitate the labour of computation. These substitutions are now common; but they remained long unknown in Europe; and 700 years after they were employed by the Arabsians, we first meet with some examples of their use in the writings of Thomas Simpson.

The zeal of the Arabsians for astronomical observations was communicated by them to the Persians and Tartars. About the year 1072, Omar Cheyam determined the length of the tropical year, and introduced the calendar which has ever since been used in Persia. Hulegu-Ilecou-Khan, who conquered that country about the year 1264, caused an observatory to be built at Maragha, near Tauris, where he assembled the most celebrated astronomers who could be found within his dominions, and employed them in forming new astronomical tables. This work was directed by the famous Nassireddin, and brought to a conclusion in the year 1269. With the exception of some trifling corrections of the mean motions, the whole of these tables are copied from Ptolemy.

Ulugh Beigh, a Tartar prince, and grandson of the great Ulugh Tamerlane, not only encouraged the study of astronomy, but was himself a diligent and successful observer. At Samarcand, the capital of his dominions, he established an academy of astronomers, and caused the most magnificent instruments to be constructed for their use. By means of a gnomon 180 feet in height, he determined the obliquity of the ecliptic to be $23^\circ 30' 20''$, the precession of the equinoxes at $1^\circ$ in 70 years, and obtained elements for the construction of tables which have been found to be scarcely inferior in accuracy to those of Tycho Brahe. The ancient astronomy had produced only one catalogue of the fixed stars, that of Hipparchus. Ulugh Beigh has the honour of having formed a second, after an interval of sixteen centuries. This learned and munificent prince, whose virtue and talents deserved the esteem of mankind, was assassinated by his own son in the 58th year of his age. After the death of Ulugh Beigh, astronomy received no further accessions in the east. But the seeds of knowledge had now begun to take root in a more propitious soil, and Europe, destined to carry the development of the human energies to its fullest extent, began to awake from the lethargy in which it had continued during so many ages. The first dawnsings of returning day appeared in Spain. In spite of the horror inspired by the Moslem religion, the Christians began to perceive and acknowledge the superiority and utility of the science of the Moors; and the schools of Cordova became the resort of all those whom curiosity, or love of knowledge, induced to seek abroad for that information which could not be obtained in their own countries. Geber, afterwards Pope Silvester II., acquired the knowledge of arithmetic from that source; and John of Halifax, better known by the name of Sacrobosco, after having studied some time in Spain, made an abridgement of the Almagest, which was long famous under the title of a Treatise of the Sphere.

The emperor Frederick II. is no less celebrated for his protection of the sciences, than for the continual struggles in which he was involved with the popes. He founded the university of Naples, and caused Latin translations to be made of the works of Aristotle and the Almagest of Ptolemy. About the same time astronomy was zealously encouraged and cultivated by Alphons X., king of Castile. This monarch, whose liberal mind seems to have been far superior to the age in which he lived, formed a college or lyceum at Toledo, the capital of his dominions, whither he assembled the most eminent astronomers that could be found, whether Christians, Moors, or Jews, and engaged them in the task of correcting the errors of the ancient tables. From their united labours were produced the Alphonsoine Tables, which obtained great celebrity, and were, in some respects, superior in accuracy to any which had preceded them. They are supposed to have been chiefly the work of Rabbi Isaac Aben Sid, surnamed Hazan, inspector of the synagogue of Toledo. They are said to have cost the king 40,000 ducats,—a sum certainly far exceeding their real value, which is confined to the correction of some epochs, and a more accurate determination of the sun's motions and the length of the year.

The same century gave birth to several other individuals distinguished by their attachment to the sciences. Campanus of Nivari translated Euclid, and left a treatise on the sphere. Vitello, a native of Poland, composed a treatise on optics, in ten books; and Albert, bishop of Ratisbon, whom his contemporaries styled the great, was the author of some works on arithmetic, geometry, astronomy, and mechanics. But the greatest luminary of that age was Roger Bacon, a Franciscan friar, whose numerous works contain many indications of a powerful and inventive genius. He made many important discoveries in optics; but his knowledge of natural philosophy and chemistry, uncommon in those days, had nearly proved fatal to him; for he was suspected of necromancy, and thrown into a dungeon, from which he did not escape till he had satisfied his superiors and the pope that he had never held unlawful intercourse with the devil. He composed a work on the utility of astrology, the places of the stars, and the aspects of the moon; and he had the merit of perceiving the necessity of reforming the calendar.

The fourteenth century produced no astronomer from Austria, where he was born in 1423, obtained great celebrity as a professor. He studied at Vienna, and after giving proofs of distinguished talents, he travelled into Italy, where he was favourably received by the cardinal of Cusa, who himself cultivated astronomy. On his return to Vienna he undertook a translation of the Almagest; and although ignorant both of Greek and Arabic, his perfect acquaintance with the subject enabled him to correct many errors which had been introduced through the carelessness or ignorance of former translators. He published a table of sines for every ten minutes to a radius of 6,000,000 parts, which was afterwards extended by his scholar Regiomontanus to every minute of the quadrant. The most celebrated of his works is his Theoria Nova Planetarum, which was published in 1460. He constructed a celestial globe, on which was represented the motion of the stars in longitude from the time of Ptolemy to the year 1450. He also measured the obliquity of the ecliptic, and is considered as the inventor of decimal arithmetic. He died in 1461, having only reached his 38th year, with the reputation of being, at that time, the first astronomer in Europe.

Purbach had the good fortune to form a disciple who executed many of the plans which had been interrupted by his premature death. This was the celebrated John Muller of Königsberg, better known by the name of Regiomontanus. Attracted in his youth to Vienna by the great reputation of Purbach, he continued to study there during ten years, and on the death of his master repaired to Rome for the purpose of acquiring the Greek language, and of making himself, through it, acquainted with the Almagest. At Rome he continued his observations, and translated into Latin the works of Ptolemy, the Conics of Apollonius, and some other treatises of ancient science. In 1471 he retired to Nuremberg, where, with the aid of Bernard Walther, a wealthy burgess, he founded an observatory, and furnished it with excellent instruments, principally of his own invention, by means of which he was enabled to detect many errors in the ancient tables. On the invitation of Pope Sixtus IV., who wished to reform the calendar, he again repaired to Rome; but after a few months' abode there he died suddenly, according to some accounts, of the plague, according to others, through the effects of poison administered to him by the sons of George of Trebizond, who adopted this excusable method of revenging the exposure which he had made of their father's errors in the translation of the Almagest. Regiomontanus was a learned and skilful man, but the great expectations which his early labours gave of future services to astronomy were disappointed by his untimely death. He paid great attention to trigonometrical calculation; and, although he did not reach the point which had been attained by the Arabians, he had the merit of introducing some useful theorems which till then were entirely unknown in Europe. His genius, however, did not enable him to rise above the prejudices of his age, for he was an astrologer as well as an astronomer, and is said to have most lamented the errors of the Alphonsoine Tables on account of the uncertainty which they occasioned in the calculation of genitures or horoscopes.

After the death of Regiomontanus, Walther continued to observe at Nuremberg during thirty years. His observations were collected by order of the senate of Nuremberg, and published by Schöner in 1544, a second time by Snellius, and, lastly, along with those of Tycho Brahe. In 1484 Walther began to make use of clocks, then a recent invention, to measure time in celestial observations. He was also the first who employed the planet Venus as a term of comparison for determining the longitudes of the stars.

Nuremberg had the honour of producing another astro- Astronomy.

erner was the first who explained the method which was afterwards brought into general use by Maskelyne, of finding the longitude at sea, by observing the distance between a fixed star and the moon. He published some mathematical and geographical treatises, and made a number of observations to determine the obliquity of the ecliptic and the precession of the equinoxes.

We are now come down to a period in the history of astronomy when the science was destined to undergo a total renovation, and the system which had been so laboriously established by Ptolemy, and blindly followed as an article of religious belief by Arabs, Persians, Tartars, and Europeans, during so many centuries, was about to be exploded for ever. In proportion as the observations became more numerous and accurate, the difficulty of representing them by the Ptolemaic system became greater; and astronomers were obliged to have recourse to the most violent and improbable suppositions in order to explain the phenomena, and, in the language of Ptolemy, were the appearances. We have mentioned that Pythagoras and his disciples entertained an idea very different from that which commonly prevailed, and supposed the sun to be the immovable centre of the celestial motions. It appears, however, from Aristotle, that this opinion was not founded on any analysis of the phenomena, but on certain metaphysical notions respecting the comparative dignity of the several elements. For example, fire, being a nobler substance than earth, ought to occupy the centre or place of honour. But such arguments could have little weight except in the schools, and accordingly were rejected by Ptolemy as too absurd to require a serious refutation. In order to give any probability to the Pythagorean doctrine, it was necessary to explain the succession of the seasons and the precession of the equinoxes on the hypothesis of the annual revolution of the earth about the sun; to show how the unequal motions of the planets in concentric orbits would give rise to the phenomena of the stations and retrogradations; to account, in short, for all the appearances, and point out their coherence and mutual connection. All this was effected by Copernicus, who had thereby the glory of first making known the true system of the universe, and of leading the way in that career of astronomical discovery in which the genius of the human race has gained its noblest trophies.

Nicholas Copernicus was born at Thorn, a city of Prussia, on the confines of Poland, according to Junctinus on the 19th of January 1472, and according to others on the 19th February 1473. From his earliest years he displayed a great fondness and aptitude for mathematical studies, and pursued them with corresponding success at the university of Cracow. Stimulated by the desire of acquiring a reputation equal to that of Regiomontanus, he set out for Italy at the age of 23 years, in order to study astronomy at Bologna under the celebrated Dominic Maria. He afterwards removed to Rome, where he employed himself in studying and teaching the mathematics, and where he made several astronomical observations about the year 1500. On his return to his native country he was made a canon of Ermeland by his uncle the bishop of Worms, and took up his residence at Frauenberg, a small Prussian town near the mouth of the Vistula, where he passed 36 years of his life in observing the heavens and meditating on the system of the world. In this retirement he composed his famous work entitled Astronomia Restaurata, sive de Revolutionibus Orbium Coelestium, in which he explained the celestial motions in a manner as simple and connected as the system of Ptolemy was complicated and incoherent. The system which Copernicus adopted in this work is now so familiar to everyone, that it is almost unnecessary to describe it. The heaven, composed of stars perfectly at rest, occupies the remotest bounds of space, then the orbit of Saturn, next Jupiter, Mars, the Earth accompanied by its moon, Venus, Mercury, and, lastly, the Sun immovable at the centre. By this arrangement the stations and retrogradations of the planets became simple mathematical corollaries, following from the differences of the radii of their orbits and their unequal motions. The diurnal rotation of the earth explained more simply and rationally the apparent daily revolution of the heavens; and the precession of the equinoxes was referred to a small variation in the inclination of the earth's axis to the plane of the ecliptic. But the simplicity of the system, and its consequent probability, were the only arguments which Copernicus was able to bring forward in proof of its reality. The motion of the earth can indeed never be made an object of sense; but after Richer's discovery of the diminution of gravity towards the equator, it was impossible to doubt longer of the existence of its rotatory motion; and when Roemer had measured the velocity of light, and Bradley observed the phenomena of the aberration, the evidences of its annual revolution were rendered equally convincing. Great, however, as were the merits of Copernicus, it must be acknowledged that he left his system in a very imperfect state. After the example of the ancients, he assumed as an axiom the uniform circular motion of the planets; and as the only motions which are observed are in a state of incessant variation, he was obliged, in order to save the inequalities, to suppose a different centre to each of his orbits. The sun was placed within the orbit of each of the planets, but not in the centre of any of them, consequently he had no other office to perform than to distribute light and heat; and, excluded from any influence on the system, he became as it were a stranger to all the motions. Yet notwithstanding these and other imperfections, the establishment of the doctrine of the earth's motion, with an evidence which dissipated the illusions of sense, was a great step towards the true knowledge of the planetary system; and when we consider the ignorance and prejudices of the age, and that Copernicus was moreover a priest, we cannot hesitate to admit his claim to a high rank among philosophers. But whether the actual services which he rendered to astronomy are commensurate with the great fame he has obtained, may admit of doubt. He revived an ancient opinion opposed to the prejudices and religious dogmas of his times, and fortified it with new and strong, though not absolutely convincing, proofs. It seldom happens, however, with regard to those sciences which ultimately appeal to experience, that general reasoning, even of the soundest kind, tends much to their real advancement; and there is little reason for thinking that astronomy would have been less perfect, or that any discoveries since made in it would have been retarded a single day, even if Copernicus had never lived. His great merit, like that of Lord Bacon, consists in the sound views which he took of nature, and in advancing so far before the general attainments of his age.

Fearing the opposition which was likely to arise from religious bigotry to opinions so much at variance with vulgar prejudice, Copernicus long delayed the publication of his great work; and it was only at the urgent request of his friends that he at last allowed it to be printed. He is said to have received the last sheet of it only on the day of his death. He was buried in the cathedral of Frauenberg, and his only epitaph consisted of some spheres cut out in relief on his tombstone. The ideas of Copernicus soon spread over Germany, where astronomy was at that time diligently cultivated; but they do not seem to have met with general favour before the commencement of the seventeenth century. The art of observing was, however, gradually receiving improvement; instruments were constructed on better principles, and more accurately divided; and the methods of computation were rendered much less laborious. Nonius, or Nunez, a born 1497, Portuguese, invented the ingenious method of subdividing the small divisions of instruments which still retains his name; Reinhold extended the table of tangents of Regiomontanus to every minute of the quadrant, reformed the tables of Copernicus, and composed many works of practical utility; but of the immediate successors of Copernicus, no one deserves to be more honourably mentioned than William IV., landgrave of Hesse. This prince built a magnificent observatory on the top of his palace at Cassel, which he furnished with excellent instruments of copper, and is said to have calculated himself the positions of no less than 400 stars. He was aided in the labours of the observatory by some astronomers of great merit whom his liberality drew to his court; among others, by Rothman, and Justus Byrgius, a distinguished artist, to whom Kepler has ascribed some idea of the logarithms.

Tycho Brahe, one of the best and most indefatigable observers of whom practical astronomy can boast, was born at Knudstorp, in Scania, the 13th of December 1546. His family was one of the most ancient and noble in Denmark; and his father, probably thinking that illustrious birth superseded the necessity of education, refused his consent to have his son instructed in Latin. Through the kindness of a maternal uncle, however, he was rescued from the state of barbarous ignorance to which he had been doomed by his parent; and, after having received the requisite preliminary instruction, he was sent to the university of Leipsic to study jurisprudence and scholastic philosophy. The tastes of the future astronomer were first excited by an eclipse of the sun which happened in 1560. Struck with astonishment at the accuracy of the prediction, he conceived a vehement desire to become acquainted with the principles of so certain a science, and exerted his utmost ingenuity to elude the obstacles which were interposed by his parents and governor to prevent him from acquiring the elementary notions of mathematics and astronomy. By placing the hinge of a common compass near his eye, he contrived to guess at the distances of the planets from the stars, and by this means, according to his own account, detected several errors in the Ephemerides of Stadius. By his persevering efforts he at last obtained the consent of his family to study according to his own inclinations; and from that moment he divided his time between the observation of the heavens, and chemical experiments. He visited the different cities of Germany where he hoped to meet with astronomers and skilful mechanicians, and was received with flattering attention by the landgrave of Hesse-Cassel, with whom he contracted an intimate friendship. On his return to Denmark he obtained from Frederick II. a grant of the small island of Huen, in the strait of Sunda, together with a pension and some presents, by means of which, and an expenditure of 100,000 crowns of his own patrimony, he was enabled to build the castle of Uraniburg, and procure a magnificent collection of the largest and most accurate instruments which could then be constructed. In this celebrated retreat he passed twenty-five years, actively employed in making observations, and attracting by his discoveries the attention of the learned throughout Europe. On the death of his protector Frederick he fell under the displeasure of the government, and a storm of persecution was raised against him, from causes which he has not explained, by a minister named Walckendorp, who, on this account, has been devoted by Lalande to the infamy and execration of all ages. He was deprived of his pension, compelled to leave the castle of Uraniburg, and to banish himself from Denmark. He retired first to Wandeshurg, near Hamburg; he afterwards sought an asylum in Bohemia, and ultimately settled at Prague under the protection of the emperor Rudolph II. Here he resumed his observations, assisted by the illustrious Kepler and Longomontanus. The causes of his exile are involved in mystery; but to whatever it was owing, it turned out fortunate for the progress of astronomy. Had he remained in his island, his observations would not probably have fallen into the hands of Kepler, and the discovery of the laws of the planetary motions might have been deferred to another age. He died at Prague on the 24th of October 1601.

As an indefatigable and skilful observer, Tycho is justly considered as far superior to any astronomer who had preceded him since the revival of the science in Europe. His ample fortune gave him the means of procuring the best instruments which the age could produce; and by his ingenuity and persevering application, he was admirably qualified to employ them to the best advantage. He computed the first table of refractions, and if it extended only to 45°, the reason was, that the effects of refraction, at a higher altitude, were altogether insensible to his instruments. His solar tables were brought to so great a degree of exactness, that he affirms he could never detect an error in them exceeding a quarter of a minute; but there is reason to suspect some exaggeration in this statement, particularly as Cassini, a century after, with much better means, could scarcely answer for errors of a whole minute. He contributed greatly to the improvement of the lunar tables, and detected a considerable inequality in the moon's motion in longitude, to which he gave the name of the Variation, by which it has ever since been distinguished. He also discovered an equation in latitude similar to the evection which had been observed by Hipparchus, and fixed its amount with great accuracy. He remarked the fourth inequality of the moon in longitude, although he failed in his attempt to ascertain its amount, or assign its law. He represented the inequalities of the motions of the nodes, and in the inclination of the lunar orbit, by the motion of the pole of that orbit in a small circle round the pole of the ecliptic. He demonstrated that the region of the comets is far beyond the orbit of the moon, and determined the relative and absolute positions of 777 fixed stars with a scrupulous attention, which gave his catalogue an immense superiority over those of Hipparchus and Ulugh Beigh; and he left to his successors a regular series of observations on the planets, amassed for the purpose of establishing the truth of his own system, but of which Kepler made a better use by employing them to establish the system of Copernicus.

These are some of the important benefits which resulted to astronomy from the labours of Tycho. As a philosopher he ranks low. Alchemy and judicial astrology, in the reveries of which he was a firm believer, engrossed as much of his attention as astronomy. Yet his errors, or rather weaknesses, ought to be viewed with indulgence. He was seated, to use the simile of Bailly, on the confines of two ages; partaking of the darkness which preceded, and the light which came after him. He rejected the simple system of Copernicus, and, whether from participating in the religious scruples of his age, or from the ambition of appearing as the author of a new system, he restored to the earth its immobility, and placed it in the centre of the motions of the sun and moon. The Tycho- nic system was an unsuccessful attempt to reconcile the incongruous hypotheses of Ptolemy and Copernicus. It never enjoyed any real estimation; and its followers were only found among those who dreaded the anathemas of the church, or who, belonging to some university or religious corporation, were deprived of the liberty of expressing their real opinions.

Tycho was assisted in his observations at Huen, during eight years, by Longomontanus, who afterwards became professor of the higher mathematics at Copenhagen. This astronomer composed a large work, entitled Astronomia Danica, in which he deduced the elements of the different theories from the observations made at Uranibourg, and gave formulae for computing the planetary motions, according to the three systems of Ptolemy, Copernicus, and Tycho.

The great mass of accurate observations accumulated by Tycho furnished the materials out of which his disciple Kepler may be said to have constructed the edifice of the world. This great man, the true founder of modern astronomy, was born at Wiel, in the kingdom of Wurtemberg, on the 27th of December 1571. He studied philosophy at Tubingen, and was instructed in mathematics and astronomy by Mestlin, whose name deserves a place in the history of science, on account of his having been one of the first who had the courage to adopt and to teach the system of Copernicus. The philosophical mind of Kepler, disgusted with the improbabilities and absurdities of the ancient system, received with transport the novel doctrines explained by Mestlin. The appointment of mathematician to the emperor, which he procured on the death of Studius, confirmed him in the resolution which he had taken to devote himself to astronomical pursuits; and the energy of his character enabled him in a very short time to make himself thoroughly master of the different hypotheses and principal discoveries which had been made prior to his time. In the year 1596 he published a Prodromus of Dissertations on the properties and causes of the celestial orbits, which procured him the friendship of Tycho, and an invitation to take part in the observations and researches of that great astronomer at Prague. On the death of Tycho, which happened soon after, Kepler obtained possession of his invaluable collection of observations, and was charged with the task of completing and publishing the Rudolphine Tables.

During his short residence with Tycho, Kepler learned to check the fanciful suggestions of his brilliant imagination, and to draw his conclusions from observations alone, by rigorous and patient induction. The observations of the Danish astronomer had furnished him with the means of establishing with certainty the truth or inaccuracy of the various hypotheses which he successively imagined; and the diligence with which he laboured in comparing and calculating these observations during 20 years, was finally rewarded by some of the most important discoveries which had yet been made in astronomy. Deceived by an opinion which had been adopted by Copernicus, and had never been called in question by the ancients, that all the celestial motions are performed in circles, he long fruitlessly endeavoured to represent, by that hypothesis, the irregular motions of Mars; and after having computed with incredible labour the observations of seven oppositions of that planet, he at length succeeded in breaking down the barrier which had so long obstructed the progress of knowledge, and found that the motions could only be accurately represented by supposing the planet to move History in an ellipse, having the sun in one of its foci. Having arrived at this important result, he next proceeded to consider the angular motion of the planet, and finding that it was not uniform in respect of any point situated within the orbit, he concluded that the uniform motion, till then universally received as an axiom, was a vain chimera, which had no existence in nature. He perceived, however, that the areas described by the radius vector of the planet, at its greatest and least distances, were equal in equal times; and subsequent observations enabled him to demonstrate that this equality extended to every point of the orbit. It was therefore discovered that Mars moves in an elliptic orbit, of which the sun occupies a focus, and in such a manner, that the area described by a line drawn from the centre of the planet to that of the sun is always proportional to the time of description. The same conclusions he found to be true in respect of the orbit of the earth; and therefore he could no longer hesitate to extend them by analogy to the other planets. These are two of the three general principles which are known by the name of the Laws of Kepler.

It was some years later before Kepler arrived at the knowledge of the analogy which subsists between the distances of the several planets from the sun, and the periods in which they complete their revolutions. To the discovery of this analogy he attached the greatest importance, and regarded all his other labours as incomplete without it. After having imagined numberless hypotheses, it at last occurred to him to compare the different powers of the numbers which express the distances and times of revolutions; and he found, to his infinite satisfaction, that the squares of the periodic times of the planets are always in the same proportion as the cubes of their mean distances from the sun. This is the third law of Kepler. He demonstrated it to be true of all the planets then known. It has been found to be equally true in regard to those which have been since discovered, and likewise to prevail in the systems of the satellites of Jupiter and Saturn. It is indeed, as can be shown mathematically, a necessary consequence of the law of gravitation, directly as the masses, and inversely as the squares of the distances.

By these brilliant discoveries, the solar system was reduced to that degree of beautiful simplicity which had been conceived by Copernicus, but from which that great astronomer had found himself constrained to depart. The sun could not occupy the common centre of the circular orbits, but his place is in the common focus of the elliptic orbits of all the planets; and it is to this focus that every motion is to be referred, and from which every distance is to be measured. The discovery of the elliptic motion, of the proportionality of the areas to the times, and the method of dividing an ellipse, by straight lines drawn from the focus to the periphery, into segments having a given ratio, formed the solution of a problem which had been the constant object of the labours of all astronomers from Ptolemy to Tycho, namely, to assign the place of a planet at any instant of time whatever. The tables which he computed for the elliptic motions form the model of those in present use. Some additions have been made in consequence of the perturbations, which the geometry of Kepler was inadequate to estimate, and which were only partially detected by the genius of Newton. It has been considered matter of surprise that Kepler did not think of extending the laws of the elliptic motion to the comets. Prepossessed with the idea that they never return after their passage to the sun, he imagined that it would only be a waste of time to attempt the calculation of the orbits of bodies which had so transitory an existence. He supposed History: the tail to be produced by the action of the solar rays, which, in traversing the body of the comet, continually carry off the most subtle particles, so that the whole mass must be ultimately annihilated by the successive detachment of the particles. He therefore neglected to study their motions, and left to others a share of the glory resulting from the discovery of the true paths of the celestial bodies.

The observations of eclipses had formed the principal object of the earliest astronomers, but it was Kepler who first showed the practical advantages which may be derived from them, by giving an example of the method of calculating a difference of meridians from an eclipse of the sun. The method extends to occultations of the stars, and is deservedly considered as the best we possess for determining geographical longitudes and correcting the tables. He composed a work on optics, replete with new and interesting views, and gave the first idea of the telescope with two convex glasses, which has since been advantageously substituted for that of Galileo. Prompt to seize every happy idea of his contemporaries, he perceived with delight the advantages which practical astronomy would derive from the new invention of the logarithms, and he immediately constructed a table, from which the logarithms of the natural numbers, sines, and tangents could be taken at once.

as not merely an observer and calculator; he inquired with great diligence into the physical causes of every phenomenon, and made a near approach to the discovery of that great principle which maintains and regulates the planetary motions. He possessed some very sound and accurate notions of the nature of gravity, but unfortunately conceived it to diminish simply in proportion to the distance, although he had demonstrated that the intensity of light is reciprocally proportional to the surface over which it is spread, or inversely as the square of the distance from the luminous body. In his famous work *De Stella Martis*, which contains the discovery of the laws of the planetary motions, he distinctly states that gravity is a corporeal affection, reciprocal between two bodies of the same kind, which tends, like the action of the magnet, to bring them together, so that when the earth attracts a stone, the stone at the same time attracts the earth, but by a force feebler in proportion as it contains a smaller quantity of matter. Further, if the moon and the earth were not retained in their respective orbits by an animal or other equilibrant force, the earth would mount towards the moon one fifty-fourth part of the interval which separates them, and the moon would descend the fifty-three remaining parts, supposing each to have the same density. He likewise very clearly explains the cause of the tides in the following passage. "If the earth ceased to attract its waters, the whole sea would mount up and unite itself with the moon. The sphere of the attracting force of the moon extends even to the earth, and draws the waters towards the torrid zone, so that they rise to the point which has the moon in the zenith." It is not difficult to imagine how much these views must have contributed to the immortal discovery of Newton.

It is afflicting to consider how frequently the just rewards of true merit are usurped by charlatanism and pretension. While the fire-eaters and astrologers of Rudolph were basking in the sunshine of imperial favour, Kepler, from whose labours the sciences derived the most signal benefits, passed his life in extreme indigence. Born without fortune, the only revenue he possessed, and out of which he had to maintain a numerous family, arose from the precarious produce of his writings, and his pension of mathematician to the emperor—a pension which, owing to the calamities of the times, was seldom duly paid.

On this account he was obliged to prefer frequent solicitations, and undertake long journeys, whereby he lost his time, always precious to genius, and exhausted his mind in anxiety. He died on the 15th of November 1630, at Ratisbon, whither he had gone to solicit the arrears that were due to him. In the present century a marble monument has been erected to his memory by an enlightened prince, Charles of Alberg. It contains his bust and the eclipse of Mars; a monument more glorious and more imperishable than brass or marble.

Contemporary with Kepler was the illustrious Galileo, whose discoveries, being of a more popular nature, and hence far more striking and intelligible to the generality of mankind, had a much greater immediate effect on the opinions of the age, and in hastening the revolution which was soon about to change the whole face of physics and astronomy. Galileo-Galilei, a Florentine patrician, was born at Florence in the year 1564. He passed his youth at Venice, where he continued till he was appointed to a professor's chair at Padua. After a residence of eighteen years in that city, he was induced to remove to Pisa by Cosmo II, who assigned him a pension, and conferred on him the title of his first mathematician. While residing at Venice, he heard it reported that Metius, a Dutch optician, had discovered a certain combination of lenses, by means of which distant objects were approximated to the sight. This vague and scanty intelligence sufficed to excite the curiosity of Galileo, who immediately set about inquiring into the means whereby such an effect could be produced. His researches were attended with prompt success, and on the following day he had a telescope which magnified about three times. It was formed by the combination of two lenses, a plano-convex and plano-concave, fitted in a leaden tube. In a second trial he obtained one which magnified seven or eight times; and subsequent essays enabled him to increase the magnifying power to 32 times. On directing his telescope to the moon, he perceived numerous inequalities on her surface, the diversified appearances of which led him to conclude almost with certainty that the moon is an opaque body similar to the earth, and reflecting the light of the sun unequally, in consequence of her superficial asperities. The planet Venus exhibited phases perfectly similar to those of the moon. These phases had been formerly announced by Copernicus as a necessary consequence of his system; and the actual discovery of their existence made it impossible to doubt of the revolution of Venus round the sun. He detected the four satellites or moons of Jupiter, and, in honour of his patron, gave them the name of the Medicean Stars. The discovery of these little bodies circulating round the huge orb of Jupiter afforded him a strong analogical proof of the annual revolution of the earth, accompanied by its moon. He perceived spots on the disk of the sun, from the motions of which he concluded the rotation of that body about its axis in the space of 27 days. The singular appearances of Saturn were beheld by him with no less pleasure than astonishment. His telescope was not sufficiently powerful to separate the ring from the body of the planet; and to explain the appearances he supposed Saturn to be composed of three stars almost in contact with one another. These discoveries proved that the substances of the celestial bodies are similar to that of the earth, and demolished the Aristotelian doctrine of their divine essence and incorruptible nature. They enlarged the ideas of mankind respecting the planetary system, and furnished the most convincing arguments in favour of the doctrines of Copernicus.

The discoveries of Galileo excited the envy of his Astronomy.

History contemporaries, and stirred up against him a persecution which embittered the last days of his life. The motion of the earth, which he had proved so triumphantly, was considered contrary to many express declarations of Scripture; it was also considered as a heresy in the schools, where the doctrines of Aristotle were followed with implicit submission. In defending the Copernican system, Galileo had incurred the bitterest resentment, both of theologians and peripatetics. His great reputation, his title of professor and first mathematician, gave them reason to dread that the new doctrines, recommended by such an advocate, would spread too rapidly, and finally overthrow the altars of Aristotle. They therefore combined to check their progress, and to procure the ruin of Galileo by injurious representatives to the Grand Duke and the court of Rome. Soon after his first telescopic discoveries he was cited to appear before the Inquisition, and a promise extorted from him that he would never, either by word or writing, support the opinion of the motion of the earth. But in a great mind the love of truth is the most imperious of all passions, and cannot be restrained by the frowns of despotism, or the persecution of a fanatical tribunal. The evidences of the motion of the earth burst forth from every point of the heavens; and Galileo, in his celebrated Dialogues on the system of the world, exposed them in so clear and forcible a manner, that, although he appeared to put them forth for the purpose of refuting them, it was not difficult to perceive that he regarded them as complete and indubitable. For this relapse into heresy he was again brought before the tribunal of the Inquisition, and, in the seventieth year of his age, condemned formally to retract and abjure the doctrine of the earth's motion, and to be imprisoned during the pleasure of the Inquisition. This scandalous proceeding has called forth the indignant reprobation of every lover of truth and freedom. "What a spectacle," exclaims Bailly, "an old man, whose hairs were blanched with study, watchings, and benefits to mankind, on his knees before the sacred Scriptures, abjuring the truth in the eyes of Italy, which he had enlightened, in opposition to the testimony of his conscience, and in spite of the manifestations given by nature through all her works."

The sentence passed on Galileo was not inflicted with great severity. At the end of a year the Grand Duke had the influence to procure his release from prison; but he was prohibited from returning to Florence, and obliged to confine himself to the Tuscan territory. He retired to the village of Arcetri, where he resumed his observations, and shortly afterwards discovered the libration of the moon. The satellites of Jupiter continued also to engage his attention, and he commenced a table of their motions, and pointed out the method of determining the longitude by means of their eclipses. The states of Holland, aware of the great benefit of his researches to commerce and navigation, sent two astronomers, Hortensius and Blaeu, to present him with a gold chain, and encourage him to persevere in his useful labours. A short time after receiving this honourable mark of esteem from a foreign country, Galileo suddenly became blind; and the task of forming the tables of the satellites was reserved for Cassini. He survived this misfortune only a few years, and expired in 1642, in the seventy-eighth year of his age.

Science is indebted to Galileo for two other discoveries of a different kind, less brilliant perhaps, but of far greater importance than those which we have yet enumerated. These are the isochronism of the vibrations of the pendulum, and the law of the acceleration of falling bodies. His telescopic discoveries could not have remained long unknown; in fact, with the exception of those of the phases of Venus, and of the triple form of Saturn, they were all History, fiercely disputed, even during his own lifetime. It is now universally admitted that he was the first who discovered the satellites of Jupiter, and the spots of the sun; but the very circumstance of other claimants to these discoveries having arisen, proves that they were within the reach of ordinary attention. No one ever thought of disputing with Kepler the discovery of the laws of the planetary motions. Those of Galileo required only eyes, and may be regarded as following of course from the discovery of the telescope; but his persecution, his condemnation, and his being compelled to retract and abjure a doctrine of which he had given the physical proof, inspire a hallowed interest in his history, and contributed much to the great reputation which he acquired throughout Europe.

While astronomy was making these rapid advances in Logarithms, the hands of Kepler and Galileo, an event occurred in 1614, which contributed, though less directly, no less powerfully, to the acceleration of its progress. This was the invention of the logarithms by Lord Napier, baron of Merchiston; "an admirable artifice," says Laplace, "which, by reducing to a few days the labour of many months, doubles the life of the astronomer, and spares him the errors and disgust inseparable from long calculations; an invention of which the human mind has the more reason to be proud, insomuch as it was derived exclusively from its own resources." It may be added, that without this, or some equivalent artifice, the computations rendered necessary by more correct observations would far exceed the limits of human patience or industry, and astronomy could never have acquired that precision and accuracy by which it is now distinguished above all the other branches of human knowledge.

The same epoch presents to us a great number of excellent observers, who, although they did not produce any different revolution in the state of astronomy, still rendered it useful service. Scheiner is celebrated for his observations on the solar spots, and his disputes with Galileo. John Bayer of Augsburg published a description of the constellations, accompanied by maps, in which the stars are marked by a Greek letter; a simple idea, which has been universally adopted. Lansberg, a Flemish mathematician, published in 1632, a set of astronomical tables, which, though filled with inaccuracies, rendered good service to science by apprising Horrox of the transit of Venus over the sun's disk, which that young astronomer and his friend Crabtree had the satisfaction of observing on the 24th of November 1639. They were the first who ever witnessed that interesting but rare phenomenon. Snellius is celebrated for his measure of the earth. Gassendi, who had the merit, along with Descartes, of hastening the downfall of the Aristotelian philosophy in France, made some useful observations, particularly one of a transit of Mercury in 1631. His works, which fill six folio volumes, abound with curious and useful researches. Riccioli, a Jesuit, born at Ferrara in 1598, contributed to the progress of astronomy, not so much by his own discoveries, as by collecting and rendering an account of those of others. He rejected the system of Copernicus, and was more zealous in maintaining the doctrines of the church than in investigating nature; but his works form a vast repertory of useful information. His Novum Almagestum is a collection of the observations, opinions, and physical explanations of the phenomena, together with all the methods of computation then known. He was assisted in his labours by Grimaldi, who discovered the inflection of light, and gave the names to the principal spots of the moon which are now used by astronomers. The most accurate observations that were ever made prior to the adaptation of the telescope to astronomical instruments were those of Hevelius, a rich citizen of Dantzig, who devoted his life and a large fortune to the service of astronomy. Having fitted up an observatory, and furnished it with the best instruments which could be procured, he commenced a course of observations, which he followed assiduously upwards of forty years. In his *Selenographia* he has given an accurate description of the face and spots of the moon, accompanied with excellent delineations of her appearance in her different phases and librations. The idea of making drawings of the different phases of the moon had previously occurred to Gassendi and Peiresc; but they had not been able to execute the project; indeed the difficulty attending it was such, that it occupied Hevelius, who was an excellent draughtsman, as well as observer, during a great number of years.

Hevelius made an immense number of researches on comets; and finding that the observations could not be represented by rectilinear or circular orbits, he supposed them to move in parabolas. During a temporary absence from Dantzig, this indefatigable astronomer had the misfortune to lose, in a great fire which occurred in the city, his observatory, instruments, manuscripts, and almost the entire edition of the second volume of his *Machina Coelestis*, which contained the results of his long labours and numerous observations. He was now in his old age, but his zeal did not give way under the terrible calamity. He patiently recommenced all his calculations, reconstructed tables of the sun, and prepared for publication his *Firmamentum Sobiescianum*, or celestial chart, which did not appear till after his death. Towards the latter part of his life, the use of telescopic sights began to be generally adopted. Hevelius, however, resisted the innovation, and continued to employ plain sights. This preference given to the ancient method by so skilful an observer induced Dr Halley to visit him at Dantzig; for the purpose of ascertaining, by a comparison of observations made at the same time and place, which of the two methods gave the most correct results. Dr Halley observed with the telescope, and Hevelius with his own instruments; but such was the dexterity he had acquired through long practice, that the difference of their observations seldom amounted to more than a few seconds, and in no case to so much as a minute. Notwithstanding this agreement, it is to be regretted that Hevelius did not adopt the new method; for, on account of the greater precision given to instruments by the use of the telescope, his observations, which were made without it, cannot now be admitted in the construction of tables, and consequently are for the most part useless to astronomy.

Few individuals have rendered more important services to science than Huygens. Born at the Hague in 1629, he studied geometry under Schooten, the commentator of Descartes, and gave early proofs of proficiency in that science by a treatise on the quadrature of the conic sections. Having passed into France, he studied law at the university of Angers; but his principal attention was directed to the physical sciences, particularly to optics. He employed himself in grinding and polishing lenses; and constructed a telescope of ten feet, with which he discovered one of the satellites of Saturn. His application of the pendulum to clocks deserves to be considered as one of the best gifts which genius has ever conferred on science. He seems to have conceived the idea of this application in 1656; and he presented the first description of the pendulum clock to the states of Holland in 1657. He endeavoured to make this invention subservient to the problem of the longitudes; and if his efforts were not attended with the desired success, it may be said, that without another invention, in which also he had a principal share, that of the spiral spring, the object would never have been accomplished so nearly as it was a hundred years later. By means of his excellent telescopes he discovered that the extraordinary appearance exhibited by Saturn was occasioned by a ring surrounding the body of the planet, and inclined to the ecliptic in an angle which he estimated at 21°. He published his observations on this planet in a work entitled *Systema Saturnium*, which still shows some traces of that species of reasoning from final causes which so greatly disfigures the writings of Kepler. For example, on discovering the satellite, he conceived that as the number of satellites now equalled the number of planets, it was in vain to look for more, the equality being necessary to the harmony of the system. He lived, however, to witness the discovery of four more satellites belonging to the same planet. Huygens was invited to settle in France by Colbert, the patriotic minister of Louis XIV., who assigned him a pension and a seat in the academy of sciences. He continued in that country till the revocation of the edict of Nantz in 1681, when he resigned his pension and retired to Holland. After this he contributed several papers to the *Philosophical Transactions*, and in 1690 published a treatise on light and gravitation. Geometry, mechanics, and optics, are indebted to the genius of Huygens for many important discoveries. His theorems on central forces, his researches on the doctrine of probabilities and continued fractions, and his theory of involutes and evolutes, raise him to the highest rank among the mathematicians of his age. He died on the 8th of June 1695, at the age of 66 years.

The application of telescopes and micrometers to graduated instruments forms an important epoch in the history of astronomy. This happy improvement was first brought into use by Picard in 1667. Morin, indeed, had applied a telescope to the quadrant so early as 1634, and perceived the stars in full day in 1635. In 1669 Picard began to observe the stars on the meridian in the daytime, with a quadrant, to which, in concert with Azoult, he had applied an astronomical telescope having cross wires in its focus. Huygens invented the plate micrometer in 1650; Malvasia that with the fixed wires in 1662; and Azoult that with the movable wire in 1666. (See Delambre, *Astronomie du Moyen Age*, p. 618. Note by Bouvard.) It is principally to the happy discoveries and ingenious inventions just referred to, and the fine application of the pendulum to clocks by Huygens in 1656, that we must attribute the rapid progress since made in practical astronomy, and the extreme precision of modern observations. Picard was also the first who introduced the modern method of determining the right ascensions of the stars, by observing their meridional passages, and employed the pendulum for that purpose. He likewise introduced the method of corresponding altitudes, and is entitled to be regarded as the founder of modern astronomy in France.

Roemer, the friend and pupil of Picard, discovered the progressive motion of light in 1673, and measured its velocity by means of the eclipses of Jupiter's satellites. He was the first who erected a transit instrument, which gave a new accuracy to observations of right ascension.

The Royal Observatory of Paris was completed in 1670, and its direction intrusted to Dominic Cassini. This celebrated astronomer was born at Perinaldo, in the county of Nice, and educated in a college of the Jesuits at Genoa. He acquired an early passion for astronomical observations; and in 1644 was invited to Bologna by the marquis Malvasia, where, in 1650, he succeeded Cavalleri. Cassini enriched astronomy with a great number of curious observations and discoveries. He determined the motions of Jupiter's satellites from observations of their eclipses, and constructed tables of them, which were found to be remarkably exact. He observed that the ring of Saturn is double, and discovered four of the satellites of that planet. He also determined the rotation of Jupiter and Mars, and made a number of observations on Venus with the same view. He observed the zodiacal light, and made a near approximation to the parallax of the sun. We also owe to him the first table of refractions, calculated on correct principles; and a complete theory of the libration of the moon. Galileo had only observed the libration in latitude; Hevelius explained the libration in longitude, by supposing that the moon always presents the same face to the centre of her orbit, of which the earth occupies a focus. Cassini made the important remark, that the axis of rotation of the moon is inclined to the ecliptic, and that its nodes coincide with those of the lunar orbit, so that the poles of the orbit, ecliptic, and equator of the moon, are on the same circle of latitude, the pole of the ecliptic being situated between the other two. The greater number of these discoveries are, however, only of secondary importance; and it must be confessed that Cassini took no part in the great and permanent improvements which astronomy received in that age. He has, nevertheless, obtained an extraordinary reputation. Lalande remarks, that in his hands astronomy underwent the most signal revolutions, and that his name is, in France, almost synonymous with that of creator of the science. Delambre has, however, taken a different and far more accurate view of the real services of Cassini. "The revolution in Astronomy," this judicious critic observes, "was brought about by Copernicus, by the laws of Kepler, by the pendulum of Huygens, the micrometers of Azoult and Picard, by the sectors and mural of Picard, and his method of corresponding altitudes, by the transit instruments of Roemer; and Cassini appears to us an entire stranger to all these innovations. He followed another route; he devoted a long life to painful observations, which at last deprived him of sight. Let us not refuse him the praise which he has so well merited, but let us reserve a place in our esteem for labours less brilliant perhaps, but of greater and more permanent utility, and which evince at least equal talent and sagacity."

Cassini was assisted in his observations by his nephew, James Philip Maraldi, who determined the regression of the nodes, and the progressive motion of the apsides of the orbit of Jupiter. This astronomer also corrected the theory of Mars, and observed the sun's parallax. He rejected the hypothesis of the progressive motion of light, as being insufficient to explain the inequalities of Jupiter's satellites; and he conceived the design of forming a new catalogue of the stars, which, however, was never executed. He died in 1729.

There is no period in the history of mankind so distinguished by great and important discoveries, or so remarkable for the rapid development of the human intellect, as the seventeenth century. We have already noticed the invention of the pendulum, and its application to regulate the motion of time-keepers; of the telescope, and some of the phenomena of the new worlds it has exposed to view; of the logarithms, by which computations are so much abridged; and of the mechanical contrivances for measuring minute angles in the heavens. The same century witnessed the application of algebra to geometry, the discovery of the laws of the planetary motions, of the infinitesimal calculus, the acceleration of falling bodies, the sublime theory of central forces, and the great principle of gravitation which connects the celestial orbs, and regulates the motions which it had been the business of the astronomer to observe since the earliest ages of the world. The different steps which conducted to this important discovery, and the immediate consequences deduced from it by its immortal author, are so fully developed in the admirable Dissertation on the Progress of the Physical Sciences, in this Encyclopedia, that it is unnecessary to enter into any detail respecting them here; we may only remark, that if observation has furnished the data for the discovery of the mechanical principle and primordial laws of the universe, the knowledge of these laws has been, in turn, of the most essential service to observation, by guiding and directing it to its most important objects. Many of the inequalities of the planetary motions, in consequence of their minuteness and the slowness with which they vary, could not have been detected by observation; others might perhaps have been perceived, but we should still have been ignorant whether their constant accumulation might not ultimately change the state of the system, and, by destroying all confidence in the tables, demolish the fabric which had been reared at such a vast expense of time and labour. But when these inequalities are detected by theory, and separated from the mean motions with which they were blended, it becomes an object of the highest interest to confirm their existence by the most delicate and accurate observations. Hence, a more refined practice has constantly followed every theoretical discovery. Besides, it is the perfection of theory, and not the mere knowledge of isolated facts, which gives astronomy its greatest value in the eyes of the philosopher. Numerous and important as its applications are, they have but a subordinate interest, in comparison of the knowledge of those general laws to which every particle of matter in the universe is subject, and by the discovery of which man has penetrated so deeply into the mysteries of nature.

By the discovery of the law of gravity Newton laid the foundations of physical astronomy; and by the consequences which he deduced from that law, proceeded far in the erection of the superstructure. He showed that the motions of all the bodies of the planetary system are regulated by its influence; he determined the figure of the earth on the supposition of its homogeneity; he gave a theory of the tides, discovered the cause of the precession of the equinoxes, and determined some of the principal lunar inequalities and planetary perturbations. Many of his theories were left in an imperfect state; for it is not in matters of science that it is given to the same individual to invent and bring to perfection; their complete development required that several subsidiary sciences should be farther advanced; but it has been the triumph of his system, that every subsequent discovery has only tended to strengthen and confirm it. This bright ornament of the human genius was born on the 25th of December 1642, the day of the death of Galileo, and died on the 20th of March 1727, in the 85th year of his age.

While physical astronomy was undergoing a complete revolution in the hands of Newton, the practical part was born 1646, receiving great improvement from Flamsteed, the first astronomer royal, who conducted the Greenwich Observatory. This celebrated institution, from which so many important discoveries have emanated, was erected under the reign of Charles II. in 1675. Flamsteed was appoint- History ed to it in 1676, and continued with indefatigable zeal to discharge the duties of the office during the long period of 33 years. In the course of this time he made an immense number of excellent observations, the results of which are given in the Historia Cælestis, the first edition of which was published in 1712, at the expense of Prince George of Denmark, the husband of Queen Anne. The second appeared in 1723, some time after the death of the author, in three volumes folio. The first volume contains the observations which he made, first at Derby, and afterwards at Greenwich, on the fixed stars, planets, comets, spots of the sun, and Jupiter's satellites. The second volume contains the transits of the planets and stars over the meridian, and the places of the planets deduced from these observations. The third contains an historical notice, in which he gives a description of the instruments used by Tycho and himself; catalogues of fixed stars by Ptolemy, Ulugh Beigh, Tycho, the landgrave of Hesse, and Hevelius; together with the British Catalogue, containing the places of 2884 stars. The labours of Flamsteed were, however, confined entirely to the practical part of astronomy. He made no improvements in theory; but he is entitled to the merit of having been the first who brought into common use the method of simultaneously observing the right ascension of the sun and a star, a method by means of which the determination of the positions of the stars is reduced to the observation of meridional transits and altitudes. He was likewise the first who explained the true principles of the equation of time; and he improved the lunar tables by introducing into them the annual equation which had been suggested by Horrox. The Atlas Cælestis, another posthumous work of Flamsteed, was published in 1753.

as succeeded in the observatory by Dr Halley, a philosopher whose inventive genius and indefatigable activity rendered him one of the brightest ornaments of his country. Halley was the son of a wealthy citizen of London, where he was born on the 8th of November 1656. From his earliest years he applied himself with ardour to the study of mathematics and astronomy; and having procured a few instruments, he began to make observations, by which he was led to remark the inaccuracy of the tables of Jupiter and Saturn. In his 19th year he published a direct and geometrical method of finding the eccentricities and aphelia of the orbits of the planets; and in the year following he undertook a voyage to St Helena, with a view to form a catalogue of the stars in the southern hemisphere. The station was unfortunately chosen, for, owing to the incessant rains and foggy atmosphere of that island, he was able to determine the places of only 360 stars in the course of a whole year. He had, however, the satisfaction of observing a transit of Mercury over the sun's disk, a phenomenon which suggested to him the important remark, that the transits of the inferior planets might be advantageously employed in determining one of the most essential elements of the planetary system, viz. the parallax of the sun, and consequently the diameters of the orbits. The method has since been successfully employed in the case of Venus: the transits of Mercury, though much more frequent, are not so well adapted to the purpose. On his return from St Helena he was commissioned by the Royal Society to visit Hevelius at Dantzig, and determine, by a direct comparison of observations, the dispute which had arisen between that astronomer and Dr Hooke respecting the relative advantages of plain and telescopic sights. After this Halley for some time travelled on the Continent, and on his return to England devoted himself entirely to scientific pursuits. The fruits of his leisure soon began to appear in the multitude of treatises which he published from time to time in every department of the mathematical and physical sciences. The most important of those connected with astronomy was his Synopsis Astronomica Cometicae,—a work abounding in profound and original views, and which, in respect of theory, formed perhaps the most remarkable accession to the science that had been made since the time of Kepler. In this work he revived an ancient opinion, that the comets belong to the solar system, and move in very eccentric orbits round the sun, returning after stated but long intervals. He also ventured to predict that the comet of 1681 would again return to its perihelion in 1759,—the first prediction of the kind that was verified. In 1720 Dr Halley was appointed to succeed Flamsteed in the Royal Observatory; and though now in the 64th year of his age, he undertook, with a view to improve the lunar theory, to observe the moon through a whole revolution of her nodes, erroneously supposing, that after such a revolution the errors of the tables would again appear in the same order. He was the first who, by a comparison of ancient and modern observations, remarked the acceleration of the mean motion of the moon, and thus called the attention of mathematicians to an important and curious phenomenon, the physical cause of which was at length detected by the powerful analysis of Laplace. He was also the first who pointed out the secular inequalities of Jupiter and Saturn, occasioned by their mutual perturbations,—a theory that formed the subject of several profound memoirs of Euler and Lagrange, and for the complete development of which astronomy is likewise indebted to Laplace. Besides these important discoveries in astronomy, the labours of Dr Halley also greatly contributed to the promotion of geometry and navigation. He undertook two long voyages, and traversed the Pacific Ocean to observe the deviations of the magnetic needle; and posterity will gratefully recollect that it was through his pressing solicitations that Newton consented to the publication of the Principia. In 1703 he succeeded Dr Wallis as professor of geometry at Oxford. He became secretary of the Royal Society in 1713, and died at Greenwich in 1742.

The discoveries of Bradley, who succeeded Halley as astronomer royal, form a memorable epoch in the history of the science. This great astronomer was born at Sherborne in Gloucestershire in 1692, and acquired from his uncle, Mr Pound, an early taste for astronomical observation. He was destined for the church, and for some time held a curacy, which he resigned in 1721, on being appointed to succeed Keill as Savilian professor of astronomy at Oxford. His first essays indicated no extraordinary talent; but an opportunity having presented itself of engaging in more important researches, he embraced it with ardour; and by his sagacity and perseverance was conducted to two discoveries which have entirely changed the face of astronomy.

A singular motion of the polar star had been observed by Picard, of which, however, that astronomer could neither assign the law nor give any satisfactory explanation. He only remarked that the inequality was annual, and amounted to about 40 seconds. Hooke, in 1674, a few years after the observations of Picard, imagined that he had discovered a parallax in some of the stars; and Flamsteed, following the ideas of Hooke, explained, by means of parallax, the minute changes of position which he had observed in Polaris and some circumpolar stars. Manfredi and Cassini demonstrated the error of Flamsteed, but were not more successful in their attempts to explain the motion in question. Samuel Molyneux conceived the idea of verifying all that had been said respecting the Bradley, anxious to verify his ingenious theory, continued his observations, and soon felt the difficulty that had so much embarrassed Picard. The places of the stars, calculated according to his formula for the aberration, could not be reconciled with the observations. The errors continued to augment during nine years, after which they went on diminishing during the nine years following. This inequality, of which the period, like that of the nodes of the moon, was 18 years, was readily explained by supposing a slight oscillation of the earth's axis, occasioned by the action of the moon on the protuberant parts surrounding the equator of the terrestrial spheroid. After assiduously observing its effects during twenty years, Bradley found that the phenomena could be accurately represented by giving the pole of the equator a retrograde motion about its mean place in an ellipse whose axes are 18° and 16°, and completing its revolution in the period of 18 years. This result was communicated to the Royal Society in 1748. To these two grand discoveries of Bradley, the aberration and nutation, modern astronomy is wholly indebted for all its accuracy and precision; and, as Delambre remarks, they assure to their author a distinguished place, after Hipparchus and Kepler, above the astronomers of all ages and all countries.

Bradley was appointed astronomer royal in 1741, and from this time to the period of his death made an immense number of observations, which were presented by his heirs to the university of Oxford, on condition that they should forthwith be published. The first volume appeared only in 1798, edited by Dr Hornsby. The rest were committed to the care of the late Dr Abraham Robertson, and appeared in 1805. Dr Bradley was chosen a corresponding member of the Academy of Sciences in 1748. In 1752, twenty-four years after his great discovery of the aberration of light, he was admitted into the Royal Society. He died on the 13th of July 1762, in the 70th year of his age.

While England was deriving so much glory from the brilliant discoveries of Bradley, France produced a multitude of excellent astronomers, by whose successful labours every department of the science was signalized promoted. Among these Lacaille was pre-eminently distinguished, both by the variety and importance of his observations, and the indefatigable zeal with which, during twenty-two years, he prosecuted the most laborious researches. He commenced his astronomical career in 1739, by assisting Cassini de Thury (grandson of the first Cassini) in the verification of the measurement of the meridian through France. In 1751 he undertook a voyage to the Cape of Good Hope, the primary objects of which were to determine the sun's parallax, by means of observations on the parallaxes of Mars and Venus, while similar observations were made in Europe; and to form a catalogue of the southern circumpolar stars. No undertaking for the benefit of science was ever more successfully executed. In the course of a single year, Lacaille, without assistance, observed upwards of ten thousand stars, situated between the tropic of Capricorn and the pole, and computed the places of 1942 of them; a labour which would scarcely be credited, if the details of his observations had not been published in the Cæstum Australis Stelliferum, a work which was given to the world in 1763. Our admiration of the rapid execution of this vast undertaking will be increased, when we consider, that during the same time he measured a degree of the meridian, and made numerous observations on the moon simultaneous with those of Lande, who observed at Berlin, in order to determine the moon's parallax, by means of direct observations made at the extremities of a meridional arc of upwards of 85 degrees. Before his return to Europe, he visited the isles of France and Bourbon, where he measured the length of the pendulum, and made numerous remarks on the natural and civil history of those countries. Astronomy is likewise indebted to Lacaille for a table of refractions which he computed from a comparison of above 300 observations made at the Cape and at Paris. In 1757 he published his Astronomie Fundamenta, in which he gave rules and tables for computing the apparent motions of the stars, which continued to be employed till Lambert supplied the corrections depending on the nutation, and Delambre those depending on the aberration. To defray the expense attending the publication of this important work, which he distributed in presents to the different observatories and astronomers in Europe, he submitted to the drudgery of calculating ephemerides during ten years. Lacaille composed several elementary works for the use of the students in the college of Mazarin, where he occupied the chair of astronomy, and inserted a great number of memoirs in the volumes of the Academy of Sciences. This great astronomer, distinguished as much by the excellence of his moral qualities as his profound knowledge and indefatigable zeal, died suddenly in 1762.

The Royal Observatory of Paris continued under the Cassini II. direction of the family of Dominic Cassini during 120 years. James Cassini, the second of that name, is principally known by a work on the magnitude and figure of the earth, and his Elements of Astronomy. He seems not to have duly appreciated the new discoveries which were daily making around him. His Elements, published in 1740, contains no mention of the aberration; and he adopted the opinion that the earth is elongated instead of being flattened at the poles. His son, Cussini de Thury, was chiefly occupied with the meridian, and the geometrical survey of France. The last astronomer of the family, the Count de Cassini, was obliged to resign the observatory at the revolution.

The question of the figure of the earth furnished ample materials for the practical as well as the speculative as... The results of the measurement of the meridian by Cassini were at variance with the theories of Newton and Huygens; and the Academy of Sciences resolved on making a decisive experiment by the actual measurement of the lengths of two degrees, one at the equator, and another in as high a latitude as could be reached. In the year 1735, three astronomers, Godin, Bouguer, and La Condamine, were commissioned by the French government to accomplish the first of these objects in Peru; and the year following, Maupertuis, Clairaut, Camus, and Lemonnier, went to Lapland to execute the second under the polar circle. Notwithstanding the greater difficulties they had to contend with, the first party were the most successful; but the result of both operations established the compression of the earth at the poles. Bouguer published the details of the Peruvian measurement in an admirable work on the Figure of the Earth, in which he also inserted an account of a great number of experiments made by him in the same country to determine the length of the seconds' pendulum, and the effects of the attraction of mountains on the plumb-line. Bouguer is likewise the author of an excellent treatise on light.

This accomplished mathematician and experimenter did not adopt the Newtonian theory of gravitation, but he was the last apostle of the Cartesian philosophy in the Academy of Sciences.

It would greatly exceed the limits of this article to give even an abridged account of the numerous observers who, about this period, contributed to the improvement of every department of practical astronomy. We must therefore content ourselves with merely noticing the names of some of the most distinguished, leaving the details of their labours to be given in the biographical articles interspersed throughout the work.

Delisle formed a school of astronomy in Russia, and has left a method of computing the heliocentric places of the sun's spots, and of Mercury and Venus in their transits over the sun's disk, and likewise of determining, by means of the stereographic projection, the directions of their path when they enter and leave the disk. Lemonnier introduced the discoveries of Bradley into his Astronomical Institutions, and was the instructor of Lacaille and Lalande. He published a Histoire Céleste, containing a collection of observations from 1666 to 1685, and a number of other works and memoirs connected with astronomy. He made an immense number of observations, but their accuracy is far inferior to those of Bradley. Wargentin, secretary of the Academy of Sciences of Stockholm, devoted the whole of his life to the correction of the tables of the satellites of Jupiter. The theory of the satellites was not then far advanced; but when theory failed him, he profited by the remarks of others and by his own reflections, and endeavoured by repeated trials to find empirical equations capable of conciliating the tables with the best observations. By confining himself almost exclusively to this subject, he acquired a high reputation, and was ranked among the first astronomers of an epoch which abounds in great names. His tables of the satellites have, on account of their superior accuracy, been employed in determining the masses, and other elements, which serve as the basis of the analytical theories.

One of the most celebrated astronomers of the last century was Lalande. He commenced his early career by a set of lunar observations at Berlin, undertaken simultaneously with those of Lacaille at the Cape, for the purpose of determining the moon's horizontal parallax. On his return to Paris he was received, though only twenty-one years of age, into the Academy of Sciences; and on the resignation of Delisle was appointed professor of astronomy in the college of France. By his extraordinary zeal, indefatigable activity, and the care which he took to have his name constantly before the public, Lalande soon became one of the most distinguished men of his day; but his reputation, acquired in a great measure from attention to subjects which had only an ephemeral interest, and not through any permanent or fundamental additions to science, has already begun to wane; and his works, many of which are of considerable utility, seem to have fallen into unmerited neglect. His character as an astronomer is fairly and impartially summed up by Delambre in the following terms:—"If Lalande did not renew the foundations of astronomical science, like Copernicus and Kepler,—if he did not, like Bradley, immortalize himself by two brilliant discoveries,—if he was not so learned and accurate a theorist as Mayer,—if he was not to the same degree as Lacaille an exact, expert, scrupulous, industrious, and indefatigable observer and calculator,—if he had not the constancy to attach himself, like Wargentin, to a single object, in order to become the first in a particular department,—and if, in all these respects, he was only an astronomer of the second rank,—he was the first of all as a professor. No one ever did more to propagate the knowledge and love of astronomy. It was his object to be useful and celebrated; and he succeeded, through his labours, his activity, his credit, and his solicitations: by means of a most extensive correspondence he incessantly laboured for the benefit of science; he even endeavoured to serve it after his death, by founding a medal, which the Academy of Sciences adjudges annually to the astronomer who has made the most interesting observation, or written the most useful memoir." (Delambre, Astronomie du Moyen Age.)

Bradley was succeeded in the Greenwich Observatory by Dr Bliss, who died in the course of a few years after his appointment. Dr Bliss was in turn succeeded by Dr Maskelyne, the late astronomer royal, under whose care the observatory maintained the high character which it acquired from the immortal labours of his illustrious predecessors. Dr Maskelyne began his astronomical career in 1761, when he was appointed to observe the transit of Venus at the island of St Helena, and endeavour to verify the existence of a small parallax of the star Sirius, which seemed to be indicated by the observations of Lacaille at the Cape. Unfortunately the state of the weather prevented him from observing the transit; and his observations on Sirius were abandoned in consequence of the discovery of a defect in the zenith sector which he had carried out with him for the purpose of making the observations in question. The main objects of his voyage were thus frustrated; but some indirect advantages resulted from it, which compensated in some measure for the disappointment. The attention of Ramsden was called to the sector, and a better method of constructing these instruments was devised. At St Helena he made several interesting observations on the tides, the variation of the compass, the moon's horary parallaxes, &c. In going out and returning home, he paid particular attention to the different methods of finding the longitude at sea, and practised that which depends on observations of the lunar distances from known stars, taken with a Hadley's sextant, or other reflecting instrument. He also composed a set of rules for the use of the seamen, which he published on his return, first in the Philosophical Transactions, and afterwards in the British Mariner's Guide. In the year 1765 he was appointed astronomer royal, and soon after recommended to the Board of Longitude the general adoption in the navy of the lunar method of finding the

as led to conclude that the whole solar system is in motion about some distant centre, and that its direction is at present towards the constellation Hercules—a conclusion, however, which succeeding observations have not verified. His observations on nebulae and double stars have opened up a new field of research, boundless in extent, and interesting by reason of the variety of the objects it presents to the attention of the observer. The extraordinary activity with which he pursued his favourite occupations is attested by 67 memoirs communicated from time to time to the Royal Society. A great part of these, however, is filled with speculations of no value to astronomy; and his taste was rather to observe astronomical phenomena than to engage in computations, or the more arduous and essential, though less fascinating, labours through which the science can be really benefited. In the course of the following chapters we shall have frequent occasion to allude to his discoveries, and his ingenious speculations concerning the constitution of the sun and the sidereal heavens.

Few individuals have contributed so eminently to the perfection of modern astronomy as Delambre, the late perpetual secretary of the Academy of Sciences. The scientific life of this illustrious astronomer did not commence till he had attained his 40th year; but from that time till his death it was occupied by a series of unremitting labours to enlarge the boundaries, and ameliorate the practice and theory of the science. Associated with Mechain, he was employed during the troubles of the revolution in measuring the meridian from Dunkirk to Barcelona—a labour which was prosecuted with admirable zeal in the face of innumerable difficulties, and even dangers of the most formidable kind. By an immense number of excellent observations he determined the constants which enter into the formulae deduced from theory by the profound researches of Lagrange and Laplace, and also formed a set of tables much more exact than any that had appeared before them. His *Astronomie Théorique et Pratique*, in three quarto volumes, contains the best rules and methods which have yet been devised for the guidance of the practical astronomer; and his *Histoire*, in six large volumes 4to, gives an account of every successive improvement which has been made in the science, and a full abstract of every work of celebrity which has been written respecting it, since the first rude observations of the Greeks to the end of the last century. It is invaluable to the historian, and will ever remain a proud monument of the profound learning and laborious research of its author.

The observatory which was established at Palermo Piazzi, about the year 1790, under the active superintendence of Piazzi, holds a distinguished rank among the similar institutions of Europe. Piazzi, born in 1746, took the habit of the religious order of the Theatins at Milan, and finished his noviciate in the convent of St Anthony. Among his preceptors he had the advantage of counting Tiraboschi, Beccaria, Le Scurl, and Jacquier; and from these illustrious masters he speedily acquired a taste for mathematics and astronomy. After filling several professors' chairs in the colleges of the Jesuits at Rome and Ravenna, he was appointed in 1780 professor of the higher mathematics in the academy of Palermo. Here his first care was to reform the general system of education; and, by the alterations which he introduced, he contributed powerfully to dissipate the shades of ignorance, which, under the double influence of the Jesuits and the Inquisition, still lowered over the soil of Sicily. After having rendered this service to literature, he obtained from the prince of Caramanico, viceroy of the island, permission to found an observatory, and undertook a voyage to France and History. England, in order to provide the instruments necessary for the new establishment. Having procured a vertical circle, a transit, and some other instruments from Ramsden, he returned to Palermo and commenced his observations. His first care was to prepare a new catalogue of stars, the exact positions of which he justly considered as the basis of all true astronomy. In prosecuting this object he did not content himself with a single observation, but before he fixed the position of any star, observed it several times successively; and, by this laborious but accurate method, he constructed his first great catalogue of 6748 stars, which was crowned by the Academy of Sciences of France, and received with admiration by the astronomers of all countries. His constant practice of repeating his observations led to another brilliant result, the discovery of an eighth planet. On the 1st of January 1801, Piazzi, searching for the star 87 of the catalogue of Mayer, cursorily observed a small star of the eighth magnitude, between Aries and Taurus. On the following day he remarked that the star had changed its position, and accordingly supposed it to be a comet. He communicated his observations to Oriani, who, seeing that this luminous point had no nebulosity like the comets, and that it had been stationary and retrograded within comparatively small limits like the planets, computed its elements on the hypothesis of a circular orbit. He found that this hypothesis agreed with the observations, and other astronomers soon confirmed its accuracy. He gave the planet the name of Ceres Ferdinandina, in honour of Ferdinand, king of Naples, in whose dominions it was discovered, and who proposed to consecrate the event by a gold medal, struck with the effigy of the astronomer; but Piazzi, nobly preferring the interests of science to vain honours which could add nothing to his glory, requested that the money destined for this purpose should be employed in the purchase of an equatorial, which was still wanting to his observatory. In 1814 he published a new catalogue, extended to 7646 stars,—a splendid monument of indefatigable zeal and activity. He made an uninterrupted series of solstitial observations from 1791 to 1815; for the purpose of determining the obliquity of the ecliptic, which, compared with those of Bradley, Mayer, and Lacaille, in 1750, give a diminution of 44" in a year. Besides these labours, sufficient to occupy a life of ordinary industry, Piazzi composed a number of memoirs for the different societies of which he was a member, and was intrusted by the Neapolitan government with several important commissions respecting the public instruction and the regulation of weights and measures in Sicily. He died in 1826, after having bequeathed his library and all his instruments to the observatory at Palermo, and assigned a liberal annuity to be devoted to the instruction of young men who evince a decided taste for astronomy.

The discovery of Ceres led to that of three other little planets, circulating at nearly the same distance from the sun,—a circumstance unique in the construction of the solar system. On account of the smallness of the new planet, and the nebulosity by which it is surrounded, the difficulty of finding it after it had for some time ceased to be observed, was so considerable, that Dr Olbers of Bremen was induced to examine, with particular care, the configurations of all the small stars situated in the vicinity of its path, in order to have the means of detecting it at any time with greater facility. While engaged in making observations for this purpose, he discovered, on the 28th of March 1802, and nearly in the same place where he had before observed Ceres, another planet similar in size and appearance, to which he gave the name of Pallas. The extraordinary circumstance of the discovery of two planets having nearly the same mean distance, and performing their revolutions nearly in the same time, led Dr Olbers to imagine that Ceres and Pallas were fragments of a larger planet which had revolved in the same place, and been shattered by some external force or internal convulsion. The immediate consequence of this hypothesis was the probability of the existence of other fragments of the original planet, hitherto undiscovered; and that if such fragments existed, the planes of their orbits would pass through the points in which the orbits of Ceres and Pallas intersect each other. This bold idea acquired some probability from the subsequent discovery of two other planets in the very quarter of the heavens to which Dr Olbers had directed the attention of astronomers. Juno was discovered by M. Harding of Lilienthal, on the 2d of September 1804; and Vesta, by Dr Olbers himself, on the 29th of March 1807. Diligent observation has not since added to their number.

Astronomy is a science which borrows the aid of several improvements in other sciences, with which its advancement is simultaneous. Instruments capable of appreciating or measuring the exceedingly minute quantities about which the observer now concerns himself, could only be produced in a very advanced state of mechanics. In the accurate graduation of large instruments, our English artists have long maintained an unrivalled superiority. Graham, the celebrated watch-maker, united to great dexterity in the mechanical arts a decided taste for observation; and the extraordinary improvements which he effected in the art of dividing form an era in the history of practical astronomy. The sector used by Bradley, in the observations which led to the discovery of the nutation, was of his construction, and he was also the inventor of the first compensation pendulum. Sisson, who succeeded him, maintained in this department the honour and pre-eminence of England. Bird was originally a weaver in Durham, and first gave proof of his mechanical genius by dividing dials for the watch-makers in a manner far superior to what had been commonly practised. He was employed by Sisson in the division of mathematical instruments, and recommended by that artist to Graham, who instructed him in his methods. He constructed the 8 feet quadrants employed in the Greenwich and Paris observatories, and another of 6 feet for Mayer at the observatory of Göttingen. Ramsden invented a machine for the more accurate division of instruments, on account of which he received a premium from the Board of Longitude. The mural quadrants of this artist were held in high estimation, and the transit instrument, sextant, and refracting micrometer, in passing through his hands, received considerable ameliorations. His astronomical instruments in general were considered the best that could be procured in Europe. Mr Troughton, who at present stands the foremost of our British opticians, has more than rivalled his predecessors. In the hands of this distinguished artist and astronomer, the division of instruments has been carried to a degree of precision and accuracy which probably will never be surpassed; while the great improvements he has introduced into the methods of constructing and mounting large instruments, and his ingenious inventions to elude natural obstacles, and guard against the accidental derangements from which it is impossible altogether to protect them, entitle him to the high place which he holds by universal consent among the most eminent philosophers and original mechanicians of the age. The numerous circles and other instruments of his manufacture, whether in the hands of private individuals or deposited in the different public observatories, are considered as the finest specimens of Having now endeavoured to give an account of the labours of those astronomers who have principally contributed to make us acquainted with the state of the heavens, and the order and succession of the various phenomena they exhibit, we will conclude this part of the article by briefly adverting to the profound researches of some illustrious mathematicians who have developed the theory of Newton, and raised the fabric of physical astronomy to its present proud elevation.

Although the law of gravitation, as proposed by Newton, had from the first been admitted by all the most eminent astronomers of Britain, it was for a long time either opposed or neglected on the Continent. In fact, great improvements were required both in analysis and mechanics before it admitted of other applications than had been made by its great author, or could be regarded as anything more than a plausible hypothesis. Newton demonstrated, that if two bodies only were projected in space, mutually attracting each other with forces proportional directly to their masses, and inversely to the squares of their distance, they would each accurately describe an ellipse round the common centre of gravity; and the spaces described by the straight line joining that centre and the moving body, would be proportional to the time of description, according to the second law of Kepler. But when it is attempted to apply Newton's law to the case of the solar system, great difficulties immediately present themselves. Any one planet in the system is not only attracted by the sun, but also, though in a greatly smaller degree, by all the other planets, in consequence of which it is compelled to deviate from the elliptic path which it would pursue in virtue of the sun's attraction alone. Now, the calculation of the effects of this disturbing force was the problem which geometers had to resolve. In its most general form it greatly transcends the power of analysis; but there are particular cases of it, and those too the cases presented by nature, in which, by reason of certain limitations in the conditions, it is possible to obtain an approximate solution to any required degree of exactness. For example, the Sun, Moon, and Earth form in a manner a system by themselves, which is very slightly affected by the aggregate attractions of the other planets. In the same way the Sun, Jupiter, and Saturn form another system, in which the motions are very little influenced by the action of any other body. In these two cases, then, the number of bodies to be taken into consideration is only three; and in this restricted form, the problem, celebrated in the history of analysis under the denomination of the Problem of the Three Bodies, is susceptible of being treated mathematically. With the hope of ameliorating the lunar tables, and of completing the investigations which Newton had commenced in the Principia, three distinguished geometers, Clairaut, D'Alembert, and Euler, about the middle of the last century, undertook, simultaneously, and without the knowledge of each other, the investigation of the problem of the three bodies, and commenced that series of brilliant discoveries which it is the glory of our own times to have seen completed.

Clairaut's solution of the problem of the three bodies was presented to the Academy of Sciences in 1747, and was applied to the case of the moon. From this solution he deduced with great facility, not only the inequality of the variation, which Newton had obtained by a more complicated, though at the same time a very ingenious method, but also the evocation, the annual equation, and many other inequalities which Newton had not succeeded in connecting with his theory. It happened, however, curiously enough, that in the calculation of one effect of the disturbing force, namely, the progression of the moon's apogee, Clairaut was led into an error which produced a result that threatened to overturn the system of gravitation. The error consisted in the omission of some of the terms of the series expressing the quantity in question, which he wrongly supposed to have only an insensible value; and by reason of this omission, his first approximation gave only half of the observed progressive motion of the apogee. As this result was confirmed by D'Alembert and Euler, who had both fallen into the same error, it seemed to follow, as a necessary consequence, either that the phenomenon depended on some other cause than the disturbing force of the sun, or that the law of gravity was not exactly proportional to the inverse square of the distance. The triumph which this result gave to the Cartesians was not of long duration. Clairaut soon perceived the cause of his error; and by repeating the process, and carrying the approximations further, he found the computed to agree exactly with the observed progression—a result which had the effect of dissipating for ever all doubt respecting the law of gravity. The researches of Clairaut were followed by a set of lunar tables, much more correct than any which had been previously computed.

The return of the comet of 1682, which Halley had predicted for the end of 1758, or beginning of 1759, afforded an excellent opportunity for putting to the test both the theory of gravity and the power of the new calculus. Clairaut applied his solution of the problem of the three bodies to the perturbations which this comet sustained from Jupiter and Saturn, and, after calculations of enormous labour, announced to the Academy of Sciences, in November 1758, that the comet would return in the beginning of the following year, and pass through its perihelion about the 15th of April. It returned according to the prediction, but passed its perihelion on the 13th of March. The correction of an error of computation reduced the difference to nineteen days; and if Clairaut had been aware of the existence of the planet Uranus, he might have come still nearer the truth.

Besides these important researches on the system of the world, Clairaut composed an admirable little treatise on the figure of the earth, in which he gave the differential equations, till then unknown, of the equilibrium of fluids, whether homogeneous or heterogeneous, supposing an attractive force, following any law whatever, to exist among the molecules. He applied these equations to the earth; demonstrated that the elliptic figure satisfies the conditions of equilibrium; assigned the ellipticity of the different strata of which the earth may be supposed to be formed, together with the law of gravitation at the exterior surface. He likewise discovered the important theorem which establishes a relation between the oblateness of the terrestrial spheroid and the increase of gravitation towards the poles, on every supposition which can be imagined relative to the interior construction of the earth. By means of this theorem the ellipticity of the spheroid is deduced from observations of the lengths of the seconds' pendulum at different points of the earth's surface.

D'Alembert, as has already been mentioned, presented D'Alembert's solution of the problem of the three bodies to the Academy of Sciences at the same time with Clairaut. In the year 1749 he published his treatise on the precession of the equinoxes—a work remarkable in the history of analysis and mechanics. By means of his newly invented calculus of Partial Differences, and the discovery of a fertile principle in dynamics, he determined from theory the rate of the precession, which is nearly 50° in a year. He also determined the nutation of the earth's axis, which had been discovered by Bradley, and assigned the ratio of the axes of the small ellipse which the true pole of the earth describes around its mean place, in the same time in which the nodes of the lunar orbit complete a revolution. The solution of this problem led to the determination of the ratio of the attractive forces of the sun and moon, which D'Alembert found to be that of seven to three very nearly; whence he inferred that the mass of the earth is 70 times greater than that of the moon. He proved likewise that the precession and nutation are the same in every hypothesis concerning the interior constitution of the earth. In 1754 he published the first two volumes of his Researches on the System of the World. In this work he applied the formulae by which he had calculated the motions of the moon to the motions of the planets disturbed by their mutual attraction, and pointed out the simplest method of determining the perturbations of the motions of a planet occasioned by the action of its own satellites. D'Alembert also treated the subject of the figure of the earth in a much more general manner than had been done by Clairaut, who had confined his investigations to the case of a spheroid of revolution. He determined the attraction of a spheroid of small eccentricity, whose surface can be represented by an algebraic equation of any order whatever, and even supposing the spheroid to be composed of strata of different densities.

The works of D'Alembert, which are extremely numerous, abound generally in profound and original views, and contributed greatly to the advancement of the physical and mathematical sciences; but it is to be regretted that they are very frequently deficient in that perspicuity and order so necessary in abstruse speculations, and that the course of his reasoning can be followed with difficulty when he descends into the detail of analytical operations. He had a horror of calculation, and delighted in general considerations and speculations which frequently turned on matters of pure curiosity.

born 1707, died 1783. Sciences in July 1747, some months before Clairaut and D'Alembert had communicated their solutions of the problem of the three bodies, and it carried off the prize which the academy had proposed for the analytical theory of the motions of Jupiter and Saturn. In this memoir Euler gave the differential equations of the elements of the disturbed planet, but suppressed the analysis by which he had been conducted to them. This analysis, however, he subsequently expanded in two memoirs, the first of which appeared in the Berlin Memoirs in 1749, and the second in those of Petersburg in 1750. Of these supplementary memoirs the first is remarkable on several accounts. It contains the first example of a method which has been fruitful of important consequences—namely, that of the variation of the arbitrary constants in differential equations, and the development of the radical quantity which expresses the distance between two planets in a series of angles, multiples of the elongations. The expressions which he gave for the several terms of this series were simple and elegant; and he demonstrated a curious relation subsisting among any three consecutive terms, by means of which all the terms of the series may be calculated from the first two. He was thus enabled to develop the perturbing forces in terms of the sines and cosines of angles increasing with the time, and thereby to surmount a very great analytical difficulty. Notwithstanding, however, the great merit of Euler's memoir, several of the formulae expressing the secular and periodic inequalities were found to be inaccurate; and in order to procure a correction of these errors, and give greater perfection to so important a theory, the academy again proposed the same subject for the prize of 1752. This prize was also carried off by Euler. In the memoir which he presented on this occasion, he considered simultaneously the motions of Jupiter and Saturn, and determined, in the first instance, the amount of their various inequalities, independently of the consideration of the eccentricities of their orbits. Pushing the approximations farther, and having regard to the inequalities depending on the eccentricities, he arrived at a most important result relative to the periodic nature of the inequalities occasioned by the mutual perturbations of the planets; which laid the foundation of the subsequent discovery by Lagrange and Laplace of the permanent stability of the planetary system. He demonstrated that the eccentricities and places of the aphelia of Jupiter and Saturn are subject to constant variation, confined, however, within certain fixed limits, which it never exceeds; and he computed that the elements of the orbits of the two planets recover their original values after a lapse of about 30,000 years. In the year 1756 the Academy of Sciences crowned a third memoir of Euler on the same subject as the two former, namely, the inequalities of the motions of the planets produced by their reciprocal attractions. This memoir, analytically considered, is also of great value. The method which he followed and illustrated has since been generally adopted in researches of the same nature, and consists in regarding as variable, in consequence of the disturbing forces, the six elements of the elliptic motion, viz. 1st, the greater axis of the orbit, which, by the law of Kepler, gives the ratio of the differential of the mean longitude to the element of the time; 2d, the epoch of this longitude; 3d, the eccentricity of the orbit; 4th, the motion of the aphelion; 5th, the inclination of the orbit to a given fixed plane; and, 6th, the longitude of the node. By considering separately the variations introduced into each of these elements by the disturbing forces, Euler obtained some important results; but even in this memoir his theory was not rendered complete. He did not consider the variation of the epoch; and the expression which he gave for the motion of the aphelion did not include that part of it which depends on the ratio of the eccentricities of the orbits of the disturbed and disturbing planet. Besides, the present memoir, like the two former, contained several errors of computation, which, by leading to results known to be wrong, probably prevented the author himself from being aware of the full value of the ingenious methods of procedure which he had exposed. Euler concluded this important memoir by making an extended application of his formulae to the orbit of the earth as disturbed by the action of the planets. From some probable suppositions, first employed by Newton, relative to the ratios of the masses of the planets to that of the sun, he determined the variation of the obliquity of the ecliptic at 48° in a century,—a result which agrees well with observation. By this determination the secular variation of the obliquity of the ecliptic, which had been regarded by Lahire, Lemonnier, D'Alembert, and other eminent astronomers, as uncertain, was placed beyond doubt. The three memoirs which we have mentioned contain the principal part of Euler's labours on the perturbations; but physical astronomy is indebted to him for many other researches. He gave a solution of the problem of the precession of the equinoxes, and made several important steps in the lunar theory, with which he seems to have explained and made use of. Lagrange had the honour of carrying off the prize; but although he treated the subject in a manner altogether new, and with extraordinary analytical skill, he did not on this occasion arrive at a complete solution of the problem. In 1766 he obtained another prize for a theory of Jupiter's satellites. In the admirable memoir which Lagrange presented to the academy on this subject, he included in the differential equations of the disturbed motion of a satellite, the attracting force of the sun, as well as of all the other satellites, and thus, in fact, had to consider a problem of six bodies. His analysis of this problem is remarkable, inasmuch as it contained the first general method which was given for determining the variations which the mutual attractions of the satellites produce in the forms and positions of their orbits, and pointed out the route which has since been so successfully followed in the treatment of similar questions.

Of all the grand discoveries by which the name of Lagrange has been immortalized, the most remarkable is that of the invariability of the mean distances of the planets from the sun. We have already mentioned that Euler had perceived that the inequalities of Jupiter and Saturn, in consequence of their mutual actions, are ultimately compensated, though after a very long period. In prosecuting this subject, which Euler had left imperfect, Laplace had discovered that, on neglecting the fourth powers in the expressions of the eccentricities and inclinations of the orbits, and the squares of the disturbing masses, the mean motions of the planets, and their mean distances from the sun, are invariable. In a short memoir of 14 pages, which appeared among those of the Berlin Academy for 1776, Lagrange demonstrated generally, and by a very simple and luminous analysis, that whatever powers of the eccentricities and inclinations are included in the calculation of the perturbations, no secular inequality, or term proportional to the time, can possibly enter into the expression of the greater axis of the orbit, or, consequently, into the mean motion connected with it by the third law of Kepler. From this conclusion, which is a necessary consequence of the peculiar conditions of the planetary system, it results that all the changes to which the orbits of the planets are subject in consequence of their reciprocal gravitation, are periodic, and that the system contains within itself no principle of destruction, but is calculated to endure for ever.

In 1780 Lagrange undertook a second time the subject of the moon's libration; and it is to the memoir which he now presented to the Berlin Academy that we must look for the complete and rigorous solution of this difficult problem, which had not been resolved before in a satisfactory manner, either on the footing of analysis or observation. In the same year he obtained the prize of the Academy of Sciences on the subject of the perturbations of comets. In 1781 he published, in the Berlin Memoirs, the first of a series of five papers on the secular and periodic inequalities of the planets, which together formed by far the most important work that had yet appeared on Physical Astronomy since the publication of the Principia. This series did not, properly speaking, contain any new discovery; but it embodied and brought into one view all the results and peculiar analytical methods which had appeared in his former memoirs, and contained the germs of all the happy ideas which he afterwards developed in the Mécanique Analytique.

On account of the brilliant discoveries and important labours which we have thus briefly noticed, Lagrange must be considered as one of the most successful of those illustrious individuals who have undertaken to perfect the theory of Newton, and pursue the principle of gravi- History, ration to its remotest consequences. But the value of his services to science are not limited to his discoveries in physical astronomy, great and numerous as they were. After Euler, he has contributed more than any other individual to increase the power and extend the applications of the calculus, and thereby to arm future inquirers with an instrument of greater power, by means of which they may push their conquests into new and unexplored fields of discovery.

Laplace. With the name of Lagrange is associated that of Laplace, whose rival labours divided the admiration of the scientific world during half a century. Like Newton and Lagrange, Laplace raised himself at an early age to the very highest rank in science. Before completing his 24th year, he had signalized himself by the capital discovery of the invariability of the mean distances of the planets from the sun, on an hypothesis restricted, indeed, but which, as we have already mentioned, was afterwards generalized by Lagrange. About the same time he was admitted into the Academy of Sciences, and thenceforward devoted himself to the development of the laws which regulate the system of the world, and to the composition of a series of memoirs on the most important subjects connected with astronomy and analysis. His researches embraced the whole theory of gravitation; and he had the high honour of perfecting what had been left incomplete by his predecessors.

Among the numerous inequalities which affect the motion of the moon, one still remained which no philosopher as yet had been able to explain. This was the acceleration of the mean lunar motion, which had been first suspected by Dr Halley, from a comparison of the ancient Babylonian observations, recorded by Hipparchus, with those of Albategnius and the moderns. The existence of the acceleration had been confirmed by Dunthorne and Mayer, and its quantity assigned at 10" in a century, but the cause of it remained doubtful. Lagrange demonstrated that it could not be occasioned by any peculiarity in the form of the earth; Bossut ascribed it to the resistance of the medium in which he supposed the moon to move; and Laplace himself at first explained it on the supposition that gravity is not transmitted from one body to another instantaneously, but successively in the manner of sound or light. Having afterwards remarked, however, in the course of his researches on Jupiter's satellites, that the secular variation of the eccentricity of the orbit of Jupiter occasions a secular variation of the mean motions of the satellites, he hastened to transfer this result to the moon, and had the satisfaction to find that the acceleration observed by astronomers is occasioned by the secular variation of the eccentricity of the terrestrial orbit. This was the last celestial phenomenon which remained to be accounted for on the principle of gravitation.

Another discovery relative to the constitution of the planetary system, which does infinite honour to the sagacity of Laplace, is the cause of the secular inequalities indicated by ancient and modern observations in the mean motions of Jupiter and Saturn. On examining the differential equations of the motions of these planets, Laplace remarked, that as their mean motions are nearly commensurable (five times the mean motion of Saturn being nearly equal to twice that of Jupiter), those terms of which the arguments are five times the mean longitude of Saturn, minus twice that of Jupiter, may become very sensible by integration, although multiplied by the cubes and products of three dimensions of the eccentricities and inclinations of the orbits. The result of a laborious calculation confirmed his conjecture, and showed him that in the mean motion of Saturn there existed a great inequality, amounting at its maximum to $48^\circ 9'3"$, and of which the period is 929 years; and that in the case of Jupiter there exists a corresponding inequality of nearly the same period, of which the maximum value is $19^\circ 46'$, but which is affected by a contrary sign, that is to say, diminishes while the first increases, and vice versa. He also perceived that the magnitude of the co-efficients of these inequalities, and the duration of their periods, are not always the same, but participate in the secular variations of the elements of the orbits.

The theory of the figures of the planets, scarcely less interesting than that of their motions, was also greatly advanced by the researches of Laplace. He confirmed the results of Clairaut, Maclaurin, and D'Alembert, relative to the figure of the earth, and treated the question in a much more general way than had been done by those three great mathematicians. From two lunar inequalities depending on the non-sphericity of the earth, he determined the ellipticity of the meridian to be $\frac{1}{317}$ very nearly.

the Principia, explained the cause of the phenomena of the tides, and laid the foundations of a theory which was prosecuted and extended by Daniel Bernoulli, Maclaurin, Euler, and D'Alembert; but as no one of these geometers had taken into account the effects of the rotatory motion of the earth, the subject was in a great measure new when it was taken up by Laplace in 1774. Aided by D'Alembert's recent discovery of the calculus of Partial Differences, and by an improved theory of hydrodynamics, he succeeded in obtaining the differential equations of the motion of the fluids which surround the earth, having regard to all the forces by which these motions are produced or modified, and published them in the Memoirs of the Academy in 1775. By a careful examination of these equations, he was led to the curious remark, that the differences between the heights of two consecutive tides about the time of the solstices, as indicated by Newton's theory, are not owing, as Newton and his successors had supposed, to the inertia of the waters of the ocean, but depend on a totally different cause, namely, the law of the depth of the sea, and that it would disappear entirely if the sea were of a uniform and constant depth. He also arrived at the important conclusion, that the fluidity of the sea has no influence on the motions of the terrestrial axis, which are exactly the same as they would be if the sea formed a solid mass with the earth. The same analysis conducted him to the knowledge of the conditions necessary to insure the permanent equilibrium of the waters of the ocean. He found that if the mean density of the earth exceeds that of the sea, the fluid, deranged by any causes whatever from its state of equilibrium, will never depart from that state but by very small quantities. It follows from this, that, since the mean density of the earth is known to be about five times greater than that of the sea, the great changes which have taken place in the relative situation of the waters and dry land must be referred to other causes than the instability of the equilibrium of the ocean.

Closely connected with the problem of the tides is that of the precession of the equinoxes, which also received similar improvements in passing through the hands of Laplace. He demonstrated, as has just been mentioned, that the fluidity of the sea has no influence on the phenomenon of precession and nutation. He considered some of the effects of the oblate figure of the earth which had not been attended to by D'Alembert, and showed that the annual variation of the precession causes a corresponding variation in the length of the tropical year, which at present is about 9 or 10 seconds shorter than it was in the time of Hipparchus. He proved that the secular inequa- Physical astronomy is also indebted to Laplace for a complete theory of the system of Jupiter's satellites, from which Delambre constructed a set of tables which represent the motions of these bodies with all desirable accuracy. And when to these numerous and most important researches we add the mathematical theories of molecular attraction, and the propagation of sound, together with many great improvements in analysis,—and reflect, besides, that he is the author of the Mécanique Céleste, the Système du Monde, and the Théorie des Probabilités,—we shall not hesitate to rank him next to Newton among the greatest benefactors of the mathematical and physical sciences.

By the brilliant discoveries of Laplace, the analytical solution of the great problem of physical astronomy was completed. The principle of gravity, which had been discovered by Newton to confine the moon and the planets to their respective orbits, was shown to occasion every apparent irregularity, however minute, in the motions of the planets and satellites; and those very irregularities which were at first brought forward as objections to the hypothesis have been ultimately found to afford the most triumphant proofs of its accuracy, and have placed the truth of the Newtonian law beyond the reach of all future cavil. Such is the state to which analysis has now attained, that the geometer embraces in his formulae every circumstance which affects the motions or positions of the different bodies of the planetary system; and the conditions of that system being made known to him at any given instant of time, he can determine its conditions at any other instant in the past or future duration of the world. He ascends to remote ages to compare the results of his theories with the most ancient observations; he passes on to ages yet to come, and predicts changes which the lapse of centuries will hardly be sufficient to render sensible to the observer. But notwithstanding the proud elevation to which the theory of astronomy has been raised, it is still far from having reached the limit beyond which further refinement becomes superfluous. The masses of the planets, and some other elements, remain to be determined with still greater precision, by a diligent comparison of the analytical formulae with good observations; and the labours of the geometer may still be beneficially employed in giving greater simplicity to the calculus, or in extending its power over subjects which have hitherto eluded its grasp. The recent discovery of two periodic comets completing their revolutions in comparatively short intervals of time, opens up an interesting field for speculation and research, and will doubtless be the means of throwing light over some curious, and as yet very obscure, points, respecting the appearances, motions, and physical constitution of those strange bodies.

In the other departments of the science, also, numerous questions still remain to be discussed, the solution of which will occupy and reward the future labours of the astronomer. The curious phenomena of double and multiple stars, some of which appear to form connected systems of bodies revolving about one another, or a common centre of motion,—the variable stars,—the proper motions of the stars,—the translation of the solar system in space,—the progressive condensation of nebulae,—are subjects still in a great measure new; for it is only of late years that observers have begun to direct the requisite attention towards them, or indeed have been in possession of instruments of sufficient power and delicacy to observe and measure the minute changes which take place beyond the boundaries of our own system. The observations of some living astronomers, particularly of Mr Herschel and Mr South, have made known some important facts regarding the nebulæ and double-stars, and laid the foundations of interesting discoveries to be realized by their successors in future ages of the world.

Among the events auspicious to the future progress of astronomy in our own country, we cannot forbear to regard the establishment of the Astronomical Society of London, an association which, though it dates only from 1820, includes in its list of Members the names of the greater part of the most distinguished cultivators of the science in Europe. The society has already given the best proofs of its energy and usefulness, by the publication of a catalogue of the reduced places of 2881 of the principal stars visible above our horizon, and four splendid volumes of Memoirs, replete with information and remarks of the most useful kind to the observer. The number of public observatories recently established, or at present in a state of progress, in various parts of the country and its colonies, also holds out a cheering prospect to the friends of astronomical discovery. Nor if we turn our eyes abroad shall we perceive less reason for congratulation. The same zeal for extension and refinement is manifested in every country of Europe; and while some of the states, among which Prussia and Denmark deserve to be honourably mentioned, are gaining themselves glory by the encouragement they hold out to the cultivation of this department of science, the exertions of individual observers appear to increase in proportion as the field is narrowed by their success. Armed with instruments to which former times had nothing to compare, the astronomer of the present day not only anticipates a more perfect knowledge than he yet possesses of the nature and number of the bodies belonging to the solar system, but aspires to set at rest the disputed question of parallax, to determine the proper motions of the stars, and trace the effects of gravity in the remotest regions of the universe.

Such are the present prospects of astronomy. The mind delights to contemplate unlimited advancement in a walk in which its efforts have been so signalily triumphant; yet in consequence of what has already been done, the future progress of the science must necessarily be slow; and it must be acknowledged that greater precision than has already been attained is unnecessary to navigation or geography, or any practical application of which astronomy admits. The future results of observation may add to the stock of speculative knowledge, but can only be remotely useful to mankind.

The following works may be consulted on the history of Works on astronomy:—Riccioli, Almagestum Novum, Bononia, 1653; the history 2 vols. folio; Sherburn's Translation of the Astronomicum of Manilius, London, 1675; Weidler, Programma de Veteris et Novae Astronomiae Discrimine, Wittemberge, 1720; Souciet, Observations Mathématiques, Astronomiques, &c. 1729, 2 vols. 4to. The second volume of this work, by Gaubil, contains a history of the Chinese Astronomy, with Dissertations. Weidler, Historia Astronomiae, Wittemb. 1741; Idem, Commentatio de Mechanica Astronomica Medii Aevi, 1742; Long's Astronomy, London, 1742; Costard's Letter to Martin Folkes concerning the Rise and Progress of Astronomy among the Ancients, London, 1746; Foulques' History of Astronomy, London, 1746; Heathcote, Historia Astronomiae, Cantab. 1747, 8vo; Esteve, Histoire Générale et Particulière de l'Astronomie, Paris, 1755, 3 vols. 12mo; Goguet, Origine des Lois, des Arts, et des Sciences, various editions; Jablonow, De Astronomiae Ortu ac Progressu, &c. Rome, 1763, 4to; Bailly, Hist. PART II.

CHAP. I.

GENERAL PHENOMENA OF THE HEAVENS.

Sect. I.—Of the Celestial Sphere.

When a spectator on a clear evening directs his attention to the sky, he perceives a concave hemisphere studded with innumerable brilliant points, all which appear to move in parallel directions, constantly preserving the same distances and relative positions. New groups of stars incessantly follow one another, rising in the east, mounting to a certain height, and then gradually sinking till they disappear in the west. On turning his face towards the north he observes some groups of stars which remain visible during the whole night; wheeling round a certain fixed point, and describing circles of greater or less magnitude, according as they are at a greater or less distance from that point. The same phenomena are observed every successive night; and as the stars present always the same configurations, and succeed each other in the same order, the mind is irresistibly led to the hypothesis of the diurnal revolution of the whole heavens about a fixed axis. This hypothesis implies that the stars move in parallel planes, and that their mutual distances remain invariable. In order, however, to be assured that such is the case, it is necessary to have recourse to more accurate observations, and to refer their successive positions to something that is fixed,—to certain points or planes which do not participate in the general motion.

What is most obviously adapted for this purpose is the Horizon, or the plane which appears extended all round us, and bounded by a circle of which the eye of the spectator occupies the centre. Let the circle SENE (fig. 1) represent the horizon, C being the centre or place of the observer. Suppose now that a star is observed to rise at a certain point of the horizon A, and, after tracing its circuit in the sky, to set at the point A'; suppose also, that another star rises at B, and sets at B'; then, if the distances AB, A'B' are measured by means of some angular instrument, it will be found that AB is equal to A'B', and that this is the case with regard to any two stars whatever. Since therefore the arcs AB and A'B' are equal, it follows that the chords AA', BB', joining their extremities, are parallel; whence we infer, that if two stars rise successively at the same point of the horizon A, they will also set at the same point A', a circumstance which is of itself strikingly indicative of the uniformity of the motion of the celestial bodies.

Through C let a perpendicular be drawn to the chord AA', meeting the boundary of the horizon in S and N points. These two points are the South and North points of the horizon; and another straight line drawn through C, parallel to AA', will meet the horizon in E and W, the East and West points. The four together, namely, S, E, N, W, are called the four Cardinal points of the horizon. The horizontal circle is itself denominated the Azimuth Circle, etc. Azimuth distances are measured from the North and South points, so that SA is the azimuth distance of a star's rising at A, and SA' is that of its setting at A'. The complement of the azimuth, or its defect from a right angle, is called the Amplitude, which is consequently measured from the East and West points; thus the arc EA' is the amplitude of a star's rising at A, or its ordinate amplitude, and WA' is that of its setting, or its abscissa amplitude.

Conceive a straight line to pass through C, perpendicular to the horizon. This line, which is called the Vertical, will meet the visible hemisphere in a point directly above the observer, which is denominated the Zenith; the Zenith point diametrically opposite, in which the prolongation of the vertical meets the invisible hemisphere, is the Nadir. The plane determined by the vertical and the straight line NS is called the Meridian, because when the sun reaches it, he is equally distant from the points at which he rises and sets, or, it is mid-day. The great circle formed by the intersection of the meridian with the celestial sphere is the Meridional circle; it divides the sphere into the eastern and western hemispheres. Any circle passing through the Zenith and Nadir is called a Vertical Circle; that which intersects the meridional circle at right angles, or which passes through the points E and W, is called the Prime Vertical.

Let HPZH (fig. 2) represent a meridional circle passing through P, that point of the sphere which appears to remain immovable; and let Z be the zenith, and HH' the horizon. Join PC, and let it meet the circle in P'. The points P and P' are denominated the Poles of the world, and the straight line PP' is the Axis about which the whole heavens appear to revolve. Let also wr, Hg, kl, AE, gH', be projections of the paths of different stars... After having recognised that the celestial bodies move in parallel planes, the next object of the astronomer is to determine whether they revolve with velocities uniform or variable. For this purpose it is necessary to compare, by means of a well-regulated clock, the angular distances between the meridians and the successive positions of a star, with the times elapsed since the star's passage over the meridian. Let \( mn \) (fig. 4) be the parallel described by a star, and \( s' \) two of its successive positions. The arcs \( ns, ns' \), intercepted between the meridional plane and the planes passing through the axis \( PP' \) and the points \( s, s' \), are measured by the arcs of the equator \( Ca, Cb \). Now, it is the result of constant experience, that the number of degrees in the arc \( Ca \) or \( Cb \) is to the number in the whole circumference, as the time employed by the star in describing \( ns \) or \( ns' \) is to the time in which it completes a revolution. The time occupied by the star in completing its revolution is called a sidereal day; and the uniformity of the diurnal revolution enables us to calculate the angle \( CPs \), which is measured by \( Ca \), without having recourse to any other observation than that of the star's meridional passage. Denoting a sidereal day by \( T \), the interval of time which elapses while the star describes the arc \( ns \) by \( t \), and the angle at the pole by \( P \), we shall have the analogy \( T:t::360:P \); therefore \( P=370^\circ\frac{t}{T} \); a formula which gives the angle at \( P \), or its measure \( Ca \).

The angles \( CPs, CPs' \), and \( sPs' \) are called Hour Angles, because the arcs which they intercept on the star's parallel of declination correspond to the hours and fractions of an hour into which the sidereal day is divided. For example, if the hour angle \( sPs' \) contain \( 15^\circ \), the two planes passing through \( s \) and \( s' \) will intercept on \( mn \), the diurnal circle of the star, an arc equal to \( 15^\circ \); consequently there will be 24 such arcs in the whole circumference, each of which will be described in the 24th part of a sidereal day, or in one hour of sidereal time. In consequence of this perfect uniformity of the diurnal motion, the arc which a star describes on its parallel is conveniently measured by the time of its description; and the sidereal day, or the interval between two successive returns of the same star to the plane of the meridian, offers the most perfect unit of time, inasmuch as it is exactly the same to all the inhabitants of the earth, and remains absolutely unalterable in all ages, being one of the very few invariable elements of the system of the world.

In fixing the position of a point on a plane surface, Method of geometers usually refer it to two straight lines at right-angles to each other, the positions of which are supposed to be known. The same practice is imitated by astronomers, who refer the position of any point on the surface of the sphere to two great circles, which are employed as a system of rectangular co-ordinates, and from which all distances are measured. The equator being a circle of the sphere not subject to any arbitrary condition, but determined by the very nature of the spherical revolution, is obviously well adapted for this purpose. It is therefore universally employed as a term of comparison in assigning the places of the celestial bodies. Suppose an hour circle passing through the star \( s \) to intersect the equator at \( a \); the arc \( sa \), which measures on the sphere the distance of \( s \) from the equator, is called the Declination of the star. All other stars situated on the same parallel have the same declination; and it is invariable for all places of the earth, because it is not affected by the diurnal motion which is performed in the parallel \( mo \). The declination is northern or southern, according as the star is situated in the northern or southern hemisphere. It is expressed in degrees from 0 to 90, proceed- Theoretical ing from the equator to the pole. The polar distance Astronony. Ps is the complement of the declination, because Ps + ss = 90°. In order to distinguish s from all other stars which may be situated on the same parallel, we must also know the point a of the equator to which it corresponds; and for this purpose it is necessary to select a point ϕ of the equator for the origin of the co-ordinates. This point ought to be independent of the diurnal motion, otherwise its position could only be determined for a single instant of time; and it must be such that it can be readily found, in whatever part of the earth the observer may be situated. Astronomers have agreed to employ for this purpose the point determined by the intersection of the equator with another very remarkable circle of the sphere, of which we shall speak hereafter, and to count the degrees on the equator, setting out from ϕ, from 0 to 360, in the direction opposite to that of the diurnal motion, or from west to east. The arc ϕa, which measures the distance between ϕ and a, is called the Right Ascension of the star s; it is also the right ascension of any other star situated in the hour circle PaP'. It is measured by the interval of time which elapses between the transit of the star s over the meridian, and that of the point ϕ, which has been chosen as the point of departure. It may be remarked that the point ϕ is also the origin of sidereal time; that is to say, the time is counted 0 hour 0 min. 0 sec. when that point passes the meridian.

From these definitions it is easy to conceive the general method of forming a catalogue of stars. A clock being regulated to sidereal time, and marking 0 hour at the instant of the transit of the point ϕ (or any star chosen arbitrarily), the hours, minutes, and seconds, are noted at which the stars successively pass the axis of the transit instrument, together with their respective altitudes at the same instant. These observations give the right ascensions and declinations; for the time is easily converted into degrees, at the rate of 15° an hour, or by the formula given above for determining the hour angle. If the star s pass the meridian one hour after ϕ, the arc a b, which measures the distance between their hourly circles, PaP', PbP', is 1 hour or 15°; and the declination of a star s which moves in the parallel H (fig. 2) is HA - HA' = HA', that is, equal to the difference between its meridional altitude and the complement of the altitude of the pole H-P.

Modern celestial charts are constructed on the principle of assigning to each star the place indicated by the values of these two co-ordinates; and as observation proves that the mutual distances and relative positions of the stars scarcely undergo any sensible variation, a globe or chart once constructed will serve to represent the state of the heavens for at least a long series of ages.

Having now considered the general phenomena of the diurnal revolution, and explained the terms that are technically employed in assigning the positions of the celestial bodies, we may next proceed to inquire how these phenomena are to be accounted for,—whether the stars are really in motion, as they seem to be, or if the apparent motion is only an illusion occasioned by the revolution of our own earth. The perfect uniformity of the motions of the different stars renders it exceedingly improbable that they are disconnected; hence the simplest view of the phenomena is obtained by substituting for each star its projection on the celestial sphere, at an infinite distance, and supposing this sphere, with the projections of all the stars, to revolve in 24 hours from east to west round an immovable axis. But it is easy to see that the phenomena will be equally well explained by supposing that the starry firmament is absolutely at rest, and that the earth revolves in the same time, round the same axis, but in an opposite direction, from west to east. In both cases the stars remain immovable, the phenomena are exactly the same, and, relatively to Spherical Astronomy, in which we only concern ourselves with the apparent motions, it is absolutely indifferent which of the two hypotheses we adopt. They are both only modes of explaining certain appearances; and the one may be employed which renders the explanation most simple and perspicuous. The proofs of the earth's motion will go on accumulating as we proceed. At present we need only remark, that as the organ of sight makes us acquainted with the existence of relative, and not of absolute motion, it is impossible to decide, merely from appearances, whether the motion we perceive is real or otherwise; for whatever motion one body may have in respect of another, it is always possible to explain the phenomena by supposing it to be perfectly at rest, and the other to move in an opposite direction. The conclusions which we draw from the optical effects of motion afford no mathematical certainty with regard to the cause of that motion. The hypothesis of the revolution of the sphere is attended with innumerable and insurmountable difficulties. The distance of the nearest fixed star from the earth is not less than 350,000 times the distance of the sun; a distance which light, prodigious as its velocity is, would not traverse in less than five years. This immense line, therefore, supposing the heavens to revolve round the earth, would form the radius of a circle, the circumference of which, six times larger, would be passed over by the star in the space of 24 hours; its velocity must therefore be $6 \times 5 \times 365 = 10950$ times greater than that of light. This velocity, which is nearly equivalent to 2100 millions of miles in a second of time, is so enormous that it baffles the utmost efforts of the imagination to form any conception of it; and the supposition of its existence will appear still more revolting when we reflect that the distance of the nearest star is probably many thousand times less than that of the Milky Way. If we ascribe the motion to the earth, its velocity, though it may still be supposed great, is moderate in comparison of many well-known phenomena of nature. A point on the equator will describe in 24 hours a circle of about 25,000 miles, or about 17 miles in a minute; a velocity somewhat exceeding that of sound, but 7400 millions of times less rapid than the preceding.

In all that has hitherto been said respecting the apparent motion of the starry sphere, we have only had regard to the innumerable multitude of stars which constantly maintain the same relative situations, and which are in consequence called Fixed Stars. There are, however, several other bodies, some of them remarkable on account of their splendour, which, besides participating in the general motion, have peculiar motions of their own, and are incessantly shifting their positions among the fixed stars. An attentive observation of the state of the sky during a few successive evenings, will suffice to show that there are some stars which have in the mean time changed their places; and, on continuing to observe them, they will be found to separate themselves from particular constellations, and gradually but imperceptibly to approach others, till they at length appear, after unequal intervals of time, in an opposite quarter of the heavens. From this circumstance they were designated by the Greeks Planets, that is, wandering stars, in contradistinction to those which obey only the law of the diurnal motion. Besides the sun and moon, there are five discernible by the naked eye, and which have consequently been known from the remotest ages. These are Mercury, Venus, Mars, Jupiter, and Saturn. Five others, namely, Uranus, Ceres, Pallas, Juno, Astronomy.

The Sun.................. ☉ Mercury.................. ♦ Venus.................... ♦ Mars..................... ♦ Juno..................... ♦ Ceres.................... ♦ Pallas................... ♦ Jupiter.................. ♦ Saturn................... ♦ Uranus.................. ♦

The Earth, which, as we shall afterwards see, is also a planet, and takes its place between Venus and Mars, has for its symbol ⊕, and the moon ♪. Venus, Jupiter, and sometimes Mars, are distinguished by their extraordinary brilliancy. Mercury, on account of his proximity to the sun, is rarely visible to the naked eye. Uranus, discovered by Sir W. Herschel in 1778, can with difficulty, by reason of his great distance, be perceived without a telescope. Ceres, Pallas, Juno, and Vesta, discovered since 1801, are extremely small in size, and can only be seen with the aid of the telescope.

Of all the celestial bodies, the most interesting to us are the sun and moon; and their peculiar motions have accordingly, in every age of astronomy, been studied with the greatest attention. The proper motion of the moon is particularly remarkable. In the course of a single night she separates herself very sensibly from the stars in her vicinity, moving over a space nearly equal to her own breadth in an hour, and completing a whole circuit in about 27 days. The sun moves with much less velocity, but his motion is still sufficiently apparent. If we take notice of the stars which immediately follow him when he sinks under the horizon, we shall find that in the course of a few nights they will be no longer visible. Others which, some time previously, did not set till long after him, have taken their places and now accompany the sun. In the morning similar appearances present themselves, but in a contrary order. The stars which appear in the eastern horizon at sun-rise, are, after a few days, considerably elevated above it at the same time. Thus the sun seems to fall daily behind the stars, by insensible degrees, till at last he appears in the east when they are about to set in the west. To account for these appearances, the ancients supposed the sun's diurnal motion to be really slower than that of the stars; hence they supposed him to be attached to a different sphere. For like reasons they ascribed particular spheres to the moon and each of the planets; and as no trace of these imaginary spheres is perceptible in the heavens, they next supposed them to be crystalline and transparent. The appearances are explained equally well, and with infinitely greater simplicity, by ascribing to the sun a proper motion, in a direction opposite to that of the diurnal rotation of the sphere, in consequence of which he advances to meet the stars, instead of falling behind them.

The greater part of the observations of the early astronomers had for their object the determination of the positions of the stars relatively to the sun at his rising and setting, by which they fixed the seasons, and regulated the operations of agriculture. They distinguished all these phenomena by technical terms, which occur very frequently in the works of the ancient poets, particularly in Hesiod and in Ovid's Fasti. A star which rises at the same time with the sun is efficacious by his light, and is said to rise heliacally (ortus heliacus). Soon after, when the sun by his proper motion has advanced so far to the east that the star can be perceived on the eastern horizon in the morning twilight, it disengages itself from the sun's rays, and is said to rise aeronomically (ortus aeronomicus). At the end of six months, the sun being diametrically opposite to the same star, it sets as the sun rises, and in this case it is said to set cosmically (occasus cosmicus): at nearly the same time it rises when the sun sets, and is said to rise aeronomically (ortus aeronomicus). The sun afterwards begins to approach the star, till he advances so near that it is again about to be effaced by his light; it is now said to set heliacally, or just so long after the sun as to be visible when he has disappeared (occasus heliacus). At the end of a year, the star again rises and sets at the same moment with the sun; it is now said to set aeronomically (occasus aeronomicus). These distinctions, and the ancients had several others of the same kind, which are all defined by Ptolemy, are now scarcely ever mentioned. They have lost the whole of their interest since, in the progress of astronomy, more certain methods have been discovered of determining the commencement of the year and the seasons.

The proper motions of the planets are in general, like Stations that of the sun, in a direction opposite to the diurnal motion, and retrogression, or from west to east, among the stars: but they do not preserve the same character of uniformity; the planet sometimes becomes stationary among the fixed stars, and even advances from east to west in the direction of the diurnal motion. In this case it is said to retrograde. This retrograde motion is, however, not of long continuance. After having been accelerated during a short time, it begins to relax. The planet again becomes stationary, and then resumes its direct motion from west to east. These phenomena were observed in the remotest antiquity, and their explanation formed the principal part of the rational astronomy of the Greeks and the Arabs.

Besides the planets, other bodies occasionally make Comets. Their appearance in the heavens, which, by reason of the extraordinary phenomena they exhibit, have been frequently contemplated with terror and dismay, and regarded by superstitious ignorance as harbingers of calamity, and precursors of the divine vengeance. These bodies shoot down, from the remote regions of space, with inconceivable velocity, towards the sun. At their first appearance they are small; their light is feeble and dusky; and they are generally accompanied by a sort of nebulosity or luminous tail, from which they have derived their appellation of Comets (coma, hair). As they approach the sun, their apparent magnitudes and brilliancy greatly increase, and the nebulosity sometimes occupies a large portion of the heavens, presenting a magnificent and astonishing spectacle. Having attained the point of their orbits nearest the sun, they again recede to enormous distances, and vanish by insensible degrees. They differ from the planets not only by the appearances they present, but also by the diversity of their motions; for, instead of being confined to a particular zone, and moving from west to east, they traverse the sky indifferently in all directions. They are visible only in a small part of their orbit, which, being near the sun, is passed over with prodigious rapidity. They seldom continue visible longer than six months. Their number is entirely unknown, but during the last two centuries upwards of 150 have been observed, and their orbits computed.

Thus we recognise three distinct classes of celestial bodies; the planets, the comets, and the fixed stars. It is the business of the astronomer to determine the positions of these bodies in the heavens; to observe their motions, and measure them with precision; to discover the laws by which their courses are regulated; and from these laws to assign the past or future state of the heavens. Sect. II.—Of the Globular Form of the Earth, Parallax, and Refraction.

Having now considered the general phenomena of the diurnal motion, we proceed next to inquire into the form of the earth, and our situation with respect to the centre round which the celestial sphere appears to revolve. The plane of the horizon seems to be stretched out indefinitely till it actually meets the sky; but this illusion is quickly dissipated by transferring ourselves from one place to another on the surface of the earth, and attending to the phenomena which such a change of place gives rise to. Let an observer, for example, set out from any given point in the northern hemisphere, and proceed directly south. In proportion as he advances, the stars in the southern region of the heavens will be elevated more above the horizon, and describe larger segments of their diurnal circles, while new ones come into view which were invisible at the station he left. On the other hand, the polar star, with those in its vicinity, will be depressed, and some stars which before continued above the horizon during the whole time of their revolution will now rise and set. The planes of the diurnal circles become also more perpendicular to the horizon, so that the aspect of the heavens is entirely changed. If, instead of advancing in the direction of the meridian, the spectator proceeded towards the east or west, he would in this case also remark that his horizon constantly shifted its position. A star will pass his meridian sooner as he advances eastward, or later as he travels westward; in short, by a change of place in any direction whatever, the perpendicular to the horizon, or the plumb-line, will correspond to a different point of the heavens. The plane of the horizon is therefore variable, and its variation by insensible degrees indicates with the greatest evidence the rotundity of the earth. Experience also shows that a spectator sees more of the terrestrial surface in proportion as he is elevated above it; and that on a mountain surrounded by the sea, or standing in the middle of a level plain, the horizon appears equally depressed all round, which is the distinctive feature of a spherical curvature. In fig. 5 let O be the place of a spectator on the summit of a mountain, the straight line H H will represent his horizon, and O h, O k the directions of visual rays to the remotest visible points on the surface of the earth. The inclinations of these lines to the plane of the horizon, or the angles H O h, H O k are called the apparent depression; and these being always observed to be equal to each other, it follows that the curve h A k is a circle. On the Peak of Teneriffe, Humboldt observed the surface of the sea to be depressed on all sides in an angle of nearly 2°. The sun arose to him 12 minutes sooner than to an inhabitant of the plain; and from the plain, the top of the mountain appeared enlightened 12 minutes before the rising or after the setting of the sun. The same phenomena, though on a smaller scale, are observed with regard to every mountain or elevation on the earth. Another familiar illustration of the globular figure of the earth is derived from the successive and gradual disappearance of a ship which leaves the shore and stands out to sea. The hull disappears before the sails and the rigging, and the top of the mast is the last part that is visible. The ship thus appears gradually to sink under the horizon, exactly in the same manner as she must necessarily do on the supposition that the surface of the sea is spherical. The appearance of the moon at the time she is eclipsed is also demonstrative of the roundness of the earth. When the moon penetrates the shadow of the earth, the line which separates the illuminated from the eclipsed portion of the disk is circular; evidently proving the conical form of the shadow, and consequently the roundness of the body by which the shadow is projected. From all these considerations, it is inferred with the highest certainty that the earth with its waters forms a round mass, isolated in space.

The globular form of the earth is a property common to it with the other bodies which compose the planetary system. The sun and moon are evidently round bodies; and when the planets are examined through a telescope their disks appear sensibly circular; and as they are known to have a motion of rotation, in consequence of which they successively present different points of their surfaces to the earth, the uniform roundness of their disks may be taken as a conclusive proof of their sphericity. We shall by and by have occasion to remark other striking analogies between the planets and the earth.

Since the figure of the earth is spherical, the horizontal plane cannot coincide with any considerable portion of its surface. It may be defined to be the plane which touches the earth at any given point, and is perpendicular to the vertical line, or the direction of gravity at that point. The Sensible Horizon of any place is the plane which passes through the eye of the spectator, perpendicular to the plumb-line at that place. The Rational Horizon is a plane parallel to this, passing through the centre of the earth. The particular phenomena of different places depend on the position of their horizon with respect to the planes of the apparent diurnal motion of the sun and stars. The rational horizon of a place on the equator passes through the poles, and divides equally the equator and its parallels. Hence the days and nights are always equal in such places, and each of the stars performs one half of its revolution above, and the other below its horizon. The circles of diurnal motion are all perpendicular to the horizon, and therefore the inhabitants are said to be under a Right Sphere. If a spectator could place himself directly under the pole, his horizon would coincide with the equator, and the whole of the northern celestial hemisphere would be within his view, while no part of the southern hemisphere would be visible to him, on account of its being always beneath the horizon. The circles of the diurnal motion being parallel to the equator, and consequently also to the horizon, the fixed stars would never either rise or set. A spectator thus situated is said to be under a Parallel Sphere. Intermediate places the circles of the diurnal motion are oblique to the horizon, one pole being always elevated above it, and the other equally depressed below it. The stars whose distances from the elevated pole are not greater than the arc of its elevation above the horizon never set, while those within the same distance of the depressed pole never rise. These phenomena belong to all places situated between the equator and the poles, and the inhabitants of such places are said to be under an Oblique Sphere. The situation of a place on the surface of the earth is determined by two co-ordinate circles, in the same manner as that of a star on the celestial sphere. Let Z (fig. 2) be a place of which the position is required to be assigned with reference to circles of the terrestrial sphere, analogous to those which we have described as belonging to the sphere of the heavens. Since the axis of the celestial sphere passes through the centre of the earth, we may suppose the poles of the world, the equatorial circle, and the meridians, to be transferred to the earth's surface, so that the same figure may represent either the celestial or terrestrial sphere. The intersection of the plane PZÆ with the surface of the earth is the terrestrial meridian of the place Z, and the straight lines CZ, CÆ, in that plane, intercept an arc of the terrestrial meridian between Z and the equator, containing as many degrees as the arc of the celestial sphere which measures the declination of the zenith of the place Z. The angle ÆCZ, which is measured by the meridional arc ÆZ is called the Geographical Latitude of Z. But in order to particularize the point at which the equator is intersected by the meridian, it is necessary to assume some point of the equator to which all the other points of that circle may be referred. The meridian passing through the assumed point is called the First Meridian; and the angular distance between the planes of the first and any other meridian measured by the equatorial arc intercepted by these planes on the equator, is called the Geographical Longitude of the place through which the last meridian passes. The longitude is said to be east or west, according as the degrees of the equator are counted from the first meridian towards the east or west. It is usual to reckon the degrees towards the east, all round the globe, or from 0 to 360°. From these definitions it is evident that the geographical latitudes and longitudes are exactly analogous to the declinations and right ascensions on the celestial sphere. The first point of the terrestrial equator cannot be determined by any star or fixed point in the heavens, on account of the diurnal motion; it is therefore necessary to fix its position by means of known places on the earth; and geographers are in the habit of assuming as the first meridian that which passes through the capital city or principal observatory of their country. The choice is of no importance; for in geography, as in astronomy, what is essential to be known is only the difference of longitudes or meridians, in order to reduce the situation of a place, or observations which have been made at any indicated meridian.

The length of the terrestrial radius is a very important element in astronomy, inasmuch as it furnishes the observer with the only scale by which he can estimate the distances of the sun, moon, and planets. On the supposition that the figure of the earth is perfectly spherical, the general principles on which its magnitude may be determined are sufficiently obvious; but the accurate determination of its actual dimensions is attended with great practical difficulties, which can only be overcome by the perfect instruments and refined science of modern times. Eratosthenes seems to have been the first who made use of astronomical methods to determine the circumference of the earth, or the length of the meridian. He remarked that, at Syene, in the Thebais, the sun on the meridian, at the time of the summer solstice, was vertical; and that at Alexandria, at the same time, its zenith distance was $7^\circ 12'$. Now let S (fig. 6) be the sun, vertical to M, and Z the zenith of Alexandria, C being the centre of the earth, and O Alexandria. The angle ZOS is the sun's zenith distance, which was observed to be $7^\circ 12'$. But $ZOS = ZCS + OSC$; whence, as the angle OSC, which is extremely small on account of the sun's great distance, may be neglected in the calculation, the angle ZCS at the centre is also equal to $7^\circ 12'$, and therefore the whole circumference equals $\frac{360^\circ}{7^\circ 12'} \times MO$. To obtain the length of the meridian, it is therefore only necessary to measure the arc MO. Eratosthenes assumed the distance between Syene and Alexandria to be 5000 stadia; hence the circumference of the earth $= \frac{360^\circ}{7^\circ 12'} \times 5000$ stadia $= 250,000$ stadia. The uncertainty which exists respecting the length of the Egyptian stade prevents us from deriving any precise information from this rude attempt to estimate the dimensions of the globe.

The method which has just been described takes for granted that the meridian is exactly circular,—an assumption which, even supposing the method perfect in all other respects, would lead to erroneous results, especially in an arc of so great a magnitude as seven degrees. In a small arc, of $1^\circ$ for example, the error arising from the non-sphericity of the earth will be insensible; for it is certain, from the phenomena before explained, that the deviation from the spherical figure is not very considerable; and besides, whatever the nature of the meridional curve may be, it will sensibly coincide with its osculating circle, throughout the extent of an arc of $1^\circ$. It has been found by numerous and accurate experiments, that the lengths of arcs of $1^\circ$ on the same meridian are longer in proportion as we advance nearer the pole. Hence, on account of the similarity of the isosceles triangles of which these arcs form the bases, their sides, or the terrestrial radii, must also be longer, and consequently the convexity of the earth is less towards the pole than at the equator. The surface of the earth is extremely irregular, even independently of the inequalities occasioned by mountains and cavities; yet it has been discovered that the meridional curves differ almost insensibly from ellipses; whence it is concluded that the figure of the earth is an ellipsoid of revolution about its shortest axis. In comparing the results of the various measurements which have been made with the formulae belonging to the dimensions of such a body, this conclusion has been fully verified; and the lengths of the arcs, the ellipticity, the distance of the pole from the equator, and, in short, all the elements of the spheroid, have been determined. The results of theory and observation give an ellipticity amounting very nearly to $\frac{1}{306}$; that is to say, the equatorial is to the polar diameter in the ratio of 306 to 305. The following may be regarded as a very near approximation to the dimensions of the earth in English miles:

- Semidiameter of the equator: 3963.7 - Semidiameter of the pole: 3949.8 - Semidiameter at the latitude of 45°: 3956.2 - Ellipticity: $\frac{1}{306}$ - Length of $1^\circ$ of the meridian: 69.06 - Quarter of the meridian of Paris: 6214.47

The figure of the earth is one of the most difficult and most important questions of astronomy; and as our limits will not permit us to treat it in this place with all the details which its importance renders necessary, we shall reserve, for a separate article, an account of the different geodetical operations which have been undertaken with a view to determine it, together with the development of the mathematical theory of its equilibrium. See Figure of the Earth.

In describing the phenomena of the diurnal revolution, we have supposed the eye of the spectator to be placed Theoretical in the centre of motion, on which supposition the apparent and true places of the celestial bodies will be the same. This, however, as is evident, can only be rigorously true in respect of one point of the earth's surface, and is therefore most probably not true of any. Astronomers have on this account agreed to refer all the motions to the centre of the earth; that is, to consider the centre of the earth as the centre of the celestial sphere, and to regard as the true geocentric place of a star, that at which it would appear to a spectator placed at the centre. But the spectator is necessarily placed at the surface. In order, therefore, to render observations made at different stations on the earth's surface comparable with one another, we must consider what changes will be produced in the apparent positions of the celestial bodies, in consequence of their being observed from a point not situated in the centre of motion.

Let O' and O (fig. 6) be two stations on the earth's surface on the same meridian, from both of which a star S is observed at the same time. As an object is always seen in the direction of the visual ray, the spectator at O will refer the star S to a certain point I on the surface of the celestial sphere, while the spectator at O' will refer it to a different point I' in the direction O'I'; and the difference of the apparent places I and I' depends on the magnitude of the angle OSO', which is subtended by the chord of the arc OO'. If a third spectator could be placed at C, the centre of the earth, the same star would appear to him in the direction CS, and be referred to its true place on the sphere at Q. Now, the difference between the angles ZOS and ZQS, that is, the angle OSC, under which the radius of the earth is seen by an observer at S, is called the parallax of S. On account of the great distance of the fixed stars, this angle is altogether insensible with regard to them; it is, however, very sensible in the case of the moon, with regard to which it amounts to about 1°. The greatest parallax of the nearest planets does not exceed 30'.

From the definition which has just been given, it appears that the parallax of a star is the angle comprised between two lines drawn from the star, the one to the centre of the earth, and the other to any point whatever on its surface. But it is evident, that while the distance of S from the earth remains constant, the angle OSC will vary with the angle ZOS, that is, with the distance of the star from the zenith, or its altitude above the horizon.

Let OH (fig. 7) represent the horizon. It is easy to see that the angle OSC will have its greatest value when OS touches the surface of the earth, or coincides with the horizon OH, that is, when the angle ZOS = 90°. It is equally evident that the parallax OSC vanishes entirely when the line CS coincides with CZ, that is, when the star is in the zenith of the observer at O; hence the parallax depends on the altitude of the star, according to a law which we now proceed to determine.

In the triangle OSC (fig. 7), we have sin. OSC : sin. SOZ :: CO : CS, whence sin. OSC = \(\frac{CO}{CS}\) sin. SOZ. Let the angle SOZ, or the zenith distance of the star = Z, the semidiameter of the earth OC = a, the distance of the star CS = r, and the parallax OSC = p, we have then sin. \(p = \frac{a}{r} \sin. Z\). On account of the smallness of the angle \(p\), which, as we have mentioned, in no case exceeds 1°, the arc may be substituted for the sine (for their difference even in an arc of 1° does not exceed 0°18): the above expression therefore becomes \(p = \frac{a}{r} \sin. Z\); that is to say, the parallax of a star is proportional to the sine of its zenith distance. At the zenith, \(Z = 0\), therefore \(\sin. Z = 0\), and the parallax vanishes: at the horizon, \(Z = 90°\), and \(\sin. Z = 1\); the parallax, therefore, becomes equal to \(\frac{a}{r}\). In this case it is at its maximum, and is denominated the Horizontal Parallax; in all other cases it is called the Parallax of Altitude. Denoting the horizontal parallax by P, we have \(P = \frac{a}{r}\), whence

\[p = P \sin. Z;\]

that is to say, the parallax of altitude is equal to the horizontal parallax, multiplied by the sine of the apparent zenith distance. It is evident that the apparent altitude SOH is always less than the true altitude SBH, by the whole amount of the parallax; the effect of the parallax is therefore to depress the object, or increase its zenith distance; hence, if \(θ\) be the apparent altitude of the star, and \(θ'\) its true altitude, \(θ'\) will be found from the equation \(θ' = θ + P \sin. Z\).

From the above results it is manifest, that if the horizontal parallax can be by any means determined, the parallax at any other altitude will be found at the same time. The determination of the horizontal parallax is, however, attended with considerable difficulty, and various methods have been proposed and practised to ascertain its exact amount, modified by particular circumstances in the cases of the different celestial bodies. The method, however, which serves as the basis of all the others is extremely simple, and exactly analogous to that by which the distance of a remote object is determined on the surface of the earth. Suppose two observers to be stationed at the points O and O' (fig. 6), of which the latitudes are known, and which are both situated on the same meridian, and let them simultaneously observe the zenith distances of the star S; these observations will give the angles ZOS, Z'O'S, whence their supplements SOC and SO'C become known at the same time. A third angle of the quadrilateral figure SOCO', namely OCO', is also known, being measured by the meridional arc OMO', the difference or sum of the latitudes of the two observers, according as they are on the same or opposite sides of the equator. The two sides also, CO and CO', being semidiameters of the earth, are supposed to be known; every part of the quadrilateral figure is therefore determined, and its diagonal CS may be calculated by the rules of plane trigonometry. But when CS is determined, the horizontal parallax is obtained immediately from the formula \(P = \frac{a}{r}\), that is, \(P = \frac{CO}{CS}\). On this principle the horizontal parallax of the moon was determined by Lacaille and Lalande, the former observing at the Cape of Good Hope, and the latter at Berlin; and the parallax of Mars by Lacaille at the Cape, and Wargentin at Stockholm.

Let us suppose two lines to be drawn from the centre of a planet touching the surface of the earth in points diametrically opposite; the inclination of these two straight lines is the double of the horizontal parallax; but the same angle also measures the diameter of the earth as seen from the planet: hence the horizontal parallax of a planet is equal to the apparent semidiameter of the earth at the distance of the planet. From this consideration the true diameter, and consequently the volume, of a planet may be found by measuring its apparent diameter, that is, the optical angle under which its diameter appears when seen from the earth. Let \(d\) be the apparent, and D the true diameter of a planet; the angle \(\frac{1}{2}d\) is comprised between two straight lines drawn from the A S T R O N O M Y.

earth's semidiameter. That the apparent place of an object must be changed in consequence of a change in the situation of the observer, is a simple geometrical truth which no experiment was required to discover. The first observers must accordingly have anticipated a parallax in all the celestial bodies; and it was doubtless only after considerable experience that they admitted the existence of any exception to the general law. There is, however, another cause of variation in the apparent positions of the celestial bodies also connected with the earth, the existence of which could not be known a priori, but must have been discovered by experience alone. We allude to the refraction of the rays of light in passing through the earth's atmosphere.

The angular distance between two stars is found to undergo very sensible variations at different hours of the day. This phenomenon cannot be explained by any proper motion of the stars, because it evidently depends on their altitude above the horizon; and the differences are found to be the same daily at the same altitude. It is most striking when we compare a star which, without setting, passes the meridian twice a day, once near the zenith, and the second time near the horizon, with another star situated very near the pole, and of which the altitude is consequently nearly invariable. It will be found that at the time of the first transit the distance between the two stars is greater than at the second by nearly half a degree. It is evident, therefore, that the phenomenon consists in diminishing the distance between a star situated in the horizon and the visible pole, that is to say, in elevating the stars, whereas the effect of the parallax is to depress them. The refraction also takes place in an equal degree with regard to the fixed stars, and even the moon and planets, without being in any degree modified by the great differences in the distances of the celestial bodies, contrary in this respect likewise to the parallax, which depends entirely on the distance. The reason of this will be evident from the consideration of the physical cause of the phenomenon.

According to the known principles of optics, a ray of light, in passing obliquely from one transparent medium into another of a different density, does not hold on in its rectilinear course, but is refracted, or bent towards the denser medium. Now, the atmosphere which surrounds the earth may be regarded as composed of an infinity of concentric spherical strata, the densities of which are greater in proportion as they are nearer to the earth's surface. When a ray of light, therefore, proceeding from a star enters the atmosphere, it is inflected towards the earth, or bent so as to form a smaller angle with a perpendicular to the surface of the earth; and this inflection will be increased by every successive stratum of the atmosphere through which the light passes. In fig. 8, let Fig. a, AA', BB', CC', represent the boundaries of the successive strata, which, for the sake of illustration, we here suppose to have a finite thickness. A ray of light proceeding from S comes in contact with the highest stratum of the atmosphere AA' at a. The molecular attraction of this atmospherical stratum, acting in the direction of a normal to AA' at a, causes the luminous ray to deviate from the direction S x, and assume another, a y, in which it would continue to move if the atmosphere were equally dense from AA' to the earth. But in the course of its progress the ray penetrates another denser stratum at b, and consequently suffers another inflection; so that instead of proceeding in the direction a y, it is bent into a new direction b z, more nearly perpendicular to the concentric strata. A similar effect is produced at c, so that the luminous ray, when it finally reaches the observer at O, has as-

From the equation \( P = \frac{a}{r} \), we have \( r = \frac{a}{P} \); that is, the distance of a planet from the earth is known in terms of its parallax and the earth's semidiameter. The parallaxes, therefore, give the ratio of the distances of all the planets from the earth, and consequently of their distances from one another, and from the sun; hence the radius of the earth furnishes the scale by which the astronomer measures the dimensions of the whole solar system, and the magnitudes or volumes of all the bodies of which it is composed. On this account the accurate determination of the parallaxes of the celestial bodies is a problem of great importance in practical astronomy.

It is evident from the mere inspection of the figure, that the plane in which the straight lines CS and OS are situated is the vertical plane passing through S; consequently the whole effect of the parallax is to diminish the altitude of a planet in its vertical circle. When the observation, therefore, is made in the meridian, the effect of the parallax is to alter the declination, without producing any change whatever in the right ascension of the planet. Out of the meridian it is necessary to have regard to the azimuth or hour angle, as well as to the altitude, in calculating the correction due to the parallax. When we suppose the earth to be spherical, the formulae for the calculation of the parallax are extremely simple, because the radius is constant, and the vertical line, or perpendicular to the surface of the earth, passes through its centre. But in the case of the true or elliptical figure of the earth neither of these circumstances takes place. The radius in this case is variable, and must be determined by a particular process of computation for every point on the meridian; and the vertical line, with reference to which the latitude of the place and the altitudes are determined, does not pass through the centre of the earth, but makes different angles with the axis. These circumstances render the calculation of the parallax a matter of much greater complication and difficulty.

The term parallax in its general signification properly denotes change of place. There are consequently various kinds of parallaxes, such as parallax of right ascension, of declination, longitude, latitude, &c. In what has preceded, we have supposed the earth to be at rest in the centre of the universe, and therefore have had regard solely to the variations produced in the apparent diurnal motions by the eccentric position of the observer on the surface of the earth, and which are comprehended generally under the denomination of the diurnal parallax.

The effects of the diurnal parallax are only sensible with regard to those bodies of which the distance from the earth is not so great as to be incomparable with the Theoretical sumed the direction e O. In its progress from a to O, it has therefore successively moved in the direction of the sides of the polygon a, b, c, O; and to the spectator at O, the star from which it proceeded, instead of appearing in its true place at S, will appear to be at S', or in the last direction of the visual ray. Now, if A A' is the most elevated stratum of the atmosphere into which the ray enters in the direction S a, it is clear that the whole effect is produced by the atmospherical strata situated below A A', and that the length of S a is perfectly indifferent; hence the refraction is entirely independent of the distance of the stars, provided they are beyond the limits of the earth's atmosphere.

The decrease of the density of the atmosphere, from the surface of the earth upwards, follows the law of continuity, or takes place by insensible degrees; so that the luminous ray, in traversing the atmosphere, enters at every instant into a denser medium, and is therefore continually brought nearer and nearer to the vertical direction. Hence the true path of the ray is curvilinear, and concave towards the earth, as represented in fig. 9. This is equivalent to the supposition that the thickness of the different concentric strata of uniform density is infinitely small, and that the light, as it successively penetrates each, deviates from its former path by an infinitely small angle, which may be considered as the differential of the refraction, the total amount of which will therefore be obtained by integration.

The direction of the ray, when it reaches the eye of the observer, is the tangent to the last portion of its curvilinear path; and the apparent zenith distance of the star will be ZOS', while the real zenith distance is ZOS. The difference of these two angles, namely S'O'S, is what is denominated the Astronomical Refraction. It is evident that the whole path of the ray is confined to the vertical plane, in which the star and the eye of the observer are situated; for the earth and its atmosphere being very nearly spherical, that plane will divide the strata symmetrically; the attracting forces will therefore be equal on each side of it, and consequently produce no effect in a lateral direction.

When the observed star is due north or south, the vertical plane is the plane of the meridian; hence, in meridional observations, the whole of the refraction, like that of the parallax, takes place in declination, while the right ascension remains unaltered.

It is evident that the amount of the refraction is greater in proportion as the observed star is nearer to the horizon; for in this case the luminous rays strike the tangent planes of the atmospherical strata more obliquely, and have besides to traverse a greater extent of atmosphere before they arrive at the eye of the observer. On determining by experiment the refraction at every altitude from zero to 90°, tables of Refraction may be constructed, which will furnish the means of discovering the law of its diminution; but as such a process would be exceedingly tedious, and likewise subject to lead to erroneous results on account of the inevitable errors of observation, it is found more convenient to assume some hypothesis for a basis of calculation, and to verify the results which it leads to by comparing them with observation. In regard to media which may be said to be permanent, such, for instance, as water and glass, the determination of the refraction is not attended with great difficulty; but the circumstances are greatly altered when we come to make experiments on the atmosphere. In this case the difficulty arises from the incessant changes which the atmosphere is undergoing relatively to its refringent powers; changes which it is impossible for the observer fully to appreciate, insomuch as he can only determine its physical state within a short distance of the earth, while that of the upper strata remains wholly unknown to him.

The refringent power of the atmosphere is affected by its density and temperature. The effects of the humidity are insensible; for the most accurate experiments seem to prove that the watery vapours diminish the density of the air in the same ratio as their refractive power is greater. It is therefore only necessary, even in delicate experiments, to have regard to the state of the barometer and thermometer at the time the observation is made. At a medium density, and at the temperature of melting ice, it was found by Biot and Arago, from a great number of exact experiments, that at any altitude between 10° and the zenith the refraction is very nearly represented by the formula \( r = \frac{60}{\tan (Z - 325 \times r)} \), in which \( r \) is the refraction corresponding to a given zenith distance \( Z \).

With the exception of the numerical co-efficients, this formula was first given by Bradley; but whether it was deduced from theory by that great astronomer, or was only empirical, is uncertain. Bradley's formula was \( r = \frac{57}{\tan (Z - 3 \times r)} \). When the direction of the luminous rays makes a smaller angle than 10° with the horizon, it becomes indispensable to take into account, in the calculation of the refraction, the law of the variation of the density of the atmosphere at different altitudes; a law which is subject to incessant variation, from the operation of winds, and other causes which agitate the atmosphere, as well as the decrease of temperature in the superior regions. For this reason all astronomical observations which have not refraction directly for their object, or which are by their nature independent of its influence, are made at an elevation exceeding 10°. For lower altitudes, it is to be feared that no theory will ever be found sufficiently exact to entitle the observations to much confidence.

The existence of the atmospherical refraction was not unknown to the ancient astronomers; but it is only in modern times that the subject has been studied with the requisite care to admit of its influence being calculated in the reduction of observations. Ptolemy does not allude to the subject in the Almagest, but he has given a sufficiently exact idea of it in his work on Optics. He mentions experiments made to ascertain how far a ray of light is bent from its rectilinear course in passing from air into glass and water; and observes that the astronomical refraction brings a star nearer to the zenith, and that it is feebler in proportion as the star is more elevated. He likewise indicates a method of measuring its effects, although he does not seem to have attempted to practise it. The same notions, but in a less precise form, were reproduced in the Optics of Alhazen. Walther began to estimate the effects of refraction near the horizon; and Tycho, a century after, found the means of measuring them with greater accuracy, and was thereby enabled to construct a table. Tycho estimated the horizontal refraction at 34°, and supposed it to vanish at the altitude of 45°; which proves that his ideas on the subject were less accurate than those of Ptolemy, who says expressly that it vanished only at the zenith. Dominic Cassini was the first who proposed an hypothesis for calculating the refraction at any altitude; and he computed a table, which was published in 1662. Since that time the subject of refraction has been investigated, theoretically and practically, with the most scrupulous and delicate attention, and tables constructed to exhibit its amount at every altitude, and at the different seasons of the year. (See Mécanique Céleste, tom. iv. p. 231; Mr Ivory's Paper in the Phil. Trans. for 1823, p. 409; Delambre, Astronomie du xviii. Siècle, p. 774.)

The refringent power of the atmosphere gives rise to a number of curious phenomena. When the sun appears in To assign the time at which it is a maximum or minimum, or of a given length at a given latitude, is a problem of pure geometry, which has been frequently solved since the time of the Bernoullis.

The scintillation or twinkling of the stars is another phenomenon produced by certain modifications of atomization of the spherical refraction. The atmosphere is at all times more or less agitated; and on this account the clusters of molecules of which it is composed are constantly undergoing momentary compressions and dilatations, which occasion minute differences of refraction, and consequent changes in the directions of the luminous rays. The minute and rapid variations thus occasioned in the apparent places of the stars produce the twinkling or tremulous appearances which, in certain states of the atmosphere, particularly on the approach of rain after a long drought, are very remarkable. The agitations of the atmosphere are manifest to the eye in the tremulous motion of the shadows cast from high towers, and in looking at objects through the smoke of a chimney, or over beds of hot sand. The scintillation of the planets is much less than that of the stars. This proceeds from the magnitude of their disks, which, although apparently very small, are still far greater than those of the fixed stars, and indeed so considerable, that the accidental variations in the directions of the rays of light are too minute to displace them entirely. The borders of their disks are affected only by a slight undulation; whereas the stars, which are merely brilliant points of insensible magnitude, undergo a total displacement.

The apparent enlargement of the sun and moon in the horizon is an optical illusion, connected in some measure with the atmosphere, of which various explanations have been given since the time of Ptolemy. According to the ordinary laws of vision, the celestial bodies, particularly the moon, which is nearest to the earth, ought to appear largest in the meridian, because their distance is then less than when they are near the horizon; and yet daily experience proves that the contrary takes place. To an observer placed at E (fig. 10), the visual angle subtended by the moon in the horizon at M is somewhat less than that under which she appears in the zenith at O; and this fact, a consequence indeed of her circular motion, is proved by accurate measurements of her diameters in those circumstances by the micrometer. The mean apparent diameter of the moon, at her greatest height, is 31' in round numbers, but in the horizon she seems to the eye two or three times larger. The commonly received explanation of this phenomenon was first given by Descartes, and after him by Dr Wallis, James Gregory, Malebranche, Huygens, and others, and may be stated as follows. The opinion which we form of the magnitude of a distant body does not depend exclusively on the visual angle under which it appears, but also on its distance; and we judge of the distance by a comparison with other bodies. When the moon is near the zenith there is no interposing object with which we can compare her, the matter of the atmosphere being scarcely visible. Deceived by the absence of intermediate objects, we suppose her to be very near. On the other hand, we are used to observe a large extent of land lying between us and objects near the horizon, at the extremity of which the sky begins to appear; we therefore suppose the sky, with all the objects which are visible in it, to be at a great distance. The illusion is also greatly aided by the comparative feebleness of the light of the moon in the horizon, which renders us in a manner sensible of the interposition of the atmosphere. Hence the moon, though seen under nearly the same angle, alternately appears very large and very small. Desaguliers illustrated the doctrine of the horizontal moon on the Theoretical supposition of our imagining the visible heavens to be Astronomy only a small portion of a spherical surface, as in fig. (fig. 10), in which case the moon, at different altitudes, will appear to be at different distances, and therefore seem to vary in magnitude, as at m, n, o. It is evident, however, that this affords no explanation of the phenomenon; for why does the sky, it may be asked, appear to be a smaller segment than a hemisphere? In the solution of this question the whole difficulty is contained.

It may be remarked that these illusions disappear as soon as the intermediate objects are concealed from view. They may be destroyed by regarding the moon through a tube, which permits her disk alone to be seen, and in this manner she will appear no larger at the horizon than near the zenith. The same effect may be produced by viewing her through a smoked glass, because the dark tint permits only the luminous object to be seen, and conceals all the rest. The only precaution necessary to be observed in making this experiment, is to place the eye in such a position that the surrounding bodies may not be visible.

CHAP. II.

OF THE SUN.

SECT. I.—Of the Apparent Circular Motion of the Sun in the Ecliptic, and Position of the Ecliptic in Space.

The consideration of the diurnal motion common to all the celestial bodies, and the succession of day and night resulting from it, forms the first object of astronomy. The second is to pass from the diurnal motion of the sphere to the proper motion of the sun, and from the vicissitudes of the day to the seasons of the year. The sun, as has already been remarked, is constantly shifting his place among the stars. If we observe the altitude of any star, or group of stars, above the eastern horizon at sun-set, we shall find, on making the same observation a few days afterwards, that its elevation is considerably increased, and that it has approached nearer to the meridian. At the end of three months it will appear at sun-set on the meridian, and from that time continue to advance nearer and nearer to the sun, till it is at last concealed by the splendour of his rays. After remaining for some time invisible, it will again make its appearance in the morning to the westward of the sun, and its distance from him will continue to increase daily, till, at the end of a year, it has made a complete circuit of the sky, and regained the position it occupied at the time of the first observation. The earliest observers explained this phenomenon by supposing the diurnal motion of the sun to be less rapid than that of the stars, in consequence of which he constantly falls behind them. This supposition would be admissible if the sun remained constantly in the plane of the equator; but as he deviates considerably from that plane, on both sides of it, it is infinitely more simple to ascribe to the sun a motion of his own, independent of the diurnal motion, and performed in a contrary direction, in virtue of which he traces an oblique route among the stars.

The oblique path of the sun may be determined by observing the positions of the stars near which he successively passes in the course of the year. The same object is however accomplished with much greater precision by the modern practice of observing his declinations and right ascensions, that is to say, his distances from the equator, and an arbitrary meridian. In this manner his place may be accurately assigned every day; and, by a repetition of similar observations, a series of points will be obtained on the surface of the celestial sphere, which mark out his annual course. The result of constant experience shows, that the declination reaches its maximum on the south side of the equator about the 22d of December, when it amounts to 23°46′ degrees. From this time it gradually diminishes till about the 21st of March, when the sun reaches the plane of the equator. At this time the days and nights are of equal length all over the earth, and the instant of time at which the sun’s centre is in the equatorial plane is called the instant of the equinox. The sun then appears on the opposite side of the equator, and his declination or meridional altitude continues to increase till about the 22d of June, when he becomes stationary, and then again shapes his course towards the equator. His maximum declination on the north side of the equator is exactly equal to that on the south, amounting to 23°46′. The sun now continues to approach the equator till about the 24th of September, when he again reaches that plane, and a second equinox succeeds. Continuing still to move in the same direction, he declines from the equator southward, till he reaches his former limit about the 22d of December, after which he resumes his former course.

The two small circles of the sphere, parallel to the equator, which pass through the two points where the declination is great, are called the Solstices, or the Tropics; that on the northern hemisphere is called the Tropic of Cancer, and the other is called the Tropic of Capricorn. These two parallels, which mark the extreme limits of the sun’s declination, are, as has just been stated, equally distant from the equator, with regard to which the variations of declination on either side are perfectly symmetrical and uniform.

If at the time of the vernal equinox we remark the stars which set in the true west while the sun is rising in the east, and which are then separated from him by a semi-circumference, it will be found that the difference of their right ascensions, and the right ascensions of those which have precisely a similar situation relatively to the sun at the time of the autumnal equinox, is exactly 180°. It follows, therefore, that the solar orbit intersects the equator in two points diametrically opposite; but we have seen that its northern and southern declinations are equal; hence the orbit projected on the sphere must be a great circle, provided it lies wholly in the same plane. Whether this is the case or not it will be easy to prove by means of a few observations, in the following manner.

Let AQ (fig. 11) be the equator, AE the orbit of the sun, PSM, PTN two circles of declination, drawn through any two points S and T of the orbit. If the sun’s path is confined to a plane, then AE must be a great circle, and we shall have the equation sin AM = tan MS according to the well-known properties of spherical triangles. Let the cotangent of the unknown but constant angle MAS = n, the declinations MS = D, NT = D′; the right ascensions AM = ξ, AN = ξ′; then, according to the above formula, we must have sin ξ = n tan D, and sin ξ′ = n tan D′, at whatever points of the orbit S and T may be situated. From these two last equations there result also cos ξ = √(1 - n² tan² D), cos ξ′ = √(1 - n² tan² D′), which, being substituted in the trigonometrical formula sin (ξ′ - ξ) = sin ξ′ cos ξ - cos ξ′ sin ξ, we shall have sin (ξ′ - ξ) = n tan D′√(1 - n² tan² D) - n tan D√(1 - n² tan² D′). The observations of the meridional altitudes will give the declinations D and D′; and the difference of right ascensions, ξ′ - ξ, will be found by comparing the time of the sun’s culmination, or transit over the meridian, with that of a star. If, therefore, it is found that, by assigning a certain constant value to \( n \), this equation will satisfy all the observations, combined by pairs, of the sun's right ascensions and declinations, it will follow that the plane determined by any two points in the sun's course and the point in which it intersects the equator, has always the same inclination to the equator; in other words, all the planes so determined are identical.

Now the observations of the sun's right ascensions and meridional altitudes, which have been made daily during so great a number of years, and under so many different meridians, are found to conform entirely with the preceding formulae: they therefore furnish so many proofs that the projection of the sun's orbit is a great circle of the celestial sphere, and that the orbit itself is wholly confined to the same plane.

The great circle which the sun describes in virtue of his proper motion is called the Ecliptic. It has received this name from the circumstance that the moon, during eclipses, is either in the same plane or very near it. These phenomena can in fact only happen when the sun, earth, and moon are nearly in the same straight line, and, consequently, when the moon is in the same plane with the earth and the sun. The angle formed by the planes of the ecliptic and equator, and which is measured by the arc of a circle of declination intercepted between the equator and tropic, is called the Obliquity of the Ecliptic. The two points in which the equator and ecliptic intersect each other are called the Equinoctial Points; they are also denominated the Nodes of the Equator; and the straight line conceived to join them is the Line of the Equinoxes, or the Line of the Nodes. The node through which the sun passes on coming from the south to the north of the equator is called the Ascending Node, and is usually distinguished by the character \( \alpha \); the opposite node is the Descending Node, and is marked by \( \beta \). A straight line passing through the centre of the earth, perpendicular to the plane of the ecliptic, is called the Axis, and the points in which its prolongation meets the sphere are called the Poles of the Ecliptic; these denominations being analogous to those of the axis and poles of the equator. The two small circles of the sphere which pass through the poles of the ecliptic, and are parallel to the equator, are called the Polar Circles.

The ecliptic has been divided by astronomers, from time immemorial, into twelve equal parts, called Signs, each of which consequently contains 30 degrees. The names and symbols by which they are characterized are as follows:

| North of the Equator | South of the Equator | |----------------------|---------------------| | Aries | Libra | | Taurus | Scorpio | | Gemini | Sagittarius | | Cancer | Capricornus | | Leo | Aquarius | | Virgo | Pisces |

In each of these signs the ancients formed groups of stars, which they denominated asterisms, constellations, animals (\( \zeta \omega \mu \alpha \)), not confined to the ecliptic, but included within an imaginary belt, extending 8° on each side of it, to which they gave the name of Zodiac (\( \zeta \omega \delta \iota \sigma \alpha \epsilon \tau \alpha \nu \iota \sigma \), circle or zone of the animals). The term sign is now employed only to denote an arc of 30°, and will probably soon be banished entirely from the astronomical tables. It is already confined to the tables of the planets. Thus, to denote that the longitude of a planet is 276° 12', it is usual to write 9° 6° 12'. Formerly it was usual to employ the characteristic symbol, and to write \( \gamma \) 6° 12', meaning that the planet was 12' in the 6th degree of Capricornus, or the tenth sign. This inconvenient practice is now laid aside, and the signs, when they are employed, are simply distinguished by the ordinal numbers.

As the greater part of the celestial phenomena connected with the planetary system take place either in the ecliptic or in planes not greatly inclined to it, it is found to be most convenient to refer the positions of the planets, and frequently those of the stars also, to that plane. The first point of Aries, which is the technical expression for the intersection of the ecliptic and equator, or the place of the sun at the vernal equinox, is assumed as the origin from which the degrees of the equator, as well as of the ecliptic, are counted from west to east, or in the direction of the sun's annual motion. The angular distance of the sun from this point is called his Longitude; and the longitude of a star is the arc intercepted on the ecliptic between the same point and a great circle passing through the star perpendicular to the ecliptic. The arc of this circle intercepted between the star and the ecliptic, or, which is the same thing, the complement of the star's distance from the pole of the ecliptic, is called the Latitude of the star; so that longitude and latitude are with regard to the ecliptic what right ascension and declination are with regard to the equator. Thus, let \( O \gamma Q \) (fig. 12) be the equator, \( L \gamma P \) the ecliptic, \( P \) and \( Q \) the north poles of these two circles respectively, and \( S \) the place of a star. Having drawn through the pole of the ecliptic and the star the great circle \( ESM \) perpendicular to the ecliptic in \( M \), then \( \gamma M \) is the longitude of \( S \), counted on the ecliptic from the vernal point \( \gamma \) towards the east; and \( SM \), the distance of the star from the ecliptic, is its latitude; and for this reason all great circles passing through the poles of the ecliptic are called Circles of Latitude. The place of a star is thus determined by its longitude \( \gamma M \), and its latitude \( SM \), as well as by its right ascension \( \gamma R \), and its declination \( SR \); and when the angle \( L \gamma Q \) is known, it is easy to pass from the one system of co-ordinates to the other by means of the formulae of spherical trigonometry. These formulae are of constant use, for it is the declinations and right ascensions only which are directly observed.

The plane of the sun's orbit will be determined completely when its inclination to the equator and the position of the line of the nodes in space have been made known by observation. The declination of the sun twice a year, namely, at the summer and winter solstice, is equal to the obliquity of the ecliptic; whence, if the solstice happened exactly at mid-day, the obliquity would be given directly by an observation of his meridional altitude. This circumstance, however, can happen only for one terrestrial meridian; but as the declination of the sun when he approaches the tropics varies little from one day to another, his greatest observed declination will be a very near approximation to the obliquity, at whatever part of the earth the observation may have been made. It is easy, however, to correct the error which results from the observation being made on a meridian different from the solstitial colure. Taking an example from Woodhouse, let us suppose the sun's declination to be observed on three successive days (the 20th, 21st, and 22nd of June), and found to be on these days respectively, 23° 27' 37", 23° 27' 41", 23° 27' 20"; then it is obvious, that if the middle observation gave the greatest inclination exactly, the other two would differ from it equally, which they do not. The maximum declination is therefore a quantity somewhat different from 23° 27' 41"; and it is easy to conclude, from the inspection of the numbers, that it is nearer to 23° 27' 41" than to either of the other two. It is obvious, therefore, that that observation must have been made within 12 hours of the Theoretical time when the sun was exactly in the solstitial point. In Astronomy, order to form an exact notion of the amount of error which may possibly arise from this circumstance, let S be the place of the sun at the time of the observation, and X the true but as yet unknown solstitial point, and Sx meridional arcs intersecting the equator in s and x. The arc SX must be less than 30°, for in 12 hours the variation of the sun's longitude does not exceed an arc of that magnitude. Suppose it 30°; then, by Napier's rules,

\[ \text{rad.} \times \sin Ss = \sin \varphi \times \sin S\varphi, \] and rad. \( \times \sin Sx = \sin \varphi \times \sin X\varphi; \)

whence, on eliminating \( \sin \varphi, \) and observing that

\[ \sin X\varphi = \sin 90° = 1, \]

we shall have

\[ \frac{\sin Xx}{\sin S\varphi} = \cos SX. \]

By taking, according to the observation, \( Ss = 23° 27' 41'' \), and \( SX = 30°, \) we shall find from the logarithmic tables \( Xx = 23° 27' 44'' 5. \) It will be observed that 30° is the maximum error in longitude; if instead of 30° it had been supposed only 3°, the corresponding error in declination would have amounted only to 0° 06' 08.5. In the example chosen, the error of longitude is about 20', whence the error of declination is 1° 5 nearly, and consequently the resulting obliquity differs little from 23° 27' 42'' 5. This result is, however, to be understood of the apparent obliquity, which is subject to slight variations, depending on the longitudes of the moon's nodes: the mean obliquity, deduced from the comparison of a great number of observations, both of the summer and winter solstice, may be regarded as amounting to 23° 27' 41'' at the commencement of the year 1830. We shall see, when we come to speak of the Nutation, in what the difference between the apparent and mean obliquity consists.

When the mean value of the obliquity of the ecliptic, as determined by the delicate instruments of the present day, is compared with that given by ancient observations, it appears to have undergone a progressive diminution, and is always greater as the observation is more remote. The ancient observers were not, indeed, possessed of the means of determining an element of this sort with great precision; but as all the observations recorded in history agree in making the obliquity greater in former times than it is now, the probability is almost infinite that the angle formed by the planes of the equator and ecliptic has really diminished; for, had the differences of the values assigned to it arisen solely from errors of observation, they would have been in excess and defect indifferently, instead of being, as they are, uniformly in excess. The various observations and traditions by which the progressive diminution of the obliquity is confirmed have been collected by Bailly; in the following table we have inserted those which appear to be the best authenticated, and have added the results of some recent observations, which exhibit the present value of the obliquity, and the rate of its diminution.

| Year | Name of Observer | Obliquity | |------|-----------------|-----------| | Before Christ | Eratosthenes, confirmed by Hipparchus and Ptolemy | 23° 51' 15" | | | The Chinese | 33° 45' 52" | | After Christ | Arabians at Bagdad | 23° 33' 52" | | | Albategnius | 23° 35' 40" | | | Almansor | 23° 33' 30" | | | The Chinese | 23° 32' 12" | | | Ulugh Beigh | 23° 31' 58" | | | Walther | 23° 29' 47" | | | Tycho | 23° 29' 52" | | | Riccioli | 23° 30' 20" | | | Hevelius | 23° 29' 10" | | | Cassini | 23° 29' 00" |

| Year | Name of Observer | Obliquity | |------|-----------------|-----------| | After Christ | Flamsteed | 23° 28' 48" | | | Bianchini | 23° 28' 35" | | | Condamine | 23° 28' 24" | | | Cassini de Thury| 23° 28' 26" | | | Lacaille | 23° 28' 19" | | | Bradley | 23° 28' 15" | | | T. Mayer | 23° 28' 16" | | | Maskelyne | 23° 28' 10" | | | Cassini | 23° 27' 54" | | | Maskelyne | 23° 27' 56" | | | Piazzi | 23° 27' 56" | | | Delambre | 23° 27' 57" | | | Pond | 23° 27' 48" | | | Bessel | 23° 27' 47" | | | Dr Brinkley | 23° 27' 49" | | | Dr Pearson | 23° 27' 44" |

Although the comparison of these observations with one another gives very discordant results relatively to the law according to which the obliquity varies, their totality places the fact of its progressive diminution beyond all manner of doubt. Lalande, who was followed by the greater number of astronomers, estimated the diminution of 44" in a hundred years; a result to which he was led chiefly by a comparison of his own observations with those of Walther. Lalande, after comparing an immense number of modern observations with those of the 17th, 16th, and 15th centuries, and also with those of the Arabians and Chinese, found the secular diminution to be 50". Bessel, rejecting the ancient observations as too uncertain, and comparing those only which have been made since the time of Bradley, has fixed the secular diminution at 45" 7", which differs inconsiderably from the determination of Lacaille. It cannot yet be determined by observation whether this diminution is uniform, or accelerated, or retarded; but so slow is the rate at which it proceeds, that it may, without any sensible error, be regarded as uniform for many centuries to come.

The gradual diminution of the obliquity of the ecliptic might lead us to suppose that a time will ultimately arrive when that plane will coincide with the equator, and the earth be deprived, in consequence, of the agreeable vicissitude of the seasons. But the theory of universal gravitation, which has revealed the cause of the diminution, has also shown that there are certain limits which the angle of the two planes can never exceed, and between which it must continue for ever to oscillate. Geometers have not yet ventured to assign the precise extent of these limits, but their existence is certain; and the planes of the ecliptic and equator, which have been approaching to each other during the last 2000 years, will, in the course of some thousands of years more, begin to recede.

In what has yet been said respecting the diminution of the obliquity of the ecliptic, no fact has been mentioned from which it can be inferred, whether the phenomenon is tic- Theoretical occasioned by the displacement of the plane of the ecliptic or that of the equator. This question may also be decided by a comparison of modern with ancient observations; for it is evident, that if the inclination of these two planes becomes less, the stars which are situated between them, particularly those near the solstitial colure, will appear to approach to that plane which changes its position; so that if the ecliptic is displaced, the latitudes of those stars will be diminished, or their declinations if the displacement belongs to the equator. It was first observed by Tycho, and the observation has been confirmed by succeeding astronomers, that the latitudes of the southern stars situated near the solstitial colure, that is, of those stars whose longitudes are nearly 90°, have diminished upwards of 20' since the time of Hipparchus and Ptolemy, while the latitudes of the northern stars have undergone a corresponding augmentation. From this fact it is proved that the diminution of the obliquity is occasioned by the displacement of the ecliptic; and theory has shown that the cause of the displacement is the action of the planets, particularly of Jupiter and Venus, on the earth, by virtue of which the plane of the earth's orbit is drawn nearer to the planes of the orbits of these two planets. This, however, though by far the most considerable, is not the sole cause of the phenomenon; for theory also shows that a slight motion of the plane of the equator is produced by the attraction of the sun and moon, but so very minute that its effects will only become appreciable after a long series of ages.

After determining the inclination of the plane of the ecliptic to that of the equator, the only element requisite to fix its position absolutely in space, is the situation of the straight line formed by its intersection with that plane, that is to say, the line of the nodes. The longitudes of the stars, as has already been mentioned, are counted on the ecliptic from the vernal equinox; and therefore, if the line of the equinoxes, which is the same as the line of the nodes, is invariable, the longitude of any star will always be the same, whatever interval of time may elapse between two observations of that longitude. But on comparing the actual state of the heavens with the observations recorded by ancient astronomers, it is perceived that the longitudes of all the stars are considerably increased; whence we must infer, either that the whole firmament has advanced in the order of the signs, or that the equinoctial points have gone backwards, or retrograded. The latter supposition is infinitely the more probable; for it is inconceivable that the innumerable multitude of stars should have a common motion relatively to points which depend solely on the motion of the earth. The phenomenon is therefore to be explained by attributing to the equinoctial points a retrograde motion from east to west, in consequence of which, the sun, whose motion is direct, arrives at them sooner than if they remained at rest; and therefore the equinoxes, spring, autumn, and the other seasons, happen before the sun has completed an entire circuit. On this account the motion has been denominated the Precession of the Equinoxes. As this motion is extremely slow, its exact amount can be discovered only by a comparison of observations separated from each other by long intervals of time; but the imperfection of instruments prior to the sixteenth century renders the ancient observations of little authority where quantities so minute are concerned, and therefore some discrepancies may be expected in the different determinations of the amount of the precession. The comparison of modern observations with those of Hipparchus gives its annual amount equal to 50\(\frac{3}{4}\) seconds, and with those of Ptolemy somewhat greater. The mean result of the observations of Tycho, compared with those of Lacaille, gives 50\(\frac{1}{2}\). On comparing modern observations with one another, we find Astronomy \(= 50°06'\). Delambre, in his solar tables, supposes the annual precession to be equal to 50°1'. According to this estimate the equinoctial points go backwards at the rate of one degree in 71-6 years nearly, and therefore will make a complete revolution of the heavens in about 25,968, or nearly 26 thousand years.

The discovery of the precession of the equinoxes is generally attributed to Hipparchus, who, on comparing his own observations with those of Timocharis, more ancient by 160 years, perceived that in this interval the longitudes of the stars had been augmented by about two degrees. It would seem, however, from many proofs collected by Bailly, that this motion, slow as it is, was known to all the ancient nations who cultivated astronomy, long before the time of Hipparchus. It is indeed easy to conceive, from the great attention which they gave to the heliacal risings of the remarkable stars, that they might observe a gradual change of the seasons at the occurrence of these phenomena, from which they would necessarily be led to conclude a variation of the star's longitude. In consequence of this regression of the equinoctial points, the sun's place among the zodiacal constellations at any given season of the year is now greatly different from what it was in remote ages. Some time prior to Hipparchus, the first points of Aries and Libra corresponded to the vernal and autumnal equinoxes; those of Cancer and Capricorn to the summer and winter solstices; at present these constellations have receded 80 degrees from the same points of the ecliptic. The vernal equinox now happens in the constellation Pisces, the summer solstice in Gemini, the autumnal equinox in Virgo, and the winter solstice in Sagittarius. Astronomers, however, still count the signs from the vernal equinox, which, therefore, always corresponds to the first point of the Sign of Aries. On this account it is necessary to distinguish carefully between the Signs of the Zodiac, which are fixed with regard to the equinoxes, and the Constellations, which are movable with respect to those points.

The diminution of the obliquity of the ecliptic arises from the displacement of the ecliptic itself; the precession of the equinoxes is, on the contrary, occasioned by the continual displacement of the plane of the terrestrial equator. This displacement results from the combined action of the sun and moon (for the influence of the planets amounts only to a fraction of a second, and is consequently scarcely sensible,) on the mass of protuberant matter accumulated about the earth's equator, or the matter which forms the excess of the terrestrial spheroid above its inscribed sphere. The attracting force of the sun and moon on this shell of matter may be resolved into two; one parallel to the plane of the equator, the other perpendicular to it. The tendency of this last force is to diminish the angle which the plane of the equator makes with that of the ecliptic; and if the earth had no motion of rotation, it would soon cause the two planes to coincide. In consequence, however, of the rotatory motion of the earth, the inclination of the two planes remains constant; but the effect produced by the action of the force in question is, that the plane of the equator is constantly shifting its place, in such a manner that the line of the equinoxes advances in the direction of the diurnal motion, or contrary to the order of the signs, its pole having a slow angular motion about the pole of the ecliptic, so slow indeed, that it requires nearly 26,000 years to complete its revolution.

If the sun and moon moved in the plane of the equator, there would evidently be no precession; and the effect of Theoretical their action in producing it varies with their distance from that plane. Twice a year, therefore, the effect of the sun in causing precession is nothing; and twice a year, namely at the solstices, it is a maximum: on no two successive days of the year is it exactly the same, and consequently the regression of the equinoctial points, which results from the sun's action, must be unequal.

On this account the obliquity of the ecliptic is subject to a semi-annual variation; for the sun's force, which tends to produce a change in the obliquity, is variable, while the diurnal motion of the earth, which prevents the change from taking place, is constant. Hence the plane of the equator is subject to an irregular motion, which is technically called the Solar Nutation. The existence of the solar nutation is, however, only a deduction from theory, for its amount is too small to be perceptible to observation; but a similar effect of the moon's action is sufficiently appreciable, and was, in fact, discovered by Dr Bradley before theory had indicated its existence. Its period, however, is different, and depends on the time of the revolution of the moon's nodes, which is performed in 18 years and about 7 months. During this time the intersection of the lunar orbit with the ecliptic has receded through a complete circumference; and the inequality of the moon's action will consequently, in the same time, have passed through all its different degrees. Bradley observed that the declinations of the stars continued to augment during nine years, that they diminished during the nine years following, and that the greatest change of declination amounted to 18°. He remarked further, that this motion was connected with an irregularity of the precession of the equinoxes, which followed exactly the same period; whence he concluded that the motion of the poles of the equator, occasioned by this vibration of its plane, was not confined to the solstitial colure. A series of observations on stars differently situated proved that all the phenomena could be explained on the hypothesis that the pole of the equator describes in 18 years a small circle of 18° diameter, contrary to the order of the signs; or that the axis of the world, following the circumference of this circle, describes the surface of a cone, the axis of which forms with its side an angle of 9°. This apparent vibratory motion is significantly denominated the Nutation of the Earth's Axis.

In consequence of the two motions which occasion the precession and the nutation, the true path of the pole of the equator round that of the ecliptic is an epicycloidal curve, which will be understood by referring to fig. 14. Let E be the pole of the ecliptic, round which the pole of the equator P describes, in virtue of the precession, and in a direction contrary to the order of the sines, the circle PQR, of which the radius EP is equal to the obliquity of the ecliptic, or the mean distance of the two poles. While P, the mean place of the pole, moves in the circle PQR, with a velocity equal to the regression of the equinoctial points, or at the rate of 50°/1 a year, the true pole p describes at the same time round P a small circle of 18° diameter in the same direction pqr. The true path of the pole is therefore along the circumference of a circle pqr, the centre of which retrogrades on the circumference of another circle, and consequently moves in an epicycloid abcdefg, the curve which results from the composition of the two motions. Suppose the mean pole at Q, the true pole at a, and aQ = 9°. In the course of nine years the mean pole will have retrograded from Q to R, making QR = 9 × 50°/1, while the true pole will have accomplished a semi-revolution in its circle. It will therefore be at e (Re being = 9°), and have described the epicycloidal arc abe, the greatest distance of which from the circle PQR is at b, and equal to 9°. At the end of the following nine years the mean pole has retrograded from R to S, or 9 × 50°/1, and the true pole has returned to the same point a of its epicycle; it will therefore be found at e, Se being = 9°. In this interval it has necessarily been within the circle PQR, its greatest distance from which at d is 9°, so that in 18 years it has traced the curve abede. But ac = QR + aQ + Re = 9 × 50°/1 + 18° = 7° 49°, and ac = RS − Re − Se = 9 × 50°/1 − 18° = 7° 13°. From a to e the motion of the true pole in the epicycle is in the same direction as that of the mean pole; from e to a it is in an opposite direction, but as it is always much slower, being only about 3° in a year, while that of the mean motion is 50°/1, the true motion which results from the combination of both will always be in the direction ae, and at d its velocity will be equal to their difference. At b and d the difference between the latitudes of the true and mean pole is a maximum, while the difference of their longitudes is nothing; in other words, the correction of the obliquity is greatest when that of the precession vanishes; at a, c, and e the correction of the obliquity vanishes, and that of the precession is a maximum.

Dr Bradley remarked that the effects of the nutation would be represented still more accurately by supposing the curve described by the pole of the equator about its mean place to be an ellipse instead of a circle, the transverse and conjugate axes being 18° and 16° respectively. This is also confirmed by theory, from which Laplace calculated the semi-axes of the ellipse at 9°'63 and 7°'17. The semi-transverse axis of the ellipse described by the pole in virtue of the sun's action alone does not exceed half a second, and is consequently totally inappreciable. The sensible part of the nutation, therefore, follows exactly the period of the revolution of the nodes of the moon. From a series of 810 observations, embracing three complete revolutions of the moon's node, the major and minor semi-axes of the ellipse have been found by M. Lindenau to be 8°'977 and 6°'682 respectively.

It is now easy to see the reason of the distinction drawn above between the mean and the true or apparent obliquity. The mean obliquity is represented by the radius EP of the deferent circle PQR, along the circumference of which the centre of the small circle or ellipse is carried, while the true obliquity is that quantity increased or diminished by the nutation. The calculation of the mean obliquity from the true is performed by the aid of the astronomical tables.

The progressive diminution of the mean obliquity and the nutation of the earth's axis are inequalities distinguished from each other, not only by their being derived from different and distinct causes, but still more by the very great difference of time required for their full development. Almost every other element of the planetary system is affected in a similar manner by inequalities of two kinds, which are distinguished by the terms secular and periodic. The secular inequalities proceed with extreme slowness, and continue progressive in the same sense during many centuries; while the periodic are much more rapid in their march, and run through the whole period of their changes in comparatively short intervals of time. The inequalities of the first kind are also periodic; but their periods are vastly longer, and may be reckoned by centuries instead of years; and from this circumstance they derive their name of secular inequalities.

Sect. II.—Of the Orbit of the Sun.

Having now considered the situation of the plane in which the sun is observed to move relatively to the fixed stars, and also the small secular and periodic variations Astronomy.

The angular velocity of the sun's motion in his orbit is not uniform, is obvious from the fact that he remains 7½ days longer in the northern than in the southern signs, or, which is the same thing, that the interval between the vernal and autumnal equinoxes is 7½ days longer than the interval between the autumnal and vernal. It is proved by numerous observations that the sun moves with the greatest velocity when at a point situated near the winter solstice; while at the opposite point of the orbit, or near the summer solstice, his velocity is the least. At the first point the diurnal motion is 1°01943, and at the second only 0°95319. It is constantly varying between these two points; and the variation is observed to be nearly proportional to the sun's angular distance from the point of his orbit where his velocity is a maximum or minimum. The mean velocity is 0°98632, or nearly 59'11", which is the rate of the sun's daily motion about the beginning of April and October.

The point of the solar orbit, which is the most remote from the earth, is called the apogee (ἀπόγειον), away from the earth); that which is nearest is called the perigee (περίγειον, near the earth). The same points are also called respectively the superior and inferior Apsis of the orbit, and the straight line which joins them is called the Line of the Apsidies.

The exact determination of the sun's diameter is a problem which engaged the earliest astronomers, but of which, before the invention of the telescope and micrometer, it was impossible to obtain a solution sufficiently accurate to make known its variations. Archimedes, by an extremely ingenious though imperfect method, demonstrated that it must be included between the limits \( \frac{1}{2} \) and \( \frac{1}{3} \) of a right angle, that is, between 27° and 33°7'. The Egyptians, by observing the time which the sun takes to rise above the horizon, found it to be between a 750th and a 700th part of a circumference, that is, between 28°48' and 30°51'5. Aristarchus of Samos supposed it to be 30°. The precision of modern observations shows that the apparent diameter is greatest about the time of the winter solstice, and least about the summer solstice; but there is some discrepancy among the results of different astronomers with respect to its actual magnitude. According to Lalande, the greatest apparent diameter, about the end of December, is 32°35'5; the least, about the end of June, 31°30'5. According to the tables of Delambre, its greatest value is 32°35'6, and its least 31°31'; the mean apparent diameter, or the diameter at the sun's mean distance, is equal to 32°2'9.

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, however, that the variation in his angular velocity may not be real, but only apparent. Thus, in fig. 15, let AMPN be the orbit of the sun, C the centre of that orbit, and E the position of the earth at some distance from the centre. It is obvious that P is the sun's perigee, and A his 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 Theoretical 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 ac 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 pp', 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 intersect 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.

But if the variation in the angular motion of the sun's motion were owing alone to the eccentric position of the earth within the solar orbit, 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.

The observation, that the sun's angular motion in his orbit is inversely proportional to the square of his distance from the earth, is due to Kepler. The discovery was made by a careful comparison of the sun's diurnal motion with his apparent diameter, which is inversely proportional to his distance from the earth. Let ASB (fig. 16) be the sun's orbit, E the earth, and S' the sun. Suppose a line ES joining the centres of the earth and sun to move round along with the sun. 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 SES'. In the same time that S performs one revolution in its orbit, the radius vector ES will describe the whole area ABS inclosed within the sun's orbit. Let SS' be the sun's angular motion during one day. It is evident that the small sector SES' 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 describes a constant quantity, whatever the angular motion of the areas produced; and the whole area SEA increases as the number proportional of days which the sun takes in moving from S to A, times. Hence results that remarkable law, first pointed out by Kepler, that the areas described by the radius vector are proportional to the times of description. Suppose the sun to describe SS' in one day, and SA in twenty days; then Theoretical 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. 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 (fig. 17) represent the earth, and Ea, Eb, Ec, Ed, Ea, &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 aei 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, and thus proved that the orbit of the sun is an ellipse, having the earth in one of its foci.

Having arrived at the knowledge of the true nature of the sun's orbit, it becomes necessary, in the next place, to determine its position on the plane of the ecliptic, that is, to assign the position of the transverse axis, or line of the apsides, with reference to some other straight line given by position on that plane. The line which it is most convenient to assume for this purpose is the line of the equinoxes; and the position of the apsides may be determined from observations of the time which the sun occupies in performing a semi-revolution, counting from different points of his orbit. Thus, if among the observations of the sun's longitude made daily during the course of a year, we compare two and two, all those which are diametrically opposite, or which differ by 180°, it will be found that the interval between them will be somewhat less than half a year, if the sun, during that interval, has passed through his perigee, or longer if he has passed through his apogee; and that the time between the two observations will differ less from half the time of a whole revolution, in proportion as the sun's place, at the time of the two observations, was nearer to the apsides. If the difference of time between the two observations is exactly half a year, then the place of the apsides would be obtained at once, because the sun must have been in those points when the observations were made. The probability is, however, infinitely small, that this can ever happen exactly; but as the position of the apsides is known very nearly from the diurnal observations of the variation of the sun's angular velocity, the necessary corrections can easily be supplied by computation.

From the comparison of observations made in different ages, it appears that the position of the apsides is not fixed on the plane of the ecliptic, but that the greater axis of the solar ellipse revolves in the direction of the sun's annual motion. The observations of Hipparchus, compared with those of the present times, show that the apsides have a direct motion at the rate of 65° in a year. According to the observations of Walther, the longitude of the apogee in 1496 was 93° 57' 57"; and according to Lacaille, the same element in 1750 was 98° 37' 28"; whence, dividing the difference by 254, the interval between the two epochs, the annual motion of the line of the apsides amounts to 66°. Delambre found, by the comparison of a great number of modern observations, that the annual motion is 61° 95' a year. The theory of attraction, which, in respect of such slow and minute variations, must be considered as giving more accurate results than can be obtained directly by observation, gives 61° 9' for the yearly progressive motion of the apsides.

In the above determination, the position of the apsides is referred to the line of the equinoxes, and their motion compared with the sun's tropical revolution. But if we wish to determine their motion with reference to the fixed stars, it is necessary to have regard to the retrograde movement of the equinoctial points; for it is obvious, that if the line of the equinoxes is not fixed, the displacement of the apsides with respect to the stars will be increased or diminished by the whole amount of its motion, according as the two motions are in the same or opposite directions. Now the motion of the line of the apsides is direct, and has just been stated to amount to 61° 9' in a solar year; that of the equinoxes is, on the contrary, retrograde, and amounts to 50° 1' in the same time. Hence the displacement of the apsides, with reference to a star or fixed point in the ecliptic, is 61° 9' - 50° 1' = 11° 8' a year, in the direction of the sun's annual motion. The time, therefore, in which the solar perigee completes a revolution in the heavens, or returns to the same star, is $\frac{360°}{11° 8'}$ equal nearly to 110,000 years.

Since the greater axis of the solar ellipse has a progressive motion on the plane of the ecliptic, it forms a variable angle with the line of the equinoxes, and at distant epochs will coincide with that line, or be perpendicular to it. The epochs at which these phenomena happen may be easily found by simple proportions, when the longitude of the perigee at any given time, and its annual motion, are known. According to the observations of Lacaille, already quoted, the longitude of the perigee in 1750 was 278° 37' 28"; but when the longitude of the perigee was 270°, the greater axis of the solar ellipse must have been perpendicular to the line of the equinoxes. The difference of these two longitudes is 8° 37' 28", and the number of years requisite to describe that arc, at the rate of 61° 9' annually, is $\frac{8° 37' 28"}{61° 9'} = 500$ years nearly; whence the major axis was perpendicular to the line of the equinoxes in the year 1250, when the perigee of the orbit coincided with the winter solstice.

Suppose it were required to assign the epoch at which the major axis coincided with the line of the equinoxes. At the occurrence of this phenomenon the longitude of the perigee was 180°, consequently from that time to 1750 the perigee had advanced 278° 37' 28" - 180° = 98° 37' 28". Now $\frac{98° 37' 28"}{61° 9'} = 5735$; therefore 5735 is the number of years intervening between the occurrence of the phenomenon and 1750. Hence about 4000 years before the commencement of our era, the transverse axis of the solar orbit coincided with the line of the equinoxes; and it is a singular coincidence that this epoch is considered by chronologists to be that of the beginning of the world, or, to speak more correctly, of the first traces of the existence of the human race; for numerous physical circumstances attest that the earth itself has existed during an infinitely longer period. The same phenomenon will again occur when the longitude of the perigee shall have reached 360°, or become zero; and by calculating as above, this will be found to take place in the year 6485 of our era. The solar perigee will then coincide with the vernal equinox; in the former case it coincided with that of autumn.

By reason of the progressive motion of the perigee, the season length of the seasons is continually varying. Let AP (figs. vary i 18, 19, 20) represent the line of the apsides, S and W the summer and winter solstices, V and O the vernal and autumnal equinoxes. When the greater axis was perpendicular to the line of the equinoxes, as happened in the year 1250 of our era, the perigee P (fig. 18) coincided Fig. 1 between the autumnal equinox O, and the winter solstice W, was equal to the time between W and V, or between the winter solstice and the vernal equinox. But in this position the equator, which is here represented by VO, divides the ellipse into two unequal portions, the smaller of which must be described in less time than the greater, because the times of description are proportional to the spaces passed over. The summer was therefore, at that time, longer than the winter, and both were divided into equal parts by the solstices. In any other position the seasons differ in length from each other. In the time of Hipparchus, the longitude of the perigee was less than in 1750 by $32^\circ - 292$, or was $246^\circ - 330$. The position of the ellipse with regard to the equinoxes was therefore such as is represented in fig. 19; the angle PEW being $270^\circ - 246^\circ = 23^\circ 40' 12''$. The interval between V and S was then $94\frac{1}{2}$ days, and that between S and O only $92\frac{1}{2}$ days. The spring was therefore at that time longer than the summer, and the winter longer than the autumn. The position of the ellipse in 1800 is represented in fig. 20. The angle PEW was then $9^\circ 29'$, and the following were nearly the lengths of the seasons:

| Days | Hours | Min | |------|-------|-----| | from V to S | 92 | 21 | 45 | | S to O | 93 | 13 | 35 | | O to W | 89 | 16 | 47 | | W to V | 89 | 1 | 42 |

The spring is therefore at present shorter than the summer, and the autumn longer than the winter.

After describing the position of the transverse axis of the solar ellipse on the plane of the ecliptic, we come next to consider the species of that ellipse, or its eccentricity, on which depend the apparent inequalities of the sun's angular motion. It is evident that this element will be made known if we can by any means determine the ratio of the two segments into which the major axis is divided by the focus. This ratio might seem to be obtainable without any difficulty by comparing the sun's apparent diameters at the perigee and apogee, to which the distances are inversely proportional; but such observations, on account of the irradiation and other difficulties, are liable to considerable uncertainty; it is therefore necessary to have recourse to other methods of estimating the eccentricity, and the variations to which it is subject. These methods are derived from the fundamental law of the elliptic motion, namely, the proportionality of the areas described by the radius vector to the times of description, by which the inequalities of motion and the ellipticity of the orbit are connected with each other; so that when the inequalities are made known by direct observation, the species of the ellipse can be computed from the principles of geometry. It will, however, be necessary to attend more closely than we have yet done to the phenomena resulting from the eccentric position of the earth in the sun's orbit, and to explain some terms technically employed by astronomers to abbreviate and simplify their descriptions.

The sun's motion may be regarded as composed of two distinct parts, namely, a circular uniform motion, which constitutes the principal part of it; and a correction depending on the deviation of the orbit from a circle, which modifies the first, and alternately accelerates and retards the mean angular velocity. To illustrate this, let E (fig. 21) be the earth, PSA the orbit of the sun, AP the line of the apsides, and $\gamma$ the intersection of the plane of the orbit with that of the equator. While the true sun moves round his elliptic orbit, describing areas proportional to the times, conceive a fictitious sun s to move round the earth in the circumference of a circle of which the radius is equal to EP the sun's distance at the perigee, and with a uniform motion such, that if the two suns set out at the same instant from P, they may also return to the same point together, after having completed each a revolution. At the perigee the radius vector of the real sun is a minimum; his velocity is consequently a maximum, and he therefore advances before the fictitious sun. But his motion relaxes in proportion as his distance from P becomes greater; and at a certain point S of the orbit it becomes equal to that of the fictitious sun, which has then only reached the point s. After passing the point S, the angular distance SEs continues to diminish on account that the motion of the real sun continues to relax, and it vanishes altogether when he reaches his apogee at A, the fictitious sun arriving at a in the same straight line with A and E at the same instant of time. From the apogee to the perigee the phenomena are reversed. The motion of the real sun being then the slowest possible, he at first falls behind the fictitious sun; but his motion continuing to be gradually accelerated, he has again acquired the same velocity when he reaches S', after which his motion is more rapid than that of the fictitious sun, which he at last overtakes at P. If, therefore, we conceive a radius vector to be drawn from the earth to each of the two suns, those lines will form with each other a variable angle, having its maximum value when the velocities are equal, and vanishing at the perigee and apogee. This angle, namely, SEs, is called the Equation of the Centre, or Equation of the Orbit. It is the correction necessary to be made to the longitude of the sun deduced from his equable motion, in order to have his real longitude. Those quantities which form the difference between the true and mean results are, in astronomy, denominated equations. The ancients employed the term Prosthaphaeresis in the same sense.

From the perigee to the apogee, the true longitude of the sun is found by adding the equation of the centre to the mean longitude, and in the other half of the orbit by taking the difference of the same quantities. The equation of the centre is a maximum when the sun is at the points S and S', or when the mean and true motions are equal; and if its value can be obtained from observation at those times, the eccentricity of the orbit may be deduced by means of the geometrical properties of the ellipse. Now the points S and S' may be found very nearly by observing when the diurnal velocity of the sun is equal to the mean motion, or to $0.985647$ parts of a degree. By these observations the angle SES, which is the difference of the real longitudes of the sun, or of his distances from $\gamma$, will be given. But sEs is also given; for it is the angle which would be described by the sun, in virtue of his mean motion, during the interval between the two observations. Hence the difference between sEs and SES is given, being equal to twice SEs the maximum equation of the centre, in consequence of the symmetrical position of the points S and S', and also of s and s', in respect of the line of the apsides AP. The accuracy of a result obtained in this manner may appear questionable, on account of the impossibility of determining by observation the exact moment at which the sun's true motion in longitude is equal to his mean motion; but as the real motion varies very little during a few days before and after that epoch, the equation of the centre will be scarcely affected by a slight error in the time. Besides, any error in the result is corrected by taking the mean of a great number of similar observations. By a comparison of numerous observations of this kind, Delambre found that the greatest equation of the centre, in the year 1776, amounted to $1^\circ 49'25''$, or $1^\circ 53' 31''$. In 1801 the same equation was $1^\circ 53' 27''$. (See Baily's Astronomical Tables, p. 66.)

Having obtained the value of the greatest equation of the centre, the eccentricity of the orbit may be computed from the first two terms of the following series,

$$e = \frac{1}{2}E - \frac{11}{2^4}E^3 + \frac{587}{2^6 \cdot 3^5}E^5 + \ldots$$

in which $e$ represents the eccentricity, and $E$ the greatest equation of the centre. By reversing the series the following expression is found for the greatest equation in terms of the eccentricity, viz.

$$E = 2e + \frac{11}{2^4}e^3 + \frac{599}{2^6 \cdot 3^5}e^5 + \ldots$$

For an investigation of the above formulae the reader may consult Biot, Astronomie Physique, tom. ii. p. 185.

It has been discovered by observation, that the equation of the centre of the sun's orbit is subject to a nearly uniform secular diminution. This fact is confirmed by theory, from which also the secular diminution has been assigned. It amounts to $8''0047$ in a century. This phenomenon implies a corresponding diminution of the eccentricity of the solar orbit amounting to $000037495$ in the same time, the semiaxis major being unit. This will be conceived more distinctly by converting the above fraction into some familiar expression of linear magnitude. Supposing the mean distance of the sun to be $96,000,000$ miles, the fraction $000037495$ will represent nearly $3700$ miles, so that the annual diminution of the eccentricity is nearly at the rate of $37$ miles,—a line which is considerable on the surface of the earth, but which has scarcely an appreciable ratio to the immense distance of the sun. If, however, this diminution, small as it is, were to be continued indefinitely, the eccentricity would ultimately vanish, and the sun's orbit would be changed into a circle; but the theory of universal gravitation proves that, like all the other variations of the elements of the solar system, the variation of the eccentricity is subject to periodic laws. After having continued during a certain time to diminish, the eccentricity will again begin to increase, and will successively pass through all its former values. Thus it will continue to oscillate within certain limits, of which the extent, though not precisely known, cannot be very great; and the solar orbit will eternally preserve its elliptic form, unless the application of some external force shall derange the system of the world, or modify the laws by which it is at present governed.

The true nature of the solar orbit being known, together with its situation on the plane of the ecliptic, and the amount of its eccentricity, as also the time of a revolution, the sun's longitude may be determined at any assigned epoch. This determination is what is usually termed Kepler's Problem, having been first proposed by that great astronomer on the hypothesis of an elliptic orbit. Its solution, which is not susceptible of being exhibited under a finite form, is derived from the principle of the equable description of areas. Let ASP (fig. 22) represent the semi-orbit of the sun, C the centre of the orbit, E the focus occupied by the earth, and let the time and motion be counted from the perigee P. On AP describe a semicircle, on the circumference of which let a point M be supposed to move uniformly, and to describe the semicircle PMA, in the same time in which the sun describes the semi-ellipse PSA. Now, suppose that at the end of a given time $t$, the sun has moved from P to S; then, denoting by T the time of a complete revolution, and putting $\pi$ for the semi-circumference, the place of M, at the end of the same time $t$, will be given by the equation $PM = \frac{T}{\pi} \cdot 2\pi$. Through S let a perpendicular be drawn to AP, meeting the semicircle in D and AP in N, and join EM, ES, ED, CM, and CD. The different angles of this figure have received certain technical names, which it is necessary to explain. The angle PES, which is the true longitude of the sun, is called the True Anomaly; PEM is called the Mean Anomaly, and PCD is denominated the Eccentric Anomaly. This last is measured by the arc PMD, which is the perigee distance of a star describing a circle circumscribed about the solar ellipse, and having constantly the same absciss PN as the sun. The word anomaly originally signified inequality; it now simply designates the angular distance of a planet from its perihelion, seen from the sun, or, as in the present case, the angular distance of the sun from his perigee.

According to the law of the equable description of areas, we have the sector PCM to PES as the surface of the semicircle to the surface of the semi-ellipse; that is, as the semi-transverse to the semi-conjugate axis, or as ND to NS, according to a well-known property of the ellipse. Hence $PCM : PES :: PED : PES$, and therefore $PCM = PED = PCD - ECD$. Now, if we express the arc PM by $z$, PD by $x$, the eccentricity EC by $e$, and make the radius equal to unity, we shall have the sector $PCM = \frac{1}{2}z$, the sector $PCD = \frac{1}{2}x$, and the triangle $ECD = \frac{1}{2}e$. Substituting, therefore, these expressions in the above equation, it will become, on multiplying both sides of it by $2$,

$$Z = x - e \sin x. \quad \ldots \quad (1)$$

where $z$ denotes the mean, and $x$ the eccentric, anomaly.

If in this equation $x$ and $e$ were both known, it would be easy to deduce from it the mean anomaly $z$; but the data are $z$ and $e$, and the value of $x$ can only be obtained in a series, the equation being a transcendental one. Its resolution gives the value of the eccentric in the mean terms of anomaly; and if another equation can be found in which the eccentric anomaly is expressed in terms of the true, it will only remain to put these two expressions equal to one another in order to have an equation between the mean and true anomaly.

For this purpose it will be necessary to equate two values of the radius vector ES, expressed in terms of the eccentric and true anomalies respectively. Putting ES = $r$, we shall have, in the first place,

$$r^2 = EN^2 + NS^2;$$

but if the semiaxis minor is represented by $b$, the common equation of the ellipse gives

$$NS^2 = \frac{b^2}{a^2}(a^2 - CN^2) = b^2 - \frac{b^2}{a^2}CN^2,$$

and we have also

$$EN^2 = EC^2 + 2EC \cdot CN + CN^2 = e^2 + 2e \cdot CN + CN^2;$$

therefore, by addition,

$$NS^2 + EN^2 = b^2 + e^2 + 2e \cdot CN + \left(1 - \frac{b^2}{a^2}\right)CN^2.$$

Now, $b^2 + e^2 = a^2$, and $1 - \frac{b^2}{a^2} = \frac{e^2}{a^2}$; therefore

$$NS^2 + EN^2 = r^2 = a^2 + 2e \cdot CN + \frac{e^2}{a^2}CN^2;$$

consequently

$$r = a + e \cos x. \quad \ldots \quad (2)$$

But $CN = a \cos x$; therefore ultimately

$$r = a + e \cos x. \quad \ldots \quad (2)$$ To obtain a second value of \( r \), let SF be drawn from S to the other focus of the ellipse, and let SF = \( r' \). The rectangular triangles ESN, FSN give \( SN^2 = r^2 - EN^2 \), and \( SN^2 = r^2 - (2e - EN)^2 \); therefore \( r^2 - EN^2 = r^2 - (2e - EN)^2 \), whence we derive \( r^2 - r'^2 = 4e \cdot EN - 4e^2 \). But by the property of the ellipse \( r + r' = 2a \), from which it is easy to deduce \( r^2 - r'^2 = 4e^2 - 4ar = 4ar - 4a^2 \); and hence, by equating these two values of \( r^2 - r'^2 \), \( e \cdot EN - e^2 = ar - a^2 \). Now, if we denote the angle PES, which is the true anomaly, by \( v \), then \( EN = r \cos v \), and consequently by substitution, \( er \cos v - e^2 = ar - a^2 \), whence

\[ r = \frac{a^2 - e^2}{a - e \cos v} \tag{3} \]

If we now compare the two values of \( r \) given by the equations (2) and (3), we shall find

\[ \cos x = \frac{a \cos v - e}{a - e \cos v}, \]

and therefore, \( 1 - \cos x = (a + e)(1 - \cos v) \)

\[ 1 + \cos x = (a - e)(1 + \cos v); \]

whence, by dividing,

\[ \frac{1 - \cos v}{1 + \cos v} = \frac{a - e}{a + e} \cdot \frac{1 - \cos x}{1 + \cos x}; \]

consequently, by the trigonometrical formulae,

\[ \tan \frac{1}{2}v = \left( \frac{a - e}{a + e} \right)^{\frac{1}{2}} \tan \frac{1}{2}x \tag{4} \]

The equations (1) and (2) were first given by Kepler; the last is due to Lacaille. The problem of finding the sun's place after any given time is solved analytically by (1) and (4).

To express \( v \) the true anomaly, in a series involving the powers of \( x \) the mean anomaly, is a problem requiring the aid of the higher analysis, and which it is unnecessary to investigate here. The following are a few of the first terms of the series, in which \( nt \) is substituted for \( x \), \( n \) being the mean motion in the unit of time, and \( t \) the time elapsed since the passage through the perihelion:

\[ v = nt - \left( 2e - \frac{1}{4}e^3 + \frac{5}{96}e^5 \right) \sin nt \]

\[ + \left( \frac{5}{4}e^3 - \frac{11}{24}e^4 + \frac{17}{192}e^6 \right) \sin 2nt \]

\[ - \left( \frac{13}{12}e^5 - \frac{43}{64}e^7 \right) \sin 3nt \]

\[ + \text{etc.} \]

The form of the sun's orbit is discovered from the variations of his apparent diameter in passing from the perigee to the apogee; but in order to obtain a knowledge of its real dimensions, or of the mean distance of the sun from the earth, it is necessary to determine his parallax, that is to say, the angle subtended by the earth's semidiameter as seen from the sun. This angle is too small to be measured directly in the manner which was pointed out when treating of parallax; it is most accurately deduced from observations of the time occupied by an inferior planet in passing over the sun's disk, as will be explained afterwards. From such observations it has been found to be less than \( 9'' \), an angle of which the sine is to radius in the ratio of 1 to 23,000 nearly; whence the distance of the sun is about 23,000 times the semidiameter of the earth. The following table exhibits more accurately the numerical results of calculation.

| Distance of the Sun from the Earth | |-----------------------------------| | Semidiameters of the Earth | | English Miles | | In Perigee | 23580 | 93280945 | | In Apogee | 24388 | 96478967 | | At Mean Distance | 23984 | 94879956 | | Greatest diameter of his orbit | 47969 | 189764357 |

When the sun's parallax is known, we are enabled not only to estimate his distance, but also to compare his diameter and magnitude with those of the earth. For example, the mean parallax of the sun is \( 8''56 \); his mean apparent semidiameter, seen from the earth, is \( 961''4 \); the semidiameter of the sun and earth are therefore to each other in the ratio of \( 961:4 : 8:56 \), or of 112 to unity nearly. Supposing the sun and earth to be both spherical bodies, which is a supposition sufficiently accurate for the present purpose, their volumes will be to one another as the cubes of their radii; hence the volume of the sun is to that of the earth as the cube of 112 to unity, or the sun's volume is about 1,300,000 times greater than that of the earth.

**Sect. III.—Of the Motion of Translation of the Earth, and the Aberration of Light.**

Hitherto we have regarded the sun as being actually in motion round the earth in an elliptic orbit, of which the earth occupies one of the foci. This supposes the motion which we observe to be real. But we have already seen that all the phenomena of the diurnal revolution of the celestial bodies may be equally well explained by supposing these motions to be only optical illusions, of the same nature as those by which a person sailing rapidly along a river, or the sea-shore, is almost irresistibly led to ascribe the motion to the banks on the shore, even when fully aware that the appearance is only occasioned by his own motion in a contrary direction. In the same manner all the phenomena of the annual motion are susceptible of an equally simple explanation, on the hypothesis that the earth revolves in an elliptic orbit, and that the sun is at rest in one of the foci of the ellipse. In fact, if there were no other bodies in the universe than the earth and the sun, it would be matter of absolute indifference which of the two hypotheses we should adopt in order to explain the appearances. Supposing, for example, the earth to revolve in a circular orbit (the eccentricity, on account of its smallness, being here neglected), and the sun to be placed at the centre; a spectator at E (fig. 23) sees the sun S in the heavens at the point A, and corresponding to the star \( a \) on the celestial sphere. If the earth now take the place of the sun, and the sun be placed on the curve at A, the spectator will still see the sun correspond to the star \( a \). Let the earth be transferred from E to E'; the spectator will then see the sun in A'. The sun therefore appears to him to have moved from A to A', which is exactly the same appearance as would be presented to a spectator at the centre by a real translation of S from A to A'; so that, in respect of the annual motion, the appearances, when referred to the celestial sphere, are precisely the same on either hypothesis.

If we now consider the phenomena with reference to the earth, we shall find that they may be equally well explained on the hypothesis of the earth's motion, and the immobility of the sun. The most remarkable phenomenon connected with the annual revolution is the variation of the seasons; and in order to explain their cause, it is only necessary to suppose that the earth, in describing its oblique orbit, always preserves its axis parallel to the same straight line. Let A, B, C, D (fig. 24) repre... Theoretical sent the earth in four different positions of its orbit, n s Astronomy being its axis, and n and s its north and south poles respectively. While the earth goes round the sun in the order of the letters A, B, C, D, its axis ns preserves its obliquity, and always continues parallel to its first direction. At A the north pole inclines towards the sun, and brings all the northern places more into the light than at any other season of the year. But when the earth is at C, the opposite point of the orbit, the north pole declines from the sun, and a less portion of the northern hemisphere enjoys the blessings of his light and heat. At B and D the axis is perpendicular to the plane of the orbit, so that the poles are situated in the boundaries of the illuminated hemisphere, and the sun, being directly over the equator, makes the days and nights equal at all places. These phenomena are illustrated in fig. 25, which represents the situation of the north pole with regard to the limits of illumination, in eight different positions of the orbit. In this figure, AE is the terrestrial 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. The spectator is supposed to be placed at the pole of the ecliptic.

When the earth is at the beginning of Libra, about the 20th of March, the sun, 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 light, and the sun is vertical to the equator, which, with all its parallels, is divided into two equal parts by the circle which forms the boundary between the dark and illuminated hemispheres, and therefore the days and nights are equal all over the earth. As the earth moves in the ecliptic according to the order of the letters A, B, C, D, &c., the north pole P comes more and more into the light, and the days increase in length at all places north of the equator AE. When the earth comes to the position between B and C, or the beginning of Capricorn, the sun, as seen from the earth, appears at the beginning of Cancer, about the 21st of June; and the north pole of the earth inclines towards the sun, so as to bring into light all the north frigid zone, and more of each of the northern parallels of latitude in proportion as they are farther from the equator. 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; the boundary of light and darkness cutting the equator and all its parallels equally. The north pole then goes into the dark, and does not emerge till the earth has completed a semi-revolution of its orbit, or from the 23rd of September till the 20th of March. All these phenomena will be readily understood from the bare inspection of the figure; and it will be perceived that what has been said of the northern hemisphere is equally true of the southern in a contrary sense, that is, at opposite seasons of the year.

The only objection against the annual motion of the earth, which at first sight creates some difficulty, is the enormous distance which, on account of the want of annual parallax, that hypothesis makes it necessary to assign to the fixed stars. Abstracting from the precession, which in the region of the poles amounts only to about 20°, the axis of the earth, in every part of its orbit, appears to be directed towards the same point of the starry sphere. It is certain, however, that the radius of the terrestrial orbit is upwards of 90 millions of miles; and therefore, when the earth is at opposite points of this immense circle, its pole ought to be directed to points in the heavens 180 millions of miles distant from each other. Suppose the sun to be at rest in S (fig. 26), the centre of the orbit Eab, and the earth to be at the point E. Suppose also A to be the projection of S on the sphere of the fixed stars, Fig. 2 and ψ the first point of the ecliptic. The longitude of A, as seen from E, is the angle AEψ; but when seen from S, the longitude of the same point is ASψ. This last is the true longitude of A; AEψ is the apparent longitude, and their difference is the angle SψE. Now, in the first half of the ecliptic, the apparent longitude will be less than the true, and in the second half greater; hence there ought to result an apparent annual inequality in the sun's longitude. But as no such inequality is indicated by observation, it becomes necessary either to abandon the hypothesis of the earth's motion, or to suppose the distance of the fixed stars to be so great, that, in comparison of it, the radius ES, which exceeds 90 millions of miles, is altogether insensible; and that the whole orbit of the earth, or the ellipse Eab, is a mere point in comparison of a great circle of the starry sphere.

The most convincing proof of the earth's motion is not to be found in any circumstance of which the senses can take immediate cognizance, but is afforded by the full development of the planetary system, and the mutual connection of all the truths of rational astronomy,—by the clearness, simplicity, and coherence which this hypothesis gives to the most complicated phenomena,—and can therefore only be fully appreciated after an attentive study of the whole series of facts which astronomy makes known. There exists, however, one direct proof of the earth's annual motion, in a phenomenon discovered by the accurate observation and patient sagacity of Bradley, although it is one which, we are almost tempted to think, ought to have been perceived a priori, after Roemer's discovery of the progressive motion of light. It is known by the name of the Aberration of Light, and is manifested in a small difference between the apparent and true places of a star, occasioned by the motion of light combined with that of the earth in its orbit.

To illustrate this effect, conceive a body to move in the direction EE' (fig. 27), and another to impinge on it in the direction SE. To find the direction of the resulting motion, take EC and EA proportional to the two velocities respectively, and, having completed the parallelogram EABC, draw the diagonal EB. The combination of the two motions produces an impression on the eye exactly similar to that which would have been produced if the eye had remained at rest in the point E, and the molecule of light had come to it in the direction ES'; the star, therefore, whose real place is at S, will appear to the spectator at E to be situated at S'. The difference between its true and apparent place; that is, the angle SES', is the aberration, the magnitude of which is obtained from the known ratio of EA to EB, or the velocity of light to that of the earth in its orbit. Now, we know from the phenomena of the eclipses of Jupiter's satellites, that a ray of light describes a line equal to the mean radius of the ecliptic in 8 min. 13.2 sec. or 493.2 seconds of time. But the arc described by the earth in that time is found from the proportion

\[ \frac{365^d}{256} : \frac{493^s}{2} :: \frac{360^o}{x}, \]

whence \( x = 20^o 25' \). It is evident, therefore, that when the directions of the two motions are at right angles, the star S always appears in advance of its real place, in a direction parallel to that of the motion of the earth, by a quantity equal to 20° 25'. This quantity, 20° 25', is called the Constant of Aberration; but as it has been obtained on the assumption that the earth moves uniformly in a circular orbit, it is evidently not altogether exact, and recourse It must be had to observation to determine its precise amount. Bradley supposed it to be 20°; Dr Brinkley, from the mean of 2633 comparisons of various stars, has deduced the value 20°37'. From a series of 2000 observations of circumpolar stars, made for the purpose with the utmost care at the Greenwich Observatory by Mr Richardson, its value was found to be 20°302'. On account of the number and accuracy of these observations, the constant of aberration may be regarded as one of the best determined elements of astronomy. (See Memoirs of the Astronomical Society, vol. iv.)

It is easy to see that, on the same principles, there ought also to be a diurnal aberration; but the diurnal rotation of the earth being sixty-five times less rapid than the orbital motion, its effect in producing aberration does not amount to more than 0°0208, when reduced to seconds of time. It is consequently altogether insensible in the observations.

The magnitude of the angle of deviation SES' depends on the relative directions of the earth and the visual ray, and may have any value from 0° to 20°3. Suppose, for example, we observe a star situated in the plane of the ecliptic. When the earth is at that point of its orbit, between the sun and the star, where the tangent to the orbit is perpendicular to the visual ray (which, on account that the star has no sensible parallax, always maintains a parallel direction), the apparent place of the star will be 20°3 to the eastward of its true place. When the earth is in the opposite point of its orbit the same star will appear to be 20°3 to the westward of its true place; so that it will appear to have an oscillatory motion on the ecliptic, the range of which is 40°6, and the period exactly a year. Half-way between these two points the tangent of the orbit is parallel to the direction of the ray of light, and consequently there is no aberration. When the star is not situated in the ecliptic, it will suffer a displacement in latitude as well as in longitude. To render this still more sensible, let EEE (fig. 28) be the ecliptic, E the earth, and A the true place of a star situated at any altitude above the ecliptic. In the direction EA take Ea to represent the velocity of light, ab that of the earth, and in a parallel direction, that is, parallel to the tangent to the ecliptic at E; the line Eb will now be the apparent visual ray, and the star will seem to be situated at B. Suppose the earth to be placed at different points of its orbit; the lines Ea will be all parallel to each other on account of the infinite distance of the star A; the lines ab will vary little in magnitude, because they are very small in comparison of Ea, but their directions will undergo every possible change, being always parallel to the tangent at E. At the two points of the orbit where the tangent is parallel to EA, the two lines Ea and ab coincide, and consequently there is no aberration. Let us next suppose the star to be situated in the pole of the ecliptic. In this case the visual ray is constantly perpendicular to the direction of the earth's motion, so that the star will always appear at a distance of 20°3 from its true place, or appear to describe a small circle about the pole of the ecliptic. In all other situations, out of the ecliptic, the star's apparent path will be an ellipse, the major axis of which, parallel to the plane of the ecliptic, is always 40°6, while the minor axis varies with the latitude.

The cause of the aberration being known, we have two methods of measuring its quantity, or the extent of the apparent oscillations, viz. by calculating the angle of deviation SAS' (fig. 27), the tangent of which is radius as AB to AE, that is, as 1 to 10313; or, by direct observation, when the star is in opposition to the sun. The two methods give exactly the same result, and the dimensions of the ellipse described by any star not situated in the Theoretical ecliptic, in consequence of the aberration, are found to be the same when computed from theory or determined by direct observation. The motion of translation of the earth, therefore, receives a mathematical demonstration from this agreement; and the phenomenon of the aberration, otherwise unimportant on account of its minuteness, thus becomes one of the most interesting discoveries ever made in astronomy. The fact of the earth's orbital motion is, however, rendered so probable by other phenomena, that it must have been universally admitted, although the direct proof had never been discovered.

Sect. IV.—Of the Measure of Time. Equation of Time.

The notion of time, or of succession, is generally said to be acquired from the aspect of natural phenomena; but from whatever source it may be derived, it is certain that time itself can only be measured by comparing it with something of which the senses can take cognizance. If we have a series of events, such as the oscillations of a pendulum, or the flux and reflux of the sea, uniformly succeeding each other, then time may be measured by the number of such events that have been observed. In like manner, if a body move uniformly in a certain direction, time may be measured by the spaces over which it successively passes. In strict language, motion and time, being heterogeneous quantities, cannot measure one another; but different times are compared with each other by means of the motions that have taken place in those times respectively; for the motions being supposed uniform, equal spaces are passed over in equal times, or, which is the same thing, the times are directly proportional to the spaces described. If, therefore, we assume for unit the time which is absolved while a certain uniform motion takes place, we may obtain, by a simple proportion, the corresponding time of any other similar motion. In this manner we find the ratio of one portion of time to another, although we can form no idea whatever of its absolute quantity.

In order that motion may be employed as a measure of time, it is indispensably requisite that it be perfectly uniform. The only motions having this property, with which we are acquainted, are those of the rotation of the celestial bodies about their own axes. The motion of the sun and planets in their orbits is irregular from various causes; even the successive returns of the fixed stars to the meridian are rendered unequal on account of the precession and nutation; but the diurnal rotation of the earth appears to be affected by no cause of irregularity whatever. Time, therefore, may be properly measured by this uniform motion; and the method of proceeding is as follows. The culmination of a star, or of a given point of the equator, marks the instant at which the day commences, and at any other instant its horary angle determines the portion of the day which has elapsed at the moment of the observation. In order, therefore, to determine the time of an observation, we must find the horary angle formed by the meridian of the observed star with that which passes through the given point where the motion is supposed to begin; in other words, it is necessary to determine the star's right ascension. This, however, when attempted by direct measurement, is a determination attended with difficulty, and liable to considerable uncertainty. Happily the invention of the pendulum has rendered it unnecessary; and astronomers, instead of deducing the time from the right ascensions, determine, on the contrary, the right ascensions by means of the time indicated by the clock. The clock, of which the motion is supposed to be perfectly regular, is adjusted in such a Theoretical manner that its index describes 24 hours, while a point of the equator describes an arc of 360°, or makes a complete revolution. This is called a sidereal day. The sidereal hour is divided in the usual manner into minutes, seconds, &c.; whence the corresponding arcs are easily found at the rate of 15° to an hour. The point whose culmination marks the origin of time is arbitrary; but astronomers have agreed to choose for that purpose the equinoctial point of Aries, from which the right ascensions are reckoned, so that the hours of the clock and the degrees of the equator may commence at the same instant. This point, it is true, is not, and cannot be, permanently marked by any star; but the right ascension in time of any star whatever being the hour of its transit over the meridian, the star will be in the plane of the meridian at the instant denoted by its right ascension in time. On this account sidereal time expresses the actual right ascension of the zenith; or, as it is frequently termed, the right ascension of the mid-heaven.

We have already seen, that on account of the precession of the equinoxes, a star employs somewhat more time than the first point of Aries in returning to the meridian. It is therefore not without some violation of language that the interval between two successive transits of a given point of the equator over the meridian is called a sidereal day, which, in its strict acceptation, denotes the time which elapses between two successive transits of the same star; but in this case the difference is so small as to be totally inappreciable. The annual precession in longitude is = 50°1', and that in right ascension is nearly the same, excepting in regard to the circumpolar stars, which, therefore, are not employed in regulating the clocks. This arc of 50°1', converted into time, gives 3-3 seconds as the time in which a star will pass the meridian later than the equinoctial point, at the end of the year; and this small quantity being distributed over the whole year, is altogether insensible in short intervals of two or three days. The successive transits of a star, therefore, if we abstract from the nutation and aberration, will mark the sidereal day within the hundredth part of a second of time; and the sidereal year, though not immediately ascertainable by observation, becomes a quantity which may be easily computed. But the regularity of the motion of the stars is deranged by the effects of aberration and nutation; so that in order to measure time with the precision required by modern observers, it is necessary to be acquainted with the minute displacements of the stars. If they seem to return to the meridian after equal portions of absolute time, it is only because our organs are unable to distinguish the hundredths of a second.

The sidereal day is a measure of time which, on account of its uniformity and the facility of observing it, is excellently well adapted for astronomical purposes; but relatively to the ordinary wants of life it is not sufficiently marked—the culmination of the stars is an event entirely unconnected with civil occupations, and which, for any given star, is even invisible during a great part of the year. The proper motion of the sun causes the sidereal day to commence sometimes by day and sometimes by night, so that great confusion and embarrassment would arise from regulating time and civil affairs by the motions of the stars. On this account the diurnal revolution of the sun has been universally adopted as the measure of time. This is called the civil day, and denotes the interval of time which elapses between two successive transits of the sun over the same hour-circle. Most nations have agreed in reckoning it from the inferior semicircle of the meridian, so that the civil day commences and terminates at midnight; but astronomers, in imitation of Hipparchus and Ptolemy, usually reckon the commencement of the day from the instant of the sun's culmi-

nation, that is, from noon; and count through the 24 hours from one noon to the following. Thus 9 o'clock in the morning of February 14th is by astronomers called February the 13th at 21 hours. The day thus determined is called the astronomical or solar day; and being regulated by the true motion of the sun, the time which is measured by it is called true or apparent time.

Astronomical or solar days are not equal. Two causes, Days in particular, conspire to produce their inequality, namely, in length the unequal velocity of the sun in his orbit, and the obliquity of the ecliptic. The effect of the first cause is sufficiently 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 have regard to the motion of the sun in reference to the equator. The sun describes every day a small arc of the ecliptic. Through the extremities of this arc suppose two meridians to pass; the arc of the equator, which they intercept, is the sun'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 cosine of the obliquity of the ecliptic to radius: at the solstices, on the contrary, it is greater in the proportion of radius to the cosine 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 second sun to move uniformly on the ecliptic, and to pass over the extremities of the axis of the sun's orbit at the same instant with the real sun. This removes the inequality arising from the inequality of the sun's motion. To remove the inequality arising from the obliquity of the ecliptic, conceive a third sun to pass through the equinoxes at the same instant with the second sun, and to move along the equator in such a manner that the angular distances of the two suns at the vernal equinox shall be always equal. The interval between two consecutive returns of this third sun to the meridian forms the mean astronomical day. Mean time is measured by the number of the returns of this third sun to the meridian; and true time is measured by the number of returns of the real sun to the meridian. The arc of the equator, intercepted between two meridian circles drawn through the centres of the true sun and the imaginary third sun, when reduced to time, is what is called the Equation of Time. This will be rendered plainer by the following diagram.

Let Zφz (fig. 29) be the earth; ZFRz its axis; Fig. 2 abode, &c. the equator; ABCDE, &c. the northern half of the ecliptic from φ to z, on the side of the globe next the eye; and MNOP, &c. the southern half on the opposite side from z to φ. Let the points at A, B, C, D, E, F, &c. mark off equal portions of the ecliptic, gone through in equal times by the real sun, and those at a, b, c, d, e, f, &c. equal portions of the equator described in equal times by the fictitious sun; and let Zφz be the meridian.

As the real sun moves obliquely in the ecliptic, and the fictitious sun directly in the equator, any point between φ and F on the ecliptic must be nearer the meridian Zφz than the corresponding point on the equator from φ to f; that is to say, than the point whose distance from φ is expressed by the same number of degrees; and the ASTRONOMY.

As the real sun moves from A towards C, the velocity of his motion increases all the way to C, where it is at its maximum. But notwithstanding this, the fictitious sun gains so much upon the real, soon after his departing from A, that the increasing velocity of the real sun does not bring him up with the equally-moving fictitious sun till the former comes to C and the latter to e, when each has gone half round its respective orbit; and then being in conjunction, the meridian EH, revolving to 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 its maximum, 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 e to g, the real sun moves through a greater arc from C to G; consequently the point K has its noon by the clock when it comes to k, but not its noon by the sun till it comes to L. And although the velocity of the real sun diminishes all the way from C to A, and the fictitious sun by an equable motion is still coming nearer to the real sun, yet they are not in conjunction till the one comes to A and the other to a, and then it is noon by them both at the same moment.

Mean solar time and sidereal time being both uniform, it is easy to compare the one with the other, and assign son of the number of degrees, minutes, &c., which the sun and a star will respectively describe in a given portion of side-time or mean solar time. In a mean solar day the sun's right ascension and mean longitude are increased by $59^\circ 8'33''$; consequently $360^\circ - 59^\circ 8'33''$ of the equator pass the meridian in 24 mean solar hours. The sidereal time corresponding to this period is 24 hours 3 min. 56'555 sec.; therefore 24 mean solar hours are equal to 24 hours 3 min. 56'555 sec. of sidereal time; and 24 hours of sidereal time are equal to 23 hours 56 min. 4'0907 sec. of mean solar time, or to 24 hours minus 3 min. 55'9093 sec. This difference of 3 min. 55'9093 sec. is called the acceleration of the fixed stars in mean solar time; and the preceding excess of 3 min. 56'555 sec. is the retardation of the sun in sidereal time. Hence the one species of time may be easily converted into the other; and the arc of the equator passed over by the meridian in a given mean time may be calculated. Thus,

$$\frac{360^\circ - 59^\circ 8'33''}{24} = 15^\circ 2'27'84708$$

is the arc described by a star in one hour of mean solar time.

But the process of converting sidereal or mean solar time into true or apparent time, or of computing from the instants of apparent time the corresponding mean solar and sidereal times, is attended with much greater difficulty. The reason is, that the interval between two successive transits of the sun over the meridian, which, in apparent time, measures the day, is a variable quantity; and hence there cannot exist any constant ratio between true and mean time, as there does between mean and sidereal time. The correction or equation by which apparent time is reduced to mean time, is technically called the Equation of Time, and is composed of the aggregate of the several variable terms which denote the inequalities of the sun's motion in longitude. Besides the eccentricity, Theoretical Astronomy.

The obliquity of the ecliptic, and the variations of that obliquity occasioned by the nutation of the earth's axis, it is affected also by the small alterations of the sun's right ascension, which result from the effect of the planetary perturbations on the earth; and hence the equation of time cannot be exactly computed without the aid of Physical Astronomy.

It will be evident, from what has preceded, that the equation of time expresses merely the difference between the true and mean right ascensions of the sun, reduced to time. Its different parts may be calculated numerically in the following manner. Let NA (fig. 31) represent the mean motion of the sun during a given interval, then NA is the sun's mean longitude, N being the first point of Aries. Let NA = M, and from this mean longitude subtract the longitude of the apogee, the remainder will be the sun's mean anomaly. From the mean anomaly let the equation of the centre be found and denoted by E. Take AB = E, then NB = M + E is the longitude corrected for the eccentricity.

Let us next suppose BC to be the small quantity by which the sun's longitude is increased in consequence of the perturbations of the planets, and let BC = P; then NC = M + E + P is the true and exact longitude of the sun. Through C let the arc CD be drawn perpendicular to the equator; the point D will be that point of the equator which passes the meridian at the same time with the sun. Let R = NC - ND = the reduction to the ecliptic; we have then the sun's right ascension = M + E + P - R.

Let NF = NA, and F will be the place which the sun would occupy in the equator at the same instant that he occupies the point A in the ecliptic, if he moved uniformly in the former circle; for NF as well as NA will represent the mean diurnal motion of 59°8'33" multiplied by the number of days elapsed in the interval between the equinox N and the time of the observation. The mean sun would therefore pass the meridian with the point F, whereas the true sun passes it with the point D; therefore, at the instant of true noon, when the sun C and the corresponding point D are on the meridian, the mean sun is at a distance from D expressed by the arc FD = M + E + P - R = M + E + P - R. Now, the arc of the equator FD measures the horary angle between the mean and the true sun, or the angle at the pole between the meridian and the hour-circle passing through D; it is therefore converted into time by the following proportion:

\[ \frac{360°}{FD} : 24 \text{ mean solar hours} : \text{time from true noon}; \]

consequently the difference between mean and true noon, or the equation of time,

\[ \frac{FD \times 24}{360°} = \frac{1}{15}(E + P - R) = dT. \]

But this equation is not yet perfectly accurate; it requires to be corrected for the effects of nutation. Now it is known that the variation of the mean longitude of the sun, arising from the unequal precession of the equinoxes in consequence of the nutation occasioned by the inclination of the lunar orbit, is expressed by the formula \(18°\sin(360° - \text{moon's node}) = 18°\sin N\). This variation, reduced to the direction of the equator, will therefore be \(18°\sin N\cos e\) (e being the obliquity of the ecliptic). The difference between the two expressions is \(18°\sin N(1 - \cos e) = 36°\sin^2\frac{1}{2}\sin N = 1°4887\sin N\), which, reduced to time, becomes 0.09925 sec. × sin N. This small correction, amounting to less than a tenth of a second, was long omitted in computing the equation of time. When it is included, that equation becomes

\[ dT = \varepsilon_0(E + P - R) + 0.09925 \text{sec.} \times \sin N. \]

The quantities E, P, R, and N, are to be computed separately from the astronomical tables; and it must be observed, that the result will be expressed in mean solar time.

The cosine of the obliquity, that is, \(\cos 23°28'\), is equal to 0.9173 very nearly. Hence, since the equation of time is equal to the sun's true right ascension, diminished by his mean longitude and the effects of nutation in right ascension (\(= ND - NA = 18°\sin N\cos e\)), it may, on denoting the true right ascension by A, be expressed as follows:

\[ dT = A - M = 18°\sin N \times \frac{1}{15}. \]

This is the form under which the equation of time was expressed by Dr Maskelyne.

The equation of time is at its maximum about the 3d of November, when it amounts to 16°16'7", and is subtractive. At four different times of the year it vanishes, namely, about the 25th of December, the 16th of April, the 16th of June, and the 1st of September. These epochs, however, do not remain constant; for, on account of the change which the line of the apsides is constantly undergoing in reference to the line of the equinoxes, the difference between the true and mean right ascensions of the sun—in other words, the equation of time—varies continually in different years.

Sect. V.—Of the Spots of the Sun, his Rotation, and Constitution.

The sun, the great source of light, heat, and animation, when beheld with the naked eye, appears only as a luminous mass of uniform splendour and brightness; but when examined with the telescope, his surface is frequently observed to be mottled over with a number of dark spots, of irregular and ill-defined forms, and constantly varying in appearance, situation, and magnitude. These spots are occasionally of immense size, so as to be even visible without the aid of the telescope; and their number is frequently so great that they occupy a considerable portion of the sun's surface. Dr Herschel observed one in 1779, the diameter of which exceeded 50,000 miles, more than six times the diameter of the earth; and Scheiner affirms that he has seen no less than 50 on the sun's disk at once. Most of them have a deep black nucleus, surrounded by a fainter shade, or umbra, of which the inner part, nearest to the nucleus, is brighter than the exterior portion. The boundary between the nucleus and umbra is in general tolerably well defined; and beyond the umbra a stripe of light appears more vivid than the rest of the sun.

The discovery of the sun's spots has been attributed to different astronomers. They appear to have been first taken notice of in a work of Fabricius, the friend of Kepler, which was published at Wittenberg in 1611, under the title of Joh. Fabricii Phrysi de Maculis in Sole Obseruatis, et Apparente carum cum Sole Conversione Narratio. It contains, however, nothing more than a few vague conjectures respecting the spots, the phenomena of which he could not have observed with any degree of accuracy, insomuch as he seems to have been unacquainted with any method of protecting the eye by intercepting a portion of the solar rays; for he recommends to those who should repeat his observations, to admit into the telescope only a small portion of the sun at once, till the eye should by degrees become able to support the full blaze of light. About the same time the discovery was warmly disputed by the illustrious Galileo, and Scheiner, a German Jesuit, professor of mathematics at Ingolstadt. The whole circumstances connected with this dispute are narrated at great length by Galileo in his work entitled Istoria e Di- Theoretical mostrazioni intorno alle Macchie Solari, e loro Accidenti, from which it appears certain that he observed the spots so early as April 1611. In a letter published by him in 1612, he remarks that the spots are situated on the surface of the sun, or that at least their distance from it is imperceptible; that the time of their continuance varies from 2 or 3 to 30 or 40 days; that their figures are irregular and variable; that some are seen to separate, and others to unite, even on the middle of the disk; that besides these peculiar motions, they have also a common motion, in virtue of which they traverse the disk in parallel lines. From this general motion he infers that the sun turns on an axis from west to east; and he adds as a curious remark, that the spots are confined within a zone extending only about 28 or 29 degrees to the north and south of the sun's equator. Galileo illustrates all these positions by mathematical reasoning, and by drawings of the spots made on many successive days.

Scheiner's observations were first announced in January 1612, in three letters addressed to his friend Marc Velser, a magistrate of Augsburg. In the first of these, the date of which is November 1611, he says that he had observed the spots seven months before, but that, having a different object in view, he had given little attention to them. He observed them again in the following October, and at that time imagined the appearance was owing to some imperfection of his telescope, till he was convinced by repeated observations that it was necessary to refer it to the sun. From these remarks it is pretty clear that Scheiner had formed no accurate notions respecting the spots before October 1611, that is, six months after they had been observed by Galileo. Scheiner made the observation of the solar spots his whole occupation during the following eighteen years, in the course of which he discovered the position of the solar equator, and formed a theory much more complete than that of Galileo. The account of his observations was published in 1630, under the title of Rosa Ursina, sive Sol ex admirando Facularum et Macularum urinarum Phænomeno Varius, &c.

The discovery of the solar spots has also been claimed for our countryman Harriot. Amidst these conflicting pretensions it is perhaps impossible to arrive at the truth; but the matter is of little importance; the discovery is one which followed inevitably that of the telescope, and an accidental priority of observation can hardly be considered as establishing any claim to merit.

The solar spots furnish an extensive subject of curious speculation, but in an astronomical point of view they are chiefly interesting on account of their establishing the fact, and affording the means of determining the period of the rotation of the sun. In order to obtain a precise idea of the position of a spot, and the path which it describes, it is necessary to project that path on the plane passing through the centre of the sun, and perpendicular to the visual ray drawn from the earth to the sun's centre.

Suppose the diameter of a circle ASB (fig. 32) to be divided into as many parts of unity as there are seconds in the apparent diameter of the sun. Let CP be taken equal to the number of seconds contained in the difference of the longitudes of the spot and the sun's centre, and the perpendicular PM equal to the number of seconds in the latitude of the spot; then M will represent its position on the surface of the sun. By repeating the same operation a number of days consecutively, a series of points M M' M'', &c. will be obtained in the apparent path of the spot on the sun's disk, or rather in the projection of that path on the plane perpendicular to the visual ray. This projection is in general an oval slightly differing from an ellipse; and it is found that all the spots observed at the same time describe similar and parallel curves. They also return to the same relative positions in the same time, and their period is about 27½ days.

The paths described by the spots undergo very considerable changes, according to the season of the year at which they are observed. About the end of November and beginning of December they appear simply as straight lines MM', M'M'', M''M''' (fig. 33), along which the spots move in the direction MM', that is, they enter on the eastern and disappear on the western edge of the sun's disk; and the points at which they disappear are more elevated, or nearer the north pole of the ecliptic, than those at which they enter. After a certain time the lines MM' begin to assume a curved appearance, and form ovals, as represented in fig. 34. During the winter and spring Fig. 34. the convexity of the ovals is turned towards the north pole of the ecliptic; but their inclination, or rather the inclination of the straight lines joining their extreme points, to the plane of the ecliptic continues to diminish, and about the beginning of March disappears; so that the points at which they seem to enter and leave the sun's disk are equally elevated, as in fig. 35. From this Fig. 35. time the curvature of the ovals diminishes; they become narrower and narrower till about the end of May or beginning of June, when they again appear under the form of straight lines (fig. 36); but their inclinations to the ecliptic are now precisely in a contrary direction to what they were six months before. After this they begin again to expand, as in fig. 37, and their convexity is now turned towards the south pole. Their inclinations also vary at Fig. 37. the same time, and about the commencement of September they are seen as represented in fig. 38; the points Fig. 38. at which they enter and disappear being again equally elevated. After this period the ovals begin to contract and become inclined to the ecliptic, and by the beginning of December they have exactly the same direction and inclination as they had the previous year.

These phenomena are renewed every year in the same order, and the same phases are always exhibited at corresponding seasons. Hence it is evident that they depend on a uniform and regular cause, which is common to all of them, since the orbits described by the various spots are exactly parallel, and subject in all respects to the same variations. The simplest method of explaining the phenomena is to suppose with Galileo that the spots are adherent to the surface of the sun, and that the sun uniformly revolves round an axis inclined to the axis of the ecliptic. If the axis of revolution were perpendicular to the plane of the ecliptic, the spots, supposing them to adhere to the sun's surface, would describe circles parallel to that plane, which, seen from the earth, would appear as so many parallel straight lines; but by supposing the axis to have a suitable inclination, all the phenomena become explicable in a very simple manner. While the sun is carried round in his orbit, his axis, constantly preserving its parallelism, will successively assume different positions relatively to the earth; and the planes of the circles described by the spots, which planes are always perpendicular to the axis, will consequently be presented to us under different inclinations; hence the variations of their apparent curvature. In two opposite points of the orbit the visual ray drawn from the earth to the centre of the sun is perpendicular to the axis of rotation. In these two positions the poles of the sun, or the points in which the axis meets the surface, are both visible at the same moment, and the spots appear to move in parallel straight lines. But as the axis retains this perpendicular position only for an instant, and declines from it very sensibly while the spot traverses the sun's disk, the path of Theoretical the spots over the entire disk is neither a straight line nor an ellipse, one of which it would necessarily be if the sun, while revolving about his axis, did not change his place in his orbit. When the axis is not perpendicular to the visual ray, the path of the spots will appear to be a curve of which the concavity is turned towards that pole which is visible from the earth. This inclination of the solar axis to the plane of the ecliptic also explains the reason why the points of the disk at which the spots appear are more elevated during one half of the year, and more depressed during the other half, than those at which they disappear. It will also follow from the same hypothesis, that the curvature of the ovals must be the greatest possible when the straight lines joining their extremities are parallel to the ecliptic; and, on the contrary, least when the same straight lines are most inclined to the ecliptic; all which is exactly conformable to observation.

The various appearances which we have now described may be accurately represented by means of a common celestial or terrestrial globe. Let the wooden horizon of the globe, which is here supposed to represent the sun, be placed horizontally in the same plane with the eye of the spectator, and the pole be inclined about 7° from the zenith. The wooden horizon will now represent the ecliptic; and if the spectator walk round the globe, always keeping his eye in the plane of the wooden horizon, the circles of latitude will appear to him as ellipses of different inclinations and eccentricities: in two opposite points they will appear as straight lines, and, in short, exhibit in their various positions all the phenomena of the oval paths described by the spots of the sun.

The consequences deduced from the hypothesis of the rotatory motion of the sun are so perfectly conformable with observation, as to render the inference inevitable, that the sun revolves from west to east, on an axis inclined about 93° to the plane of the ecliptic. The plane which passes through the centre of the sun, perpendicular to the axis of rotation, is the Equator of the sun; the straight line joining the points in which it intersects the ecliptic is called the Line of the Nodes of the equator. The Nodes themselves are the two opposite points in which this straight line, produced indefinitely, meets the celestial sphere.

In order to determine the situation of the solar axis in space, it is necessary to find its inclination to the ecliptic, and the angle which the line of the nodes makes with any given line on the plane of the ecliptic, for example, with the line of the equinoxes. The requisite data for the solution of this problem are three different positions of the same spot, which must be obtained by observation.

Let S (fig. 39) be the centre of the sun's disk, AB a parallel to the terrestrial equator, and M the place of a spot, the co-ordinates of which, as referred to AB, are SX and MX. In this figure the earth is supposed to be situated in a straight line passing through S perpendicular to the plane of the paper, and it is to be recollected that SX and MX are in fact arcs of the solar globe, though so small when seen from the earth that they may be regarded as straight lines. By comparing the times of the transits of B, the border of the disk, and the spot M, we shall find BX, from which, as SB the semidiameter of the sun is known, we shall have SX the difference of the right ascensions of the spot and sun's centre. The line CY, which is the difference of the declination of the border of the disk and that of the spot, is measured by the micrometer. This will give MX the declination of the spot.

Now, tan. MSX = \frac{MX}{SX}, and SM = \frac{SX}{\cos. MSX}; therefore SM is also a known quantity. Let EE be the ecliptic, e = obliquity of the ecliptic, and O = the longitude of the sun. The angle BSE, which is technically called the Angle of Position, is the complement of the angle made by the ecliptic with the circle of declination passing through the sun, and is therefore given by the formula tan. BSE = tan. e cos. O; hence MSE (= MSX — BSE) is also given. On SE let fall the perpendicular Mm, then Ma = SM sin. MSE is the geocentric latitude of the spot, and Sm = SM cos. MSE is the difference between its geocentric longitude and that of the centre of the sun: the object is now to determine the angles subtended by these lines at the centre of the sun, that is, to convert the geocentric into heliocentric latitudes and longitudes.

Suppose two straight lines to be drawn from the earth, one to the centre of the sun, and the other to the centre of the spot M, and let \( \phi \) be their inclination, which is measured by SM, and is consequently known, being the geocentric distance of the spot from S, the centre of the sun's disk. Let R = distance of the sun's centre from the earth, \( r \) = semidiameter of the sun, \( \varphi \) = the angle made at the centre of the sun by the straight lines drawn from it to the earth and the spot M, and let \( \psi \) be the remaining angle of the triangle formed by the lines joining the earth, the centre of the sun, and the spot. We have then \( r : R :: \sin. \phi : \sin. \psi \), whence \( \sin. \psi = \frac{R}{r} \sin. \phi \). But \( R : r \) as radius to the sine of half the sun's true diameter; therefore \( \sin. \psi = \frac{\sin. \phi}{\sin. \frac{1}{2} \text{sun's diameter}} \).

Now \( \varphi = 180^\circ - \psi - \delta \); consequently we have the number of degrees, minutes, and seconds in \( \varphi \), the heliocentric distance of the spot from the straight line which joins the centres of the earth and sun. Having thus obtained the hypothenuse SM in parts of a great circle of the solar orb, the sides Sm and Mm will be obtained in similar parts from the common trigonometrical formulae for the resolution of a right-angled spherical triangle. These formulæ give

\[ \tan. Sm = \tan. SM \cos. MSE, \] \[ \sin. Mm = \sin. SM \sin. MSE. \]

It will be remarked that Mm is the heliocentric latitude of the spot, Sm the difference of its longitude and that of the earth; and as the longitude of the earth is equal to 180° + that of the sun, the heliocentric longitude of the spot will be 180° + O — SM if the spot is behind or to the east of the centre of the sun, and 180° + O + Sm if it precedes the centre.

In this manner the heliocentric longitudes and latitudes of the spots are deduced from their observed right as, solar ascensions and declinations. The next step is to show how they are employed in determining the position of the solar axis. This problem is in practice somewhat laborious, although the principles on which its solution rests are sufficiently simple. The planes of the circles described by the spots are parallel to the sun's equator: if, therefore, the position of one of them can be found, the position of the equator, and consequently of the axis, will be found at the same time. Now, the position of a plane is determined by three given points through which it is required to pass; consequently, by three observations of the same spot, we shall have three points in its plane, and thence the plane itself. The problem is therefore one of pure geometry, and may be solved in various ways. The results of the most accurate observations make the inclination of the solar equator to the ecliptic amount to 7° 19' 23" or, 7° 19' very nearly; and the heliocentric longitude of the ascending node, that is, the point in which the equator of the sun intersects the ecliptic, in passing from Theoretical south to north, $80^\circ 7' 4''$. The position of the node seems to undergo no variation, except such as may be supposed to arise from the precession of the equinoxes.

It has already been mentioned that the mean time in which a solar spot returns to the same position, relatively to the earth, is 27.3 days. This, however, is not the time in which the sun makes a revolution about his axis. In the interval of 27.3 days the sun describes in the ecliptic an arc equal to $26^\circ 91'$ (for $365^\circ - 25 \cdot 27 \cdot 3 = 360^\circ - 26^\circ 91'$), and by virtue of this motion alone he exhibits every day different points of his surface, the whole of which would be successively shown to the earth in the course of a year, independently of the motion of rotation. Hence results an apparent annual rotation round an axis perpendicular to the ecliptic, and we must abstract the effects of this optical illusion in order to arrive at the time of a real rotation. This apparent rotatory motion will be easily understood by referring to fig. 40, in which S is the sun, E the earth, and C the point in which the visual ray ES intersects the surface of the sun. Suppose the sun to have advanced in his orbit from S to S', the visual ray drawn from the sun's centre to the earth will now meet the surface in a different point C', and the angular distance between C and C' will be found by drawing SE' parallel to SE; for the point e in which SE' meets the surface will correspond to C, and the arc C'e, or the angle ES'E', will be the apparent rotation, while the sun advances in his orbit from S to S'. Now, the angle ES'E' is equal to SES, or the apparent rotatory motion is equal to the angular motion of the sun in his orbit; hence, since the real rotation is in a contrary direction, it is obvious that, if the axis were perpendicular to the plane of the ecliptic, when a spot appears to have made one revolution, it has in reality passed over an arc equal to $360^\circ + 26^\circ 91'$, or $386^\circ 91'$ degrees. By a simple proportion, therefore,

$$\frac{386^\circ 91'}{360^\circ} = \frac{27^\circ 3' 25^\circ 4'}{\text{real time}}$$

and consequently $25^\circ 4'$ days is the real time of the sun's revolution. This result is, however, not quite accurate, on account of its having been supposed that the axis of rotation is perpendicular to the ecliptic, whereas it differs from a perpendicular to that plane by $7^\circ 4'$; but the correction necessary on account of the difference is so small as to fall far within the limits of the errors to which the observations are liable; it is therefore unnecessary to have regard to it.

Observations of the spots, from which the elements of the sun's rotation are deduced, are attended with a very considerable degree of uncertainty. The semidiameter of the sun, which at the surface of the earth subtends an angle of only $16'$, or $960''$, is equivalent to $90^\circ$ at the centre of the sun. Hence the error of a single second (and it is impossible to answer for one or two seconds in observations of this sort) corresponds to about $338''$, or $5' 38''$, on the surface of the solar globe. An observer may, therefore, notwithstanding the greatest care, be mistaken to the extent of $10'$ with regard to the length of a heliocentric arc; and when to this we add, that the margins of the spots are ill defined, and even changeable, so that their centres, which it becomes necessary to observe, are not always the same, it will not appear surprising that a considerable discordance exists among the results of different observations. The uncertainty respecting the time of rotation amounts to no less than 10 or 12 hours. From a careful discussion of a numerous set of observations, Delambre found the time of rotation to be only $25 \cdot 01154$ days, instead of $25 \cdot 4$ given above; and he remarks that, in order to render the last-mentioned number admissible, it is necessary to suppose either an error of $2^\circ 22'$, which is scarcely possible, or else that the spot has a proper motion in the same direction as the motion of rotation of the earth and all the planets.

From four different combinations of equations, derived from eleven observations of the same spot, Delambre computed the following table of the elements of rotation:

| Node | Inclination | Revolution | Synodic Revol. | |------|-------------|------------|----------------| | 1 | $80^\circ 45'$ | $7^\circ$ | $7^\circ 19' 17''$ | $25^\circ 0' 17''$ | | 2 | $79^\circ 21' 35''$ | $7^\circ 12' 37''$ | $26^\circ 4' 17''$ | | 3 | $80^\circ 33' 40''$ | $7^\circ 16' 33''$ | $26^\circ 4' 17''$ | | 4 | $79^\circ 47' 55''$ | $7^\circ 29' 4''$ | $26^\circ 4' 17''$ |

With regard to observations on the sun's spots, Delambre remarks that he attaches little value to them, first, because it is impossible to make them well; and, secondly, because, even if they were sure, they only lead to results of little importance to astronomy. He discussed a hundred different spots, each observed at least three times by Messier, and deduced thirty different determinations of the elements of rotation. The more he multiplied his calculations, the more certain he became of the impossibility of a good solution; of which, indeed, there is no other chance than in a compensation of errors, little probable on account of their enormous magnitudes. These discrepancies render it probable that the spots, besides partaking of the general motion of the solar globe, have also proper motions either of displacement, or occasioned by a change of form, which may long prevent this part of astronomy from reaching a greater degree of exactness than it has already attained.

From the circumstance that the sun is incontestably endowed with a rotatory motion, Lalande concluded that, according to the rules of probability, he ought also to have a motion of translation in space. This idea was adopted by Herschel, and is now generally received as probable. It explains some, though not all, of the proper motions which have been supposed to be observed among the stars. If, indeed, the stars are so many suns, revolving like ours each on its own axis of rotation, it is extremely probable that they have also motions of translation; and thus, there being no fixed points in the heavens, the problem of their proper motions becomes so complicated as to be altogether insoluble.

The only interesting fact which has been deduced from the observations of the solar spots is the rotation of the sun. The curious appearances which they exhibit have, however, attracted great attention, and given rise to numerous theories respecting the constitution of that immense body which governs and vivifies the planetary system. We will now proceed to give a brief account of the appearances, and mention some of the theories that have been founded on them; premising, that after all that has been written on this subject, we are not yet, and probably never will be, in possession of any definite knowledge. The nature of the spots, and the physical constitution of the sun, afford fruitful subjects of harmless conjecture and speculation, but form no part of science.

The phenomena of the solar spots, as delivered by Scheiner and Hevelius, may be summed up in the following particulars. 1. Every spot which has a nucleus, or the spots comparatively dark part, has also an umbra, 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. Theoretical 5. The exterior boundary of the umbra never consists of sharp angles, but is always curvilinear, how irregular soever the outline of the nucleus may be. 6. The nucleus of a spot, whilst on the decrease, often changes its figure by the umbra encroaching irregularly upon it, 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 umbrae are often seen without nuclei. 10. An umbra of any considerable size is seldom seen without a nucleus in the middle of it.

When a spot which consisted of a nucleus and umbra is about to disappear, if it is not succeeded by a facula or spot brighter than the rest of the disk, the place where it was is soon after not distinguishable from the rest.

In the Philosophical Transactions, vol. lxiv. Dr Wilson, late professor of astronomy at Glasgow, has given a dissertation on the nature of the solar spots, in which he 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 only 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, the Rev. Dr Wollaston observes that the appearances mentioned by Dr Wilson are not uniform. He positively affirms that the facula or bright spots on the sun are often converted into dark ones. "I have many times," says he, "observed near the eastern limb a bright facula just come on, which has the next day shown itself as a spot, though I do not recollect to have seen such a facula near the western one after a spot's disappearance. Yet, I believe, both these circumstances have been observed by others, and perhaps not only near the limbs. The circumstance of the facula being converted into spots, I think I may be sure of. That there is generally, perhaps always, a mottled appearance over the face of the sun, when carefully attended to, I think I may be as certain. It is most visible towards the limbs, but I have undoubtedly seen it in the centre; yet I do not recollect to have observed this appearance, or indeed any spots, towards 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 observes, that the nuclei of the spots are not always in the middle of the umbrae, and gives the figure of one seen on the 13th of November 1773, which is a remarkable instance to the contrary.

The facula, or bright spots, were observed with particular attention by Messier, who frequently saw them enter on the eastern limb of the sun, disappear as they approached the centre of the disk, re-appear on the opposite limb, and continue visible, as they had done at the time of their first appearance, for about three days, till they were carried off the disk by the rotation of the sun. Spots frequently broke out in these faculae, and when this did not happen, they were succeeded by spots which generally became visible on the following day; and from the regularity of this occurrence, he was enabled to predict the appearance of a spot 24 hours before it entered on the sun's disk. He observed, also, that the magnitude of the spots was proportional to the brightness of the antecedent faculae. Like the spots, the faculae are generally confined to the equatorial regions of the sun; but they have been occasionally observed by Schroeter on every part of the disk.

To explain these singular appearances, numerous theories, more or less plausible, have been proposed, but all resting on many gratuitous assumptions, and subject to great difficulties. Scheiner imagined that the spots do not belong to the sun, but supposed them to be inferior planets revolving at no great distance from the central luminary. Galileo, Hevelius, and others, have supposed them to be scoria floating in the inflammable liquid matter of which they imagined the sun to be composed. This opinion, although it accounts for the appearance of the spots in the equatorial regions of the sun, to which such scoria would be carried by the centrifugal force resulting from its rotation, cannot be reconciled with the regularity with which the spots frequently re-appear on the eastern limb of the sun. Dr Wilson, having observed that the spots situated near the edge of the disk are narrow, and without a penumbra on the side next the centre, and that only the central spots are completely surrounded by a penumbra (appearances which would be exactly represented by a conical gulf or cavity presented to us under different aspects by the revolution of the sun), was led to adopt the opinion, that the appearances of the spots are occasioned by real excavations in the solar globe. He supposed the sun to consist of a dark nucleus, covered only to a certain depth by a luminous matter, not fluid, through which openings are occasionally made by volcanic or other energies, permitting the solid nucleus of the sun to be seen; and that the umbra which surrounds the spot is occasioned by a partial admission of the light upon the shelving sides of the precipice opposite to the observer. It is evident, that, in proportion as these excavations are seen obliquely, their apparent dimensions will be diminished; one of the edges will disappear as it approaches the sun's limb, or come more into view as it advances towards the middle of the disk; when the spot is about to leave the disk, the bottom of the excavation, or the nucleus seen through it, will first disappear, but a sort of faint or obscure spot will remain visible as long as the visual ray penetrates the cavity. These appearances are all conformable to the laws of perspective; but Dr Wilson, wishing to give a still more palpable demonstration of the accuracy of his theory, fitted up a large globe, into which holes of the proper dimensions were inserted; and this machine being placed at a distance, and made to revolve, was found, when examined through a telescope, to exhibit in the course of its revolution all the phenomena of the solar spots. Dr Wilson's theory was keenly combated by Lalande, who adduced several observations of his own, and some by Cassini, that could not be explained by means of it; and urged with reason, that an hypothesis, founded on the uniformity of appearances which in reality are exceedingly variable, was entitled to little consideration. Lalande himself supposed the spots to be scoriae which have settled or fixed themselves on the summits of the solar mountains; an opinion which he grounded on this circumstance, that some large spots which had disappeared for several years were observed to form themselves again at the identical points at which they had vanished.

The late Sir William Herschel, with a view to ascertain more accurately the nature of the sun, made frequent observations upon it from the year 1779 to the year 1794. He imagined the dark spots on the sun to be mountains, 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 thought might 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, when it evidently appeared that the dark spots are the opaque ground or body of the sun, and that the luminous part is an atmosphere, through which, when interrupted or broken, we obtain a view of the sun itself. Hence he concluded that the sun has a very extensive atmosphere, consisting of elastic fluids that are more or less lucid and transparent, and of which the lucid ones furnish us with light. This atmosphere, he thought, cannot be less than 1843, nor more than 2765 miles in height; and he supposed that the density of the luminous solar clouds needs 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, disseminating its light and heat to all the bodies with which it is connected.

Herschel supposed that there are two regions or strata of solar clouds; that the inferior stratum is opaque, and probably not unlike our own atmosphere, while the superior is the repository of light, which it darts forth in vast quantities in all directions. The inferior clouds act as a curtain to screen the body of the sun from the intense brilliancy and heat of the superior regions, and, by reflecting back nearly one half of the rays which they receive from the luminous clouds, contribute also greatly to increase the quantity of light which the latter send forth into space, and thereby perform an important function in the economy of the solar system. The luminous clouds prevent us in general from seeing the solid nucleus of the sun; but in order to account for the spots, he supposes an empyreal elastic gas to be constantly forming at the surface, which, carried upwards by reason of its inferior density, forces its way through the planetary or lower clouds, and mixing itself with the gases which have their residence in the superior stratum, causes decompositions of the luminous matter, and gives rise to those appearances which he describes under the name of corrugations. Through the openings made by this accidental removal of the luminous clouds, the solid body of the sun becomes visible, which, not being lucid, gives the appearance of the dark spots or nuclei seen through the telescope. The length of the time during which the spots continue visible renders it evident that the luminous matter of the sun cannot be of a liquid or gaseous nature; for, in either case, the vacancy made by its accidental removal would instantly be filled up, and the uniformity of appearance invariably maintained. Herschel supposed the luminous clouds to be phosphorescent.

It would be a needless waste of time to enter into any discussion of a theory so entirely vague and fanciful, respect and so destitute of all solid foundation. Till we are better acquainted with the nature of light, fire, and heat, and have attained to the knowledge of every possible mode in which these elements can be produced and propagated, all hypotheses respecting the construction of the sun can only be gratuitous and conjectural. Some interesting questions, however, arise out of this subject. Whether the inferior stratum of solar clouds is sufficiently dense, as Herschel imagined, to protect the body of the sun from the scorching effects of the surrounding regions of light and heat, and render it a fit habitation for human beings, is a question of no importance to man, or to anything pertaining to his planet; but it is interesting for him to know whether the light and heat dispensed by the sun are liable to any variation or secular diminution, either connected with the spots, or resulting from a decrease of the sun's volume.

Those philosophers who adopt the hypothesis of the production of light and heat by the vibrations of an ethereal fluid, consider the mass of the sun to be invariable. Those, on the contrary, who attribute these effects to an emanation from the sun, think his mass and volume must be diminished by the incessant discharge of torrents of luminous particles from his surface. During the two thousand years which have elapsed since the first astronomical observations, no diminution of the sun's volume has been perceived; but it must be remarked that such an effect may have taken place, though not yet sensible to our instruments. The sun's diameter is nearly 2000"; and at the distance of 95,000,000 miles a second corresponds to 460 miles. Now, supposing the solar diameter to suffer a daily diminution of two feet, which may be considered as enormous, considering the vast magnitude of the sun, and the excessive rarity of light, the diminution would amount to 800 feet in a year, and to 460 miles, or 1", in 3000 years. Thus, after thirty centuries, the diminution would still be imperceptible, insomuch as our instruments are not sufficiently accurate to enable us to appreciate, in an observation of this sort, so small a variation as one second.

Some astronomers, after Herschel, have imagined that the existence of the solar spots has an influence on the temperature of the seasons. In 1823 the summer was cold and wet; the thermometer at Paris rose only to 23° of Reaumur, and the sun exhibited no spots; whereas in the sea the summer of 1807 the heat was excessive, and the spots of vast magnitude. The relation, however, between the temperature and the appearance of the solar spots is not so uniform as to give much weight to this opinion. Warm summers, and winters of excessive rigour, have happened in the presence or absence of the spots. The year 1783 was remarkable for its fertility and the magnitude of the solar spots; a dry fog enveloped the greater part of Europe, and was followed by the earthquake of Calabria. Another opinion entertained by Herschel was, that one hemisphere of the sun emits less light than the other, so that when viewed at a great distance he will resemble some stars of which the brilliancy is subject to periodical variations.

By reason of the globular figure of the sun, his surface towards the border of the disk is seen obliquely, and therefore a much greater portion of it is comprehended. Theoretical ed under a given visual angle, than when the ray proceeds from his centre. Now, as every point of the sun's surface is supposed to emit an equal quantity of light in all directions, it follows that the light ought to be much more intense near the circumference of the disk, because a greater number of rays will proceed from the larger surface, which forms the oblique base of the luminous cone.

Bouguer, deceived by some imperfect experiments thought the light more intense at the centre of the disk than towards the limb; a circumstance which could only be explained by supposing the light to be diminished by some cause which acts most powerfully with regard to the borders of the disk. Such would be the effect of a dense atmosphere surrounding the sun; for in this case the rays which proceed from the border must traverse a much greater extent of the solar atmosphere, and consequently be absorbed in greater proportion than those which proceed from the central parts and traverse it directly; just as the atmosphere of the earth renders the light of the stars at the horizon much feebler than at the zenith.

It has been found, however, that Bouguer's experiments were inaccurate, and that the light is equally intense at the border and the centre. The existence of a solar atmosphere cannot therefore be demonstrated in this manner; but it is clearly indicated by the faint light which is observable round the sun's limb during a total eclipse.

Another very curious phenomenon connected with the sun, is the faint nebulous aurora which accompanies him, known by the name of the Zodiacaal Light. This phenomenon was first observed by Kepler, who described its appearance with sufficient accuracy, and supposed it to be the atmosphere of the sun. Dominic Cassini, however, to whom its discovery has been generally but erroneously attributed, was the first who observed it attentively, and gave it the name which it now bears. It is visible immediately before sunrise, or after sunset, in the place where the sun is about to appear, or has just quitted, in the horizon. In total eclipses it is seen surrounding the sun's disk, and resembling the beard of a comet. It has a flat lenticular form, and is placed obliquely on the horizon, as represented in fig. 41, the apex extending to a great distance in the heavens. Its direction is always in the plane of the sun's equator, and for this reason it is scarcely visible in our latitudes, excepting at particular seasons, when that plane is nearly perpendicular to the horizon. When its inclination is great, it is either concealed altogether under the horizon, or at least rises so little above it, that its splendour is effaced by the atmosphere of the earth. The most favourable time for observing it is about the beginning of March, or towards the vernal equinox. The line of the equinoxes is then situated in the horizon, and the arc of the ecliptic SS' S" (fig. 42) is more elevated than the equator SEQ by an angle of $23\frac{1}{2}$ degrees; so that the solar equator, which is slightly inclined to the ecliptic, approaches nearer to the perpendicular to the horizon, and the pyramid of the zodiacal light is consequently directed to a point nearer the zenith, than at any other season of the year. For example, at the summer solstice, S'T', a tangent to the ecliptic, is parallel to EQ the tangent to the equator, and the luminous pyramid is in a plane less elevated by $23\frac{1}{2}$ than at the time of the vernal equinox.

Numerous opinions have been entertained respecting the nature and cause of this singular phenomenon. Cassini thought it might be occasioned by the confused light of an innumerable multitude of little planets circulating round the sun, in the same manner as the milky way owes its appearance to the light of agglomerated myriads of stars. Its resemblance to the tails of comets has been noticed by Cassini, Fatia Duillier, and others; and Euler endeavoured to prove that they are both owing to similar causes. Mairan, like Kepler, ascribed it to the atmosphere of the sun; and this hypothesis was generally adopted, till it was shown by Laplace to be untenable for the following reasons. The atmosphere of any planet, endowed with a motion of rotation, cannot extend to an indefinite distance: it can only reach to such a height that the centrifugal force is exactly balanced by the force of gravity. Beyond this height the atmosphere would be dissipated by the superior energy of the centrifugal force. Now the height above the sun at which the two forces are equal is that at which a planet, if placed there, would revolve about the sun in the same time in which the sun performs a revolution on his axis. But the orbit of such a planet would be greatly inferior to the orbit of Mercury; for the time in which Mercury makes a revolution in his orbit is eighty-eight days, while the sun revolves about his axis in twenty-five: it is therefore certain that the atmosphere of the sun cannot extend to the orbit of Mercury. Now, the greatest elongation of Mercury does not exceed $28^\circ$, and the zodiacal light has been observed to extend to above $100^\circ$, reckoning from the sun to the apex of the luminous pyramid. Hence the phenomenon cannot proceed from the sun's atmosphere. Laplace further remarks, that the ratio of the equatorial and polar axes of the solar atmospherical spheroid cannot exceed that of three to two; whence its form would not correspond with the lenticular appearance of the zodiacal light. But to whatever cause this luminous matter is to be attributed, it is certain that it is of extreme rarity, insomuch as it does not intercept the light of the smallest stars which are seen through it without any diminution of splendour.

On the subject of the solar spots and zodiacal light the following works may be consulted:—Galileo, *Istoria e Dimostrazioni intorno alle Macchie Solari*, Rome, 1613; Scheiner, *Rosa Ursina*, Braccianoi, 1630; Hevelius, *Selenographia*, Gedani, 1647; Reuschii, *De Maculis et Faculis Solaribus*, Wittemb. 1661, 4to; Cassini, *Nouv. Observ. des Taches du Soleil*; Hooke, *Tractatus de Maculis Solaribus, et Lunae Zodiacali*, in Oper. Posth. Lond. 1705; Weidler, *De Coloribus Mac. Sol. in his Observ. Meteorol.* 1728–9, Wittemb. 1729; Bosovich, *De Mac. Sol. Rom.* 1736, 4to; De Lisle, *Memoire pour servir à l'Histoire de l'Astr. Petersb.* 1738; Bernoulli, *Lettres Astronomiques*, 1771; Wilson, *Phil. Trans.* 1774, vol. xix.; *Ibid.* 1783; Wollaston, *Phil. Trans.* 1774; Lalande, *Phil. Trans.* 1776; *Mém. Acad.* 1776, and *Astronomie*, tom. iii. p. 277; Herschel, *Phil. Trans.* 1795 and 1801; Woodward on the Substance of the Sun, Washington, 1801; Biot, *Traité de l'Astronomie Physique*, tom. ii.; Bohm, *Disput. Astron. de Fascia Zodiacaal*; Cassini, *Mém. Acad. Par.*, tom. vii. p. 119, and viii. p. 193; Mairan, *Traité Physique et Historique de l'Aurore Boreale*, 1731; Lalande, *Astronomie*, tom. i. p. 276; Laplace, *Exposition du Système du Monde*; Delambre, *Hist. de l'Astronomie Moderne*, tom. ii. p. 742.

**CHAP. III.**

**OF THE MOON.**

Next to the sun, the moon is to mankind the most important and interesting of all the celestial bodies. Her conspicuous appearance in the heavens, the variety of her phases, and the rapidity with which she changes her place among the fixed stars, have rendered her at all times an object of admiration to the vulgar; while her proximity to the earth, her physical effects on the ocean, the intricacy of the theory of her motions, and the vast importance of that theory to navigation and geography, have equally Sect. I.—Of the Phases, Parallax, and Magnitude of the Moon.

The different appearances or phases of the moon were probably the first celestial phenomena observed with any degree of attention. When the moon, after having been for some days invisible, is again seen on the eastern side of the sun, and at a distance of 20 or 30 degrees from him, she appears as a curved thread of silvery light; and her form is that of a crescent, the horns of which are turned towards the east. The breadth of the crescent increases continually in proportion as she separates herself from the sun, till, having obtained a distance of 90°, she appears under the form of a semicircle. At this point she is said to be in her first quarter. Continuing her motion to the eastward, the line which terminates the eastern side of her disk assumes the curvature of an elliptic arc, and her visible portion continues to increase till she has attained the distance of 180° from the sun, when she appears perfectly round. She is then full, and is said to be in opposition, rising as the sun sets, and consoling us by her pale light for the absence of the great luminary. Having passed this point, she begins to approach the sun; her western side now takes the form of the elliptic arc, and her luminous portion diminishes exactly in the same proportion as it increased through the first half of her orbit. About seven days after the full she again appears as a semicircle, the diameter of which is turned towards the west, and she is now at her third quarter, and at a distance of 90° from the sun. The semicircle after this changes into a crescent, and she continues to approach the sun, till, having advanced to within 20° or 30° from him, she again disappears, being lost in the splendour of his rays. These phases regularly succeed each other, and the time in which they run through all their changes is about 29½ days. When the moon passes the meridian at the same time with the sun, she is said to be in Conjunction. The two points of her orbit in which she is situated when in opposition or conjunction are called the Syzygies; those which are 90° distant from the sun are called the Quadratures; and the intermediate points between the syzygies and quadratures are called the Octants.

A slight attention to the lunar phases during a single revolution will be sufficient to prove that they are occasioned by the reflection of the sun's light from the opaque spherical surface of the moon. This fact, which was recognised in the earliest ages, will be made obvious by the help of a diagram. If the moon is an opaque body we can only see that portion of her enlightened side which is towards the earth. Therefore, when she arrives at that point of her orbit A (fig. 43) where she is in conjunction with the sun S, her dark half is towards the earth, and she disappears, 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 her conjunction, 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. Theoretical At G we see just one half of her enlightened side, and she appears as a semicircle, as at g. At H we only see a quarter of her enlightened side, being in her fourth octant, when 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, exhibiting like phases at equal distances from A and E.

The magnitude of the visible portion of the moon's disk thus depends on the situation of the moon relatively to the sun and the earth, and is easily determined geometrically from her elongation or angular distance from the sun. Let ADBC (fig. 44) be the projection of the lunar orb on the plane which passes through the centres of the sun, moon, and earth; let S be the place of the sun, E that of the earth, M the centre of the projection, and AB, CD two diameters perpendicular to SM and EM respectively, and let EM meet the circumference in G. It is evident that AGDB represents the hemisphere illuminated by the sun, and CAGD that which is visible from the earth; the whole portion of the visible disk, therefore, is represented by AGD, or by AG and DG. Now, if we conceive the moon's surface to be projected on the plane which passes through her centre perpendicular to the visual ray, the illuminated portion of the disk DG will appear as a semicircle, while the part GA will appear as a semi-ellipse, the minor axis of which is to its major as MF to MC, AF being perpendicular to MC. The eye at E will therefore see the semicircle DMG, together with the semi-ellipse GMA, and the visible part will be to the entire disk as DF : DC, that is, as \(1 + \sin AME : 2\), or, as \(1 + \cos \phi : 2\); \(\phi\) being equal to EMS the elongation of the earth from the sun as seen from the moon. The geocentric elongation of the moon from the sun, or the angle MES = \(\psi\), may be substituted instead of \(\phi\), when we know the ratio of the distances ES = r and MS = R; for \(\sin \phi = \frac{r}{R} \sin \psi\), whence \(\cos \phi = \sqrt{1 - \frac{r^2}{R^2} \sin^2 \psi}\), and the above proportion becomes

\[DF : DC :: 1 + \sqrt{1 - \frac{r^2}{R^2} \sin^2 \psi} : 2.\]

When the moon is in opposition the angle \(\phi = 0\), whence \(1 + \cos \phi = 2\), and the whole disk is visible. At the conjunction \(\phi = 180°\), and \(1 + \cos \phi = 0\); the moon is consequently invisible.

The triangle EMS likewise furnishes the means of finding the distance of the moon when that of the sun is known; or reciprocally, the distance of the sun from that of the moon; but for this purpose it is necessary to observe accurately the limit of illumination, which is not easily done. Let ADBE (fig. 45) be the illuminated part Fig. 45 of the moon ACBE. The distance between the cusps A and B, measured by the micrometer, gives the major axis of the ellipse ADB, and the semiconjugate axis DM is found by measuring in the same manner the distance ED, and subtracting EM, half the transverse AB. This gives \(\frac{DM}{AM} = \cos EMS\) (fig. 44); this gives the angle EMS or M; and from SEM, which is given by observation, we have ESM or S = 180° — E — M; whence M and S are given angles; and from the property of the triangle we have \(EM = ES \frac{\sin S}{\sin M}\), or \(ES = EM \frac{\sin M}{\sin S}\). The calculation would be greatly facilitated if the observation... Theoretical were made at the exact instant when the moon is dichotomized, or in her first or third quarter, for the elliptic arc then becomes a straight line and the angle EMS = 90°; so that sin. M = 1. In this case therefore we have EM = ES sin. S = ES cos. E, and ES = \(\frac{EM}{\cos. E}\), which gives the ratio of the distances of the sun and moon. In this manner Aristarchus of Samos attempted to compare the distance of the two luminaries; and although his result was extremely erroneous, inasmuch as he found the distance of the sun to be only between eighteen and twenty times greater than that of the moon, whereas it is actually 380 times greater, yet the idea was an ingenious one, and the method perfectly accurate in theory. The error arose from the inaccuracy of the ancient instruments, and the impossibility of observing to within a few minutes the exact time of the dichotomy. Besides, as the angle at the sun is extremely small, a very slight error either with regard to the time or the angle has a great influence on the result.

Parallax of the moon:

The moon's absolute distance from the earth is obtained by means of her parallax, which, on account of her proximity, is very considerable, and can therefore be determined by the methods which were explained in chap. i. sect. 2. On comparing her parallaxes observed at different times, they are found to differ considerably in value. Thus, according to Lalande, the greatest horizontal parallax at Paris is 61' 29", the smallest 53' 51"; and it assumes every possible value between these two extremes. The horizontal parallax being the angle which the earth's radius subtends at the moon, and consequently equal to the radius of the earth divided by the moon's distance, the perigean and apogean distances corresponding to the extreme parallaxes are respectively 55'916 and 63'842 semidiameters of the earth. The ratio of these two numbers, or of the extreme values of the parallax, is 1:1417, and denotes the greatest variations of the moon's distance from the earth. The ratio of the corresponding quantities in the case of the sun is only 1:034; hence the moon's distance is subject to much greater variations than that of the sun.

These differences in the value of the parallax arise from the variations of the moon's distance from the earth; but it is also observed to differ sensibly at different points of the earth's surface, even at the same instant of time. If the earth were spherical, the horizontal parallax of the moon, supposing her distance to be invariable, would be the same at whatever part of the earth it might be observed. The case is, however, different; for, on account of the spheroidal figure of the earth, the section made by every vertical plane gives a different ellipse, no one of which even passes through the centre of the earth, excepting indeed the cases in which the place of the observer is on the equator or under the pole; for in all other cases the perpendicular to the surface of the earth does not pass through its centre. Hence it is necessary, in speaking of the horizontal parallax, to specify the place of the observation.

Since the parallax of the moon is subject to incessant variation, it is necessary to assume a certain mean value of it, about which the true and apparent values may be conceived to oscillate. This is denominated the constant parallax. It is evident that it cannot be found by taking the arithmetical mean of the extreme values; for by reason of the disturbing force of the sun, the eccentricity of the lunar orbit, and consequently the perigean and apogean distances, are constantly varying; and as the quantity by which one of these distances is increased is not equal to that by which the other is diminished, it follows that the mean of the perigean and apogean parallaxes will not be a constant quantity, but different in successive revolutions of the moon. If we abstract from all the inequalities of the lunar orbit, and suppose the moon to be at her mean distance and mean place, the constant parallax will be the angle under which a given semidiameter of the earth is seen by a spectator at the moon in such circumstances. According to Lalande, the following are the values of the constant parallax:

- Under the equator: 57° 5' - Under the pole: 56° 58' 2" - For the latitude of Paris: 56° 58' 3" - For the radius of a sphere equal in volume to the earth: 57° 1"

The mean equatorial parallax being 57° 5', its double is Magn. 1° 54' 10", which expresses the angle subtended by the tide of diameter of the earth at the distance of the moon. The lunar angle subtended by the moon at the same distance is 31° 26"; whence the diameter of the moon is to that of the earth as 31° 26" is to 1° 54' 10"; that is, as 524 to 1899, or as 3 to 11 nearly. According to the tables of Burg, the accurate expression of the above ratio is 1:0-27293; hence the true diameter of the moon is 0-27293 diameters of the terrestrial equator. The surface of the moon is consequently (0-27293)^2 = 0-07449908 = \( \frac{1}{13} \) of that of the earth, and its volume (0-27293)^3 = 0-02033008 = \( \frac{1}{39} \) of that of the earth, or, in round numbers, \( \frac{1}{40} \)th of the volume of the earth.

Sect. II.—Of the Elements of the Lunar Orbit.

1. Nodes and Inclination of the Lunar Orbit.—A few simple observations of the right ascensions and declinations of the moon suffice to show that her path is confined nearly to a plane, having only a small inclination to the plane of the ecliptic. If her path were rigorously confined to this plane, a single revolution would be sufficient to determine its inclination; but as numerous disturbing causes exist, which produce inequalities in all the elements of the lunar motion, it is only by taking the mean value of a great number of observations that these elements can be astronomically determined. In fixing the position of the plane of the orbit, the first object is to determine the straight line formed by its intersection with the plane of the ecliptic; that is to say, to determine the line of the Nodes. This would be at once accomplished if it were possible to observe the instant at which the moon's latitude is nothing; for at this instant her centre is in the plane of the ecliptic, and consequently her longitude is the same as the longitude of the node. But as the accuracy of modern practice requires that all important observations be made in the plane of the meridian, and as it could only happen by a very rare coincidence that the latitude deduced from the meridional right ascensions and declinations would be exactly zero, it is necessary to have recourse to a process of calculation to find the precise place of the node.

Let LNL' (fig. 46) be a portion of the ecliptic, MN'M' Fig. 46 of the lunar orbit, N the place of the node, and ML = l, ML' = l' be two latitudes on opposite sides of the ecliptic. The two right-angled triangles give, according to Napier's Rules,

\[ \tan. N = \frac{\tan. l}{\sin. NM} = \frac{\tan. l}{\sin. NM'}; \]

therefore,

\[ \frac{\sin. NM}{\sin. NM'} = \frac{\tan. l}{\tan. l'}; \]

whence,

\[ \frac{\sin. NM - \sin. NM'}{\sin. NM + \sin. NM'} = \frac{\tan. l - \tan. l'}{\tan. l + \tan. l'}; \] that is, by the trigonometrical formulæ,

\[ \tan \frac{1}{2}(NM - NM') = \frac{\sin (l - l')}{\tan \frac{1}{2}(NM + NM')} \]

or

\[ \tan \frac{1}{2}(NM - NM') = \tan \frac{1}{2}MM' \cdot \frac{\sin (l - l')}{\sin (l + l')} \]

Having therefore obtained an expression for the difference of \(NM\) and \(NM'\), and knowing also their sum \(MM'\), which is the difference of the longitudes of the moon at the time of the two observations, it is easy to deduce \(NM\) or \(NM'\); consequently the longitude of the node is determined.

The Ascending Node of the lunar orbit is that point of the ecliptic through which the moon passes when she rises above the ecliptic towards the north pole; it is distinguished by the character \(Q\). The Descending Node, \(Q'\), is the opposite point of the ecliptic, through which she passes when she descends below that plane towards the south pole. The nodes of the lunar orbit were anciently called the head and tail of the Dragon.

The position of the nodes is not fixed in the heavens. They move in a retrograde direction, or contrary to the order of the signs; and their motion is so rapid that its effects become very apparent after one or two revolutions. Regulus, a star of the first magnitude, is sometimes eclipsed by the moon; and as the latitude of this star is only about 27° or 28°, it is certain that when that phenomenon occurs the moon is very near her node. Suppose it to be the ascending node: the month following it will be observed that the moon passes to the north of Regulus; and every succeeding month she will pass farther to the north of the same star, till at the end of four or five years, when her meridional altitude will be about 5° greater than that of Regulus. Having attained this elevation, her latitude begins to diminish; and, after a period of about nine years, Regulus is again eclipsed by the moon in her descent to the southern side of the ecliptic. The moon after this passes to the south of the star for the following nine years; and at the end of 18½ years the nodes have returned to their first position, after accomplishing a complete revolution. The same result is found from the observation of eclipses, the magnitude of which depends on the moon's latitude; and therefore, if the nodes remained fixed, the eclipses would always be of the same magnitude in the same quarter of the heavens. This, however, is not the case; and it is only after about 18½ years that they begin to return in the same order.

The mean retrograde motion of the nodes is found by the comparison of observations made at distant epochs to amount to 19° 19' 42"·316 in a mean solar year, and the time in which they make a complete revolution is consequently 6798 days, 4 hours, 17 min. 43·18 sec. The longitude of the ascending node on the 1st of January 1801 was 13° 53' 17"·7; hence its position at any other epoch is easily deduced, since we know the rate of its mean motion.

The inclination of the lunar orbit may be determined either by observing the moon's greatest latitudes, or it may be computed from the formula \(N = \frac{\tan l}{\sin NM'}\) when her latitude and distance from the node are known. The angle \(N\), or the inclination, is observed to vary between 5° and 5° 17' 35". The mean inclination may therefore be taken at 5° 8' 48".

The retrograde motion of the moon's nodes, as well as all the other inequalities of the lunar motion, is occasioned by the attracting power of the sun, which varies in intensity with the sun's longitude, or distance from his perigee. The principal inequality to which the inclination of the lunar orbit is subject, is proportional to the cosine of double the sun's distance from the moon's ascending node. In fact, the sun and earth being always in the plane of the ecliptic, when the moon is also in that plane the action of the sun will have no tendency to increase or diminish her latitude; it will only affect her radius vector, or distance from the earth. But if the moon is not in the plane of the ecliptic, the sun's attracting force will not only affect her distance from the earth, but also tend to bring her nearer to the ecliptic; and this action will be greater in proportion to her deviation from that plane. When the nodes are in the quarters, and the limits of latitude in the syzygies, the inclination is 5°, as was found by Ptolemy; but when the nodes are in the syzygies, and the limits in the quarters, the inclination is 5° 17' 35". It has therefore a mean value of 5° 8' 48", when the nodes and limits coincide with the octants. Denoting therefore the greatest deviation from the mean values by \(x\), this deviation will pass through all its values between \(+ x\) and \(- x\) while the sun's distance from the ascending node varies from 0° to 90°. It may therefore be represented by a function which varies as the cosine of twice that distance, for the cosine of an angle runs through all its changes from \(+ 1\) to \(- 1\) between 0 and 180°. Hence the equation which expresses the principal inequality of the moon's latitude is

\[ (8° 48") \cos 2 Q \text{ 's distance from } P \text{ 's } Q. \]

The co-efficient 8° 48" is called, in astronomy, the Co-efficient of the Argument, the argument itself being the function by which the inequality is represented.

This periodic inequality of the moon's latitude was discovered by Tycho Brahe, from a comparison of the greatest latitudes at different epochs with the corresponding positions of the moon with reference to her nodes. He observed that these latitudes oscillate about the mean value of 5° 8' 48"; and as the greatest latitudes give immediately the inclination of the moon's orbit to the plane of the ecliptic, it necessarily followed that the inclination is subject to similar variations. The motion of the node of the lunar orbit is also subject to an inequality, depending on the same angle as the preceding, but proportional to its sine. Tycho explained these two inequalities by a very simple hypothesis, similar to that by which we have explained the oscillations of the terrestrial equator, namely, a slight nutation of the earth's axis. In the present case it is only necessary to place the mean pole of the lunar orbit at the distance of 5° 8' 48" from the pole of the ecliptic, and to suppose the true pole to describe a small ellipse about the mean pole in the same time that the sun occupies in making a semi-revolution with regard to the nodes of the moon's orbit, that is, in 173·309 days.

There are various other inequalities which affect the latitude of the moon and the inclination of her orbit, to correct which, no fewer than twelve equations are given in the recent and most accurate tables. The greater part of them are, however, only known from theory, and are so small that even their accumulated effects are scarcely perceptible to observation.

The inclination of the lunar orbit to the plane of the terrestrial equator occasions considerable differences in the intervals between the moon's rising or setting on successive harvest-days, and gives rise to the phenomenon of the Harvest Moon. As the daily motion of the moon is about 13 degrees from west to east, it follows that if she moved in a plane parallel to the equator, she would rise 50 minutes later every successive evening, because her orbit would then make the same angle with the horizon at all seasons of the year, and the intervals between her consecutive risings would be constant. For the sake of explanation, we may here suppose the moon to move in the plane of Theoretical Astronomy. Now, the time in which a given arc of the ecliptic rises above the horizon depends on its inclination to the horizon. In our latitudes the inclination of the ecliptic at different points to the horizon varies so much, that at the first point of Aries an arc of 13° becomes visible in the short space of 17 minutes, while at the 23d of Leo the same arc will only rise above the horizon in one hour and 17 minutes. Hence, when the moon is near the first point of Aries, the difference of the times of her rising on two successive evenings will be only about 17 minutes; and as this happens in the course of every revolution, she will rise for two or three nights every month at nearly the same hour. But the rising of the moon is a phenomenon which attracts no attention, excepting about the time when she is full, that is, when she rises at sunset. In this case she is in opposition to the sun, and consequently, if she is in Aries, the sun must be in Libra, which happens during the autumnal months. At this season of the year, therefore, the moon, when near the full, rises for some evenings at nearly the same hour. This circumstance affords important advantages to the husbandman, on which account the phenomenon attracts particular attention.

It is obvious, that as this phenomenon is occasioned by the oblique position of the lunar orbit with regard to the equator, the effect will be greater than what has just been described if the plane of that orbit makes a greater angle with the equator than the plane of the ecliptic does. But we have seen that the plane of the moon's orbit is inclined to the ecliptic in an angle exceeding 5°; consequently, when her ascending node is in Aries, the angle which her orbit makes with the horizon will be 5° less than that which the ecliptic makes with the horizon; and the difference of time between her risings on two successive evenings will be less than 17 minutes, as would have been the case had her orbit coincided with the ecliptic. On the contrary, when the descending node comes to Aries, the angle which her orbit makes with the horizon will be greater by 5°, and consequently the difference of the times of her successive risings will be greater than if she moved in the plane of the ecliptic. If when the full moon is in Pisces or Aries the ascending node of her orbit is also in one of those signs, the difference of the times of her rising will not exceed one hour and forty minutes during a whole week; but when her nodes are differently situated, the difference in the time of her rising in the same signs may amount to 3½ hours in the space of a week. In the former case the harvest-moons are the most beneficial, in the latter the least beneficial, to the husbandman. All the variations in the intervals between the consecutive risings or settings take place within the period in which the line of the nodes makes a complete revolution.

2. Dimensions, Eccentricity, and Apsidal Lines of the Lunar Orbit; and the Inequalities of the Moon's motion.—Since the moon moves in a plane orbit, the projection of her path on the surface of the celestial sphere will be a great circle. But the variations observable in the magnitude of her apparent diameter, and in the velocity of her motion, prove that if her orbit is a circle, the earth is at least not placed at its centre. The numerous perturbing causes which affect her motion render the exact determination of her orbit a matter of great difficulty; but observation shows that it deviates very little from an ellipse, of which the centre of the earth occupies one of the foci. It has also been found that the spaces passed over by the radius vector of the moon are very nearly proportional to the times of description; which is the distinctive character of the elliptic motion.

In order to obtain a correct idea of the figure and position of the orbit, it is necessary to know its major axis, its eccentricity, and the position of its apsides. The major axis is equal to the sum of the greatest and least distances, and these have already been stated to be respectively 63'842 and 59'916 semidiameters of the earth; whence, supposing the earth's radius to be 4000 miles, the major axis of the lunar orbit will amount in round numbers to 480,000 miles. It may be here remarked that the moon's distance is an element which may be practically determined with great exactness. A variation of 1" in the parallax would occasion an error of only about 67 miles in the determination of the distance; therefore, since the parallax is certainly known to within 4", the greatest error in the distance deduced from it cannot exceed 280 miles out of about 240,000 miles.

An approximation to the eccentricity of the orbit of the moon is easily obtained from the variations of her apparent diameter. The moon's apparent diameter is observed to vary between 29°30' and 33°30' very nearly. Now, let D denote the mean, D' the apogee, D" the perigee diameter, and e the eccentricity. We shall then have

\[ D' = \frac{D}{1+e} = 29°30' \]

\[ D'' = \frac{D}{1-e} = 33°30', \]

\[ D' = \frac{1-e}{1+e} = 29°30' \]

whence we deduce

\[ e = \frac{33°30' - 29°30'}{33°30' + 29°30'} = \frac{4'}{63'} = 0.0635. \]

This eccentricity corresponds to an equation of the centre amounting to 7°16', and is much more considerable than that of the solar orbit, which, as we have before seen, amounts only to 0°0168. The result just given is, however, only to be considered as approximate, and is, in fact, considerably too large. In the best and most recent tables the value assigned to the eccentricity of the lunar orbit is 0°0548442, and the corresponding equation of the centre 6°17'12"7'. (See Astronomical Tables and Formulas, by Francis Bailly, Esq. London, 1827.)

The place of the apsides is likewise found from observations of the moon's apparent diameter, because at these points of the orbit the apparent magnitude has its maximum and minimum values. But as the apparent magnitude varies very slowly at the apsides, it is preferable, as Lalande remarks, to choose the points of mean distance where its variations are most rapid. Having selected two observations which give the same apparent diameter at two opposite points of the orbit, we shall have two points evidently at equal distances from the perigee, the longitude of which will therefore be given at the middle time between the two observations. In comparing the positions of the apsides thus determined at different epochs, it is found that they have a rapid progressive motion, that is, according to the order of the signs. In Mr Bailly's tables the motion of the perigee is stated to be 4069°-046278, or 11 revolutions + 109°2'46"6, in 365325 mean solar days. Its mean motion in a mean solar day is consequently 6°41"; and in 365 mean solar days, 40°39'46"36. Hence the time in which it completes a sidereal revolution is 32321575343 mean solar days. The period of a tropical revolution of the apsides is 32314751 mean solar days = 3231 days, 11 hours, 24 minutes, 08 seconds, or nearly 9 years. This mean motion of the lunar apsides is, however, subject to periodic inequalities of considerable magnitude, and therefore can only be determined accurately by means of observations separated from In the tables just quoted, the epoch is the commencement of the present century, when the longitude of the perigee was in 266° 10' 7" 5". From this and the rate of the mean motion, the longitude of the perigee at any other epoch may be easily computed.

If the moon's place in her orbit were not subject to the influence of any disturbing force, her true longitude, or distance from the apsis, would be found at any instant from the theory of the elliptic motion, supposing her mean motion, or the time of a return to the same perigee or apogee, to be accurately known. Thus, supposing $E$ to denote the maximum elliptic equation, which, as we have already stated, amounts to 6° 7' 12" 7', we should have the true longitude = mean longitude + $E \sin A$, $A$ being the moon's mean anomaly, and the longitudes counted from the perigee. This first equation or correction of the mean motion results solely from the circumstance that the moon moves in an elliptic orbit, according to the laws of Kepler, in the same manner as the sun and all the planets. Its argument is the mean anomaly $A$, and its period the anomalistic month. But the moon's longitudes, as determined in this manner, are far from agreeing with observation; and several other corrections must be applied, the accurate determination of which forms the principal object of the lunar theory.

After the equation of the centre, the most considerable of the inequalities of the moon's longitude is that to which Boulliau gave the name of the Evection. It was noticed by Hipparchus; but it was Ptolemy who discovered its law, and gave a construction which represents its general effects with great accuracy. These effects are to diminish the equation of the centre when the line of the apsides lies in syzygy, and to augment it when the same line lies in quadratures. Thus, supposing the apsides to lie in syzygy, and that it is sought to compute the moon's true longitude about seven days after she has left the perigee, by adding the equation of the centre to the mean anomaly, the resulting longitude will be found to be above 80' less than that which is given by observation. But if the line of the apsides lies in quadratures, the place of the moon at about the same distance, that is, 90° from the perigee, computed in the same manner, will be found to be before the observed place by above 80'; that is, the computed will be greater than the observed longitude, by more than 80 minutes. After a long series of observations, astronomers have found that the inequality in question is represented with great exactness, by supposing it proportional to the sine of twice the mean angular distance of the moon from the sun, minus the mean anomaly of the moon. In Baily's tables, its maximum value is stated to be 1° 20' 29" 9'; whence, representing the angular distance of the sun and moon by $\varphi - \Theta$, and the mean anomaly as before by $A$, the correction due to the evection will be

$$\left(1° 20' 29" 9'\right) \sin \left[2(\varphi - \Theta) - A\right].$$

A third inequality of the lunar motion, called the Variation, was discovered by Tycho Brahe, who found that the moon's place, calculated from her mean motion, the equation of the centre, and the evection, does not always agree with the true place, and that the variations are greatest in the octants, or when the line of the apsides makes an angle of 45° with that of the syzygies and quadratures. Having observed the moon at different points of her orbit, he found that this correction has no dependence on the position of the apsides, but only on the moon's elongation from the sun. Its maximum value is additive in the octants which come immediately after the syzygies, where the elongation is 45°, or 180° ± 45°, and subtractive in the octants which precede the syzygies, where the Theoretical elongation is 360° - 45°, and 180° - 45°. It vanishes altogether in the syzygies and quadratures where the elongation is 0°, 180°, 90°, or 270°; and on this account it was not perceived by the ancient astronomers, who only observed the moon in those positions. It will be readily perceived, from the limits between which it varies, that it is proportional to the sine of twice the angular distance between the sun and moon. Its maximum value, or the coefficient of its argument, is 35' 41" 9'; hence the correction necessary on account of it is represented by the formula

$$\left(35' 41" 9'\right) \sin 2(\varphi - \Theta).$$

When this equation is added to the two preceding Annual ones, the differences between the computed and observed places of the moon are brought within narrower limits; but they do not yet disappear altogether except in the months of June and December, and about the time of the equinoxes they are found to amount to eleven or twelve minutes. This inequality was observed by Tycho and Kepler, though neither of them determined its magnitude. They supposed it to be an equation of time peculiar to the moon; and under this form Horrox inserted it in his tables, assigning its maximum value at 11' 46"; which differs only by four seconds from that which is obtained from the most accurate modern observations. On account that it is regulated by the seasons, and depends not on the lunar orbit, but the anomaly of the sun, Kepler gave it the name of the Annual Equation. By reason of this inequality the motion of the moon is slower than the mean motion during the winter months, when the motion of the sun is most rapid; and, on the contrary, is most rapid in summer, when the sun's motion is slowest. Hence its argument is the same as that of the equation of the centre of the sun, but with a contrary sign. It is therefore proportional to the sine of the sun's mean anomaly; its period is the anomalistic year, and its maximum value is found by observation to be 11' 11" 9'; hence it is expressed by the formula

$$\left(11' 11" 9'\right) \sin \Theta's \text{ mean anomaly}.$$ Theoretical book of the Mécanique Céleste, for a great number of new Astronomy, inequalities derived from the theory of attraction, form the basis of the lunar tables of Burg and Burckhardt.

It is to these theoretical researches that we are indebted for the precision with which the moon's motion is actually known, and the great advantages which thence result to geography and navigation. The recent tables contain no fewer than twenty-eight equations of longitude, all which may be regarded as corrections of the mean values of the four which we have explained, and which are themselves corrections of the mean motion. It is thus that astronomy approaches nearer and nearer, by every successive step, to the last term of a series which would represent the orbits and motions of the celestial bodies with absolute accuracy.

The lunar inequalities which we have as yet considered are all of a periodic nature, are compensated in the course of a comparatively small number of years, and have passed through many complete revolutions since the commencement of the history of astronomical observations. But there are, as in the case of the sun, others of a different kind, the periods of which are so long that, with reference to the duration of human life, they may be considered as permanently affecting the elements of the lunar orbit. These are the Secular Inequalities, the most remarkable of which is the acceleration of the moon's mean motion.

On comparing the lunar observations made within the last two centuries with one another, there results a mean secular motion greater than that which is given by comparing them with those made by Ebn-Jounis, near Cairo, towards the end of the 10th century, and greater still than that which is given by comparing them with observations of eclipses made at Babylon in the years 719, 720, and 721 before our era, and preserved by Ptolemy in the Almagest. These comparisons, which have been made by different astronomers with the utmost care, prove incontestably the acceleration of the motion of the moon from the Chaldeans to the Arabians, and from the Arabians to the present times.

The acceleration of the moon's mean motion was first remarked by Dr Halley, and mentioned in his Notes on the observations of Albategnius, inserted in the Philosophical Transactions for 1693. It was fully confirmed by Dunthorne, who was led by the discussion of a great number of ancient observations of eclipses, to suppose that it proceeded uniformly at the rate of $10''$ in a hundred years; a supposition which consequently gave him a correction for the mean longitude of the moon proportional to the square of the time, or of $10''$ multiplied by the square of the number of years elapsed between 1700 and the epoch of the calculation. This was the first attempt to estimate the value of the secular equation, which had hitherto been confounded with the mean secular motion. By a similar discussion of ancient observations, Mayer was likewise led to a secular equation proportional to the square of the time, which, in his first Lunar Tables, published in 1753, he valued at $7''$ for the first century, counting from the year 1700, but which he advanced to $9''$ in his last tables, published in 1770. Lalande found it to amount to $9''886$, and therefore agreed with Dunthorne in estimating the acceleration at $10''$ for the first century after 1700. Delambre subsequently undertook to determine accurately the actual motion of the moon in a century, from a comparison of the best modern observations. He found it to amount to $307''53''9''$ (rejecting the circumferences), while the most ancient observations agree in making it less by 3 or 4 minutes. The equation, which, in consequence of these comparisons, was empirically introduced into the tables, has been confirmed by the theory of gravitation; and the discovery of the physical cause, and of the law of the acceleration, due to Laplace, forms one of the most brilliant triumphs of modern science. In 1786 Laplace demonstrated that the acceleration is one of the effects of the attraction of the sun, and connected with the variations of the eccentricity of the earth's orbit in such a manner that the moon will continue to be accelerated while the eccentricity diminishes, but that it will disappear when the eccentricity has reached its maximum value; and when that element begins to increase, the mean motion of the moon will be retarded.

In order to take into account the effects of the acceleration in the determination of the mean longitude, M. Damoiseau has given the following formula for the secular variation: $10''7232 n^2 + 0''019361 n^3$, where $n$ is the number of centuries from 1801; so that if $L$ is the mean longitude of the moon on the 1st of January 1801, and $m$ is the secular motion at that epoch, the mean longitude will be $L + mn + 10''7232 n^2 + 0''019361 n^3$ after $n$ centuries. By means of this equation the tables satisfy the most ancient observations, and may be extended to at least a thousand years from the present epoch. It is only necessary to observe, that, in applying the formula, $n$ must be taken negatively, if the epoch is anterior to 1801.

The same cause which gives rise to the acceleration of the mean motion, namely, the diminution of the eccentricity of the earth's orbit, also occasions secular inequalities in motion of the perigee and nodes of the orbit of the moon. These two inequalities are, however, affected with opposite signs to that of the former; that is, while the mean motion of the moon is accelerated, the motion of her perigee and that of her nodes are retarded. They were deduced by Laplace from theory, and the equations by which they are expressed are connected with one another by a very simple ratio. If we take $A$ to represent the secular acceleration of the mean motion, the secular variation of the perigee found by Laplace is $-4''00052 A$, and that of the nodes $-0''735452 A$. From this Laplace concluded that the three motions of the moon, with respect to the sun, to her perigee, and to her nodes, are accelerated, and that their secular equations are in the ratio of the above numbers. By pushing the approximations to a great length, MM. Plana and Carlini, and M. Damoiseau, in Memoirs which obtained the prize of the Academy of Sciences for 1820, have found different numbers; those of Damoiseau are $1, 4''702$, and $0''612$. These important results of theory are all confirmed by observation.

The three secular inequalities which have been pointed out will obviously occasion others; for all quantities depending on the mean motion, the motion of the perigee, or of the nodes, must be in some degree modified by them. Thus the mean anomaly, which is the difference of the mean longitude of the moon and the mean longitude of the perigee, is subject to a secular equation equal to the difference of the secular equations affecting the longitudes of the moon and the perigee. The radius vector, the eccentricity and inclination of the orbit, are affected by the secular inequality of the mean motion, which, although too minute to have been hitherto appreciable, will acquire sensible values in the course of ages. The major axis of the ellipse is the only element exempted from inequalities of this sort. It is not probable, however, that the utmost efforts of science will ever make us acquainted with all the irregularities to which the moon's motion is subject. They can only be developed by the complete integration of the differential equations of motion; an integration which is laboriously performed term by term, and which, when attempted to be carried beyond a certain point, transcends The following table exhibits in one view the different elements of the lunar orbit. The epoch is the commencement of the present century.

| Mean inclination to the plane of the ecliptic | 5° 8' 47" 9 | |-----------------------------------------------|------------| | Longitude of ascending node | 12 53 17 7 | | Motion of node in 365 mean solar days | 19 19 42 316 | | Longitude of perigee | 266 10 7 5 | | Mean motion of perigee in 365 mean solar days | 40 39 45 36 | | Mean longitude | 118 17 8 3 | | Greatest equation of the centre | 6 17 12 7 | | Eccentricity | 0.05484142 | | Mean distance from the earth in diameters of the terrestrial equator | 29-982075 |

Sect. III.—Of the different Species of Lunar Months.

In treating of the sun, we took notice of three different species of revolutions or years, namely, the mean solar year, or the interval of time which the sun employs in performing a complete revolution with regard to the equinoxes; the sidereal year, or the time in which he returns to the same fixed star; and the anomalistic year, or the time in which he returns to the same point of his ellipse.

In like manner, if we understand by the term month the time which the moon employs to make an entire revolution relatively to any given point, movable or fixed, we shall have as many different species of months as there are different motions with which that of the moon can be compared. For example, if we estimate her revolution relatively to the sun, the month will be the time which elapses between two consecutive conjunctions or oppositions. This is called the synodic month, lunar month, or lunation.

If we consider her revolution as completed when she has gone through 360° of longitude counted from the movable equinox, we shall have the tropical or periodic month.

The interval between two successive conjunctions with the same fixed star is the sidereal month. A revolution with regard to the apsides of her orbit, that is to say, the time in which she returns to her perigee or apogee, gives the anomalistic month; and finally, the revolution with regard to the nodes is the nodical or draconic month.

Of these different periods the most important to mankind is the synodic month. It is also that which, by reason of the striking manner in which it is marked out by the phases of the moon, would first offer itself to the attention of the observer; and when its period is accurately determined, the other months may be deduced from it without difficulty, when the relative motions of the sun and moon are known with sufficient precision. The eclipses furnish a simple means of determining the synodic revolution with a great degree of accuracy. A few rude observations suffice to show that the period of a lunation is very nearly 29\(\frac{1}{4}\) days; and with this knowledge we are in a condition to compare two distant eclipses without running any risk of mistaking the number of revolutions that have taken place in the interval. The most ancient observation recorded by Ptolemy is an eclipse of the moon, which happened 720 years B.C., on the 19th of March, at 6 hours 48 minutes mean time at Paris, according to Lalande. In order to make use of this observation directly for the purpose of determining the synodic revolution, it is necessary to compare it with another of the same kind in which the moon occupied the same point of her ellipse, or the same position relatively to her apsides; for as it is the true place of the moon which is observed, and the mean motion which we are in quest of, the equation of the centre ought to be the same in both observations. Now, a similar observation is furnished by an eclipse which happened in 1717, on the 9th of September, at 6 hours 2 minutes, the moon's anomaly being very nearly the same as in the Babylonian observation. In the interval between the two observations the moon had therefore completed a whole number of revolutions, with regard to her mean as well as her true motion. The interval between the eclipses is 2437 years and 174 days minus 46 minutes, which expressed in days is 890287-968055... days. In this interval it is found, from the approximate value of 29\(\frac{1}{4}\) days, that 30148 synodic revolutions had happened; hence the mean length of the synodic month is \( \frac{890287-968055}{30148} = 29 \text{ days } 12 \text{ hours } 44 \text{ min. } 2^{\circ}497 \text{ sec.} \)

To deduce the other revolutions, let \( N \) = the number of days in a synodic month, \( n \) = the number of synodic revolutions in the interval between the two observations, \( m \) = the mean motion of the sun, and \( T \) = the time of a tropical revolution. We shall then have the proportion \( T : N :: 360° : n360° + m \); whence

\[ T = \frac{N \cdot 360°}{n \cdot 360° + m} = \frac{N}{n + \frac{m}{360°}} = \frac{N}{1 + \frac{m}{n \cdot 360°}} = \frac{N}{1 - \frac{m}{n \cdot 360°}} + \left( \frac{m}{n \cdot 360°} \right)^2 + \text{&c.} \]

Let \( S \) = the time of a sidereal revolution. The value of \( S \) is immediately found from \( T \); for let \( p \) = the precession of the equinoxes in the time \( T \), we shall have \( S : T :: 360° : 360° - p \); whence

\[ S = \frac{T \cdot 360°}{360° - p} = \frac{T}{1 - \frac{p}{360°}} = \frac{T}{1 + \frac{p}{360°}} + \left( \frac{p}{360°} \right)^2 + \text{&c.} \]

In like manner, if we represent by \( u \) the motion of the apsidal, and by \( v \) that of the node, in the time \( T \), the times of the anomalistic and nodical revolutions will be respectively obtained by the substitution of \( u \) and \(-v\) (the motion of the node being retrograde) in the last formula in place of \( p \). Instead, however, of proceeding in this manner, it is better to assume as the basis of the calculation the mean motion of the moon in longitude, which is accurately known by the observations of 2000 years, and thence to deduce the time of the mean tropical revolution.

The mean motion of the moon in 100 Julian years, or 36525 days, is found to be 1336 circumferences \(+307°52'43''\); or, by reducing the degrees, &c. to the fraction of a circumference, 1336-85521875 circumferences. The periodic month is therefore \( \frac{35625}{1336-85521875} \) days. In order to reduce the denominator of this fraction to a smaller number of digits, we may divide its terms by 3, multiply them by 32, then divide them by 900, and there will result \( \frac{432888}{1584421} \), the denominator of which contains only 7 digits in place of 12. On performing the division, we shall find the periodic month or tropical revolution \(=27-321582388 \text{ days } = 27 \text{ days } 7 \text{ hours } 43 \text{ min. } 4^{\circ}7183 \text{ sec.} \) Theoretical The mean motion of the moon in 36525 days

hence the relative motion is.............1236° 307° 6' 58" 5

= 445267-11625 degrees. The time of a synodic revolution is therefore 360 x 36525 days; or, multiplying numerator and denominator by 800,

whence the synodic month is 29-5305885391 days.

In order to obtain the sidereal revolution, we subtract the secular motion of the equinoctial points = 5010° = 1° 23' 30" from the moon's tropical revolution; the remainder, which gives the sidereal motion of the moon in 36525 days, is 1336° 306° 29' 13" 5 = 481266-48708833 degrees. Hence the time of a sidereal revolution is

On multiplying the terms of this fraction by 2400, it is reduced to

which gives the sidereal month = 27 days 7 hours 43 min. 11-544375 sec.

In like manner, by subtracting the motion of the perigee in 100 years from the secular motion of the moon, we shall find the anomalistic revolution to be 27 days 13 hours 18 min. 34-9488 sec.; and by adding the retrograde secular motion of the node to the secular motion of the moon, the revolution in respect of the nodes is found to be performed in 27 days 5 hours 5 min. 35-60769 sec.

The following table exhibits the different kinds of lunar periods and motions:

| Days | Hr. | Min. | Sec. | Days | |------|-----|------|------|------| | Synodic revolution | 29 | 12 | 44 | 2-84 | 29-5305887 | | Tropical | 27 | 7 | 43 | 4-71 | 27-3215824 | | Sidereal | 27 | 7 | 43 | 11-54 | 27-3216614 | | Anomalistic | 27 | 13 | 18 | 37-40 | 27-5545995 | | Nodical | 27 | 5 | 5 | 35-60 | 27-2122176 | | Tropical revolution of node | 6728 | 4 | 17 | 43-18 | 6798-1789720 | | Sidereal | 6793 | 6 | 59 | 15-34 | 6798-2911498 |

According to Ptolemy, the synodic month is 29 days 12 hours 44 min. 3½ sec., which differs from the above only by half a second. The same great astronomer made the tropical month to consist of 27 days 7 hours 43 min. 7½ sec. which exceeds the true time by 2½ seconds; an error into which he was led by assigning too great a value to the mean motion of the sun.

The ancient astronomers paid great attention to these different revolutions, for the purpose of regulating their lunisolar calendar, and of avoiding the calculation of eclipses, which is attended with difficulties that to them must have proved almost insuperable. Their object was therefore to assign composite periods, after the revolution of which the eclipses would again return in the same order. Now, it is easy to see that a period which will bring back eclipses of the same magnitude and duration, on the same day of the year and at the same longitude, must be an exact multiple of the different lunar months. The return of the moon to the same distance from her node will give an eclipse of the same magnitude; if she returns at the same time to the same point of her orbit, the eclipse will also be of the same duration; and if, in addition to these circumstances, she has also returned to the same longitude, the eclipse will take place on the same day of the year. But the numbers in the above table being incommensurable, it is impossible to find any period, however long, that will embrace all these conditions; the ancients therefore formed periods of different lengths, according as they aimed at satisfying the different conditions with a greater or less degree of precision. For an account of some of the most remarkable of the ancient lunisolar periods, see CALENDAR.

On account of the acceleration of the mean motion of the moon, the ratios of the different species of months are constantly undergoing alterations, and therefore the different cycles, supposing them exact at the time of their formation, cannot continue so for an indefinite length of time. This circumstance is, however, little to be regretted; for, in the present state of astronomical science, they are not of any great use, inasmuch as we are in possession of surer methods of predicting eclipses, the calculation of which, from the ephemerides, is now a matter of comparative facility. They are, however, interesting in an historical point of view, and their formation was a principal object of the labours of the early astronomers. (On this subject see Lalande, Astronomie, tome ii. p. 185; Delambre, Astronomie Théorique et Pratique, tome ii. p. 319; Schubert, Traité d'Astronomie Théorique, tome ii.; Woodhouse's Astronomy, p. 665.)

END OF VOLUME THIRD. The mean motion of the moon is 29 days 12 hours 44 minutes 34 seconds, which differs from the tropical motion by half a second. The same great astronomer, however, found that the sidereal month consists of 27 days 7 hours 43 minutes 48 seconds, which exceeds the true time by 21 seconds, owing to the motion of the sun.

The ancient astronomers paid great attention to the different revolutions, for the purpose of regulating the lunar calendar, and of avoiding the calculation of eclipses, which is attended with difficulties that must have proved almost insuperable. They were therefore obliged to assign composite periods, after the end of which the eclipses would again recur in the same order. Now, it is easy to see that a period which will bring back eclipses of the same magnitude and duration on the same day of the year and at the same longitude, must be an exact multiple of the different lunar periods. The return of the moon to the same distance from the earth will give an eclipse of the same magnitude; if she returns at the same time to the same point of her orbit, the eclipse will also be of the same duration; and if, in addition to these circumstances, she has also returned to the same longitude, the eclipse will take place on the same day of the year. But the numbers in the above table being incommensurable, it is impossible to find any period, however long, that will embrace all these conditions. The ancients, therefore, formed periods of different lengths according as they aimed at satisfying the different conditions with a greater or less degree of precision. For account of some of the most remarkable of these lunisolar periods, see CALENDAR.

On account of the acceleration of the mean motion of the moon, the ratios of the different species of motion are constantly undergoing alterations, and therefore the different cycles, supposing them exact at the time of formation, cannot continue so far as indefinitely. This circumstance is, however, little regarded; for, in the present state of astronomical science, they are not of any great use, inasmuch as the possession of super methods of predicting eclipses, calculation of which, from the ephemerides, is necessary for navigational purposes. They are, however, of interest in an historical point of view, and their forms are a principal object of the labours of the early astronomers.

(On this subject see Lalande, Astronomie, tome ii., Delambre, Astronomie Théorique et Pratique, tome ii., Schubert, Traité d'Astronomie Théorique, tome iii., and Herschel's Astronomy, p. 665.)