ence, or 7° 12'*. The distance between Syene and Alexandria he estimated at 5000 stadii: he found thus, for the length of the circumference, 250,000 stadii. To judge of the accuracy of this measure we ought to know what was the value of the stadii employed, about which there is great uncertainty; but it is at once obvious that there are several sources of inaccuracy. Syene, we know, is not under the meridian of Alexandria, but nearly three degrees to the eastward of it; it is also about 50' to the north of the tropic; lastly, the distance between this place and Alexandria seems to have been estimated, not measured.

About two centuries later, Posidonius made an attempt to verify the measure of Eratosthenes. He observed, that in the island of Rhodes the star Canopus just grazed the horizon, while its meridian altitude at Alexandria was 7 degrees. The distance of the two places he estimated at 5000 stadii; hence he got the length of the circumference, 240,000 stadii. But this measure is perhaps still more inaccurate than the former: the distance being across the sea, could only be most roughly estimated; and there is also more than a degree of difference in the longitudes of the extreme points.

But to return to Eratosthenes: he seems to have observed the winter as well as the summer solstice at Alexandria, for we possess a valuable determination of the obliquity of the ecliptic by him he is said to have fixed the angle between the tropics at parts of the circumference, or 47° 42′ 27", whence we get for the obliquity of the ecliptic 23° 51′ 13"; the theory of universal gravitation would make it about 7' less, which, under all the circumstances, is an inconsiderable difference.

Among the distinguished men produced by the School of Alexandria, Hipparchus stands pre-eminent. He has been called the Father of Astronomy; and it is unquestionable that by his labours were laid the foundations of the science. Far surpassing his predecessors, he has been equalled by few of his successors, perhaps by none except Kepler and Bradley. Of all his writings only one,

and that one of the least importance, has descended to posterity; a commentary on the Phenomena of Aratus. This astronomical poem is in fact a description of the sphere, the materials of which Aratus seems to have taken from Eudoxus; the positions of the stars are given in a very rough and often in a very inaccurate manner, and scarcely deserve a commentary from an astronomer so eminent as Hipparchus. This commentary seems to have been written when Hipparchus was a young man; at all events before he had discovered the general motion of the stars in longitude, to which it contains no allusion. The most interesting fact that can be elicited from it is, that Hipparchus was then in possession of a method for the resolution of spherical triangles.* As we find no traces of the science of spherical trigonometry in any preceding author, Delambre concludes, and apparently with reason, that Hipparchus was the inventor of it. This is certainly not the least of the obligations we owe him; for it is evident that astronomy could make little progress without the assistance of trigonometry. But though the works of Hipparchus, with this exception, are lost, we are able to ascertain pretty exactly the extent and nature of his discoveries from the great Syntaxis of Ptolemy. We see there that the foundations of nearly all the theories developed by Ptolemy, were laid by Hipparchus: the additions made by the former will be examined in a subsequent part of this treatise.

The astronomers of Greece for several centuries had supposed the exact length of the solar year to be 365 days and a quarter. Hipparchus by comparing one of his own observations of the summer solstice, with one made 145 years previously by Aristarchus of Samos, discovered this to be too great. He found that the solstice arrived 12 hours sooner at the end of these 145 years than it ought to have done, on the supposition of the solar year being 365 days; 12 hours divided by 145 gave him the diminution to be made on the length of the year. In this way he found for the length of the tropical year 365 days 5 hours 55 minutes 12 seconds.

The sun appearing to move in a circle round the earth, it was natural to suppose that his motion in the ecliptic was uniform; and¡ such was probably the

* V. Delambre, Astron. Anc. Vol. i. p. 142.

opinion of the early astronomers of Greece. However when they began to make observations with the gnomon, they could not help perceiving a considerable difference between the intervals of the equinoxes and solstices; intervals which must be equal were the motion of the sun round the earth uniform. Hipparchus undertook to investigate this point. He observed that the interval between the vernal equinox and the summer solstice was 944 days; between the summer solstice and the autumnal equinox 924. Thus the sun took 187 days to describe the northern half of the ecliptic, and only 178 for the southern half; indicating a considerable increase of velocity during the latter. To explain this irregularity, Hipparchus supposed the sun to move round the earth in an excentric circle; that is in a circle, whose centre did not coincide with that of the earth. It is clear, that in this case the sun, though moving uniformly in its orbit, would appear to a spectator at the earth to move with an unequal velocity, on account of the variation of its distance. The question was to determine the quantity of this excentricity, that is to say, the distance of the earth from the centre of the solar orbit; and the position of the apogee and perigee, or of the points of greatest and least

distance. Let ADBF represent the circle in which the sun is supposed to revolve; let the centre of this circle be at C, and the earth at E: according to Hipparchus the sun revolves with an uniform motion round C: it is evident that, seen from E, his motion will appear unequal it will be fastest at the point B or the perigee; slowest at A, the apogee: let M N be the line joining the sun's places, at the two solstices; PQ at the equinoxes: the object of Hippar

Ptol. Syntax. Lib. iii.

chus was to ascertain the ratio of EC to BC, and the arc Q A which determines the position of the apogee. By combining his observations of the equinoxes and solstices, he found the excentricity equal to th part of the radius, and the longitude of the apogee, or the arc Q A, equal to 65° 30'. This value of the excentricity is, however, too great by about one-sixth. The excentricity and place of the apogee being once known, it was easy to construet Tables which should give the sun's position at any time. For, suppose the sun to be at S, then as he is supposed to revolve uniformly round C, we can find from the time taken to describe the are AS, the value of the angle ACS, and therefore S CE; and in the triangle SC E, CE, and CS are known, whence we may find CSE, which is the difference between the angles ACS and A ES, or between the mean and true anomaly. This difference is called the equation of the centre. From what has been just said, we may see how Hipparchus calculated the values of the equation of the centre corresponding to successive values of AS, or the angle A CS, in his solar Tables. We must recollect that the instant of the sun's passage through the equinox at Q may always be supposed known: the arc QS is proportional to the time elapsed since the equinox, and is soon found: QA is known: hence we find A S, and looking into the Tables, find the corresponding equation of the centre. This gives us AES, and consequently Q ES, or the sun's apparent longitude for any given time.

From the theory of the sun Hipparchus proceeded to that of the moon. By comparing some ancient eclipses with those observed by himself, and dividing the interval of time by the number of revolutions, he obtained the value of a synodic revolution of the moon. By methods similar to those employed for the sun, he determined the excentricity of the lunar orbit, and its inclination to the ecliptic, which latter he fixed at 5°. Finally, he is said to have measured the motions of the lunar apogee and node. With these data he calculated the first Tables of the sun and moon of which history makes mention. This alone would have secured for him the gratitude and admiration of posterity. The want of observations, and perhaps the difficulty of their theory in his system, prevented

him from attempting a similar under taking with regard to the planets.

But the most important, perhaps, of all the services rendered to astronomy by Hipparchus, was the formation of a catalogue of the fixed stars. If we consider the boldness of the attempt, the labour of the execution, and the importance of the result, the author of it seems not undeserving the enthusiastic praises of Pliny. Such a catalogue is, in fact, the foundation of all astronomy, The fixed stars are so many standard points to which the celestial motions are referred, and the determination of their relative distances is of the utmost importance. By comparing their positions at distant periods, we may detect those small variations which require centuries to become sensible; and there is every reason to believe that, if we possessed a really accurate catalogue of twenty or thirty centuries back, we should be in possession of many valua ble discoveries, which perhaps are destined to lie hid for ages. It was, indeed, in this way that Hipparchus was led to his great discovery of the precession of the equinoxes. On comparing his own observations with those of Aristillus and Timocharis, made 150 years previously, he perceived that all the fixed stars, while they retained their latitudes sensibly unaltered, had advanced about two degrees in longitude; or what comes to the same, the equinoctial points appeared to have retrograded along the ecliptic by the same quantity. It was reserved for Newton to explain the causes of this singular phenomenon.

Such is a brief account of the astronomical discoveries of Hipparchus: we have already seen that he was the inventor of trigonometry; it also appears that he was the first who suggested the method of fixing the positions of places on the earth's surface by their longitudes and latitudes, and that he proposed to determine the former by means of lunar eclipses; a method excellent in its principle, though now abandoned on account of some practical objections.

As nothing connected with astronomy seems to have escaped the sagacity of Hipparchus, he did not overlook the correction of the Calendar. We have seen that the period of Callippus was far from exact: according to the cal culations of Hipparchus, the error at

Hipparchus nunquam satis laudatus, ut quo nemo magis comprobaverit cognationem cum homine syderum, animasque nostras partem esse çœli,..... ausus rem etiam Deo improbam, an numerare posteris stellas.-Hist. Nat. ii, 26.

the end of a period was about one-fourth of a day. He proposed to quadruple the period of Callippus, and then to subtract a day. This new period brought the moon again to the same place pretty exactly: the error on the sun's motion was about a day and a quarter, which is one-fourth of the error of Callippus in the same time.

We have some reason to be surprised that the discoveries of Hipparchus were not followed up by succeeding astrohomers. One might have imagined that such brilliant success would have stimulated others to the further development of the science; but, extraordinary as it may appear, history records not one astronomer of note in the three centuries between Hipparchus and Ptolemy. The attempt made by Posidonius to measure a degree of the meridian has been already noticed: a few authors on spherical trigonometry flourished in this interval, among whom may be distinguished Theodosius and Menelaus; but astronomy itself seems to have made no progress till the time of Ptolemy. This eminent and laborious philosopher felt the necessity of uniting all the scattered materials existing in the works of Hipparchus and others, which, combined with his own discoveries, formed, as far as the knowledge of the time allowed, a complete system of astronomy: by so doing he rendered a distinguished service to science; and the publication of his palnparixn oúvražis forms an important epoch. This work, which has fortunately survived the barbarism of the middle ages, formed the basis of all the astronomy of the Arabians, and for a considerable time that of modern Europe. Its importance requires here a concise analysis.

Ptolemy begins his work with a discussion of the relative positions of the earth, sun, and planets. We have already seen that the Greek astronomers were divided on the subject of the earth's motion. Though many distinguished philosophers held the opinions of Pythagoras, the majority seem to have embraced the opposite doctrines. Ptolemy followed these latter, and, unfortunately for him, his name has become attached to a system now universally admitted to be erroneous. It is true that the ancients wanted some decisive and convincing proofs of the earth's motion, which we possess; but though much has been said to excuse Ptolemy, his justification remains very incomplete. The arguments that he urges

against the earth's motion, such as that in this case the poles would not be immoveable points on the celestial sphere, that the fixed stars would not always preserve the same apparent distances from one another, and other objections of a similar kind, are all obviated by the single remark made by Aristarchus, four centuries previously, that the earth's orbit was a point in comparison with the distance of the fixed stars. On the other hand, the motions of the planets, so complicated and almost inexplicable in the one hypothesis, are accounted for so simply in the system of Pythagoras, that one cannot but feel astonished that Ptolemy should have felt so little hesitation in rejecting it. "The same reasons," says he, which show that the earth is a point in magnitude compared with the heavens, will show the impossibility of its having a motion of translation:" and the only argument he combats at any length, is that which appears to have been urged by some Pythago reans, that the earth being spherical and unsupported, could not remain at rest in the centre of the heavenly motions. Having discussed this point, with a singular mixture of truth and error, he adds these remarkable words: "But if there were any motion of the earth common to it and all other heavy bodies, it would certainly precede them all by the excess of its mass, being so great; and animals and a certain portion of heavy bodies would be left behind, riding upon the air, and the earth itself would very soon be completely carried out of the heavens. But such things are most ridiculous, even only to imagine." This passage is remarkable, because it shows how little the Greeks had studied natural and experimental philosophy, and how falsely their geometers could reason on purely physical subjects. A heavy body in vacuo does not, as Ptolemy supposes, move faster than a lighter one, as may be verified by direct experiment; yet this he clearly considered a self-evident truth, and founded on it arguments which must be classed among the weakest ever urged against the Pythagorean system of the world.

After rejecting the motion of translation assigned by some to the earth, he proceeds to examine the probability of its diurnal motion on its axis. This system he confesses simplifies very much the appearances of the heavens; but it appears to him equally ridiculous with the former; as in this case, the earth revolving with great rapidity from west

to east, would leave behind it the clouds, birds flying in the air, and, generally, all objects suspended in the atmosphere. A stone thrown to the east would not advance, the earth constantly preceding it by the excess of its velocity. These objections are all founded on an ignorance of the principles of mechanics, and seem to be on a par with those urged by some against the roundness of the earth, and the possibility of the existence of Antipodes; arguments which he has himself successfullyrefuted.

The earth then, according to Ptolemy, was fixed and motionless in the centre of the heavens; he supposed the different planets to revolve round it, arranged in the following order, according to their distances: first, the Moon, then Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and, lastly, the sphere of the fixed stars. With regard to Venus and Mercury, Ptolemy remarks, that some astronomers had placed them beyond the sun, while others made them nearer: the most ancient writers had adopted the latter opinion, which had been rejected by subsequent authors, because these two planets had never been seen on the sun's disk. This reason Ptolemy rightly rejects as insufficient; for such passages over the sun's disk would not happen, unless the planes of the orbits coincided with the ecliptic, or else the nodes happened to coincide nearly with the sun's place at the time of inferior conjunction. He does not seem to be aware that these passages or transits really do take place; and sufficiently often in the case of Mercury, though but rarely in that of Venus. But the difficulty of observing these phenomena renders it by no means extraordinary that they should not have been noticed, though it might have taught caution to those who affirmed positively their non-existence. It is much more remarkable that Ptolemy should not have perceived that it was possible to conciliate the two hypotheses in question, by making these two planets revolve round the sun; in which case it is clear they would be sometimes more distant, and sometimes nearer, than that body. And this inadvertence is the more singular, as the doctrine just mentioned is said to have been maintained by the ancient Egyptians. It seems probable that the systematic ideas of Ptolemy made him unwilling to place the sun in the centre of any of the heavenly motions; or he might have been repugnant to consider any of the planets