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thought necessary in the orbits of the heavenly bodies.

For the inferior planets the same hypothesis of an excentric and epicycle was employed. In this case the centre of the epicycle always coincided with the mean place of the sun, while the planet described its circumference with a velocity proportional to the time employed in going from one point of greatest digression to another. As the ellipse of Mercury is much more sensibly excentric than that of the larger planets, Ptolemy found the hypotheses which satisfied the others insufficient in this case. He was compelled to suppose that the point, which was the centre of the uniform motion called the centre of the equant, instead of remaining fixed, revolved in a small circle round the centre of the excentric; the radius of this circle being equal to the distance between these two centres, and the direction of the motion against the order of the signs. But it is impossible, in this place, to follow him into all the artifices he was forced to have recourse to in explaining the irregularities of the planetary motions.

The extreme complication of this system arose in a great measure from the law he had imposed upon himself of admitting none but circular motions in the heavens: "uniform and circular motions," says he, "belonging by their nature to celestial bodies." (Syntax. lib. ix. c. ii.) That astronomers should have attempted to represent all the celestial motions by circles, was natural enough in the infancy of the science; and as long as the apparent inequalities could be represented by a combination of these circles, they were justified in so doing but it is lamentable to observe that men of talent could mistake gratuitous and arbitrary assumptions of their own for laws of nature;-these metaphysical fancies, principally borrowed from Aristotle, about the perfection and incorruptibility of circular motion, long retarded the progress of


But though the Aristotelian physics of Ptolemy form a strange contrast with the geometrical knowledge displayed in his work, it would be unfair to charge him with having admitted the monstrous doctrine of solid transparent spheres, revolving the one within the other, and each carrying a planet attached to it, which was promulgated by Eudoxus. To these he makes no allusion; and it is but justice to him to suppose that he

himself considered his system of defer-
ents and epicycles merely as a means of
determining mathematically the posi-
tions of the heavenly bodies for any
given time.

Ptolemy was unsuccessful in his researches on the quantity of precession. Hipparchus had supposed it to be about one degree in 75 years: Ptolemy, undertaking to correct this determination, went much wider from the truth; he fixed it at one degree in 100 years, whereas the real value is one degree in 72 years. But there is a heavier charge against him: that of having appropriated and published as his own the catalogue of fixed stars, formed by Hipparchus. This seems to be but too well proved. He states the quantity of precession in the 265 years between himself and Hipparchus at 2° 40': this alone would show that he had not observed, as he would have found it considerably more ; but if we subtract from all his longitudes 2° 40', the precession he supposed for 265 years, we get exactly the longitudes, such as they were in the time of Hipparchus, and such, in fact, as that astronomer seems to have fixed them, judging from the positions given in the commentary on Aratus.

As all astronomy must be founded on observation, Ptolemy has not neglected to describe the instruments used for that purpose at Alexandria. To determine the sun's altitude, the Eastern nations had long been in the habit of measuring the shadow of a vertical gnomon; and, if a few simple and obvious precautions The Greeks be attended to, this method may give very accurate results. learned the use of the gnomon from the Chaldeans at a very early period; and we have seen that it was employed by Meton, Pytheas, and others. But at Alexandria it seems to have been but little used; the astronomers of that place substituting for it armillary spheres of different kinds. To observe the sun's passage through the equinox they used two circles, firmly attached to each other, and placed one in the plane of the meridian, the other in that of the equator: at the moment of the equinox this latter was not illuminated by the sun on either side. For the solstice they used two concentric circles in the plane of the meridian, the one revolving within the other, and carrying two small prisms at right angles to the limb, and fixed at points diametrically opposed on the circle. To observe the sun's meridian altitude with this, the inner circle was

turned till the shadow of one prism completely covered the other: the shadow of this second fell on the graduated limb of the outer circle, and the middle of it being marked, gave the altitude of the sun's centre. For this solstitial circle Ptolemy substituted a quadrant, on which the observation was made in a manner very similar: but his most important invention was that of the parallactic rulers. These rulers formed an isosceles triangle, susceptible of being opened to any angle at the vertex; one of the equal sides was always vertical, the other being pointed on the star; the observer read off on the graduation of the base the length of the chord: a table of chords gave him the value of the angle at the vertex, that is, of the zenith distance.

The construction of the astrolabium, with which the longitudes and latitudes of the planets or fixed stars were observed, was rather more complicated than that of the solstitial or equatorial armillæ. They carried circles representing the equator, the ecliptic, the meridian, &c., and placed respectively in the planes of the celestial great circles they represented. Two other circles, moveable on the poles of the ecliptic, were made to pass through two stars: the observer then read off on the graduation of the ecliptic and circles in question respectively, the latitudes of the two stars and their difference in longitude.

As the Greeks had no means of measuring time with any accuracy, they were obliged, when they wished to compare the place of the sun with that of the fixed stars, to measure in the daytime the difference of longitude between the sun and moon, and at night that between the moon and a fixed star. The moon's rapid and variable proper motion necessarily rendering this method very inexact, the Arabs improved it considerably by substituting the planet Venus for the moon. A still greater improvement will be noticed when we consider the observations of El-Batani.

Ptolemy was the author of a most important discovery not recorded in the Syntaxis, the effect of refraction in augmenting the apparent altitudes of the heavenly bodies. This is clearly shewn in his Optics*, where he investigates the theory of refraction in general. He was

Vid. Delambre, Astr. Anc., vol. ii. This work of Ptolemy, though known to the Arabs and to Roger Bacon, was for a long time lost in Europe. A Latin translation of it was fortunately disco vered by La Place in the Royal Library at Paris,

aware of the existence of a certain constant relation between the angle of incidence and that of refraction, and made several experiments to determine the value of the latter when a ray passes from air into water. Though he perceived clearly the nature of the effect produced on the altitudes of the stars, and that it diminished with the zenith distance, yet he declared himself unable to determine the absolute quantity of refraction, from not knowing the height of the terrestrial atmosphere. However, this treatise is extremely remarkable, and one of those that reflect the greatest honour on its author. To point out the existence of refraction, even without measuring it, was to render an important service to astronomy; to which we must add, that this is the only work of the ancients in which there is anything resembling the experimental philosophy of the moderns. We also find here an ingenious explanation of the optical illusion which makes the disks of the sun and moon apparently much larger when near the horizon; and this explanation is the one generally received at present, though there still seems to be some doubt on the subject.


Astronomy of the Arabs.-The Per sians.-The Chinese.

WITH the Syntaxis we take our leave of the astronomy of the Greeks. The interval between the publication of this work and the conquest of Egypt and Syria by the Arabs did not produce a single astronomer; for we cannot give that name to one or two commentators on Ptolemy, of whom Theon is the most generally known. But when the Arabs had firmly established themselves in the East, they began to cultivate all the branches of mathematical science, and astronomy in particular, with extraor dinary zeal. This revolution in the character of the Arabs, the beginning of which dates from the Caliphs, El-Mansour and Haroun-el-Reschid, at the end of the eighth century after Christ, was finally accomplished under El-Mamoun, who reigned in the beginning of the ninth. Ibu Jounis has recorded several observations made by the astronomers of this prince, the most interesting of which are those instituted to determine the obliquity of the ecliptic. This was found by some 23° 33', by others 23° 33′ 52", which is exact within 34 minutes, and more correct than any of the

determinations made by the Greeks. Justly dissatisfied with the rough attempts of the Greek astronomers to measure an arc of the meridian, ElMamoun ordered his astronomers to proceed to a new measurement. The method they followed is sufficiently simple. Having chosen a large plain in Mesopotamia, they divided themselves into two parties; then, starting from a given point, each party measured in a right line an arc of one degree, the one towards the north, the other towards the south. The former found for the length of a degree fifty-six miles, the latter fiftysix and two-thirds; the mile being equal to 4000 cubits. But here arises the question, what was the length of these cubits? Unfortunately this is not easy to decide. Two Arabian authors agree, that the cubit employed was the black cubit of twenty-seven inches; but one says that the inch was determined by six grains of barley placed in contact sideways; the other makes it equivalent to five similar grains *. The latter seems to agree better with the real length of the degree; but the error is still very considerable, being between three and four miles in excess. But if we suppose the cubit employed to be the royal cubit of twenty-four similar inches, the length of the degree will then be brought within about a third of a mile of its real value.

The two centuries immediately following the reign of El-Mamoun were extremely fertile in astronomers, and particularly in observers; forming thus an advantageous contrast with the Greeks, who seem, with very few exceptions, to have had little taste for observation and experiment in any of the sciences. In this respect the Arabs effected a complete reform in astronomy. They have left behind them an immense mass of recorded observations, of which the greater part has never been printed; and which might be of great service to astronomy, did not the superiority of our instruments render the modern observations so much more accurate, as to compensate for the smaller interval of time existing between them.

The most distinguished of the Arabian astronomers is Albategnius or El-Batani, who rectified, in many points, the determinations of Ptolemy, and added the important discovery of the motion

According to Thevenot, 144 grains of Oriental barley, placed side by side, are exactly equal to one foot and a half of the old French measure.

of the solar apogee*. Ptolemy had fixed the precession at one degree in one hundred years, instead of seventy-two, the real value: El-Batani corrected this mistake, but made it, on the other hand, a little too rapid; namely, one degree in sixty-six years. Similarly he made the length of the solar year about two minutes and a half too small; but it is just to remark, that the errors of El-Batani proceed from the confidence he placed in the observations of Ptolemy,-observations which, as we have seen before, appear to be fictitious; had the Arabian astronomer compared his observations directly with those of Hipparchus, he would have approached much nearer to the truth. The excentricity of the solar orbit was determined by him with great accuracy, the equation of the centre fixed at 1° 58', and the obliquity of the ecliptic at 23° 35'. The observations used to determine these quantities seem to have been made with great care, and are much superior to any recorded by the Greeks. El-Batani, who gives in his writings many proofs of a sound judgment, rejects, with reason, a pretended motion of the fixed stars, by which they appeared to oscillate about a certain point, their motion in longitude becoming sometimes direct, and sometimes retrograde. To explain this pretended motion, which was called trepidation, the equinoctial points were supposed to revolve in a circle of 4° 18′ 43′′ radius round their mean places, which retrograded along the ecliptic, according to the laws of precession. This, at least, is the way in which the theory was represented subsequently. El-Batani merely states that the stars were supposed to move directly through 8°, then to retrograde through the same arc. The Arabian astronomer, while refuting this theory, attributes it distinctly to Ptolemy. It is remarkable that none of the extant works of Ptolemy make the slightest allusion to trepidation; the first mention

This is not expressly stated by El-Batani, but it is an evident consequence, from his discovery, that the apogee, which Ptolemy found to be in 65° 30', was now in 82° 17'. This gives an annual motion of 79". Now the Arabian astronomer allowed 54" for the annual effects of precession; there would remain, therefore, about 25" for the annual proper motion of the apogee.

The equinox taken by El-Batani to compare with his own observations, is recorded by Ptolemy with the mistake of a whole day on its date. On the observations of Ptolemy in general, Halley has expressed a severe but just opinion. V. Delamb. Ast. du Moyen Age, pp. 61, 62. Delambre is of opinion that Ptolemy never observed at all. For his catalogue of fixed stars, see what has been said above, p. 31.


of it is found in Theon, whose commentary on the Almagest has been noticed above. It is very unjustly that Thebit ben Corah, an Arabian astronomer, has been considered as the inventor of trepidation; we may see from Theon, that it was a fancy of the Greeks, probably anterior to Ptolemy, though never noticed by that author in the Almagest. However, Thebit adopted it, and even wrote a treatise purposely to establish it; at least if he be really the author of the work on the eighth sphere generally attributed to him. But there are some reasons for doubting of this; as Ibn Jounis has preserved an original letter of Thebit, in which he expresses himself as far from convinced of the existence of trepidation.

The limits of this treatise do not allow us to enter into an examination at length of the writings of the numerous Arabian astronomers to be found in the libraries of Europe, nor would a bare catalogue of names offer any interest for the reader. It will be enough to notice shortly the tables of Ibn Jounis and Arsachel. The former, an Egyptian astronomer of great merit, has left behind him a considerable mass of observations, and a treatise on astronomy, in which are some remarkable improvements in trigonometrical calculation. His tables are, in fact, those of Ptolemy, with many ameliorations in the constants and epochs. It appears rather singular, that though coming after ElBatani, he does not admit any other motion for the solar apogee than that of precession: we have seen that the observa tions of El-Batani indicated very clearly an annual proper motion. The tables of Arsachel, like all the Arabian tables, are, in substance, those of Ptolemy; in the numerical determinations they seem inferior to those of El-Batani; nor would they deserve mention here, were they not supposed to have been of great assistance to the composers of the famous Alphonsine tables.

The instruments of the Arabs were essentially the same as those of the Greeks; the gnomon, various kinds of armillary spheres, and a sort of mural quadrant. But they added (and we owe this apparently to Ibn Jounis), the

• xavoves Teoxrigal. V. Delambre, Astr. Anc., vol. ii. p. 625. Indeed, Theon says distinctly, that Ptolemy did not admit these alterations in the precession of the fixed stars; ὅπερ Πτολεμάνω ότ δοκεῖ.

† V. Delambre, Astr. du Moyen Age. Iba Jounis.

method of determining the time by observing the absolute altitude of a fixed star or planet. This was probably the best method that could be employed before the invention of pendulum clocks.

The science of trigonometry is necessarily and inseparably connected with astronomy. The Arabs, who cultivated the latter so 'zealously, made considerable additions to the former. The most important of these was the substitution of the sine instead of the chord of the double are employed by the Greeks, We owe this very important amelioration to El-Batani. It enabled him to simplify very much the solutions of several cases of oblique angled spherical triangles; particularly that in which the two sides and the included angle are given to find the third side, or either of the remaining angles. Both El-Batani and Ibn Jounis make use of tangents and cotangents in their treatises on dialling; and even give tables of these quantities; but it was reserved for Aboul Wéfa, an astronomer of Bagdad, of the eleventh century after Christ, to introduce them into trigonometry. This was a second important improvement. The same author is also the first who treats of secants and cosecants; but the Arabs do not seem to have been aware of the advantage of introducing the cosine into trigonometry, till a century later, when Geber, a Mahometan Spaniard, gave for the first time a formula into which it enters. Ibn Jounis was the first author who made use in trigonometrical problems of the tangents, cosines, and secants of subsidiary ares. This elegant method, which in many cases simplifies extremely numerical calculations, seems to have been unknown to the mathematicians of Europe till the middle of the eighteenth century, when it was reinvented by Simson.

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Chrysoccas, a Greek physician, has translated others brought from Persia by Chioniades, which seem to have been composed in the eleventh century. They offer, however, little interest, being evidently borrowed from the Greek.

The descendants of Timour were as much attached to astronomy as those of Gent-Chis. Ulugh Beg, grandson of Timour, and sovereign of Samarcand, devoted himself with extraordinary zeal to the cultivation of this science. Having erected an immense observatory, and procured the assistance of a number of mathematicians, he published a collection of tables and a catalogue of the fixed stars, which acquired, and still continue to enjoy a great reputation in the east. As far as we can judge this reputation seems well deserved; but we only know that part of the tables containing the motions of the sun, and the catalogue of the stars,-the rest has never been translated, or at least never published. The exactitude of the solar tables is very creditable to Ulugh Beg, and shows that his observations, which are said to have been made with the gnomon, were very good. The epoch of these tables is the 4th of July, 1433, A.D. The epoch of the catalogue is the year 841 of the Hegira, or 1447, A.D. The stars were observed with a quadrant of enormous dimensions; but though superior in accuracy to the Greek catalogues, the errors in longitude sometimes amount to half a degree*.

We have noticed the protection given to astronomy by the Tahtar princes in Persia and Bokhara: equal favour was shown to the science by the successors of Gent-Chis Khan on the throne of China. Though the observations of the Chinese go back to the earliest antiquity, it is not the less certain that their knowledge was extremely limited and confined to the most elementary parts of astronomy. But when the Tahtar conquest brought them into contact with the nations of Western Asia, a very sensible amelioration took place. The thirteenth century may be considered as the most brilliant epoch of Chinese astronomy. Cocheou-King who had been appointed by Kubla-Khan, the descendant of Gent-chis, to the presidency of the tribunal of mathematics, undertook to rectify from his own observa

• V. Delamb., Ast. du Moyen Age, p. 207.

tions the principal elements. This he effected with considerable success. His observations of the sun were made with a gnomon of forty feet, and appear very accurate. He fixed the length of the solar year at 365d 5h 49m 12'; and the obliquity of the ecliptic at 23° 33′ 39′′. The date of his tables is A.D. 1280. It is probable that at this time the Chinese astronomy borrowed a good deal from the Arabs. We now hear, for the first time in China, of spherical trigonometry; and the invention of it is attributed to Cocheou-King; but there is every appearance that he learnt it from the astronomers of the west; for it is known that under Kubla-Khan Persia and China were in frequent communication *. It is extraordinary that subsequently to Cocheou-King the mathematicians of China should have degenerated to such a point, that at the arrival of the Jesuits the president of the mathematical tribunal was unable to solve a plane right angled triangle. This, indeed, is the more singular, since we are told that before the Christian era the Chinese could calculate the lengths of the shadow of the gnomon, and even had methods for the prediction of eclipses.


Astronomy of the Middle Ages. IT is impossible to pay any attention to the history of the Romans without per ceiving that in that nation there prevailed at all times a singular indisposition to the pursuit of mathematical and physical science. The poets and orators of Greece, and her metaphysicians were studied with ardour in Italy, but her geometers and astronomers were totally neglected; and it appears that these sciences, so highly estimated in one country, were thought in the other to be beneath the notice of a man of good birth and liberal education. This dif ference, so little creditable to his countrymen, is remarked by Cicero; nor does the Roman character seem to have changed in this respect in subsequent ages. The extent to which astronomy was neglected is evident from the circumstance, that the difference between the beginning of the civil and of the solar year, amounted in the time of Julius Casar to three months. During the whole existence of the republic we hear but of one Roman who attained any emi

V. Montucla, vol. i., p. 463.
† Delamb., Ast. Ane, vol. i. p. 361.

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