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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 be hind 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

* κανόνες προχειροι. 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, ὅπες Πτολεμάνω ἐν


+ V. Delambre, Astr. du Moyen Age. Ibn 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 arc 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, 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.

The Tahtar conquerors who succeeded the Arabs in the east, seem to have been as zealous as the caliphs in their attachment to astronomy. The grandson of Gent-Chis Khan founded in Persia an observatory, which he fitted up with the best instruments of the time, and all the most valuable works on astronomy existing in the east. Under his protection the astronomer Nasseerad-Deen published tables, which are supposed to have been entirely borrowed from Ptolemy. These are not the only tables existing in Persia:

• Delambre, Astr, du Moyen Age, p. 78.

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.

nence in astronomical studies. C. Sulpitius Gallus is mentioned by Cicero as an indefatigable calculator of eclipses*; and he is known to have predicted an eclipse of the moon on the night preceding a decisive battle between the Romans and the king of Macedonia t. On this occasion his science may be said to have rendered his countrymen an essential service; for, from the wellknown superstition of the ancients, the Roman soldiery would have been much terrified by what was supposed to be an unfavourable omen. At the celebrated siege of Syracuse an unexpected eclipse of the moon deterred the Athenian commanders from commencing their march at the proper time, and caused eventually the destruction of a fine army. Besides Sulpitius Gallus, we find the names of one or two Romans, who seem to have written on astronomical subjects (among whom is Varro); but their works are lost, and we have no means of judging of the extent of their scientific acquirements. Perhaps Cicero himself ought to be quoted here, as he has translated into Latin verse the Phenomena of Aratus, a poem which has been already noticed. One of the books of this translation is still extant, containing a description of the celestial sphere, and in particular of the constellations of the zodiac. Nothing is to be found in this poem beyond the most elementary doctrines of astronomy: the same may be said of the astronomical poem of Manilius, the date of which is a little later.

Of the last-mentioned poem a small part is devoted to the description of the sphere; by far the greater part being occupied with astrological precepts. This superstition, the influence of which has been so long and so widely felt, and the traces of which are not yet extinct in Europe, is in all probability of Chaldæan origin. It seems to have been unknown in Greece before the conquests of Alexander; and, indeed, it is but justice to the astronomers of that country, to say that in general no traces of belief in astrology are to be found in their writings. It is true that there exists an astrological treatise attributed to Ptolemy, but it may be doubted whether it is really one of his productions.

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His great and very complete work the μεγάλη Σύνταξις contains nothing which could lead us to believe that he was infected with these ideas. At Rome, under Augustus, and the succeeding emperors, the city was inundated with Syrians*, Chaldæans, and other natives of the East; and numerous passages in ancient authors show that a belief in judicial astrology was extremely prevalent. From this time, down to the seventeenth century, the whole of the civilized world in the West as well as the East was enslaved by this childish superstition, against which philosophy and religion seem to have combated in vain. It is humiliating to know that some of the great restorers of astronomy in Europe, were not superior to the follies of their age; and that even the bold and acute Kepler could dispute on the best manner of drawing up a scheme of nativity. Our posterity will, perhaps, be equally astonished to think that in the nineteenth century, in the country of Newton and Bradley, almanacks in general circulation should contain predictions founded on the aspect of the planets, and all the mummery of an art, the existence of which is a disgrace to any nation possessing claims to civilization, or even common sense. It is difficult to imagine that there can still be persons who believe that the distances between Jupiter and Saturn can influence the fate of empires, or that on the sign of the ecliptic which is just rising, at the moment of a man's birth, depend the principal events of his life. But vain and futile as is this pretended art, it has exercised too great an influence on the cultivation of astronomy to be passed over here without an allusion. There can be no doubt that an attachment to astrology was at least one of the reasons that induced the Arabs to study with so much ardour the phenomena of the heavens; and it is quite certain that the encou ragement given at the revival of letters by many European princes to astronomy, was entirely owing to their wish to read the future by means of the stars.


But to return to the Romans. calendar, in the time of the republic, fell as has been already noticed, into great confusion. This, Julius Cæsart, who

Jampridem Syrus in Tiberim defluxit Orontes. -Juv. Sat. iii.

This sign was called the horoscope; a term famliar to most persons, though perhaps at this day few are acquainted with its exact meaning.

It appears from Pliny, vii., 25, that Cæsar was himself an observer, and had composed some

was a man of extraordinary knowledge and universal genius, endeavoured to correct, with the assistance of a Greek astronomer, named Sosigenes. At this time, Hipparchus had already shown that the length of the solar year was something less than 365 days and a quarter, and Sosigenes could not have been ignorant of this fact: but, in all probability, he considered the difference too small to be worth taking into account. Be the reasons what they may, he proposed a method of intercalation which, to be quite rigorous, supposes the solar year to be 365 days and a quarter; namely, he proposed to make the civil year of 365 days, and every fourth year to insert an additional day. The year in which this intercalation took place was called bissextile, because the additional day inserted, bore the designation of bis sexto calendas Martii; the day named sext. calend. Mart. (corresponding to our 28th of February) being repeated. It is easy to see that this intercalation, and in general the Roman form of year is the same as that now in use; but it has been found necessary to introduce some modifications, the nature of which will be explained in a subsequent part of this treatise. But without going further at present, it may be noticed that the tropical year, not having the value supposed by this method, but being in fact, at the time of Cæsar, about ten minutes shorter, at the end of about a century and a half, the beginning of the civil year would be a day behind that of the astronomical; and this difference would, of course, go on increasing. We shall see that this was in reality what took place, and that ultimately the error became pretty


The great division of the Roman empire under the sons of Theodosius, probably exercised considerable influence on the fate of letters and science. The Greek language, which for a long time had been familiar to the Romans, ceased to be cultivated in the West; and this circumstance is of some importance in the history of astronomy; for we have seen that this science was almost completely neglected in Italy: those who wished to cultivate it were necessarily obliged to have recourse to Greek authors. But, during the middle ages, the little learning that existed in Western Europe was confined entirely to a knowledge of the

works on astronomical subjects which are now lost.

Latin language: the works of Ptolemy were a sealed book for the few learned men (if they may deserve the name) of those days. However, the conquests of the Arabs having brought them into immediate contact with the Gothic nations in Spain, they began to impart to their neighbours some of their zeal for science; and the works of Euclid and Ptolemy became known in Europe, through the means of Arabic translations. Gerbert, a man of great talent and very superior to the age in which he lived, found himself obliged to attend the Moorish universities in Spain, to acquire some knowledge of mathematics and astronomy, which it was impossible at that time to study in any Christian country. On his return to France, he composed several treatises, which show a knowledge of geometry remarkable for that time, and a considerable familiarity with the works of Euclid and Archimedes.

At a later period the example of Gerbert was imitated by an English monk named Adhelard, who for the sake of acquiring scientific knowledge, travelled in Spain and Egypt, and having become acquainted with Arabic, turned it to account, by translating from that language into Latin the Elements of Euclid. This appears to be the first translation of Euclid executed in the west; but it seems to have been little known, and has remained in manuscript till the present day. The translation of Euclid by Campanus, who lived about a century later, was the first that acquired any popularity; and all the early editions were printed from his text, which was like that of Adelhard, a translation from the Arabic, a language apparently

known to several of the literati of that day, while they still remained in profound ignorance of Greek.

In the thirteenth century many symptoms of a reviving love for letters and science began to show themselves; and knowledge which had been hitherto confined in monasteries, now spread into cities and courts. The barbarous monarchs of the middle ages were succeeded by princes who cultivated and protected letters; and though it is to be feared that the encouragement given by some of them to astronomy, was founded on a superstitious belief in the influence of the stars, we may view with some indulgence a weakness which led to such beneficial effects. The emperor Frederick the Second, a prince distinguished as well by a very cultivated mind, as a

generous protection of learned men, deserves particular mention here, as to his encouragement we owe the first translation into Latin of the Almagest of Ptolemy. This translation it is to be noticed was made from the Arabic, (Greek being still unknown in the west,) and consequently was a good deal disfigured; the Mahometan writers by no means piquing themselves on a scrupulous adherence to the text of the authors they translate: but still the service thus rendered was important; and no doubt had a favourable influence on the progress of science. The writer* who had executed the translation of the Almagest, added a translation of the commentary of Geber on the same work, and of the treatise of Alhazen on twilight. But in this latter respect he seems to have been anticipated by Vitellion, a Pole, whose voluminous treatise on optics is little more than a translation of Alhazen.

To Alphonso X., king of Castile, astronomy owes still more than to Frederick; and his reign will always form a memorable epoch in the annals of that science. His situation in a country bordering on the Arabs, who then occupied the south of the Spanish Peninsula, was very favourable for collecting about him able astronomers, who were then only to be found at the Moorish universities. Of this he profited to draw to his court a number of learned men, whom he employed for four years in constructing new and complete tables; those of Ptolemy having become in the lapse of time quite insufficient. These tables, generally called the Alphonsine, cost the prince an immense sum; and if they fell short of the degree of perfection that might have been expected, it was not from any want of munificence and zeal. Their principal defect was the introduction of an inequality in the motion of the fixed stars in longitude, by which this motion appeared to be sometimes accelerated and sometimes retarded: as the equinoctial point was supposed to describe the circumference of a small circle, the centre of which moved along the ecliptic according to the ordinary law of preces sion. This pretended inequality, known by the name of the trepidation of the fixed stars, or the motion of the eighth sphere, has been already noticed in speaking of the Arabs. It seems to have originated in Greece; but it was cer

Gerard of Cremona.

fainly not admitted by Ptolemy, or the most judicious of the oriental astronomers. The introduction of it into the Alphonsine tables seems owing to the Jews, who had a large share in their formation; at least this may be conjectured from the numbers 7000 and 49,000 years, in which the small circle above-mentioned and the ecliptic were respectively described. These are cabalistic numbers for which the Jews from fanciful ideas felt great veneration; and, indeed, the astronomers of that nation seem to have been singularly attached to the doctrine of trepidation, since the time of Thebit-ben-Corah, unjustly accused of being the inventor of the system. But it reflects little credit on the Alphonsine astronomers, to have admitted an inequality founded on no observations, and rejected by every author of eminence.

It seems but reasonable to suppose that Alphonso, who protected so zealously astronomy, was himself versed in the science. Of this we have no direct proof, but there is on record a saying of his, which has been accused of impiety, though it would be fairer to regard it as an expression of the disgust caused to a sound judgment, by the complication of the Ptolemaic system. "Had the Deity," said Alphonso, "consulted me at the creation of the universe, I could have given him some good advice." If this exclamation may be justly blamed as irreverent in its expression, it certainly conveys a condemnation of the theories of his time, and the monstrous combination of

Cycle and epicycle, orb on orb, then generally received as the system of the world.

On comparing the Alphonsine tables with those of Ptolemy, we see how little progress astronomy had made in eleven centuries. Some ameliorations in the elements of the syntaxis, are perhaps more than counterbalanced by the introduction of the imaginary inequality called trepidation; which continued to disfigure the best tables as late as the time of the celebrated Copernicus. Nor to judge from the slowness with which physical science recovered from its long torpor, could any one have guessed at the rapid progress it was about to make in the sixteenth and subsequent centuries. The real restoration of astronomy in Europe can scarcely be placed earlier than two hundred years subsequent to the publication of the Alphonsine tables. At this time Pur

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