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 of constructing a celestial globe; it also treats of the configuration of the stars, first with regard to the sun, moon and planets, and then with regard to the horizon, and likewise of the different aspects of the stars and of their rising, culmination and setting simultaneously with the sun.

The remainder of the work is devoted to the planets. The ninth book commences with what concerns them all in general. The planets are much nearer to the earth than the fixed stars and more distant than the moon. Saturn is the most distant of all, then Jupiter and then Mars These three planets are at a greater distance from the earth than the sun. So far all astronomers are agreed. This is not the case, he says, with respect to the two remaining planets, Mercury and Venus, which the old astronomers placed between the sun and earth, whereas more recent writers have placed them beyond the sun, because they were never seen on the sun. He shows that this reasoning is not sound, for they might be nearer to us than the sun and not in the same plane, and consequently never seen on the sun. He decides in favour of the former opinion, which was indeed that of most mathematicians. The ground of the arrangement of the planets in order of distance was the relative length of their periodic times; the greater the circle, the greater, it was thought, would be the time required for its description. Hence we see the origin of the difficulty and the difference of opinion as to the arrangement of the sun, Mercury and Venus, since the times in which, as seen from the earth, they appear to complete the circuit of the zodiac are nearly the same—a year. Delambre thinks it strange that Ptolemy did not see that these contrary opinions could be reconciled by supposing that the two planets moved in epicycles about the sun; this would be stranger still, he adds, if it is true that this idea, which is older than Ptolemy, since it is referred to by Cicero, had been that of the Egyptians It may be added, as strangest of all, that this doctrine was held by Theon of Smyrna, who was a contemporary of Ptolemy or somewhat senior to him. From this system to that of Tycho Brahe there is, as Delambre observes, only a single step.

We have seen that the problem which presented itself to the astronomers of the Alexandrian epoch was the following&thinsp;: it was required to find such a system of equable circular motions as would represent the inequalities in the apparent motions of the sun, the moon and the planets. Ptolemy now takes up this question for the planets; he says that “&thinsp;this perfection is of the essence of celestial things, which admit of neither disorder nor inequality,” that this planetary theory is one of extreme difficulty, and that no one had yet completely succeeded in it. He adds that it was owing to these difficulties that Hipparchus—who loved truth above all things, and who, moreover, had not received from his predecessors observations either so numerous or so precise as those that he has left—had succeeded, as far as possible, in representing the motions of the sun and moon by circles, but had not even commenced the theory of the five planets. He was content, Ptolemy continues, to arrange the observations which had been made on them in a methodic order and to show thence that the phenomena did not agree with the hypotheses of mathematicians at that time. He showed that in fact each planet had two inequalities, which are different for each, that the retrogradations are also different, whilst other astronomers admitted only single inequality and the same retrogradation; he showed further that their motions cannot be explained by eccentrics nor by epicycles carried along concentric, but that it was necessary to combine both hypotheses. After these preliminary notions he gives from Hipparchus the periodic motions of the five planets, together with the shortest times of restitutions, in which, moreover, he has made some slight corrections. He then gives tables of the mean motions in longitude and of anomaly of each of the five planets, and shows how the motions in longitude of the planets can be represented in a general manner by means of the hypothesis of the eccentric combined with that of the epicycle. He next applies his theory to each planet and concludes the ninth book by the explanation of the various phenomena of the planet Mercury. In the tenth and eleventh books he treats, in like manner, of the various phenomena of the planets Venus, Mars, Jupiter and Saturn.

Book xii. treats of the stationary and retrograde appearances of each of the planets and of the greatest elongations of Mercury and Venus. The author tells us that some mathematicians, and amongst them Apollonius of Perga, employed the hypothesis of the epicycle to explain the stations and retrogradations of the planets. Ptolemy goes into this theory, but does not change in the least the theorems of Apollonius; he only promises simpler and clearer demonstrations of them. Delambre remarks that those of Apollonius must have been very obscure, since, in order to make the demonstrations in the Almagest intelligible, he (Delambre) was obliged to recast them. This statement of Ptolemy is important, as it shows that the mathematical theory of the planetary motions was in a tolerably forward state long before his time. Finally, book xiii. treats of the motions of the planets in latitude, also of the inclinations of their orbits and of the magnitude of these inclinations.

Ptolemy concludes his great work by saying that he has included in it everything of practical utility which in his judgment should find a place in a treatise on astronomy at the time it was written, with relation as well to discoveries as to methods. His work was justly called by him undefined, for it was in fact the mathematical form of the work which caused it to be preferred to all others which treated of the same science, but not by “&thinsp;the sure methods of geometry and calculation.” Accordingly, it soon spread from Alexandria to all places where astronomy was cultivated; numerous copies were made of it, and it became the object of serious study on the part of both teachers and pupils. Amongst its numerous commentators may be mentioned Pappus and Theon of Alexandria in the 4th century and Proclus in the 5th. It was translated into Latin by Boetius, but this translation has not come down to us. The Syntaxis was translated into Arabic at Bagdad by order of the enlightened caliph Al-Mamūm, who was himself an astronomer, about 827, and the Arabic translation was revised in the following century by Tobit ben Korra. The Almagest was translated from the Arabic into Latin by (q.v.). In the 15th century it was translated from a Greek manuscript in the Vatican by George of Trebizond. In the same century an epitome of the Almagest was commenced by Purbach (d. 1461) and completed by his pupil and successor in the professorship of astronomy in the university of Vienna, Regiomontanus. The earliest edition of this epitome is that of Venice (1496), and this was the first appearance of the Almagest in print. The first complete edition of the Almagest is that of P. Liechtenstein (Venice, 1515)—a Latin version from the Arabic. The Latin translation of George of Trebizond was first printed in 1528, at Venice. The Greek text, which was not known in Europe until the 15th century, was first published in the 16th by Simon Grynaeus, who was also the first editor of the Greek text of Euclid, at Basel (1538). This edition was from a manuscript in the library of Nuremberg—where it is no longer to be found—which had been presented by Regiomontanus, to whom it was given by Cardinal Bessarion.

Other works of Ptolemy, which we now proceed to notice very briefly, are as follow. (1) undefined, On the Apparitions of the Fixed Stars and a Collection of Prognostics. It is a calendar of a kind common amongst the Greeks under the name of undefined, or a collection of the risings and settings of the stars in the morning or evening twilight, which were so many visible signs of the seasons, with prognostics of the principal changes of temperature with relation to each climate, after the observations of the best meteorologists, as, for example, Meton, Democritus, Eudoxus, Hipparchus, &c. Ptolemy, in order to make his Parapegma useful to all the Greeks scattered over the enlightened world of his time, gives the apparitions of the stars not for one parallel only but for each of the five parallels in which the length of the longest day varies from 13 hours to 15 hours—that is, from the latitude of Syene to that of the middle of the Euxine. This work was printed by Petavius in his Uranologium (Paris, 1630), and by Halma in his edition of the works of Ptolemy, vol. iii. (Paris, 1819). (2) undefined, On the Planetary Hypothesis. This is a summary of a portion of the Almagest, and contains a brief statement of the principal hypotheses for the explanation of the motions of the heavenly bodies. It was first published (Gr., Lat.) by Bainbridge, the Savilian professor of astronomy at Oxford, with the Sphere of Proclus and the undefined (London, 1620), and afterwards by Halma, vol. iv. (Paris, 1820). (3) undefined, A Table of Reigns. This is a chronological table of Assyrian, Persian, Greek and Roman sovereigns, with the length of their reigns, from Nabonasar to Antoninus Pius. This table (cf. G. Syncellus, Chronogr. ed. Dind. i. 388 seq.) was printed by Scaliger, Calvisius, Petavius, Bainbridge and by Halma,