PTOLEMY 91 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 a single inequality and the same retrogradation ; he showed further that their motions cannot be explained by eccentrics nor by epicycles carried along concentrics, 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, 1 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 demonstra- tions 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 toler- ably 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. Those who wish to go into details and learn the mathematical explanation of this celebrated system of "eccentrics" and "epicycles" are referred to the Almagest itself, which can be most conveniently studied in Halma's edition, 2 to Delambre's Histoiredel'Astronomie Ancienne, the second volume of which is for the most part devoted to the Almagest,^ or to Narrien's History of Astronomy,* in which the subject is treated with great clearness. Ptolemy concludes his great work by saying that he has included in it every- thing 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 dis- coveries as to methods. His work was justly called by him M.a0rj/naTiK7] 2 (Wafts, 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 "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 Baghdad by order of the enlightened caliph Al-Mamun, who was himself an astronomer, about 827 A.D., and the Arabic translation was revised in the following century by Thabit ibn Korra. The emperor Frederick II. caused the Almagest to be trans- lated from the Arabic into Latin at Naples about 1230. 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 (died 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 Grynseus, 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. The last edition of the Almagest is that of Halma, Greek with French translation, in two vols., Paris, 1813-16. On the manuscripts of the Almagest and its biblio- graphical history, see Fabricius, Bibliotheca Graeca, ed. Harles, vol. v. p. 280, and Halma's preface. An excellent summary of the bibliographical history is given by De Morgan in his article on Ptolemy already quoted. Other works of Ptolemy, which we now proceed to notice very briefly, are as follows. (1) Qdfffis a.ira.v<j}v affrlpuv Kal ffvvayuyij ^TrioTj/iao'itDj'., On the Apparitions of the Fixed Stars antl a Collection of Prognostics. It is a calendar of a kind common amongst the Greeks under the name of irapdirrryfj.^ 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, <fcc. 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 1 Delambre compares these mean motions with those of our modern tables and finds them tolerably correct. By " motion in longitude " must be under- stood the motion of the centre of the epicycle about the eccentric, and by "anomaly" the motion of the star on its epicycle. 2 In this edition the Greek text and the French translation are given in parallel columns ; the latter, however, should not be read without reference to the former. 3 Delambre begins his analysis of the Almagest thus" L' Astronomic des Grecs est toute entiere dans la Syntaxe mathematique de Ptolemee." 4 Narrien, An Historical Account of the Origin and Progress of Astronomy, London, 1833. in which the length of the longest day varies from 13J hours to 15J hours, that is, from the latitude of Syene to that of the middle of the Euxine. This work has been printed by Petavius in his Uranologium, Paris, 1630, and by Halma in his edition of the works of Ptolemy, vol. iii., Paris, 1819. (2) 'fTroOtcreis TUV irXavtiifj^vuv 1) ru>v oupaviuv KVKUV /civTjcreis, On the Planetary Hypothesis. This is a summary of a portion of the Almagest, and con- tains a brief statement of the principal hypotheses for the explanation of the motions of the heavenly bodies. It was first published (Gr., Lat.)by Bainl.ridge, the Savilian professor of astronomy at Oxford, with the Sphere of Proclus and the Kavwv J3affitiwv, London, 1620, and afterwards by Halma, vol. iv., Paris, 1820. (3) K.avwv /3ct(7iuJ' , 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 (comp. G. Syncellus, Chronogr., ed. Dind., i. 388 sq.) has been printed by Scaliger, Calvisius, Petavius, Bainbridge (as above noted), and by Halma, vol. iii., Paris, 1819. (4) 'Apfj.ovi- K&V |3i/3ta y. This Treatise on Music was published in Greek and Latin by Wallis at Oxford, 1682. It was afterwards reprinted with Porphyry's com- mentary in the third volume of Wallis's works, Oxford, 1699. (5) Ter/xl/3i^3Xos criVrafts, Tetrabiblon or Quadripartitum. This work is astrological, as is also the small collection of aphorisms, called Ka/>7r<5s or Centiloquium, by which it is followed. It is doubtful whether these works are genuine, but the doubt merely arises from the feeling that they are unworthy of Ptolemy. They were both published in Greek and Latin by Camerarius, Nuremberg, 1535, and by Melanchthon, Basel, 1553. (6) De Analemmate. The original of this work of Ptolemy is lost. It was translated from the Arabic and published by Com- mandine, Rome, 1562. The Analemma is the description of the sphere on a plane. We find in it the sections of the different circles, as the diurnal parallels, and everything which can facilitate the intelligence ofgnomonics. This de- scription is made by perpendiculars let fall on the plane ; whence it has been called by the moderns "orthographic projection." (7) Planisphierium, The Planisphere. The Greek text of this work also is lost, and we have only a Latin translation of it from the Arabic. The "planisphere" is a projection of the sphere on the equator, the eye being at the pole, in fact what is now called " stereographic " projection. The best edition of this work is that of Commandine, Venice, 1558. (8) Optics. This work is known to us only by imperfect manuscripts in Paris and Oxford, which are Latin translations from the Arabic ; some extracts from them have been recently published. The Optics consists of five books, of which the fifth presents most interest : it treats of the refraction of luminous rays in their passage through media of different densities, and also of astronomical refractions, on which subject the theory is more complete than that of any astronomer before the time of Cassini. De Morgan doubts whether this work is genuine on account of the absence of allusion to the Almagest or to the subject of refraction in the Almagest itself; but his chief reason for doubting its authenticity is that the author of the Optics was a poor geometer. (G. J. A.) Geography. Ptolemy is hardly less celebrated as a geographer than as an astronomer, and his great work on geography exer- cised as great an influence on the progress of that science as did his Almagest on that of astronomy. It became indeed the paramount authority on all geographical ques- tions for a period of many centuries, and was only gradu- ally superseded by the progress of maritime discovery in the 15th and 16th centuries. This exceptional position was due in a great measure to its scientific form, which rendered it very convenient and easy of reference ; but, apart from this consideration, it was really the first attempt ever made to place the study of geography on a truly scientific basis. The great astronomer Hipparchus had indeed pointed out, three centuries before the time of Ptolemy, that the only way to construct a really trust- worthy map of the Inhabited World would be by observa- tions of the latitude and longitude of all the principal points on its surface, and laying down a map in accordance with the positions thus determined. But the materials for such a course of proceeding were almost wholly wanting, and, though Hipparchus made some approach to a correct division of the known world into zones of latitude, or " climata," as he termed them, trustworthy observations even of this character were in his time very few in number, while the means of determining longitudes could hardly be said to exist. Hence probably it arose that no attempt was made by succeeding geographers to follow up the im- portant suggestion of Hipparchus. Marinus of Tyre, who lived shortly before the time of Ptolemy, and whose work is known to us only through that writer, appears to have been the first to resume the problem thus proposed, and lay down the map of the known world in accordance with the precepts of Hipparchus. His materials for the execu- tion of such a design were indeed miserably inadequate, and he was forced to content himself for the most part with determinations derived not from astronomical obser- vations but from the calculation of distances from itineraries and other rough methods, such as still continue to be em- ployed even by modern geographers where more accurate
