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 vol. iii. (Paris, 1819). (4) undefined. This Treatise on Music was published in Greek and Latin by Wallis at Oxford (1682). It was afterwards reprinted with Porphyry’s commentary in the third volume of Wallis’s works (Oxford, 1699). (5) undefined, Tetrabiblon or Quadripartitum. This work is astrological, as is also the small collection of aphorisms, called undefined 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 Commandine (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 of gnomonics. This description is made by perpendiculars let fall on the plane; whence it has been called by the moderns “orthographic projection.” (7) Planisphaerium, 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. 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.

The publication of a new edition of Ptolemy’s works under the title, Claudii Ptolemaei opera quae exstant omnia, was recently undertaken at Leipzig. The first volume (in two parts, 1898, 1903) contains the Greek text of the Almagest edited by J. L. Heiberg. Consult also J. E. Mentucla, Histoire des mathématiques, i. 293; J. B. J. Delambre, Connaissance des temps (1816); and Histoire de l'astronomie ancienne, vol. 2; J. J. A. Caussin, ''Nouvelles mémoires de l'acad. des inscriptions, t. vi.; P. Tannery, Recherches sur l'histoire de l'astronomie ancienne'', ch. vi.–xv.; Narrieu, History of Astronomy (1833); Fabricius, Bibliotheca graeca, ed. Harles, vol. 5; Halma’s 1813–1816 edition of his Almagest (Greek with French translation); A. Berry, A Short History of Astronomy, pp. 62–73; British Museum Catalogue.

Geography. Ptolemy is hardly less celebrated as a geographer than as an astronomer, and his Geographikē syntaxis exercised as great an influence on geographical progress (especially during the period of the Classical Renaissance), as did his Almagest on astronomical. This exceptional position was largely due to its scientific form, which rendered it convenient and easy of reference; but, apart from this, it was really the most considerable attempt of the ancient world to place the study of geography on a scientific basis. The astronomer Hipparchus had indeed pointed out, three centuries before Ptolemy, that the only way to construct a trustworthy map of the inhabited world would be by observations of the latitude and longitude of all the principal points on its surface. But the materials for such a map were almost wholly wanting, and, though Hipparchus made some approach to a correct division of the known world into zones of latitude, “climates” or klimata, as he termed them, trustworthy observations of latitude were then very few, while the means of determining longitudes hardly existed. Hence probably it arose that no attempt was made to follow up the suggestion of Hipparchus until Marinus of Tyre, who lived shortly before Ptolemy, and whose work is known to us only through the latter. Marinus’ scientific materials being inadequate, he contented himself mostly with determinations derived from itineraries and other rough methods, such as are still employed where more accurate means of determination are not available. The greater part of Marinus’ treatise was occupied with the discussion of his authorities, and it is impossible, in the absence of the original work, to decide how far his results attained a scientific form. But Ptolemy himself considered them, on the whole, so satisfactory that he made his predecessor’s work the basis of his own in regard to all the Mediterranean countries, that is, in regard to almost all those regions of which he had definite knowledge. In the more remote regions of the world, Ptolemy availed himself of Marinus’ information, but with reserve, and himself explains the reasons that induced him sometimes to depart from his predecessor’s conclusions. It is unjust to term Ptolemy a plagiarist from Marinus, as he himself fully acknowledges his obligations to that writer, from whom he derived the whole mass of his materials, which he undertook to arrange and present to his readers in a scientific form. It is this form, unique among those ancient geographical treatises which have survived, that constitutes one great merit of Ptolemy’s work. At the same time it shows the increased knowledge of Asia and Africa acquired since Strabo and Pliny.

1. Mathematical Geography.—As an astronomer, Ptolemy was of course better qualified to explain the mathematical conditions of the earth and its relations to the celestial bodies than most preceding geographers. His general views had much in common with those of Eratosthenes and Strabo. Thus he assumed that the earth was a globe, the surface of which was divided by certain great circles— the equator and the tropics—parallel to one another, dividing the earth into five zones, the relations of which with astronomical phenomena were of course clear to his mind as a matter of theory, though in regard to the regions bordering on the equator, as well as to those adjoining the polar circle, he could have had no confirmation of his conclusions from actual observation. He adopted also from Hipparchus the division of the equatorial circle into 360 parts (degrees, as they were subsequently called, though the word does not occur in this sense in Ptolemy), and supposed other circles to be drawn through this, from the equator to the pole, to which he gave the name of meridians. He thus, like modern geographers, conceived the whole surface of the earth as covered with a network of parallels of latitude and meridians of longitude, terms which he himself was the first extant writer to employ in this technical sense. Within the network thus constructed it was his task to place the outline of the world, so far as known to him.

But at the very outset of his attempt he fell into an error vitiating all his conclusions. Eratosthenes (276–196 ) was the first who had attempted scientifically to determine the earth’s circumference, and his result of 250,000 (or 252,000) stadia, i.e. 25,000 (25,200) geographical miles, was generally adopted by subsequent, including Strabo. Poseidonius, however (c. 135–50 ), reduced this to 180,000, and the latter computation was inexplicably adopted by Marinus and Ptolemy. This error made every degree of latitude or longitude (measured at the equator) equal to only 500 (50 geographical miles), instead of its true equivalent of 600. The mistake would have been somewhat neutralized had there existed a sufficient number of points of which the position was fixed by observation; but we learn from Ptolemy himself that such observations for were very few, while the means of determining s were almost wholly wanting. Hence the positions laid down by him were, with few exceptions, the result of computations from itineraries and the statements of travellers, liable to much greater error in ancient times than at the present day, from the want of any accurate mode of observing bearings, of measuring time (by portable instruments), or of estimating distances at sea, except by the rough estimate of the time employed in sailing from point to point. Even the use of the log was unknown to the ancients. But, great as were the errors resulting from such imperfect means of calculation, they were increased by the permanent error arising from Ptolemy’s system of graduation. Thus if he concluded (from itineraries) that two places were 5000 stadia distant, he would place them 10° apart, and thus in fact separate them by 6000 stadia.

Another source of permanent error (though of less importance), which affected all his longitudes, arose from his prime meridian. Here also he followed Marinus, who, supposing that the Fortunate Islands (vaguely answering to our plus the  group) lay farther west than any part of Europe or Africa, had taken the meridian through the (supposed) outermost of this group as his prime meridian, from whence he calculated his s eastwards to the. But as both Marinus and Ptolemy had no exact knowledge of the islands in question the line thus assumed was purely imaginary, drawn through the supposed position of an island which they placed 25° (instead of 9° 20′) west of the Sacred Promontory (i.e. Cape St Vincent, regarded by Marinus and Ptolemy, as by previous geographers, as the westernmost point of Europe). Hence all Ptolemy’s longitudes, reckoned eastwards, were about 7° less than they would have been if really measured from the meridian of, which continued so long in use. This error was the more unfortunate as the longitude was really calculated, not from this imaginary line, but from Alexandria, westwards as well as eastwards (as Ptolemy himself has done in his eighth book), and afterwards reversed, so as to suit the supposed method of computation.