George Johnston Allman, Edward Herbert Bunbury, and Charles Raymond Beazley

“ Ptolemy gives an exposition of the most important properties of each parallel, commencing with the equator, which he considers as the southern limit of the habitable quarter of the earth. For each parallel or climate, which is determined by the length of the longest day, he gives the latitude, a principal place on the parallel, and the lengths of the shadows of the gnomon at the solstices and equinox. ”

“ The fifth book commences with the description of the astrolabe of Hipparchus, which Ptolemy made use of in following up the observations of that astronomer, and by means of which he made his most important discovery, that of the second inequality in the moon’s motion, now known by the name of the “evection.” In order to explain this inequality he supposed the moon to move on an epicycle, which was carried by an eccentric whose centre turned about the earth in a direction contrary to that of the motion of the epicycle. ”

“ In order to understand the difficulties of this question Ptolemy says one should read the books of the ancients, and especially those of Hipparchus, whom he praises “as a lover of labour and a lover of truth” (ἀνδρὶ φιλοπόνῳ τε ὁμοῦ καὶ φιλαληθεῖ) . He begins by telling us how Hipparchus was led to discover the precession of the equinoxes; he relates the observations by which Hipparchus verified the eccentricity of the solar orbit imperfectly known to his Chaldaean predecessors, and gives the hypothesis of the eccentric by which he explained the inequality of the sun’s motion. ”

“ It was the work of an astronomer rather than a geographer. Not only did its plan exclude all description of the countries with which it dealt, their climate, natural productions, inhabitants and peculiar features, but even its physical geography proper is treated in an irregular and perfunctory manner. While Strabo was fully alive to the importance of the rivers and mountain chains which (in his own phrase) " geographize " a country, Ptolemy deals with this part of his subject in so careless a manner as to be often worse than useless. ”

“ The starting-point is the well known theorem of plane geometry concerning the segments of the sides of a triangle made by a transversal: The segments of any side are in a ratio compounded of the ratios of the segments of the other two sides. This theorem, as well as that concerning the inscribed quadrilateral, was called after Ptolemy—naturally, indeed, since no reference to its source occurs in the Almagest. This error was corrected by Mersenne, who showed that it was known to Menelaus, an astronomer and geometer who lived in the reign of the emperor Trajan. ”

“ It thus appears that the principle laid down here by Ptolemy can be traced to Eudoxus and Plato; and it is probable that they derived it from the same source, namely, Archytas and the Pythagoreans. We have indeed the direct testimony of Geminus of Rhodes that the Pythagoreans endeavoured to explain the phenomena of the heavens by uniform and circular motions. [12]
Books iv., v. are devoted to the motions of the moon, which are very complicated; the moon in fact, though the nearest to us of all the heavenly bodies, has always been the one which has given the greatest trouble to astronomers. ”

“ 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 Μαθματικὴ σύνταξις, 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 ”

“ The distances west of the Euphrates were of course comparatively well known, nor did Ptolemy’s calculation of the length of the Mediterranean differ very materially from those of previous Greek geographers, though still greatly exceeding the truth, after allowing for the permanent error of graduation. This last, it must be remembered, would be cumulative, the longitudes being computed from a fixed point in the west, instead of being reckoned east and west from Alexandria, which was undoubtedly the mode in which they were really calculated. ”

“ Ptolemy, however, was the first writer of antiquity who showed some relations between the Tanais or Don (usually considered by the ancients as the boundary between Europe and Asia) and the Rha or Volga, which he correctly described as flowing into the Caspian. He was also the first geographer after Alexander to return to the correct view (found in Herodotus and Aristotle) that the Caspian was an inland sea, without communication with the ocean. ”

“ That of Massilia had been determined, within a few miles, by Pytheas, and those of Rome, Alexandria and Rhodes were approximately known, all having been observation-centres for distinguished astronomers. The fortunate accident that Rhodes lay on the same parallel with the Straits of Gibraltar enabled Ptolemy to connect the two ends of the Inland Sea on the famous parallel of 36° N. Unfortunately Ptolemy, like his predecessors, supposed its course to lie almost uniformly through the open sea, ignoring the great projection of Africa towards the north from Carthage westward. ”

“ On the southern shores of Asia Ptolemy’s geography is especially faulty, though he shows a greatly increased general knowledge of these regions. For more than a century the commercial relations between western India and Alexandria, the chief eastern emporium of the Roman Empire, had become more important and intimate than ever before. ”

“ Though Ptolemy himself has distinctly pointed out in his first book the defective nature of his materials, and the true character of the data furnished by his tables, few readers studied this portion of his work, and his statements were generally received with undoubting faith. It is only in modern times that his apparently scientific work has been shown to be in most cases a specious edifice resting upon no adequate foundations.
There can be no doubt that the work of Ptolemy was from the time of its first publication accompanied with maps, which are regularly referred to in the eighth book. ”

“ 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. ”

“ Ptolemy’s work as a geographer is discussed below, and an account of the discoveries in astronomy of Hipparchus and Ptolemy is given in the article Astronomy: History. Their contributions to pure mathematics, however, require to be noticed here. Of these the chief is the foundation of trigonometry, plane and spherical, including the formation of a table of chords, which served the same purpose as our table of sines. ”

“ 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. ”

“ By “ motion in longitude” must be understood the motion of the centre of the epicycle about the eccentric, and by “ anomaly” the motion of the star on its epicycle. ↑ Hipparchus pointed out the mode of determining longitudes by observations of eclipses, but the instance to which he referred (of the celebrated eclipse before the battle of Arbela, which was also seen at Carthage) was a mere matter of popular observation, of no scientific value. Yet Ptolemy seems to have known of no other. ”

“ We find in the same page another principle, or rather practical injunction, that in investigations founded on observations where great delicacy is required we should select those made at considerable intervals of time in order that the errors arising from the imperfection which is inherent in all observations, even in those made with the greatest care, may be lessened by being distributed over a large number of years. ”

“ He then enters on a summary exposition of the general principles on which his Syntaxis is based, and adduces arguments to show that the heaven is of a spherical form and that it moves after the manner of a sphere, that the earth also is of a form which is sensibly spherical, that the earth is in the centre of the heavens, that it is but a point in comparison with the distances of the stars, and that it has not an motion of translation. ”

“ These causes of error combined to make Ptolemy allow 180° long., or 12 hours’ interval, between the Fortunate Islands meridian and Sera (really about 130°) .
But in thus estimating the length and breadth of the known world, Ptolemy attached a very different sense to these terms from that which they had generally borne. Most earlier Greek geographers and “ cosmographers ” supposed the inhabited world to be surrounded on all sides by sea, and to form a vast island in the midst of a circumfluous ocean. ”

“ Theon gives us three reasons for the selection of that parallel by Ptolemy: the first is that the height of the pole at Rhodes is 36°, a whole number, whereas at Alexandria he believed it to be 30° 58′; the second is that Hipparchus had made at Rhodes many observations; the third is that the climate of Rhodes holds the mean place of the seven climates subsequently described. Delambre suspects a fourth reason, which he thinks is the true one, that Ptolemy had taken his examples from the works of Hipparchus, who observed at Rhodes and had made these calculations for the place where he lived. ”

“ The theorem now bears the name of Menelaus, though most probably it came down from Hipparchus; Chasles, indeed, thinks that Hipparchus deduced the property of the spherical triangle from that of the plane triangle, but throws the origin of the latter farther back and attributes it to Euclid, suggesting that it was given in his Porisms. [3] Carnot made this theorem the basis of his theory of transversals in his essay on that subject. It should be noticed that the theorem is not given in the Almagest in the general manner stated above ”

“ Marinus had, however, learned from itineraries both by land and sea the fact of this extension, of which he had conceived so exaggerated an idea that even after Ptolemy had reduced it by more than half it was still much in excess of the truth. The eastern coast of Africa was indeed tolerably well known, being frequented by Greek and Roman traders, as far as a place called Rhapta (opposite to Zanzibar?) ”

“ 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. ”

“ Bibliography.—Ptolemy's Geographia was printed for the first time in a Latin translation, accompanied with maps, in 1462 (?) , and numerous other editions followed in the latter part of the 15th and earlier half of the 16th centuries, but the Greek text did not make its appearance till 1533, when it was published at Basel in quarto, edited by Erasmus. All these early editions, however, swarm with textual errors, and are critically worthless. The same may be said of the edition of (Gr. and Lat., Leiden, 1618, typ. Elzevir) , which was long the standard library edition. ”