established. Ptolemy does not give them, but in each case when by Eratosthenes and used by Hipparchus. This "is followed by spherical geometry and trigonometry enough for the determination of the connexion between the sun's right ascension, declination, and longitude, and for the formation of a table of declinations to each degree of longitude. Delambre says he found both this and the table of chords very exact." 3 In book ii., after some remarks on the situation of the habitable parts of the earth, Ptolemy proceeds to make deductions from the He We now proceed to notice briefly the contents of the Almagest. It is divided into thirteen books. The first book, which may be regarded as introductory to the whole work, opens with a short preface, in which Ptolemy, after some observations on the distinction between theory and practice, gives Aristotle's division of the sciences and remarks on the certainty of mathematical knowledge, "inasmuch as the demonstrations in it proceed by the incontrovertible ways of arithmetic and geometry." He concludes his preface with the statement that he will make use of the discoveries of his predecessors, and relate briefly all that has been sufficiently explained by the ancients, but that he will treat with more care and develop ment whatever has not been well understood or fully treated. Ptolemy unfortunately does not always bear this in mind, and it is sometimes difficult to distinguish what is due to him from that which he has borrowed from his predecessors. Ptolemy then, in the first chapter, presupposing some preliminary notions on the part of the reader, announces that he will treat in order-what is the relation of the earth to the heavens, what is the position of the oblique circle (the ecliptic), and the situation of the inhabited parts of the earth; that he will point out the differences of climates; that he will then pass on to the consideration of the motion of the sun and moon, without which one cannot have a just theory of the stars; lastly, that he will consider the sphere of the fixed stars and then the theory of the five stars called "planets." All these things-i.c., the phenomena of the heavenly bodies-he says he will endeavour to explain in taking for principle that which is evident, real, and certain, in resting everywhere on the surest Book iii. treats of the motion of the sun and of the length of the observations and applying geometrical methods. He then enters year. In order to understand the difficulties of this question on a summary exposition of the general principles on which his Ptolemy says one should read the books of the ancients, and especi Syntaris is based, and adduces arguments to show that the heaven ally those of Hipparchus, whom he praises "as a lover of labour is of a spherical form and that it moves after the manner of a and a lover of truth” (ἀνδρὶ φιλοπόνῳ τε ὁμοῦ καὶ φιλαληθεί). He sphere, that the earth also is of a form which is sensibly spherical, begins by telling us how Hipparchus was led to discover the prethat the earth is in the centre of the heavens, that it is but point cession of the equinoxes; he relates the observations which led in comparison with the distances of the stars, and that it has not any motion of translation. With respect to the revolution of the Hipparchus to his second great discovery, that of the eccentricity earth round its axis, which he says some have held, Ptolemy, while of the solar orbit, and gives the hypothesis of the eccentric by admitting that this supposition renders the explanation of the which he explained the inequality of the sun's motion. Ptolemy phenomena of the heavens much more simple, yet regards it as concludes this book by giving a clear exposition of the circumAll this the altogether ridiculous. Lastly, he lays down that there are two stances on which the equation of time depends. principal and different motions in the heavens-one by which all reader will find in the article ASTRONOMY (vol. ii. p. 750). Ptolemy, moreover, applies Apollonius's hypothesis of the epicycle to explain the equator; the other, which is peculiar to some of the stars, is in the stars are carried from east to west uniformly about the poles of the inequality of the sun's motion, and shows that it leads to the a contrary direction to the former motion and same results as the hypothesis of the eccentric. He prefers the ferent poles. These preliminary notions, which takes place round latter lay both as equally fit to clear up the difficulties. hypothesis as more simple, requiring only one and not two Ptolemy, form the subjects of the second and following chapters. second chapter there are some general remarks to which attention next proceeds to that for the Practire have given an account, and which is indisp choruse of shplanation of phenomena one should adopt the simplest hypothesis practical astronomy. The employment of this table presupposes that it is possible to establish, provided that it is not contradicted In the the evaluation of the obliquity of the ecliptic, the knowledge of by the observations in any important respect. This fine principle, which is indeed the foundation of all astronomical science. Ptolemy Comte, Système de Politique Positive, iii. 324. Cantor, Vorlesungen über Geschichte der Mathematik, p. 356. 8. v. 3 De Morgan, in Smith's Dictionary of Greek and Roman Biography, "Ptolemæus, Claudius." Kaipiral, temporal or variable. These hours varied in length with the seasons; they were used in ancient times and arose from the division of the natural day (from sunrise to sunset) into twelve parts. Alm., ed. Halma, i. 159. XX. 12 book commences with a similar catalogue of the stars in the con- which is of universal application, may, we think-regard being paid | tudes, arranged according to their constellations; and the eighth 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.5 Book iv., in which Ptolemy follows Hipparchus, treats of the first and principal inequality of the moon, which quite corresponds to the inequality of the sun treated of in the third book. As to the observations which should be employed for the investigation of the motion of the moon, Ptolemy tells us that lunar eclipses should be preferred, inasmuch as they give the moon's place without any error on the score of parallax. The first thing to be determined is the time of the moon's revolution; Hipparchus, by comparing the observations of the Chaldeans with his own, discovered that the shortest period in which the lunar eclipses return in the same order was 126,007 days and 1 hour. In this period he finds 4267 lunations, 4573 restitutions of anomaly, and 4612 tropical revolutions of the moon less 7 q.p.; this quantity (73) is also wanting to complete the 345 revolutions which the sun makes in the same time with respect to the fixed stars. He concluded from this that the lunar month contains 29 days and 31′ 50′′ 8" 20""" of a day, very nearly, or 29 days 12 hours 44' 3" 20"". These results are of the highest importance. (See ASTRONOMY.) In order to explain this inequality, or the equation of the centre, Ptolemy makes use of the hypothesis of an epicycle, which he prefers to that of the eccentric. 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. This is the first instance in which we find the two hypotheses of eccentric and epicycle combined. The fifth book treats also of the parallaxes of the sun and moon, and gives a description of an instrument-called later by Theon the "parallactic rods"-devised by Ptolemy for observing meridian altitudes with greater accuracy. The subject of parallaxes is continued in the sixth book of the Almagest, and the method of calculating eclipses is there given. The author says nothing in it which was not known before his time. Books vii., viii. treat of the fixed stars. Ptolemy verified the fixity of their relative positions and confirmed the observations of Hipparchus with regard to their motion in longitude, or the precession of the equinoxes. (See ASTRONOMY.) The seventh book concludes with the catalogue of the stars of the northern hemisphere, in which are entered their longitudes, latitudes, and magni1 Système de Politique Positive, iv. 173. This Sosigenes, as Th. H. Martin has shown, was not the astronomer of that name who was a contemporary of Julius Cæsar, but a Peripatetic philosopher who lived at the end of the 2d century A.D. 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 samea year. 10 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. 12 It may be added, as strangest of all, that this doctrine was held by Theon of Smyrna, 13 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. 3 Brandis, Schol. in Aristot. edidit Acad. Reg. Borussica (Berlin, 1836), p. 498. 4 Eloaywyn els тà pairóμeva, c. i. in Halma's edition of the works of Ptolemy, vol. iii. ("Introduction aux Phénomènes Célestes, traduite du Grec de Géminus," p. 9), Paris, 1819. 5 This has been noticed by Pliny, who says, "Multiformi hæc (luna) ambage torsit ingenia contemplantium, et proximum ignorari maxime sidus indignantium" (N. H., ii. 9). We have seen that the problem which presented itself to the astronomers of the Alexandrian epoch was the following: 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 "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 6 Delambre, Histoire de l'Astronomie Ancienne, ii. 264. 7 This is true of their mean distances; but we know that Mars at opposition is nearer to us than the sun. 8 Eratosthenes, for example, as we learn from Theon of Smyrna. 9 Transits of Mercury and Venus over the sun's disk, therefore, had not beea observed. 10 This was known to Eudoxus. Sir George Cornewall Lewis (An Historical Survey of the Astronomy of the Ancients, p. 155), confusing the geocentric revolu tions assigned by Eudoxus to these two planets with the heliocentric revolu tions in the Copernican system, which are of course quite different, says that "the error with respect to Mercury and Venus is considerable"; this, however, is an error not of Eudoxus but of Cornewall Lewis, as Schiaparelli has remarked. 11 "Hunc [solem] ut comites consequuntur Veneris alter, alter Mercurii cursus (Somnium Scipionis, De Rep., vi. 17). This hypothesis is alluded to by Pliny, N. H., ii. 17, and is more explicitly stated by Vitruvius, Arch., ix. 4. 12 Macrobius, Commentarius ex Cicerone in Somnium Scipionis, i. 19. 13 Theon (Smyrnans Platonicus), Liber de Astronomia, ed. Th. H. Martin (Paris, 1849), pp. 174, 294, 296. Martin thinks that Theon, the mathematician, four of whose observations are used by Ptolemy (Alm., ii. 176, 193, 194, 195, 196, ed. Halma), is not the same as Theon of Smyrna, on the ground chiefly that the latter was not an observer. 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,' 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. 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 Histoire de l'Astronomie Ancienne, the second volume of which is for the most part devoted to the Almagest,3 or to Narrien's History of Astronomy,+ in which the subject is treated with great clearness. 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 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) Υποθέσεις τῶν πλανωμένων ἢ τῶν οὐρανίων κύκλων κινήσεις, 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 Kaviv Bariλeiŵv, London, 1620, and afterwards by Halma, vol. iv., Paris, 1820. (3) Kavov ßaσiλeiŵv, 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) Apuoi Kv Bißia y. This Treatise on Music was published in Greek and Latin by mentary in the third volume of Wallis's works, Oxford, 1699. (5) Teтpáßißλos Wallis at Oxford, 1682. It was afterwards reprinted with Porphyry's comouvragis, Tetrabiblon or Quadripartitum. This work is astrological, as is also the small collection of aphorisms, called Kapós 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 de scription is made by perpendiculars let fall on the plane; whence it has been called by the moderns "orthographic projection.' (7) Planisphærium, 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. (S) 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. 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. De Ptolemy is hardly less celebrated as a geographer than as an astronomer, and his great work on geography exercised 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 questions for a period of many centuries, and was only gradually 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, 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 MaoŋμаTIKỲ Zúvrais, 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 Syntaris was translated into Arabic at Baghdad by order of the lightened caliph Al-Mamún, who was himself an astronomer, about 524 apart from this consideration, it was really the first attempt 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 translated 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 astronomy in the university of Vienna, Regiomontanus. The earliest The Latin Astra 1461) and completed by his pupil and successor in the professorship of Other works of Ptolemy, which we now proceed to notice very briefly, are In this elition the Greek text and the French translation are given in parallel columns; the latter, however, should not be read without reference to the former. Delambre begins his analysis of the Almagest thus-"L'Astronomie des Gres est toute entière dans la Syntaxe mathématique de Ptolémée." Narrien, An Historical decount of the Origin and Progress of Astronomy, London, 188 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 trustworthy map of the Inhabited World would be by observations 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 important 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 execution 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 employed even by modern geographers where more accurate XV. means of determination are not available. The greater part It will be convenient to consider separately the two was determined by actual observation; but we learn from Ptolemy Unfortunately at the very outset of his attempt to realize this conception he fell into an error which had the effect of vitiating all his subsequent conclusions. Eratosthenes was the first writer who had attempted in a scientific manner to determine the circumference of the earth, and the result at which he arrived, that it amounted to 250,000 stadia or 25,000 geographical miles, was generally adopted by subsequent geographers, including Strabo. Posidonius, however, who wrote about a century after Eratosthenes, had made an independent calculation, by which he reduced the circumference of the globe to 180,000 stadia, or less than threefourths of the result obtained by Eratosthenes, and this computation, on what grounds we know not, was unfortunately adopted by Marinus Tyrius, and from him by Ptolemy. The consequence of this error was of course to make every degree of latitude or longitude (measured at the equator) equal to only 500 stadia (50 geographical miles), instead of its true equivalent of 600 stadia. effects would indeed have been in some measure neutralized had there existed a sufficient number of points of which the position Its Another source of permanent error (though one of much less im- With Marinus and Ptolemy, as with all preceding Greek geo- 1 Hipparchus had indeed pointed out long before 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 seen also at Carthage, was a mere matter of popular observation, of no scientific value. Yet Ptolemy seems to have known of no other. The The great problem that had attracted the attention and exercised the ingenuity of all geographers from the time of Dicæarchus to that of Ptolemy was to determine the length and breadth of the Inhabited World, which they justly regarded as the chief subject of the geographer's consideration. This question had been very fully discussed by Marinus, who had arrived at conclusions widely dif ferent from those of his predecessors. Towards the north indeed there was no great difference of opinion, the latitude of Thule being generally recognized as that of the highest northern land, and this was placed both by Marinus and Ptolemy in 63° lat., not very far beyond the true position of the Shetland Islands, which had come in their time to be generally identified with the mysterious Thule of Pytheas. The western extremity, as already mentioned, had been in like manner determined by the prime meridian drawn through the supposed position of the Fortunate Islands. But towards the south and east Marinus gave an enormous extension to the continents of Africa and Asia, beyond what had been known to or suspected by the earlier geographers, and, though Ptolemy greatly reduced his calculations, he still retained a very exaggerated estimate of their results. The additions thus made to the estimated dimensions of the known world were indeed in both directions based upon a real extension of knowledge, derived from recent information; but unfortunately the original statements were so perverted by misinterpretation in applying them to the construction of a map as to give results differing widely from the truth. The southern limit of the world as known to Eratosthenes, and even to Strabo (who had in this respect no further knowledge than his predecessor more than two centuries before), had been fixed by them at the parallel which passed through the eastern extremity of Africa (Cape Guardafui), or the Land of Cinnamon as they termed it, and that of the Sembrite (corresponding to Sennaar) in the interior of the same continent. This parallel, which would correspond nearly to that of 10° of true latitude, they supposed to be situated at a distance of 3400 stadia (340 geographical miles) from that of Meroe (the position of which was accurately known), and 13,400 to the south of Alexandria ; while they conceived it as passing, when prolonged to the eastward, through the island of Taprobane (Ceylon), which was universally recognized as the southernmost land of Asia. Both these geographers were wholly ignorant of the vast extension of Africa to the south of this line and even of the equator, and conceived it as trending away to the west from the Land of Cinnamon and then to the north-west to the Straits of Gibraltar. Marinus had, however, learned from itineraries both by land and sca the fact of this great extension, of which he had indeed conceived so exaggerated an idea that even after Ptolemy had reduced it by more than a half it was still materially in excess of the truth. The castern 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, and this is placed by Ptolemy not far from its true position in 7° S. lat. But he added to this a bay of great extent as far as a promontory called Prasum (perhaps Cape Delgado), which he placed in 15° S. lat. At the same time he assumed the position in about the same parallel of a region called Agisymba, which was supposed to have been discovered by a Roman general, whose itinerary was employed by Marinus. Taking, therefore, this parallel as the limit of knowledge to the south, while he retained that of Thule to the north, he assigned to the inhabited world a breadth of nearly 80°, instead of less than 60°, which it had occupied on the maps of Eratosthenes and Strabo. It had been a fixed belief with all the Greek geographers from the earliest attempts at scientific geography not only that the length of the Inhabited World greatly exceeded its breadth, but that it was more than twice as great,-a wholly unfounded assumption, but to which their successors seem to have felt themselves bound to conform. Thus Marinus, while giving an undue extension to Africa towards the south, fell into a similar error, but to a far greater degree, in regard to the extension of Asia towards the east. Here also he really possessed a great advance in knowledge over all his predecessors, the increased trade with China for silk having led to an acquaintance, though of course of a very vague and general kind, with the vast regions in Central Asia that lay to the cast of the Pamir range, which had formed the limit of the Asiatic nations previously known to the Greeks. But Marinus had learned that traders proceeding eastward from the Stone Tower-a station at the foot of this range-to Sera, the capital city of the Seres, occupied seven months on the journey, and from thence he arrived at the enormous result that the distance between the two points was not less than 36,200 stadia, or 3620 geographical miles. Ptolemy, while he justly points out the absurdity of this conclusion and the erroneons mode of computation on which it was founded, had no means of correcting it by any real anthority, and hence reduced it summarily by one half. The effect of this was to place Sera, the easternmost point on his map of Asia, at a distance of 45° from the Stone Tower, which again he fixed, on the authority of itineraries cited by Marinus, at 24,000 stadia or 60° of longitude from the Euphrates, reckoning in both cases a degree of longitude as equivalent to 400 | stadia, in accordance with his uniform system of allowing 500 stadia to 1° of latitude. Both distances were greatly in excess of the truth, independently of the error arising from this mistaken system of graduation. 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. The effect of this last cause, it must be remembered, would unfortunately be cumulative, in consequence of 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. The result of these combined causes of error was to lead him to assign no less than 180°, or 12 hours, of longitude to the interval between the meridian of the Fortunate Islands and that of Sera, which really amounts to about 130°. All 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 with preceding writers. former Greek geographers, with the single exception of Hipparchus, had agreed in supposing the Inhabited World to be surrounded on all sides by sea, and to form in fact a vast island in the midst of a circumfluous ocean. This notion, which was probably derived originally from the Homeric fiction of an ocean stream, and was certainly not based upon direct observation, was nevertheless of course in accordance with the truth, great as was the misconception it involved of the extent and magnitude of the continents included within this assumed boundary. Hence it was unfortunate that Ptolemy should in this respect have gone back to the views of Hipparchus, and have assumed that the land extended indefinitely to the north in the case of Europe and Scythia, to the cast in that of Asia, and to the south in that of Africa. His boundary-line was in each of these cases an arbitrary limit, beyond which lay the But in the last case he was not Unknown Land, as he calls it. content with giving to Africa an indefinite extension to the south: he assumed the existence of a vast prolongation of the land to the cast from its southernmost known point, so as to form a connexion with the south-eastern extremity of Asia, of the extent and position of which he had a wholly erroneous idea. In this last case Marinus had derived from the voyages of recent navigators in the Indian Seas a knowledge of the fact that there lay in that direction extensive lands which had been totally unknown to previous geographers, and Ptolemy had acquired still more extensive information in this quarter. But unfortunately he had formed a totally false conception of the bearings of the coasts thus made known, and consequently of the position of the lands to which they belonged, and, instead of carrying the line of coast northwards from the Golden Chersonese (the Malay Peninsula) to China or the land of the Sina, he brought it down again towards the south after forming a great bay, so that he placed Cattigarathe principal emporium in this part of Asia, and the farthest point known to him-on a supposed line of coast, of unknown extent, but with a direction from north to south. The hypothesis that this land was continuous with the most southern part of Africa, so that the two enclosed one vast gulf, though a mere assumption, is stated by him as definitely as if it was based upon positive information; and it was long received by medieval geographers as an unquestioned fact. This circumstance undoubtedly contributed to perpetuate the error of supposing that Africa could not be circumnavigated, in opposition to the more correct views of Strabo and other earlier geographers. On the other hand, there can be no doubt that the undue extension of Asia towards the east, so as to diminish by 50° of longitude the interval between that continent and the western coasts of Europe, had a material influence in fostering the belief of Columbus and others that it was possible to reach the Land of Spices (as the East Indian islands were then called) by direct navigation towards the west. It is not surprising that Ptolemy should have fallen into considerable errors respecting the more distant quarters of the world; but even in regard to the Mediterranean and its dependencies, as well as the regions that surrounded them, with which he was in a certain sense well acquainted, the imperfection of his geographical Here he had indeed some knowledge is strikingly apparent. well-established data for his guidance, as far as latitudes were concerned. That of Massilia had been determined many years before by Pytheas within a few miles of its true position, and the latitude of Rome, as might be expected, was known with approximate Those of Alexandria and Rhodes also were well known, having been the place of observation of distinguished astronomers, and the fortunate accident that the Island of Rhodes lay on the same parallel of latitude with the Straits of Gibraltar at the other end of the sea enabled him to connect the two by drawing the parallel direct from the one to the other. The importance attached to this line (36' N. lat.) by all preceding geographers has been already mentioned. Unfortunately Ptolemy, like his predecessors, supposed its course to lie almost uniformly through the open sea, wholly ignoring the great projection of the African coast towards accuracy. |