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established. Ptolemy does not give them, but in each case when required applies the theorem of Menelaus for spherics directly. This atly increases the length of his demonstrations, which the modern reader finds still more cumbrous, inasmuch as in each case it was necessary to express the relation in terms of chords—the equivalents of sines—only, cosines and tangents being of later invention. Such, then, was the trigonometry of the Greeks. Mathematics, indeed, has ever been, as it were, the handmaid of astronomy, and many important methods of the former arose from the needs of the latter. Moreover, by the foundation of trigonometry, astronomy attained its final general constitution, in which calculations took the place of diagrams, as these latter had been at an earlier period substituted for mechanical apparatus in solving the ordinary problems." Further, we find in the application of trigonometry to astronomy frequent examples and even a systematic use of the method of approximations,—the basis, in fact, of all application of mathematics to practical questions. There was a disinclination on the part of the Greek geometer to be satisfied with a mere approximation, were it ever so close; and the unscientific agrimensor shirked the labour involved in acquiring the knowledge which was indispensable for learning trigonometrical calculations. Thus the development of the calculus of approximations fell to the lot of the astronomer, who was both scientific and practical.” Wu now ". to notice briefly the contents of the Almagest. It is dividel 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 distinetion between theory and practice, gives Aristotle's division of the wiences 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 so relate briefly all that has been susliciently explained y the ancients, but that he will treat with more care and development whatever has not been well understood or fully treated. I'tolemy unfortunately does not always bear this in mind, and it is sometimes listicult to distinguish what is due to him from that which he has borrowed from his predecessors. Ptolemy then, in the first .." 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 |-sition of the oblique circle (the ecliptic), and the situation of the inhabited parts of o 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 o of the fivel stars and then the theory of the five stars called “planets." All these things—i.e., 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 ol-ervations and applying geometrical methods. . He then enters on a summary exposition of the general principles on which his syntoris is based, and adduces arguments to show that the heaven is of a spherical form and that it moves after the manner of a “phon, that the earth also is of a form which is sensibly spherical, i. it the earth is in the centre of the heavens, that it is but a loint in comparison with the distances of the stars, and that it has not inv motion of translation. With respect to the revolution of the , orth round its avis, which he says some have held, Ptolemy, while winnitting that this supposition renders the explanation of the ph, nomena of the heavens much more simple, yet regards it as altoother ridi, ulous. Lastly, he lays down that there are two Prin, ipal and different motions in the heavens - one by which all the stars are carriesl from east to west uniformly about the poles of the router: the other, which is peculiar to some of the stars, is in * , on trary direction to the former motion and takes play round off rent roles. These preliminary notions, which are all older than

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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.” " In book ii., after some remarks on the situation of the habitalle parts of the earth, Ptolemy proceeds to make deductions from the principles established in the preceding book, which he does by means of the theorem of Menelaus. The length of the longest day being given, he shows how to determine the arcs of the horizon intercepted between the equator and the ecliptic- the amplitude of the eastern point of the ecliptic at the solstice—for different degrees of obliquity of the sphere; hence he finds the height of the pole and reciprocally. From the same data he shows how to find at what places and times the sun becomes vertical and how to !alculate the ratios of gnomons to their equinoctial and solstitial shadows at noon and conversely, pointing out, however, that the latter method is wanting in precision. All these matters he considers fully and works out in detail for the parallel of Rhodes. 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. I)elambre 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. In chapter vi. 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 arth. 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. In the next chapter he enters into par. ticulars and inquires what are the arcs of the equator which cross the horizon at the same time as given arcs of the ecliptic, or, which comes to the same thing, the time which a given are of the ecliptic takes to cross the horizon of a given place. He arrives at a formula for calculating ascensional differences and gives talles of ascensions arranged by 10° of longitude for the different climates from the equator to that where the longest day is seventeen hours. He then shows the use of these tables in the investigation of the length of the day for a given climate, of the manner of reducing temporal to equinoctial hours and rice rersa, and of the nonagesimal point and the point of orientation of the ecliptic. In the following chapters of this book he determines the angles formed by the intersections of the ecliptic - first with the meridian, then with the horizon, and lastly with the vertical circle- and concludes by giving tables of the angles and arcs formed by the intersection of these circles, for the seven climates, from the louallel of Meroe thirteen hours to that of the mouth of the Borysthenes sixteen hours . These tables, he adds, should be completed by the situation of the chief towns in all countries according to their latitudes and longi. tudes; this he promises to do in a separate treatise and has in fatt done in his for orophy. look iii. treats of the motion of the sun and of the length of the vear. In order to understand the lisheulties of this uuestion Ptolemy says one should read the books of the ancients, ml especi ally those of Hippouchus, whom he prais' s “as a lover of labour and a lover of truth" idio, i ;t\omoro to ouni Aai ova Motori . He begins by telling us how Hippar hus was led to di- over the precession of the quinoxes: he relates the ol-rivations which l, ol o to his second great discovery, that of the contii, ity of the solar orbit, and gives the hypoth. -i- of the to entric ly which he explained the In lity of the son's motion. I'toleno concludes this look by civil: a in win-sition of the circumstan es on which the equation of time i. p. nil-. All this the reader will find in the arti le A-1 losovoy vol. ii. p. 750. I'tolemy, moreover, applies Apollonius's hypothesis of the livolo to explain the inequality of the sun's motion, and shows that it li al- to the same results as the hypoth, “is of the . . . no rio. He or f is the lattor hypothesis as more simpo * in-l not two motions, and as q; ally fit to ~. In the s: oud chapt, r there are son, general to in k- : h attention should be diro to il. We finil the i.l. i. own that for the is nono. 1: on -}: st ho

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which is of universal application, may, we think—regard being paid to its place in the Almagest—be justly attributed to Hipparchus. It is the first law of the “philosophia prima” of Comte.” We find in the same page another ..". 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. In the same chapter we find also the principle laid down that the object of mathematicians ought to be to represent all the celestial phenomena by uniform and circular motions. This principle is stated by Ptolemy in the manner which is unfortunately too common with him, that is to say, he does not give the least indication whence he derived it. We know, however, from Simplicius, on the authority of Sosigenes,” that Plato is said to have proposed the following problem to astronomers: “What regular and determined motions being assumed would fully account for the phenomena of the motions of the planetary bodies?” We know, too, from the same source that Eudemus says in the second book of his History of Astronomy that “Eudoxus of Cnidus was the first of the Greeks to take in hand hypotheses of this kind,” “that he was in fact the first Greek astronomer who proposed a geometrical hypothesis for explaining the periodic motions of the planets—the famous system of concentric spheres. 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.” 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.” 13ook 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 Chaldaeans 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 (7 °) is also wanting to o 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" S''' 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 eccentrie. 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. 2 This Sosigenes, as Th.H. Martin has shown, was not the astronomer of that name who was a contemporary of Julius Caesar, but a Peripatetic philosopher who lived at the end of the 2d century A.D. 3 Brandis, Schol. in Aristot. edidit Acad. Reg. Borussica (Berlin, 1836), p. 498. 4 Elaaywyh eis patvöueva, 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 ha-c (luna) ambage torsit ingenia contemplantium, et proximum ignorari maxime sidus indignantiun” (N.H., ii. 9).

tudes, arranged according to their constellations; and the eighth book commences with a similar catalogue of the stars in the constellations of the southern hemisphere. This catalogue has been the subject of keen controversy amongst modern astronomers. Some, as Flamsteed and Lalande, maintain that it was the same catalogue which Hipparchus had drawn up 265 years before Ptolemy, whereas others, of whom Laplace is one, think that it is the work of Ptolemy himself. The probability is that in the main the catalogue is really that of Hipparchus altered to suit Ptolemy's own time, but that in making the changes which were necessary a wrong precession was assumed. This is Delambre's opinion ; he says, “Whoever may have been the true author, the catalogue is unique, and does not suit the age when Ptolemy lived ; by subtracting 2° 40' from all the longitudes it would suit the age of Hipparchus; this is all that is certain.” It has been remarked that Ptolemy, living at Alexandria, at which city the altitude of the pole is 5° less than at Rhodes, where Hipparchus observed, could have seen stars which are not visible at Rhodes; none of these stars, however, are in Ptolemy's catalogue. The eighth book contains, moreover, a description of the milky way and the manner 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 ogyptians.” 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 13rahe 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: 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 disliculty, 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, | 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, 11 istoire de l'.[stronomie Anci, none, 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.

1. sole of Mercury and Venus over the sun's disk, therefore, had not been

to loservedi.

10 This was known to Eudoxus. Sir George Cornewall Lewis (An Historical Survey of the Astronomy of the Ancients, p. 155), confusing the geocentric revolutions assigned by Eudoxus to these two planets with the heliocentric revolu tions in the Collernican 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.”

(Sounism, Scipionis, Le Rep., vi. 17). This hypothesis is alluded to by Pliny, N.H., ii. 17, and is more explicitly stated by Vitruvius, Arch., ix. 4.

18 Macrobius, Comment trius or Cicerone in Sonnium Scipionis, i. 19.

13. Theon (Smyrnaous Platonicus), Liber de Astronomist, ed.Th.H. Martin (Paris, 1849), pp. 174, 294, 2. 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,

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in which the length of the longest day varies from 133 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 Uromologium, Paris, 16:50, and by Halma in his edition of the works of Ptolemy, vol. iii., Paris, 1819. (2) "Tiroffégets Tāv TNava'uévay Töv otpaviwu kūk\wu kıv gets, on the Planetary Hypothesis. This is a summary of a portion of the Almagust, 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 Bainlridge, the Savilian professor of astronomy at Oxford, with the Sphere of Proclus and the Kavdov Baqi)\etav, London, 1620, and afterwards by Halina, vol. iv., Paris, 1820. (3) Kavčov Baat Metáv, A Table of Reigns. This is a chronological table of Assyrian, Persian, Greek, and Roman sovereigns, with the length of their reigns, from Nahonasar to Antoninus Pius. This table (comp. Syncellus, Chronour., ed. Dind., i. 388 sq.) has been printed by Scaliger, Calvisius, Petavius, IRainbridge (as above noted), and by Halma, vol. iii., Paris, 1s10. (4) 'Apuovokāv B3\ta Y. 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, 1690. (5) Terpá,3:3\os asovrašas, Tetrabiblon or Quadripartitum. This work is astrological, as is also the small collection of aphorisms, called KapTós 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 1'tolemy. They were both published in Greek and Latin by Camerarius, Nuremberg, 15:5, and by Melanchthon, Basel, 15 (6) I), Analemmate. The original of this work of Ptolemy is lost. It was translated from the Arabic and published by Commandine, Rome, 1562. The small muu is the description of the sphere on a plane. We find in it the sections of the different circles, as the dinrnal parallels, and everything which can facilitate the intelligence of gnomonies. "I his ol. scription is made by perpendiculars let fall on the plane ; whence it has born called by the moderns “orthographic projection." (7) I'lunisplar, on, The Plunisphere. 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 nov: called “stereographic" projection. The lost edition of this work is that of Commandine, Venice, 1558. (S) opties. 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 thory is more complete than that of any astronomer before the time of Cassini. I), Morgan doubts whether this work is genuine on account of the al-ence of allusion to the Almost or to the subject of refraction in the .slouri, of itself: but his chief reason for doubting its authenticity is that the author of the optics was a poor geometer. (G. J. A.)

(,, o/ra/y. 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 Almatosest 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, 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 observations of the latitude and longitude of all the |rincipal 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 latitudo, or “climata," as he termed them, trustworthy observations even of this character were in his time very few in numlor, while the means of determining longitudes could hardly he 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 Tyro, who lived shortly before the time of Ptol, Iny, and whose work is known to us only throich that writer, appears to have been the first to resume the prollem thus propol, and lay down the map of the known world in ar, or love with the precepts of Hi His matrials for the ex. otion of such a do-ion wore irol, il nois, rally ina and he was fore...! to onto ut himself for

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means of determination are not available. The greater part of the treatise of Marinus was occupied with the discussion of these authorities, and it is impossible for us, in the absence of the original work, to determine how far he had succeeded in giving a scientific form to the results of his labours; but we are told by Ptolemy himself that he considered them, on the whole, so satisfactory that he had made the work of his predecessor the basis of his own in regard to all the countries bordering on the Mediterranean, a term

which would comprise to the ancient geographer almost

all those regions of which he had really any definite knowledge. With respect to the more remote regions of the world, Ptolemy availed himself of the information imparted by Marinus, but not without reserve, and has himself explained to us the reasons that induced him in some instances to depart from the conclusions of his predecessor. It is very unjust to term Ptolemy a plagiarist from Marinus, as has been done by some modern authors, as he himself acknowledges in the fullest manner 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 and arrangement that constitute the great merit of Ptolemy's work and that have stamped it with a character wholly distinct from all previous treatises on geography. But at the same time it possesses much interest, as showing the greatly increased knowledge of the more remote portions of Asia and Africa which had been acquired by geographers since the time of Strabo and Pliny. It will be convenient to consider separately the two different branches of the subject, (1) the mathematical portion, which constitutes his geographical system, properly so termed; and (2) his contributions to the progress of positive knowledge with respect to the Inhabited World. 1. Mathematical Geography. —As a great astronomer, Ptolemy was of course infinitely better qualified to comprehend and explain the mathematical conditions of the earth and its relations to the celestial bodies that surround it than any preceding writers on the special subject of geography. But his general views, except on a few points, did not differ from those of his most eminent precursors Eratosthenes and Strabo. In common with them, 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, and dividing the earth into five great 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. IIe adopted also from Hipparchus the division of the equator and other great circles into 360 parts or “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 these, from the equator to the pole, to which he gave the name of “meridians.” He thus conceived the whole surface of the earth (as is done by modern geographers) to be covered with a complete 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 the task of the scientific geographer to place the outline of the world, so far as it was then ionown by cxperience and observation. nfortunately 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 o: 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 goographical miles), instead of its true equivalent of 600 stadia. Its effects would indeed have been in some measure neutralized had there existed a suslicient number of points of which the position

was determined by actual observation; but we learn from Ptolemy himself that this was not the case, and that such observations for latitude were very few in number, while the means of determining longitudes were almost wholly wanting.” Hence the positions laid down by him were really, with very few exceptions, the result of computations of distances from itineraries and the statements of travellers, estimates which were liable to much greater error in ancient times than at the present day, from the want of any accurate mode of observing bearings, or portable instruments for the measurement of time, while they had no means at all of determining distances at sea, except by the rough estimate of the time employed in sailing from point to point. The use of the log, simple as it appears to us, was unknown to the ancients. But, great as would naturally be the errors resulting from such imperfect means of calculation, they were in most cases increased by the permanent error arising from the erroneous system of graduation adopted by Ptolemy in laying them down upon his map. Thus, if he had arrived at the conclusion from itineraries that two places were 5000 stadia from one another, he would place them at a distance of 10° apart, and thus in fact separate them by an interval of 6000 stadia. Another source of permanent error (though one of much less importance) which affected all his longitudes arose from the erroneous assumption of his prime meridian. In this respect also he followed Marinus, who, having arrived at the conclusion that the Fortunate Islands (the Canaries) were situated farther west than any part of the continent of Europe, had taken the meridian through the outermost of this group as his prime meridian, from whence he calculated all his longitudes eastwards to the Indian Ocean. But, as both Marinus and Ptolemy were very imperfectly acquainted with the position and arrangements of the islands in question, the line thus assumed was in reality a purely imaginary one, being drawn through the supposed position of the outer island, which they placed 24° west of the Sacred Promontory (Cape St Vincent), which was regarded by Marinus and Ptolemy, as it had been by all previous geographers, as the westernmost point of the continent of Europe, while the real difference between the two is not less than 9° 20'. Hence all Ptolemy's longitudes, reckoned eastwards from this assumed line, were in fact about 7°iess than they would have been if really measured from the meridian of Ferro, which continued so long in use among geographers in modern times. The error in this instance was the more unfortunate as the longitude could not of course be really measured, or even calculated, from this imaginary line, but was in reality calculated in both directions 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. It must be observed also that the equator was in like manner placed by Ptolemy at a considerable distance from its true geographical position. The place of the equinoctial line on the surface of the globe was of course well known to him as a matter of theory, but as no observations could have been made in those remote regions he could only calculate its place from that of the tropic, which he supposed to pass through Syene. And as he here, as elsewhere, reckoned a degree of latitude as equivalent to 500 stadia, he inevitably made the interval between the tropic and the equator too small by one-sixth ; and the place of the former on the surface of the earth being fixed by observation he necessarily carried up the supposed place of the equator too high by more than 230 geographical miles. But as he had practically no geographical acquaintance with the equinoctial regions of the earth this error was of little importance. With Marinus and Ptolemy, as with all preceding Greek geographers, the most in portant line on the surface of the globe for all practical purposes was the parallel of 36° of latitude, which passes through the Straits of ël. at one end of the Mediterranean, and through the Island of Rhodes and the Gulf of Issus at the other. It was thus regarded by Dicaearchus and almost all his successors as dividing the regions around the inland sea into two portions, and as being continued in theory along the chain of Mount Taurus till it joined the great mountain range north of India; and from thence to the Eastern Ocean it was regarded as constituting the dividing line of the Inhabited World, along which its length must be measured. 13ut it susliciently shows how inaccurate were the observations and how imperfect the materials at his command, even in regard to the best known portions of the earth, that Ptolemy, following Marinus, describes this parallel as passing through Caralis in Sardinia and Lilybaeum in Sicily, the one being really in 39° 12' lat., the other in 37° 50'. It is still more strange that he places so important and well known a city as Carthage 1' 20" south of the dividing parallel, while it really lies nearly 1° to the north of it.

* 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.

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The great problem that had attracted the attention and exercised the ingenuity of all geographers from the time of Dicaearchus to that of Ptolemy was to determine the length and breadth of the Inhabited §. which they justly regarded as the chief subject of the floo. consideration. This question had been very fully liscussed by Marinus, who had arrived at conclusions widely different 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 o 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 liff ring 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 castern extremity of Africa (Cape Guardafui), or the Land of Cinnamon as they termed it, and that of the Sembritte ‘corresponding to Sennaar) in the interior of the same continent. This parallel, which would correspond o to that of 10° of true latitude, they supposed to be situated at a distance of 3400 stadia (340 o 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. 13oth 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 sea 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 is a promontory callel l'rasum (perhaps Cape Delgado), which he placed in 155° 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 tiiterary 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 so, instead of less than 60', which it had occupied on the maps of Eratosthenes and Strabo. It had been a fived belief with all the Greek geographers from the earliest attempts at scientific geography not only that the 1. moth of the Inhibited 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 lound to conform. Thus Marinus, while giving an undue extension to Africa towarls the south, fell into a similar error, but to a sar creater degree, in regarl to the extension of Asia towards the east. 11-real-o 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 east of th. I'amir range, which hail formel the limit of the Asiatic nations Previously known to the Greeks. But Marinus had learned that trul...rs prolin: eastwarl from the Stone Tower—a station at the foot of this range —-to S.r.l. the capital city of the Seres, occupied -v, n months on the journey, and from thence he arrived at the to mons result that the distance between the two points was not !--- than 35.200 -tailin, or 3520 Fo miles. Ptolemy, while He jostly points ont the alourdity of this conclusion and the erroneous no is of computation on which it was founded, had no means of orrecting it by any real anthority, and hence reduced it summarily 1 r one half. The effect of this was to place Sora, the easternmost *int on his map of Asia, at a distance of 45}. from the stor F. whi. h again he fixel. on the authority of itinorari's rit, i ly Marinus, at 24.0% stalia or 60° of longitude from the Euphrat S.

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°. But in thus estimating the length and breadth of the known world Ptolemy attached a very disserent sense to these terms from that which they had generally borne with preceding writers. All 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 circumsluous 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 I'tolemy 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 east 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 Unknown Land, as he calls it. 13ut in the last case he was not content with giving to Africa an indefinite extension to the south: he assumed the existence of a vast prolongation of the land to the east 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. 13ut 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 Cattigara— the 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 hypothosis that this land was continuous with the most southern part of Africa, so that the two enclosed one vast gulf, though a more assumption, is stated by him as definitely as if it was based upon positive information; and it was long received by media val geographers as an unquestioned fact. This circumstance undoubtedly contributed to perpetuate the error of supposing that Africa could not he 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 unlue extension of Asia towards the east, so as to liminish by 50' of longitude the interval between that continent and the westein coasts of Europe, had a material influence in foster. ing the belief of Columbus and others that it was possible to reach the Land of Spies was the East Indian islands were then callel, by direct navigation towards the west. It is not son prising that l'tolemy should have fallen into considerable errors to sloting the more distant qui uters of the world; but even in log ord to the Meditorian, an and its dependenties, as well as the regions that surroundlel them, with whi. h he was in a rtain sense well a quaintrol, the importe ion of his go nigraphi al knowledge is strikingly appa: no. Heir he hul ind, , , some well-establisool dita to his -ui lan: “, as far as latitudes were run. , , incl. That of Missilia had to n is to ruin d many years 1...fore by l’ythras within a few mi’ s of its to losition, and the titude of Rome, as mi-kt lo vo. i, w is kowo, with all-roxim to a uray. Those of A. v iiid:ii and 1: ...l. also vote w, I known, having lo on to of o- 1 v.1:ion o' l 1sh. I as: 1. onors, anol the . . . . ... tha: ti... I - 1 !!!... i - loy on the s one lo i. i.... with th: - toi"; alth: at the otl, r end of lila t is | in all . t; to this alr, ally no n: supposed 1's w i.e. y iglor:l.

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