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PTOLEMY (CLAUDIUS PTOLEMAEUS)
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See also:PTOLEMY (See also:CLAUDIUS See also:PTOLEMAEUS) , the celebrated mathematician, astronomer and geographer, was a native of See also:Egypt,' but there is an uncertainty as to the See also:place of his See also:birth. Some See also:ancient See also:manuscripts of his See also:works describe him as of See also:Pelusium, but See also:Theodorus Meliteniota, a See also:Greek writer on See also:astronomy of the
2 The See also:Ptolemies were not in antiquity distinguished by the ordinal See also:numbers affixed to their names by See also:modern scholars and represented according to. the usual See also:convention by See also:Roman'figures. This is merely done for our convenience. In the See also:case of the later Ptolemies different systems of notation prevail according as the problematic Eupatpr and Philopator Neos are reckoned in or not.
See also:MATHEMATICS]
12th See also:century, says that he was See also:born at Ptolemais Hermii, a Grecian See also:city of the Thebaid. It is certain that he observed at See also:Alexandria during the reigns of See also:Hadrian and See also:Antoninus See also:Pius, and that he survived Antoninus. See also:Olympiodorus, a philosopher of the Neoplatonic school who lived in the reign of the See also:emperor Justinian, relates in his scholia on the See also:Phaedo of See also:Plato that Ptolemy devoted his See also:life to astronomy and lived for See also:forty years in the so-called Hrepa rob' Kavm(3ou, probably elevated terraces of the See also:temple of See also:Serapis at See also:Canopus near Alexandria, where they raised pillars with the results of his astronomical discoveries engraved upon them. This statement is probably correct; we have indeed the See also:direct See also:evidence of Ptolemy himself that he made astronomical observations during a See also:long See also:series of years; his first recorded observation was made in the See also:eleventh See also:year of Hadrian, 127 A.D.,i and his last in the fourteenth year of Antoninus, 151 A.D. Ptolemy, moreover, says, " We make our observations in the parallel of Alexandria." St Isidore of See also:Seville asserts that he was of the royal See also:race of the Ptolemies, and even calls him See also: 61. Mathematics. Ptolemy's See also:work as a geographer is discussed below, and an See also:account of the discoveries in astronomy of See also:Hipparchus and Ptolemy is given in the See also:article ASTRONOMY: See also:History. Their contributions to pure mathematics, however, require to be noticed here. Of these the See also:chief is the See also:foundation of See also:trigonometry, See also:plane and spherical, including the formation of a table of chords, which served the same purpose as our table of sines. This See also:branch of mathematics was created by Hipparchus for the use of astronomers, and its exposition was given by Ptolemy in a See also:form so perfect that for 1400 years it was not surpassed. In this respect it may be compared with the . See also:doctrine as to the See also:motion of the heavenly bodies so well known as the Ptolemaic See also:system, which was See also:paramount for about the same See also:period of See also:time. There is, however, this difference, that, whereas the Ptolemaic system was then overthrown, the theorems of Hipparchus and Ptolemy, on the other See also:hand, will be, as See also:Delambre says, for ever the basis of trigonometry. The astronomical and trigonometrical systems are contained in the See also:great work of Ptolemy, 'H yaOrlµariKi7 o vraEts, or, as Fabricius after See also:Syncellus writes it, Meyak17 vuvra is rC7c avrpovoµias; and in like manner Suidas says ouros [Hro11.] eypailte See also:TOP dyav avrpovoµov ijroi vuvraEiv. The Syntaxis of Ptolemy was called '0 0yas avrpovoµos to distinguish it from another collection called '0 j. wphs avrpovoµos, also highly esteemed by the Alexandrian school, which contained some works of See also:Autolycus, See also:Euclid, See also:Aristarchus, See also:Theodosius of Tripolis, Hypsicles and See also:Menelaus. To designate the great work of Ptolemy the See also:Arabs used the superlative µeylvrn, from which, the article al being prefixed, the hybrid name Almagest, by which it is now universally known, is derived. We proceed now to consider the trigonometrical work of Hipparchus and Ptolemy. In the ninth See also:chapter of the first See also:book of the Almagest Ptolemy shows how to form a table of chords. He sup-poses the circumference divided into 36o equal parts (TSohiara), and then bisects each of these parts. Further, he divides the See also:diameter into 120 equal parts, and then for the subdivisions of these he employs the sexagesimal method as most convenient in practice, i.e. he divides each of the sixty parts of the See also:radius into sixty equal parts, and each of these parts he further subdivides into sixty equal parts. In the Latin See also:translation these subdivisions become " partes minutae primae " and " partes minutae secundae," whence our " minutes " Weidler and Halma give the ninth year; in the account of the See also:eclipse of the See also:moon in that year Ptolemy, however, does not say, as in other similar cases, he had observed, but it had been observed (Almagest, iv. 9).619 and " seconds " have arisen. It must not be supposed, however, that these sexagesimal divisions are due to Ptolemy; they must have been See also:familiar to his predecessors, and were handed down from the Chaldaeans. Nor did the formation of the table of chords originate with Ptolemy; indeed, See also:Theon of Alexandria, the See also:father of See also:Hypatia, who lived in the reign of Theodosius, in his commentary on the Almagest says expressly that Hipparchus had already given the doctrine of chords inscribed in a circle in twelve books, and that Menelaus had done the same in six books, but, he continues, every one must be astonished at the ease with which Ptolemy, by means of a few See also:simple theorems, has found their values; hence it is inferred that the method of calculation in the Almagest is Ptolemy's own. As starting-point the values of certain chords in terms of the diameter were already known, or could be easily found by means of the Elements of Euclid. Thus the See also:side of the hexagon, or the chord of 6o°, is equal to the radius, and therefore contains sixty parts. The side of the decagon, or the chord of 36°, is the greater segment of the radius cut in extreme and mean ratio, and therefore contains approximately 37" 4' 55" parts, of which the diameter contains 120 parts. Further, the square on the side of the See also:regular pentagon is equal to the sum of the squares on the sides of the regular hexagon and of the regular decagon, all being inscribed in the same circle (Eucl. XIII. io); the chord of 72° can therefore be calculated, and contains approximately 70P 32' 3". In like manner, the square on the chord of 90°, which is the side of the inscribed square, is twice the square on the radius; and the square on the chord of 120°, or the side of the equilateral triangle, is three times the square on the radius; these chords can thus be calculated approximately. Further, from the values of all these chords we can calculate at once the chords of the arcs which are their supplements. This being laid down, we now proceed to give Ptolemy's exposition of the mode of obtaining his table of chords, which is a piece of See also:geometry of great elegance, and is indeed, as De See also:Morgan says, " one of the most beautiful in the Greek writers." He takes as basis and sets forth as a lemma the well-known theorem, which is called after him, concerning a See also:quadrilateral inscribed in a circle: The rectangle under the diagonals is equal to the sum of the rectangles under the opposite sides. By means of this theorem the chord of the sum or the difference of-two arcs whose chords are given can be easily found, for we have only to draw a diameter from the common vertex of the two arcs the chord of whose sum or difference is required, and See also:complete the quadrilateral; in one case a See also:diagonal, in the other one of the sides is a diameter of the circle. The relations thus obtained are See also:equivalent to the fundamental formulae of our trigonometry See also:sin (A+B) =sin A See also:cos B+cos A sin B, sin (A—B)=sin A cos B—cos A sin B, which can therefore be established in this simple way. Ptolemy then gives a geometrical construction for finding the chord of See also:half an arc from the chord of the arc itself. By means of the foregoing theorems, since we know the chords of 72° and of 60°, we can find the chord of 12°; we can then find the chords of 6°, 3°, I2° and three-fourths of 1°, and lastly, the chords of 41° 71°, 9°, toa°, &c.—all those arcs, namely, as Ptolemy says, which being doubled are divisible by 3. Performing the calculations, he finds that the chord of II° contains approximately 1p 34' 55", and the chord of three-fourths of 1° contains op 47' 8". A table of chords of arcs increasing by I a° can thus be formed; but this is not sufficient for Ptolemy's purpose, which was to See also:frame a table of chords increasing by half a degree. This could be effected if he knew the chord of one-half of 1°; but, since this chord cannot be found geometrically from the chord of II°, inasmuch as that would come to the trisection of an See also:angle, he proceeds to seek in the first place the chord of I°, which he finds approximately by means of a lemma of great elegance, due probably to See also:Apollonius. It is as follows: If two unequal chords be inscribed in a circle, the greater will be to the less in a less ratio than the arc described on the greater will be to the arc described on the less. Having proved this theorem, he proceeds to employ it in See also:order to find approximately the chord of I °, which he does in the following manner chord 6o' < 62 i.e. < .. chord I° < 4 chord 45'; chord 45' 45 3' 3 again chord 6o' 60' i.e. < 2, chord I ° > 3 chord 90'. For brevity we use a modern notation. It has been shown that the chord of 45' is op 47' 8" q.p., and the chord of 90' is I' 34' 15" q.p.; hence it follows that approximately chord I° < 1p 2' 50" 40'" and > 1p 2' 50". Since these values agree as far as the seconds, Ptolemy takes 1p 2' 50" as the approximate value of the chord of O. The chord of I° being thus known, he finds 'the chord of one-half of a degree, the approximate value of which is op 31' 25", and he is at once in a position to complete his table of chords for arcs increasing by half a degree. Ptolemy then gives his table of chords, which is arranged in three columns; in the first he has entered the arcs, increasing by half-degrees, from o° to 180°; in the second he gives the values of the chords of these arcs in parts of which the diameter contains 12o, the subdivisions being sexagesimal; and in the third he has inserted the thirtieth parts of the See also:differences of these chords for each half-degree, in order that the chords of the intermediate arcs, which do not occur in the table, may be calculated, it being assumed that the increment of the chords of arcs within the table for each See also:interval of 3o' is proportional to the increment of the arc). Trigonometry, we have seen, was created by Hipparchus for the use of astronomers. Now, since spherical trigonometry is directly applicable to astronomy, it is not surprising that its development was See also:prior to that of plane trigonometry. It is the subject-See also:matter of the eleventh chapter of the Almagest, whilst the See also:solution of plane triangles is not treated separately in that work. To resolve a plane triangle the Greeks supposed it to be inscribed in a circle; they must therefore have known the theorem—which is the basis of this branch of trigonometry: The sides of a triangle are proportional to the chords of the See also:double arcs which measure the angles opposite to those sides. In the case of a right-angled triangle this theorem, together with Eucl. I. 32 and 47, gives the complete solution. Other triangles were resolved into right-angled triangles by See also:drawing the perpendicular from a vertex on the opposite side. In one place (Alm. vi. ch. 7; i. 422, ed. Halma) Ptolemy solves a triangle in which the three sides are given by finding the segments of a side made by the perpendicular on it from the opposite vertex. It should be noticed also that the eleventh chapter of the first book of the See also:Alma gest contains incidentally some theorems and problems in plane trigonometry. The problems which are met with correspond to the following: See also:Divide a given arc into two parts so that the chords of the doubles of those arcs shall have a given ratio; the same problem for See also:external See also:section. Lastly, it may be mentioned that Ptolemy (Alm. vi. ch. 7; i. 421, ed. Halma) takes 3P 8' 3o", i.e. 3-{ 6—0+3600=3.1416, as the value of the ratio of the circum- ference to the diameter of a circle, and adds that, as had been shown by See also:Archimedes, it lies between 3; and 3',$. The foundation of spherical trigonometry is laid in chapter xi. on a few simple and useful lemmas. The starting-point is the well-known theorem of plane geometry concerning the segments of the sines 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 See also:error was corrected by See also:Mersenne, who showed that it was known to Menelaus, an astronomer and geometer who lived in the reign of the emperor See also:Trajan. The theorem now bears the name of Menelaus, though most probably it came down from Hipparchus; See also:Chasles, indeed, thinks that Hipparchus deduced the See also: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.2 See also:Carnot made this theorem the basis of his theory of transversals in his See also:essay on that subject. It should be noticed that the theorem is not given in the Almagest in the See also:general manner stated above; Ptolemy considers two cases only of the theorem, and Theon, in his commentary on the Almagest, has added two more cases. The proofs, however, are general. Ptolemy then See also:lays down two lemmas: If the chord of an arc of a circle be cut in any ratio and a diameter be See also:drawn through the point of section, the diameter will cut the arc into two parts the chords of whose doubles are in the same ratio as the segments of the chord; and a similar theorem in the case when the chord is cut externally in any ratio. By means of these two lemmas Ptolemy deduces in an ingenious manner—easy to follow, but difficult to discover—from the theorem of Menelaus for a plane triangle the corresponding theorem for a spherical triangle: If the sides of a spherical triangle be cut by an arc of a great circle, the chords of the doubles of the segments ofany one side will be to each other in a ratio compounded of the ratios of the chords of the doubles of the segments of the other two sides. Here, too, the theorem is not stated generally; two cases only are considered, corresponding to the two cases given in plane. Theon has added two cases. The proofs are general. By means of this theorem four of See also:Napier's formulae for the solution of right-angled spherical triangles can be easily established. Ptolemy does not give them, but in each case when required applies the theorem of Menelaus for spherics directly. This greatly increases the length of his demonstrations, which the modern reader finds still more cumbrous, inasmuch as in each case it was necessary to See also: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 impo;•tant methods of the former arose 3Ideler has examined the degree of accuracy of the numbers in these tables and finds that they are correct to five places of decimals. 2 On the theorem of Menelaus and the See also:rule of six quantities, see Chasles,, Aperpu historigue sur l'or+gine et developpement See also:des methodes en geometrie, See also:note v1. p. 291.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 See also:mechanical apparatus in solving the See also:ordinary problems.3 Further, we find in the application of trigonometry to astronomy frequent eke amples and even a systematic use of the method of approximations—the basis, in fact, of all application of mathematics to See also:practical questions. There was a disinclination on the See also:part of the Greek geometer to be satisfied with a See also:mere approximation, were it ever so See also: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 See also:fell to the See also:lot of the astronomer, who was both scientific and practical' We now proceed to See also:notice briefly the contents of the Almagest. It is divided into thirteen books. The first book, which may be regarded as See also:introductory to the whole work, opens with a See also:short preface, in which Ptolemy, after some observations on the distinction between theory and practice, gives See also:Aristotle's See also:division of the sciences and remarks on the certainty of mathematical knowledge, " inasmuch as the demonstrations in it proceed by the incontrovertible ways of See also: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 development whatever has not been well understood or fully treated. Ptolemy unfortunately does not always See also: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 See also:earth to the heavens, what is the position of the oblique circle (the See also: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 See also:consideration of the motion of the See also:sun and moon, without which one cannot have a just theory of the stars; lastly, that he will coneider the See also:sphere of the fixed stars and then the theory of the five stars called "See also: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 observations and applying geometrical methods. He then enters on a See also:summary exposition of the general principles on which his Syntaxis is based, and adduces arguments to show that the See also: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 any motion of translation. With respect to the revolution of the earth See also:round its See also:axis, ,which he says some have held, Ptolemy, while admitting that this supposition renders the explanation of the phenomena of the heavens much more simple, yet regards it as altogether ridiculous. Lastly, he lays down that there are two See also:principal and different motions in the heavens—one by which all the stars are carried from See also:east to See also:west uniformly about the poles of the See also:equator; the other, which is See also:peculiar to some of the stars, is in a contrary direction to the former motion and takes place round different poles. These preliminary notions, which are all older than Ptolemy, form the subjects of the second and following chapters. He next proceeds to the construction of his table of chords, of which we have given an account, and which is indispensable to practical astronomy. The employment of this table presupposes the evaluation of the obliquity of the ecliptic, the knowledge of which is indeed the foundation of all astronomical See also:science. Ptolemy in the next chapter indicates two means of determining this angle by observation, describes the See also:instruments he employed for that purpose, and finds the same value which had already been found 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 See also:ascension, See also:declination and See also: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 habitable 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 See also:day being given, he shows how to determine the arcs of the See also:horizon intercepted between the equator and the ecliptic—the See also:amplitude of the eastern point of the ecliptic at the See also:solstice—for different
8 See also:Comte, Systeme de politique See also:positive, iii. 324.
4 Cantor, Vorlesungen fiber Geschichte der Mathematik, p. 356.
$ De Morgan, in See also: Ptolemy gives an exposition of the most important properties of each parallel, commencing with the equator, which he considers as the See also:southern limit of the habitable See also:quarter of the earth. For each parallel or climate, which is determined by the length of the longest day, he gives the See also:latitude, a principal place on the parallel, and the lengths of the shadows of the See also:gnomon at the solstices and See also:equinox. In the next chapter he enters into particulars and inquires what are the arcs of the equator which See also: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 arc of the ecliptic takes to cross the horizon of a given place. He arrives at a See also:formula for calculating ascensional differences and gives tables of ascensions arranged by 10° of longitude for the different climates from the equator to that where the longest day is seventeen See also: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 See also:vice versa. and of the nonagesimal point and the point of See also: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 See also: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 parallel 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 longitudes; this he promises to do in a See also:separate See also:treatise and has in fact done in his See also:Geography. Book iii. treats of the motion of the sun and of the length of the year. 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 See also:lover of labour and a lover of truth " (&v&pl ¢rXoaovw m 0µo6 sal cktX 046). 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 See also:solar See also:orbit imperfectly known to his Chaldaean predecessors, and gives the See also:hypothesis of the See also:eccentric by which he explained the inequality of the sun's motion. Ptolemy concludes this book by giving a clear exposition of the circumstances on which the See also:equation of time depends. Ptolemy, moreover, applies Apollonius's hypothesis of the See also:epicycle to explain the inequality of the sun's motion, and shows that it leads to the same results as the hypothesis of the eccentric. He prefers the latter hypothesis as more simple, requiring only one and not two motions, and as equally See also:fit to clear up the difficulties. In the second chapter there are some general remarks to which See also:attention should be directed. We find the principle laid down that for the explanation of phenomena one should adopt the simplest hypothesis that it is possible to establish, provided that it is not contradicted by the observations in any important respect? This See also:fine principle, 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 See also:law of the " philosophia prima " of Comte.' We find in the same See also:page another principle, or rather practical See also: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 See also:object of mathematicians ought to be to represent all the See also:celestial phenomena by See also: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 See also:Simplicius, on the authority of See also:Sosigenes,' that Plato is said to have proposed the following 1 Kac.pucai, 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. Systbme de politique positive, iv. 173.
' This Sosigenes, as Th. H. See also: These results are of the highest importance. 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 See also: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 See also:discovery, that of the second inequality in the moon's motion, now known by the name of the " See also: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 See also: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 See also: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. The seventh book concludes with the See also:catalogue of the stars of the See also:northern hemisphere, in which are entered their longitudes, latitudes and magnitudes, 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 See also:Flamsteed and See also:Lalande, maintain that it was the same catalogue which Hipparchus had drawn up 265 years before Ptolemy, whereas others, of whom See also:Laplace is one, think that it is the work of Ptolemy himself. The See also:probability is that in the See also: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 See also: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 See also:Brandis, Schol. in Aristot. edidit acad. reg. borussica (See also:Berlin, 1836), p. 498. e Eieayuy"} cis ra 4acv6Eteva, c. i. in Halma's edition of the works of Ptolemy, vol. iii. (" Introduction aux phenomenes celestes, traduite du grec de Geminus," p. 9), See also:Paris, 1819. 7 This has been noticed by See also:Pliny, who says, " Multiformi haec (See also:luna) ambage torsit ingenia contemplantium, et proximum ignorari maxime sidus indignantium " (N.H., ii. 9). Delambre, Histoire de l'astronomie ancienne, ii. 264. 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, See also:culmination and setting simultaneously with the sun. The See also: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. See also:Saturn is the most distant of all, then See also:Jupiter and then See also: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, See also:Mercury and See also:Venus, which the old astronomers placed between the sun and earth, whereas more See also:recent writers2 have placed them beyond the sun, because they were never seen on the sun.' He shows that this reasoning is not See also: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 See also:circuit of the See also:zodiac are nearly the same—a year.4 Delambre thinks it See also: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 See also:idea, which is older than Ptolemy, since it is referred to by See also:Cicero,' had been that of the Egyptians.' It may be added, as strangest of all, that this doctrine was held by Theon of Srnyrna,7 who was a contemporary of Ptolemy or somewhat See also:senior to him. From this system to that of Tycho See also:Brahe there is, as Delambre observes, only a single step. We have seen that the problem which presented itself to the astronomers of the Alexandrian See also: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 See also: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 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 shower that in fact each See also:planet had two inequalities, which are different for each, that the retrogradations are also different, whilst other astronomers admitted only 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 ,6 i This is true of their mean distances; but we know that Mars at opposition is nearer to us than the sun. 2 Eratosthenes, for example, as we learn from Theon of See also:Smyrna. ' Transits of Mercury and Venus over the sun's disk, therefore, had not been observed. 4 This was known to Eudoxus. See also:Sir See also:George Cornewall See also:Lewis (An See also:Historical Survey of the Astronomy of the Ancients, p. 155), confusing the See also:geocentric revolutions assigned by Eudoxus to these two planets with the See also:heliocentric revolutions 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 See also:Schiaparelli has remarked. " Hunc [solem] ut comites consequuntur Veneris alter, alter Mercurii cursus " (Somnium Scipionis, De See also:rep. vi. 17). This hypothesis is alluded to by Pliny, N.H. ii. 17, and is more explicitly stated by See also:Vitruvius, See also:Arch. ix. 4. 6 See also:Macrobius, Commentarius ex See also:Cicerone in somnium Scipionis, i. 19. ' Theon (Smyrnaeus Platonicus), See also:Liber de astronomic, 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. ' Delambre compares these mean motions with those of our modern tables and finds them tolerably correct. By " motion inand 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 ail. treats of the stationary and See also: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 See also:Perga, employed the hypothesis of the epicycle to explain the stations and retrogradations of the planets. Ptolemy goes into this theory, but does not See also: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 See also: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. Ptolemy concludes his great work by saying that he has included in it everything of practical utility which in his See also: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 Maerraar,,o vuvrai;Ls, 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 Pappas and Theon of Alexandria in the 4th century and See also:Proclus in the 5th. It was translated into Latin by See also:Boetius, but this translation has not come down to us. The Syntaxis was translated into Arabic at See also:Bagdad by order of the enlightened See also:caliph Al-See also:Mamun, who was himself an astronomer, about 827 A.D., and the Arabic translation was revised in the following century by See also:Tobit See also:ben Korra. The Almagest was translated from the Arabic into Latin by See also:Gerard of See also:Cremona (q.v.). In the 15th century it was translated from a Greek See also:manuscript in the Vatican by George of See also:Trebizond. In the same century an See also:epitome of the Almagest was commenced by Purbach (d. 1461) and completed by his See also:pupil and successor in the professorship of astronomy in the university of See also:Vienna, See also:Regiomontanus. The earliest edition of this epitome is that of See also:Venice (1496), and this was the first appearance of the Almagest in See also:print. The first complete edition of the Almagest is that of P. See also:Liechtenstein (Venice, 1515)—a Latin version from the Arabic. The Latin translation of George of Trebizond was first printed in 1528, at Venice. The Greek See also:text, which was not known in See also:Europe until the 15th century, was first published in the 16th by See also:Simon See also:Grynaeus, who was also the first editor of the Greek text of Euclid, at See also:Basel (1538). This edition was from a manuscript in the library of See also:Nuremberg—where it is no longer to be found—which had been presented by Regiomontanus, to whom it was given by See also:Cardinal See also:Bessarion. Other works of Ptolemy, which we now proceed to notice very briefly, are as follow. (I) slsaQELr iurXavwv avrEpwv Kal euvaywyi Arranyau,mv, On the See also:Apparitions of the Fixed Stars and a Collection of Prognostics. It is a cal'ndar of a kind common amongst the Greeks under the name of ,rapa,rim.ra, or a collection of the risings and settings of the stars in the See also:morning or evening See also:twilight, which were so many visible signs of the seasons, with prognostics of the principal changes of temperature with relation to each climate, after the observations of the best meteorologists, as, for example, Meton, See also:Democritus, Eudoxus, Hipparchus, &c. Ptolemy, in order to make his Parepegma useful to all the Greeks scattered over the enlightened See also:world of his time, gives the apparitions of the stars not for one parallel only but for each of the five See also:parallels in which the length of the longest day varies from 132 hours to 152 hours—that is, from the latitude of Syene to that of the See also:middle of the Euxine. This work was printed by Petavius in his Uranologium (Paris, 163o), and by Halma in his edition of the works of Ptolemy, vol. iii. (Paris, 1819). (2) 'T,roOta,,S DCiv 7rXaVw(1EVwp , TWV OUpaeLuV KUKAuV KLV7t0'ELS, 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., See also:Lat.) by See also:Bainbridge, the Savilian See also:professor of astronomy at See also:Oxford, with the Sphere of Proclus and the Kavrav Sae,X,u.w (See also:London, 162o), and afterwards by Halma, vol. iv. (Paris, 1820). (3) Kavwv f aoiXELwv, A Table of Reigns. This is a See also:chronological table of See also:Assyrian, See also:Persian, Greek and Roman sovereigns, with the length of their reigns, from Nabonasar to Antoninus'Pius. This table (cf. G. Syncellus, Chronogr. ed. Dind. i. 388 seq.) was printed by See also:Scaliger, See also:Calvisius, Petavius, Bainbridge and by Halma, longitude " must be understood the motion of the centre of the epicycle about the eccentric, and by " anomaly " the motion of the See also:star on its epicycle. vol. iii. (Paris, 1819). (4) 'ApgOPCKmv fit3X(a y'. This Treatise on See also:Music was published in Greek and Latin by See also:Wallis at Oxford (1682). It was afterwards reprinted with See also:Porphyry's commentary in the third See also:volume of Wallis's works (Oxford, 1699). (5) TerpaithAos eiura is, Tetrabiblon or Quadripartitum. This work is astrological, as is also the small collection of aphorisms, called Kapa6s 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 l are unworthy of Ptolemy. They were both published in Greek and Latin by See also:Camerarius (Nuremberg, 1535), and by See also:Melanchthon (Basel, 1553). (6) De analemmate. The See also:original of this work of Ptolemy is lost. It was translated from the Arabic and published by Commandine (See also:Rome, 1562). The Analemma is the description of the sphere on a plane. We find in it the sections of the different circles, as the diurnal parallels, and everything which can facilitate the intelligence of gnomonics. This description is made by perpendiculars let fall on the plane; whence it has been called by the moderns " orthographic See also:projection." (7) Planisphaerium, The Planisphere. The Greek text of this work also is lost, and we have only a Latin translation of it from the Arabic. The " planisphere " is a projection of the sphere on the equator, the See also: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) See also:Optics. This work is known to us only by imperfect manuscripts in Paris and Oxford, which are Latin See also:translations from the Arabic. The Optics consists of five books, of which the fifth presents most See also:interest: it treats of the See also:refraction of luminous rays in their passage through See also: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 See also:Cassini. De Morgan doubts whether this work is genuine on account of the See also: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.) The publication of a new edition of Ptolemy's works under the See also:title, Claudii Ptolemaei See also:opera quae exstant omnia, was recently undertaken at See also:Leipzig. The first volume (in two parts, 1898, 1903) contains the Greek text of the Almagest edited by J. L. See also:Heiberg. Consult also J. E. Menturla, Histoire des mathematiques, i. 293; J. B. J. Delambre, Connaissance des temps (1816); and Histoire de l'astronomie ancienne, vol. 2; J. J. A. Caussin, Nouvelles memoires de l'acad. des See also:inscriptions, t. vi.; P. Tannery, Recherches sur l'histoire de l'astronomie ancienne, ch. vi.-xv.; Narrieu, History of Astronomy (1833); Fabricius, Bibliotheca graeca, ed. Harles, vol. 5; Halma's 1813–1816 edition of his Almagest (Greek with See also:French translation) ; A. See also:Berry, A Short History of Astronomy, pp. 62–93; See also:British Museum Catalogue. Geography. Ptolemy is hardly less celebrated as a geographer than as an astronomer, and his Geographike syntaxis exercised as great an See also:influence on See also:geographical progress (especially during the period of the Classical See also:Renaissance), as did his Almagest on astronomical. This exceptional position was largely due to its scientific form, which rendered it convenient and easy of reference; but, apart from this, it was really the most considerable See also:attempt of the ancient world to place the study of geography on a scientific basis. The astronomer Hipparchus had indeed pointed out, three centuries before Ptolemy, that the only way to construct a trustworthy See also:map of the inhabited world would be by observations of the latitude and longitude of all the principal points on its See also:surface. But the materials for such a map were almost wholly wanting, and, though Hipparchus made some approach to a correct division of the known world into zones of latitude, " climates " or klimata, as he termed them, trustworthy observations of latitude were then very few, while the means of deter-See also:mining longitudes hardly existed. Hence probably it arose that no attempt was made to follow up the See also:suggestion of Hipparchus until See also:Marinus of See also:Tyre, who lived shortly before Ptolemy, and whose work is known to us only through the latter. Marinus' scientific materials being inadequate, he contented himself mostly with determinations derived from itineraries and other rough methods, such as are still employed where more accurate means of determination are not available. The greater part of Marinus' treatise was occupied with the discussion of his authorities, and it is impossible, in the absence of the original work, to decide how far his results attained a scientific form. But Ptolemy himself considered them, on the whole, so satisfactory that he made his predecessor's work the basis of his own in regard to all the Mediterranean countries, that is, in regard to almost all those regions of which he had definite knowledge. In the more remote regions of the world, Ptolemy availed himself of Marinus' See also:information, but with reserve, and himself explains the reasons that induced him sometimes to depart from his predecessor's conclusions. It is unjust to See also:term Ptolemy a plagiarist from Marinus, as he himself fully acknowledges his obligations to that writer, from whom he derived the whole See also:mass of his materials, which he undertook to arrange and See also:present to his readers in a scientific form. It is this form, unique among those ancient geographical See also:treatises which have survived, that constitutes one great merit of Ptolemy's work. At the same time it shows the increased knowledge of See also:Asia and See also:Africa acquired since See also:Strabo and Pliny. 1. Mathematical Geography.—As an astronomer, Ptolemy was of course better qualified to explain the mathematical conditions of the earth and its relations to the celestial bodies than most preceding geographers. His general views had much in common with those of Eratosthenes and Strabo. Thus m. assumed that the earth was a globe, the surface of which was divided by certain great circles—the equator and the tropics—parallel to one another, dividing the earth into five zones, the relations of which with astronomical phenomena were of course clear to his mind as a matter of theory, though in regard to the regions bordering on the equator, as well as to those adjoining the polar circle, he could have had no See also:confirmation of his conclusions from actual observation. He adopted also from Hipparchus the division of the See also:equatorial circle into 36o parts (degrees, as they were subsequently called, though the word does not occur in this sense in Ptolemy), and supposed other circles to be drawn through this, from the equator to the pole, to which he gave the name of meridians. He thus, like modern geographers, conceived the whole surface of the earth as covered with a network of parallels of latitude and meridians of longitude, terms which he himself was the first extant writer"to employ in this technical sense. Within the network thus constructed it was his task to place the outline of the world, so far as known to him. But at the very outset of his attempt he fell into an error vitiating all his conclusions. Eratosthenes (276–196 B.c.) was the first who had attempted scientifically to determine the earth's circumference, and his result of 250,000 (or 252,400) stadia, i.e. 25,000 (25,200) geographical See also:miles, was generally adopted by subsequent geographers, including Strabo. Poseidonius, however (c. 135–50 B.c.), reduced this to 18o,00o, and the latter computation was inexplicably adopted by Marinus and Ptolemy. This error made every degree of latitude or longitude (measured at the equator) equal to only 5oo stadia (50 geographical miles), instead of its true equivalent of 600 stadia. The See also:mistake would have been somewhat neutralized had there existed a sufficient number of points of which the position was fixed by observation; but we learn from Ptolemy himself that such observations for latitude were very few, while the means of deter-mining longitudes were almost wholly wanting.' Hence the positions laid down by him were, with few exceptions, the result of computations from itineraries and the statements of travellers, liable to much greater error in ancient times than at the present day, from the want of any accurate mode of observing See also:bearings, of measuring time (by portable instruments), or of estimating distances at See also:sea, except by the rough estimate of the time employed in sailing from point to point. Even the use of the See also:log was unknown to the ancients. But, great as were the errors resulting from such imperfect means of calculation, they were increased by the permanent error arising from Ptolemy's system of See also:graduation. Thus if he concluded (from itineraries) that two places were 5000 stadia distant, he would place them 1o° apart, and thus in fact separate them by 6000 stadia. Another source of permanent error (though of less importance), which affected all his longitudes, arose from his See also:prime meridian. Here also he followed Marinus, who, supposing that the Fortunate Islands (vaguely answering to our Canaries plus the See also:Madeira See also:group) See also:lay farther west than any part of Europe or Africa, had taken the meridian through the (supposed) outermost of this group as his prime meridian, from whence he calculated his longitudes eastwards to the See also:Indian Ocean. But as both Marinus and Ptolemy had no exact knowledge of the islands in question the See also:line thus assumed was purely imaginary, drawn through the supposed position of an See also:island which they placed 22° (instead of 9° 20') west of the Sacred Promontory (i.e. Cape St See also:Vincent, regarded by Marinus and Ptolemy, as by previous geographers, as the westernmost point of Europe). Hence all Ptolemy's longitudes, reckoned eastwards, were about 7° less than they would have been if really measured from the meridian of Ferro, which continued so long in use. This error was the more unfortunate as the longitude was really calculated, not from this imaginary line, but from Alexandria, westwards as well as eastwards (as Ptolemy himself has done in his eighth book), and afterwards reversed, so as to suit the supposed method of computation. ' 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 See also:battle of See also:Arbela, which was also seen at See also:Carthage) was a mere matter of popular observation, of no scientific value. Yet Ptolemy seems to have known of no other. 624. The equator was in like manner placed by Ptolemy at a consider-able distance from its true geographical position. The place of the equinoctial line was well known to him as a matter of theory, but as no observations could have been made in those 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 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 this error was of little importance. With Marinus and Ptolemy, as with preceding Greek geographers, the mpst important line for practical purposes was the parallel of 36° 1Y, which, passing through the Straits of See also:Gibraltar, Rhodes Island and the Gulf of Issus, and thus dividing the Mediterranean (as See also:Dicaearchus and his successors usually regarded it) into two, was continued in theory along the See also:chain of Mt See also:Taurus till it joined the mountains See also:north of See also:India; thence to the Eastern Ocean it was regarded as constituting the dividing line of the inhabited world, along which the length of the latter must be measured. But so inaccurate were the observations and so imperfect the materials at command, even in regard to the best known regions, that Ptolemy, following Marinus, describes this parallel as passing through Caralis in See also:Sardinia and Lilybaeum in See also:Sicily, the one being really in 39° 12' lat., the other in 370 50'. Still more strangely he places Carthage 1° 2o' See also:south of the dividing parallel, while it really lies nearly t ° north of it. The problem that had especially attracted the attention of geographers from Dicaearchus to Ptolemy was to determine the length and breadth of the inhabited world. This question had been fully discussed by Marinus, who had arrived at conclusions widely different from his predecessors. Towards the north, indeed, there was no great difference of opinion, the latitude of See also:Thule being generally recognized as that of the highest northern See also:land, and this was placed both by Marinus and Ptolemy in 63° N., not far beyond the true position of the See also:Shetland Islands, which had come to be generally identified with the mysterious Thule of See also:Pytheas. The western extremity, as already mentioned, had been in like manner determined by the prime meridian drawn through the supposed position of the outermost of the Fortunate Islands. But towards the south and east Marinus gave an enormous See also:extension to Africa and Asia, beyond what had been known to or suspected by earlier geographers, and, though Ptolemy reduced Marinus' calculations, he retained an 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 the original statements were so perverted by misinterpretation as to give results (in map-construction) differing widely from the truth. The southern limit of the world had been fixed by Eratosthenes and even by Strabo at the parallel which passed through the eastern extremity of Africa (Cape Guardafui), the See also:Cinnamon Region (Somali-land) and the See also:country of the Sembritae (Sennaar). This parallel, which would correspond nearly to that of to° 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 See also:pretty accurately known) and 13,400 to the south of Alexandria; while they conceived it as passing eastward through Taprobane (See also:Ceylon, often Ceylon plus See also:Sumatra?), universally recognized as the southernmost land of Asia. Both these geographers were ignorant of the vast extension of Africa to the south of this line and even of the equator, and conceived it as trending away west from the Cinnamon Land and then north-west to the Straits of Gibraltar. 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 See also:coast of Africa was indeed tolerably well known, being frequented by Greek and Roman traders, as far as a place called Rhapta (opposite to See also:Zanzibar?), placed by Ptolemy not far from 7° S. To this he added a See also:bay extending to Cape Prasum (Delgado?), which he placed in 151° S. At the same time he assumed the position in about the same parallel of a region called Agisymba, inhabited by Ethiopians and abounding in rhinoceroses, which was supposed to have been discovered by a Roman general, Julius Maternus, 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, Ptolemy assigned to the inhabited world a breadth of nearly 8o°, instead of less than 60°, as in Eratosthenes and Strabo.
It had been a common belief among 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, an unfounded See also:assumption to which their successors seem to have See also:felt themselves See also:bound to conform. Thus IMfarinus, while extending his Africa unduly southward, exaggerated Asia still more grossly eastward. Here also he really possessed a great advance in knowledge over all his predecessors, the See also:silk See also:trade with See also:China having led to an acquaintance, though of[GEOGRAPHY
a vague and general kind, with regions east of the Pamir and Tian Shan, the limits of Asia as previously known to the Greeks. Marinus had learned that traders proceeding eastward from the See also: These causes of error combined to make Ptolemy allow 18o° 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 See also: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. This notion (perhaps derived from the Homeric " ocean stream," and certainly not based upon direct observation) was nevertheless in accordance with truth, great as was the misconception involved of the continents included. But Ptolemy in this respect went back to Hipparchus, and assumed that the land extended indefinitely north in the case of eastern Europe, east, south-east and north in that of Asia, and south, south-west and south-east 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. But in Africa he was not content with this extension southward; he also prolonged the See also:continent eastward from its southernmost known point, so as to form a connexion with south-east Asia, the extent and position of which he wholly misconceived. In this last case Marinus derived from the voyages of recent navigators in the Indian seas a knowledge of extensive lands hitherto unknown to the Helleno-Roman world, and Ptolemy acquired more information in this quarter. But he formed a false conception of the bearings of the coasts thus made known, and of the position of the lands to which they belonged, and, instead of carrying the line of coast northwards from the See also:Golden See also:Chersonese (See also:Malay See also:Peninsula) to the Land of the Sinae (sea-coast China), he brought it down again towards the south after forming a great bay, so that he placed Cattigara—the principal See also:emporium in this part of Asia, and the farthest' point known to him—on a supposed coast of unknown extent, but with a direction from north to south, and facing west. The hypothesis that this land was continuous with southernmost Africa, so as to enclose the Indian Ocean as one vast See also:lake, though a mere assumption, is stated by him as definitely as if based upon positive information. It must be noticed that Ptolemy's extension of Asia eastwards, so as to diminish by 50° of longitude the interval between easternmost Asia and westernmost Europe, fostered See also:Columbus' belief that it was possible to reach the former from the latter by direct See also:navigation, See also:crossing the See also:Atlantic. Ptolemy's errors respecting distant regions are one thing; it is another thing to discover, in regard to the Mediterranean See also:basin, the striking imperfections of his geographical knowledge. Here he had indeed some well-established data for latitudes. 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 See also: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 See also:lie almost uniformly through the open sea, ignoring the great projection of Africa towards the north from Carthage westward. The erroneous position assigned to Carthage being supposed to See also:rest upon astronomical observation, doubtless determined that of all North Africa. Thus Ptolemy's Mediterranean, from Massilia to the opposite point of Africa, had a width of over i I ° of latitude (really 6i°). He was still more at a loss in respect of longitudes, for which he had no trustworthy observations; yet he came nearer the truth than previous geographers, all of whom had greatly exaggerated the length of the Inland Sea. Their calculations, like those of Marinus and Ptolemy, could only be founded on the imperfect estimates of mariners; and Ptolemy, in translating these conclusions into scientific form, vitiated his results by his system of graduation. Thus while Marinus calculated 24,800 stadia as the Golden Chersonese, but pushed considerably farther east, to Cattilength of the Mediterranean from the Straits to the Gulf of Issus, this was stated by Ptolemy at 62°, or about 20° too much. Even after correcting the error due to his computation of 500 stadia to a degree, there remains an excess of nearly 500 geographical miles. Another error which disfigured the eastern portion of Ptolemy's Mediterranean map was the position of See also:Byzantium, which Ptolemy (misled by Hipparchus) placed in the same latitude with Massilia, thus carrying it up more than 2° above its true position. This pushed the whole Euxine—with whose general form and dimensions he was fairly well acquainted—too far north by the same amount; besides this he enormously exaggerated the extent of the Palus Maeotis (the Sea of See also:Azov), which he also represented as having its direction from south to north; by the combined effect of these two errors he carried up its northern extremity (with the Tanais See also:estuary and city) as high as 54° 30' (the true south See also:shore of the Baltic). Ptolemy, however, was the first writer of antiquity who showed some conception of the relations between the Tanais or See also:Don (usually considered by the ancients as the boundary between Europe and Asia) and the Rha or See also:Volga, which he correctly described as flowing into the See also:Caspian. He was also the first geographer after See also: Ptolemy also applies the term Imaus to a section of the backbone range which in his system crosses Asia from west to east. This section lies east of the Indian See also:Caucasus, and forms an angle with the other Imaus See also:running north. 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 See also:Empire, had become more important and intimate than ever before. The tract called the Periplus of the Erythraean Sea, about A.D. 8o, contains sailing directions for merchants from the Red Sea to the See also:Indus and See also:Malabar, and even indicates that the coast from Barygaza (Baroch) had a general southward direction down to and far beyond Cape Komari (See also:Comorin), which, taken together with its account of the shore-line as far as the See also:Ganges, affords some suggestions at least of a See also:peninsular See also:character for south India. But Ptolemy, following Marinus, not only gives to the Indian coasts, from Indus to Ganges, an undue extension in longitude, but practically denies anything of an Indian peninsula, placing capes Komaria and Rory (his southernmost points in India) only 4° S. of Barygaza, the real interval being over 800 geographical miles, or, according to Ptolemy's system of graduation, 16° of latitude. This error, distorting the whole appearance of south Asia, is associated with another as great, but of opposite tendency, in regard to Taprobane (in which ancient ideas of Ceylon and Sumatra are confusedly mingled). The See also:size of this was exaggerated by most earlier Greek geographers; but Ptolemy extended it through 15° of latitude and 12° of longitude, so as to make it about fourteen times as large as the reality, and bring down its southern extremity more than 2° south of the equator. Similar distortions in regions beyond the Ganges, concerning which Ptolemy is our only ancient authority, are less surprising. Between the date of the Periplus and that of Marinus it seems probable that Greek mariners had not only crossed the Gangetic gulf and visited the land on the opposite side, which they called the gara. But these commercial voyagers either brought back inaccurate notions, or Ptolemy's preconceptions destroyed the value of the new information, for nowhere does he distort the truth more wildly. After passing the Gieat Gulf, beyond the Golden Chersonese, he makes the coast trend southward, and thus places Cattigara (perhaps one of the south China ports) 8a° south of the equator. In this he was perhaps influenced by his notion of a junction of Asia and Africa in a terra incognita south of the Indian Ocean. In regard to West Africa, we may notice that he conceives this coast as running almost due north and south to to° N., and then (after forming a great bay) as bending away to the unknown south-west. Though the Fortunate Islands were so important to his system as his prime meridian, he was entirely misinformed about them, and extended the group through more than 5° of latitude, so as to bring down the most southerly of them to the real parallel of the Cape Verde Islands. In regard to the mathematical construction or projection of his maps, not only was Ptolemy greatly in advance of all his predecessors, but his theoretical skill was altogether beyond the nature of the materials to which he applied it. The methods by which he obviated the difficulty of transferring the delineation of different countries from the spherical surface of the globe to the plane surface of an ordinary map differed little from those in use at the present day, and the errors arising from this cause (apart from those produced by his fundamental error of graduation) were really of little consequence compared with the defective character of his information and the want of anything approaching to a survey of the countries delineated. He himself was well aware of his deficiencies in this respect, and, while giving full directions for the scientific construction of a general map, he contents himself, for the See also:special maps of different countries, with the simple method employed by Marinus of drawing the parallels of latitude and meridians of longitude as straight lines, assuming in each case the proportion between the two, as it really stood with respect to some one parallel towards the middle of the map, and neglecting the inclinations of the meridians to one another. Such a course, as he himself repeatedly affirms, will not make any material difference within the limits of each special map. Ptolemy especially devoted himself to the mathematical branch of his subject, and the arrangement of his work, in which his results are presented in a See also:tabular form, instead of being at once embodied in a map, was undoubtedly designed to enable the student to construct his maps for himself. This purpose it has abundantly served, and there is little doubt that we owe to the peculiar form thus given to his results their transmission in a comparatively perfect See also:condition to the present day. Unfortunately the specious appearance of these results has led to the belief that what was stated in so scientific a form must necessarily be based upon scientific observations. 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 See also: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. But how far those which are now extant represent the original series is a disputed point. In two of the most ancient See also:MSS. it is expressly stated that the maps which accompany them are the work of one See also:Agathodaemon of Alexandria, who ' See also:drew them according to the eight books of Claudius Ptolemy." This expression might equally apply to the work of a contemporary draughtsman under the eyes of Ptolemy himself, or to that of a skilful geographer at a later period, and nothing is known from any other source concerning this Agathodaemon. The attempt to identify him with a grammarian of the same name who lived in the 5th century is wholly without foundation. But it appears, on the whole, most probable that the maps appended to the MSS. still extant have been transmitted by uninterrupted tradition from the time of Ptolemy. 2. Progress of Geographical Knowledge in Certain Special Regions.—Ptolemy records, after Marinus, the penetration of Roman expeditions to the land of the Ethiopians and to Agisymba, clearly some region of the See also:Sudan beyond the See also:Sahara See also:desert, perhaps the basin of Lake See also:Chad. But while this name was the only recorded result of these expeditions, Ptolemy also gives much other information concerning the interior of North Africa (whence derived we know not) to which nothing similar is found in any earlier writer. Unfortunately this new information was of so crude a character, and is presented in so embarrassing a form, as to perplex rather than assist. Thus Ptolemy's statements concerning the See also:rivers Gir and Nigir, and the lakes and mountains with which they were connected, have baffled successive generations of interpreters. It may safely be said that they present no resemblance to the real features of the country as now known, and cannot be reconciled with them except by arbitrary conjecture. As to the See also:Nile, both Greeks and See also:Romans had long endeavoured to discover the See also:sources of this See also:river, and an expedition sent out for that purpose by the emperor See also:Nero had undoubtedly penetrated as far
as the marshes of the See also: But neither seems to have taken full See also:advantage of these; and the tables of the Alexandrian geographer abound with mistakes —even in countries so well known as See also:Gaul and See also:Spain—which might easily have been obviated by a more judicious use of such Roman authorities. In spite of the merits of Ptolemy's geographical work it cannot be regarded as a complete or satisfactory treatise upon the subject. It was the work of an astronomer rather than a geographer. Not only did its See also:plan exclude all description of the countries with which it dealt, their climate, natural productions, inhabitants and peculiar features, but even its See also: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. In Gaul, for instance, the few notices he gives of the rivers that See also:play so important a part in its geography are disfigured by some astounding errors; while he does not notice any of the great tributaries of the See also:Rhine, though mentioning an obscure streamlet, otherwise unknown, because it happened to be the boundary between two Roman provinces. Additional information and CommentsThere are no comments yet for this article.
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