.
PTOLEMY (CLAUDIUS PTOLEMAEUS), celebrated as a mathematician, astronomer, and geographer. He was a native of Egypt, but there is an uncertainty as to the place of his birth ; some ancient manuscripts of his works describe him as of Pelusium, but Theodorus Meliteniota, a Greek writer on astronomy of the 12th century, says that he was born at Ptolemais Hermii, a Grecian city of the Thebaid. It is certain that he observed at Alexandria during the reigns of Hadrian and Antoninus Pius, and that he survived Antoninus. Olympiodorus, a philosopher of the Neoplatonic school who lived in the reign of the emperor Justinian, relates in his scholia on the Phsedo of Plato that Ptolemy devoted his life to astronomy and lived for forty years in the so-called Ilrcpd rod Kaviofiov, probably elevated terraces of the temple of Serapis at 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 direct evidence of Ptolemy himself that he made astronomical observations during a long series of years; his first recorded observation was made in the eleventh year of Hadrian, 127 A.D., 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 Seville asserts that he was of the royal race of the Ptolemies, and even calls him king of Alexandria; this assertion has been followed by others, but there is no ground for their opinion. Indeed Fabricius shows by numerous instances that the name Ptolemy was common in Egypt. Weidler, from whom this is taken, also tells us that according to Arabian tradition Ptolemy lived to the age of seventy-eight years ; from the same source some description of his personal appearance has been handed down, which is generally considered as not trustworthy, but which may be seen in Weidler, Historia Astronomic, p. 177, or in the preface to Halma's edition of the Almagest, p. lxi. Ptolemy's work as a geographer is treated of below (p. 91 sq.), and an account of the discoveries in astronomy of Hipparchus and Ptolemy has been given in the article ASTRONOMY. Their contributions to pure mathematics have not yet been noticed in the present work. Of these the chief is the foundation of trigonometry, plane and spherical, including the formation of a table of chords, which served the same purpose as our table of sines. This branch of mathematics was created by Hipparchus for the use of astronomers, and its exposition was given by Ptolemy in a form so perfect that for 1400 years it was not surpassed. In this respect it may be compared with the doctrine as to the motion of the heavenly bodies so well known as the Ptolemaic system, which was paramount for about the same period of time. There is, however, this difference, that, whereas the Ptolemaic system was then overthrown, the theorems of Hipparchus and Ptolemy, on the other hand, will be, as Delambre says, for ever the basis of trigonometry. The astronomical and trigonometrical systems are contained in the great work of Ptolemy _____, or, as Fabricius after Syncellus writes it, _____; and in like manner Suidas says _____. The Syntaxis of Ptolemy was called ______ to distinguish it from another collection called _____, also highly esteemed by the Alexandrian school, which contained some works of Autolycus, Euclid, Aristarchus, Theodosius of Tripolis, Hypsicles, and Menelaus. To designate the great work of Ptolemy the Arabs used the superlative _____, from which, the article al being prefixed, the hybrid name Almagest, by which it is now universally known, is derived.

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eclipse of the moon in that year Ptolemy, however, does not say, as in other similar eases, he had observed, but it had been observed

We proceed now to consider the trigonometrical work of Hipparchus and Ptolemy. In the ninth chapter of the first book of the Almagest Ptolemy shows how to form a table of chords. He supposes the circumference divided into 360 equal parts (T/xrnjura), and then bisects each of these parts. Further, he divides the 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 radius into sixty equal parts, and each of these parts he further subdivides into sixty equal parts. In the Latin translation these subdivisions become " partes minute prima;" and "partes minut.Ee secundse," whence our "minutes" aud "seconds" have arisen. It must not be supposed, however, thatthese sexagesimal divisions are due to Ptolemy ; they must have been familiar to his predecessors, and were handed down from the Chaldeans. Nor did the formation of the table of chords originate with Ptolemy ; indeed, Theon of Alexandria, the father of 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 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 he easily found by means of the Elements of Euclid. Thus the side of the hexagon, or the chord of 60°, 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 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. 10) ; the chord of 72° can therefore he 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 he 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 pjiece of geometry of great elegance, and is indeed, as De 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 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 of 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 diiference is required, and complete the quadrilateral ; in one case a diagonal, in the other one of the sides is a diameter of the circle. The relations thus obtained are equivalent to the fundamental formula? of our trigonometry— sin (A + B) = sin A 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 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°, 14°, and three-fourths of 1°, and lastly, the ch»rds of 4J°, 7J°, 9°, 10J°, &c,—all those arcs, namely, as Ptolemy says, which being doubled are divisible by 3. Performing the calculations, he finds that the chord of lh° contains approximately IP 34' 55", and the chord of three-fourths of 1° contains 0» 47' 8". A table of chords of arcs increasing by 1J° can thus be formed ; but this is not sufficient for Ptolemy's purpose, which was to 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 1J°, inasmuch as that would come to the trisection of an angle, he proceeds to seek in the first place the chord of 1°, which he finds approximately by means of a lemma of great elegance, due probably to 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 order to find approximately the chord of 1°, which he does in the following manner—

chord 60' 60 . 4 , , , 0 4 , , .,, " j^,i.e.,<—, .".chord 1 < — chord 4o ;
chord 60' 60' ' 2: For hrevity we use modern notation. It has been shown that the chord of 45' is OP 47' 8" q.p., and the chord of 90' is 1P 34' 15" q.p.; hence it follows that approximately

chord 1° <1P 2' 50" 40"' and > 1P 2' 50". Since these values agree as far as the seconds, Ptolemy takes 1P 2' 50"
chord 45' 45' chord 90' 90
,\ chord 1° > -J chord !

as the approximate value of the chord of 1°. The chord of 1° being thus known, he finds the choid 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 0° to 180° ; in the second he gives the values of the chords of these arcs in parts of which the diameter contains 120, the subdivisions being sexagesimal ; and in the third he has inserted the thirtieth parts of the 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 bo calculated, it being assumed that the increment of the chords of arcs within the table for each interval of 30' 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 prior to that of plane trigonometry. It is the subject-matter of the eleventh chapter of the Almagest, whilst the 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 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 drawing the perpendicular from a vertex on the opposite side. In one place (Aim., vi. c. 7 ; vol. i. p. 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 Almagest contains incidentally some theorems and problems in plane trigonometry. The problems which are met with correspond to the following : 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 external section. Lastly, it may be mentioned that Ptolemy (Aim., vi. 7 ; vol. i.

p. 421, ed. Halma) takes 3P 8' 30", i.e., S + ^ + J^L = 3-1416, as
60 ooOO

the value of the ratio of the circumference to the diameter of a circle, and adds that, as had been shown by Archimedes, it lies between 3} and 3ff.

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 sides of a triangle made by a transversal : The segments of any side are in a ratio compounded of the ratios of the segments of the other two sides. This theorem, as well as that concerning the inscribed quadrilateral, was called after Ptolemy—naturally, indeed, since no reference to its source occurs in the Almagest. This error was corrected by Mersenne, who showed that it was known to-Menelaus, an astronomer and geometer who lived in the reign of the emperor Trajan. The theorem now bears the name of Menelaus,. though most probably it came down from Hipparchus ; Chasles, indeed, thinks that Hipparchus deduced the property of the spherical triangle from that of the plane triangle, but throws the origin of the latter further hack and attributes it to Euclid, suggesting that it was given in his Purisms.' Carnot made this theorem the basis of his theory of transversals in his essay on that subject. It should be noticed that the theorem is not given in the Almagest in the general manner stated above ; 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 lays down two lemmas : If the chord of an a.rc of a circle be cut in any ratio and a diameter be 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, hut 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 of any 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 piano. Theon has added two cases. The proofs are general. By means of this theorem four of 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 leader 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.

We now proceed to notice briefly the contents of the Almagest. It is divided into thirteen books. The first book, which may be regarded as introductory to the whole work, opens with a short preface, in which Ptolemy, after some observations on the distinction between theory and practice, gives Aristotle's division of the sciences and remarks on the certainty of mathematical knowledge, "inasmuch as the demonstrations in it proceed by the incontrovertible ways of arithmetic and geometry." He concludes his preface with the statement that he will make use of the discoveries of his predecessors, and relate briefly all that has been sufficiently explained by the ancients, but that he will treat with more care and development whatever has not been well understood or fully treated. Ptolemy unfortunately does not always bear this in mind, and it is sometimes difficult to distinguish what is due to him from that which he has borrowed from his predecessors.

Ptolemy then, in the first chapter, presupposing some preliminary notions on the part of the reader, announces that he will treat in order—what is the relation of the earth to the heavens, wdiat is the position of the oblique circle (the ecliptic), and the situation of the inhabited parts of the earth ; that he will point out the differences of climates ; that he will then pass on to the consideration of the motion of the sun and moon, without which one cannot have a just theory of the stars ; lastly, that he will consider the sphere of the fixed stars and then the theory of the five stars called " planets." All these things—i.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 summary exposition of the general principles on which his Syntaxis is based, and adduces arguments to show that the heaven is of a spherical form and that it moves after the manner of a sphere, that the earth also is of a form which is sensibly spherical, that the earth is in the centre of the heavens, that it is but a point in comparison with the distances of the stars, and that it has not any motion of translation. With respect to the revolution of the earth round its 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 principal and different motions in the heavens—one by which all the stars are carried from east to west uniformly about the poles of the equator ; the other, which is 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 science. Ptolemy in the next chapter indicates two means of determining this angle by observation, describes the 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 ascension, decimation, and longitude, and for the formation of a table of declinations to each degree of longitude. Delambre says he found both this and the table of chords very exact." 3

In book ii., after some remarks on the situation of the habitable parts of the earth, Ptolemy proceeds to make deductions from the 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 calculate 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 polo &i Rhodes is 36°, a whole number, whereas at Alexandria he believed it to be 30° 58'; the second is that Hipparchus had made at Ii bodes many observations ; the third is that the climate of Rhodes holds the mean place of the seven climates subsequently described. Delambre suspects a fourth reason, which he thinks is the true one, that Ptolemy had taken his examples from the works of Hipparchus, who observed at Rhodes and had made these calculations for the place where 1*3 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 earth. For each parallel or climate, which is determined by the length of the longest day, he gives the latitude, a principal place on the parallel, and the lengths of the shadows of the gnomon at the solstices and equinox. In the next chapter ho enters into particulars 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 arc of the ecliptic takes to cross the horizon of a given place. He arrives at a 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 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 temporal4 to equinoctial hours and vice versa, and of the nonagcsimal 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 parallel of Meroe (thirteen hours) to that of the mouth of the Borysthenes (sixteen hours). These tables, he adds, should he completed by the situation of the chief towns in all countries according to their latitudes and longitudes ; this he promises to do in a separate treatise and has in fact done in his 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 lover of labour and a lover of truth " (dvdpl <pikoirbvu> re 6/J.OV KCU (piXakw&e?). He begins by telling us how Hipparchus was led to discover the precession of the equinoxes ; he relates the observations wdiich led Hipparchus to his second great discovery, that of the eccentricity of the solar orbit, and gives the hypothesis of the 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 equation of time depends. All this the reader will find in the article ASTRONOMY (vol. ii. p. 750). Ptolemy, moreover, applies Apollonius's hypothesis of the 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 fit to clear up the difficulties. In the second chapter there are some general remarks to which 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.5 This 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 law of the "philosophia prima" of Comte.1 Wefindin the same page another principle, or rather practical injunction, that in investigations founded on observations where great delicacy is required we should select those made at considerable intervals of time in order that the errors arising from the imperfection wdiich 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 wdience 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. Book iv., in which Ptolemy follows Hipparchus, treats of the first and principal inequality of the moon, which quite corresponds to the inequality of the sun treated of in the third hook. 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 Chaldreans 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 7J° q.p. ; this quantity (7^°) is also wanting to complete the 345 revolutions wdiich the sun makes in the same time with respect to the fixed stars. He concluded from this that the lunar month contains 29 days and 31' 50" 8"' 20"" of a day, very nearly, or 29 days 12 hours 44' 3" 20"'. These results are of the highest importance. (See ASTRONOMY. ) In order to explain this inequality, or the equation of the centre, Ptolemy makes use of the hypothesis of an epicycle, which he prefers to that of the eccentric. The fifth book commences with the description of the astrolabe of Hipparchus, which Ptolemy made use of in following up the observations of that astronomer, and by means of wdiich 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 wdiich 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 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 Flamsteed and Lalande, maintain that it was the same catalogue wdiich 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, hut that in making the changes wdiich 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 wdiich city the altitude of the pole is 5° less than at Khodes, 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 writers8 have placed them beyond the sun, because they were never seen on the sun.9 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 wdiich, as seen from the earth, they appear to complete the circuit of the zodiac are nearly the same— a year.10 Delambre thinks it strange that Ptolemy did not see that these contrary opinions could be reconciled by supposing that the two planets moved in epicycles about the sun ; this would be stranger still, he adds, if it is true that this idea, wdiich is older than Ptolemy, since it is referred to by Cicero, had been that of the Egyptians.12 It may be added, as strangest of all, that this doctrine was held by Theon of Smyrna,13 who was a contemporary of Ptolemy or somewhat senior to him. From this system to that of Tycho Brahe there is, as Delambre observes, only a single step.

We have seen that the problem which presented itself to the astronomers of the Alexandrian epoch was the following : it was required to find such a system of equable circular motions as would represent the inequalities in the apparent motions of the sun, the moon, and the planets. Ptolemy now takes up this question for the planets ; he says that ' ' this perfection is of the essence of celestial things, which admit of neither disorder nor inequality," that this planetary theory is one of extreme difficulty, and that no one had yet completely succeeded in it. He adds that it was owdng 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 ho has left—had succeeded, as far as possible, in representing the motions of the sun and moon by circles, hut 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 showed that in fact each planet had two inequalities, which are different for each, that the rétrogradations are also different, whilst other astronomers admitted only a single inequality and the same rétrogradation ; he showed further that their motions cannot be explained by eccentrics nor by epicycles carried along concentrics, but that it was necessary to combine both hypotheses. After these preliminary notions he gives from Hipparchus the periodic motions of the five planets, together with the shortest times of restitutions, in which, moreover, he has made some slight corrections. He then gives tables of the mean motions in longitude and of anomaly of each of the five planets, and shows how the motions in longitude of the planets can be represented in a general manner by means of the hypothesis of the eccentric combined with that of the epicycle. He next applies his theory to each planet and concludes the ninth book by the explanation of the various phenomena of the planet Mercury. In the tenth and eleventh books he treats, in like manner, of the various phenomena of the planets Venus, Mars, Jupiter, and Saturn.

Book xii. treats of the stationary and retrograde appearances of each of the planets and of the greatest elongations of Mercury and Venus. The author tells us that some mathematicians, and amongst them Apollonius of Perga, employed the hypothesis of the epicycle 10 explain the stations and rétrogradations of the planets. Ptolemy goes into this theory, but does not change in the least the theorems of Apollonius ; lie only promises simpler and clearer demonstrations of them. Delambrc remarks that those of Apollonius must have been very obscure, since, in order to make the demonstrations in the Almagest intelligible, he'(Delambre) was obliged to recast them. This statement of Ptolemy is important, as it shows that the mathematical theory of the planetary motions was in a tolerably forward state long before his time. Finally, book xiii. treats of the motions of the planets in latitude, also of the inclinations of their orbits and of the magnitude of these inclinations.

Those who wish to go into details and learn the mathematical explanation of this celebrated system of "eccentrics" and "epicycles" are referred to the Almagest itself, which can he most conveniently studied in Halma's edition,*to Delambre'a Histoire de V Astronomie Ancienne, the second volume of which is for the most part devoted to the Almagest,* or to Narrien's History of Astronomy,* in which the subject is treated with great clearness.

Ptolemy concludes his great work by saying that he has included in it everything of practical utility which in his judgment should find a place in a treatise on astronomy at the time it was written, with relation as well to discoveries as to methods. His work was justly called by him MadgfxaTtKT] ^vvraCis, for it was in fact the mathematical form of the work which caused it to be preferred to all others which treated of the same science, but not by "the sure methods of geometry and calculation." Accordingly, it soon spread from Alexandria to all places where astronomy was cultivated ; numerous copies were made of it, and it became the object of serious study on the part of both teachers and pupils. Amongst its numerous commentators may be mentioned Pappus and Theon of Alexandria in tlie 4th century and Procliis in the 5th. It was translated into Latin by Boetius, but this translation has not come down to us. The Syntaxes was translated into Arabic at Baghdad by order of the enlightened caliph Al-Mamun, who was himself an astronomer, about 827 A.D., and the Arabic translation was revised in the following century by Thâbit ibn Korra. The emperor Frederick II. caused the Almagest to be translated from the Arabic into Latin at Naples about 1230. In the 15th century it was translated from a Greek manuscript in the Vatican by George of Trebizond. In the same century an epitome of the Almagest was commenced by Purbach (died 1461) and completed by his pupil and successor in the professorship of astronomy in the university of Vienna, Regiomontauus. The earliest edition of this epitome is that of Venice, 149(5, and this was the first appearance of the Almagest in print. The first complete edition of the Almagest is that of P. 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 text, which was not known in Europe until the 15th century, was first published in the 16th by Simon Grynaeus, who was also the first editor of the Greek text of Euclid, at Basel, 153S. This edition was from a manuscript in the library of Nuremberg—where it is no longer to be found—which had been presented by Regiomontanus, to whom it was given by Cardinal Bessarion. The last edition of the Almagest is that of Halma, Greek with French translation, in two vols., Paris, 1813-16. On tlie manuscripts of the Almagest and its bibliographical history, see Fabricius, BiMiotheca Orœca, ed. Harles, vol. v. p. 2S0, and Halma's preface. An excellent summary of tlie bibliographical history is given by De Morgan in his article on Ptolemy already quoted.

Other works of Ptolemy, which we now proceed to notice very briefly, are as follows. (1) <$âcreL$ àirXauQu àcrépwv teal <rvvayayyr) êwio-Tj^ao'iCjv, On the Apparitions of the Fixed Stars and a Collection of Prognostics. It is a calendar of a kind common amongst the Greeks under the name of irapdirrufxa, or a collection of the risings and settings of the stars in the morning or evening 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, Democritus. Eudoxus, Hipparchus, &c. Ptolemy, in order to make his Parapegma useful to all the Greeks scattered over the enlightened world of his time, gives the apparitions of the stars not for one parallel only but for each of the five parallels in which the length of the longest day varies from 13£ hours to 15| hours,—

that is, from the latitude of Byene to that of the middle of the Euxine. This work has been printed by Petavius in his Uranologium, Paris, 1630, and by Halma in his edition of the works of Ptolemy, vol. iii., Paris, 1819. (2) 'Tirodeo'ets r&v irXavixifjAv^v ?) TCJV ovpavioiv K(>K\WV KivrjaeLs, On the Planetary Hypothesis. This is a summary of a portion of the Almagest, and contains a brief statement of the principal hypotheses for the explanation of the motions of the heavenly bodies. It was first published (Gr., Lat.)by Bainbridge, the Savilian professor of astronomy at Oxford, with the Sphere of Proclus and the \vavojv fiafftketiuv, London, 1620, and afterwards by Halma, vol. iv., Paris, 1820. (3) Kavwv (HacriXeiQv, A Table of Reigns. This is a chronological table of Assyrian, Persian, Greek, and Roman sovereigns, with the length of their reigns, from Nabonasar to Antoninus Pius. This table (comp. G. Syncellus, Chronogr., ed. Dind., i. 3SS sfl.)has been printed by Scaliger, Calvisius, Petavius, Bainbridge (as above noted), and by Halma, vol. iii., Paris, 1S19. (4) ' ApfxovL-Ku)v (SifiXia y. This Treatise on Music was published in Greek and Latin by Wallis at Oxford, 16S2. It was afterwards reprinted with Porphyry's commentary in the third volume of Wallis's works, Oxford, 1699. (5) TerpapLfiXos vvra^is, Tetrahiblon or Quadripartitum. This work is astrological, as is also the small collection of aphorisms, called Kap7ros or Centiloquium, by which it is followed. It is doubtful whether these works are genuine, but the doubt merely arises from the feeling that they are unworthy of Ptolemy. They were both published in Greek and Latin by Camerarius, Nuremberg, 1535, and by Melanchthon, Basel, 1553. (6) De Analcmmate. The original of this work of Ptolemy is lost. It was translated from the Arabic and published by Cornmainline, Rome, 1562. The Analemww 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 tlie intelligence of gnomonics. This description is made by perpendiculars let fall on the plane ; whence it has been called by the moderns " orthographic projection." (7) Planispliterium, The Planisphere. Tlie 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 tlie equator, the eye being at the pole,—in fact what is now called " stereographic " projection. The best edition of this work is that of Cominandine, Venice, 155S. (S) Optics. This work is known to us only by imperfect manuscripts in Paris and Oxford, which are Latin translations from the Arabic some extracts from them have been recently published. The Optics consists of five books, of which the fifth presents most interest: it treats of the refraction of luminous rays in their passage through media of different densities, and also of astronomical refractions, on which subject the theory is more complete than that of any astronomer before the time of Cassini. De Morgan doubts whether this work is genuine on account of the absence of allusion to the Almagest or to the subject of refraction in the Almagest itself; but his chief reason for doubting its authenticity is that the author of the Optics was a x>oor geometer. (G. J. A.)





Geography.

Ptolemy is liardly less celebrated as a geographer than as an astronomer, and his great "work on geography exercised as great an influence on the progress of that science as did his Almagest on that of astronomy. It became indeed the paramount authority on all geographical questions for a period of many centuries, and was only gradually superseded by the progress of maritime discovery in the 15th and 16th centuries. This exceptional position was due in a great measure to its scientific form, which rendered it very convenient and easy of reference \ but, 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 trustworthy map of the Inhabited World would be by observations of the latitude and longitude of all the principal points on its surface, and laying down a map in accordance with the positions thus determined. But the materials for such a course of proceeding were almost wholly wanting, and, though Hipparchus made some approach to a correct division of the known world into zones of latitude, or "climata,"as he termed them, trustworthy observations even of this character were in his time very few in number, while the means of determining longitudes could hardly be said to exist. Hence probably it arose that no attempt was made by succeeding geographers to follow up the important suggestion of Hipparchus. Marinus of Tyre, who lived shortly before the time of Ptolemy, and whose wTork is known to us only through that writer, appears to have been the first to resume the problem thus proposed, and lay down the map of the known world in accordance with the precepts of Hipparchus. His materials for the execution of such a design "were indeed miserably inadecpuate,. and he was forced to content himself for the most part with determinations derived not from astronomical observations but from the calculation of distances from itineraries and other rough methods, such as still continue to be employed even by modern geographers where more accurate 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. See Plate I. Mathematical Geography.—As a great astronomer, Ptolemy VII., vol. was of course infinitely better qualified to comprehend and explain KV. 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. He 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 known by _experience and observation.

Unfortunately at the very outset of his attempt to realize this conception he fell into an error which had the effect of vitiating all his subsequent conclusions. Eratosthenes was the first writer who had attempted in a scientific manner to determine the circumference of the earth, and the result at which he arrived, that it amounted to 250,000 stadia or 25,000 geographical miles, was generally adopted by subsequent geographers, including Strabo. Posidonius, however, who wrote about a century after Eratosthenes, had made an independent calculation, by which he reduced the circumference of the globe to 180,000 stadia, or less than three-fourths of the result obtained by Eratosthenes, and this computation, on what grounds we know not, was unfortunately adopted by Marinus Tyrius, and from him by Ptolemy. The consequence of this error was of course to make every degree of latitude or longitude (measured at the equator) equal to only 500 stadia (50 geographical miles), instead of its true equivalent of 600 stadia. Its _effects would indeed have been in some measure neutralized had ithere existed a sufficient 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 2J° 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 iact about 7° less 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 geagraphical 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 important line on the surface of the globe for all practical purposes was the parallel of 36° of latitude, which passes through the Straits of Gibraltar 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 DicEearchus and almost all hi: successors as dividing the regions around the inland sea into twi 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. But it sufficiently 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 Lilybajum 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.

The great problem that had attracted the attention and exercised the ingenuity of all geographers from the time of Dicsearchus to that of Ptolemy was to determine the length and breadth of the Inhabited World, which they justly regarded as the chief subject of the geographer's consideration. This question had been very fully discussed by Marinus, who had arrived at conclusions widely 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 laud, and this was placed both by Marinus and Ptolemy in 63° lat, not very far beyond the true position of the Shetland Islands, which had come in their time to be generally identified with the mysterious Thule of Pytheas. The western extremity, as already mentioned, had been in like manner determined by the prime meridian drawn through the supposed position of the Fortunate Islands. But towards the south and east Marinus gave an enormous extension to the continents of Africa and Asia, beyond what had been known to or suspected by the earlier geographers, and, though Ptolemy greatly reduced his calculations, he still retained a very exaggerated estimate of their results.

The additions thus made to the estimated dimensions of the known world were indeed in both directions based upon a real extension of knowledge, derived from recent information ; but unfortunately the original statements were so perverted by misinterpretation in applying them to the construction of a map as to give results differing widely from the truth. The southern limit of the world as known to Eratosthenes, and even to Strabo (who had in this respect no further knowledge than his predecessor more than two centuries before), had been fixed by them at the parallel which passed through the eastern extremity of Africa (Cape Guardafni), or the Land of Cinnamon as they termed it, and that of the Sembrita; (corresponding to Senuaar) in the interior of the same continent. This parallel, which would correspond nearly to that of 10° of true latitude, they supposed to be situated at a distance of 3400 stadia (310 geographical miles) from that of Meroe (the position of which was accurately known), and 13,400 to the south of Alexandria ; while they conceived it as passing, when prolonged to the eastward, through the island of Taprobane (Ceylon), which was universally recognized, as the southernmost land of Asia. Both these geographers were wholly ignorant of the vast extension of Africa to the south of this line and even of the equator, and conceived it as trending away to the west from the Land of Cinnamon and then to the north-west to the Straits of Gibraltar. Marinus had, however, learned from itineraries both by land and 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 eastern coast of Africa was indeed tolerably well known, being frequented by Greek and Roman traders, as far as a place called Rhapta, opposite to Zanzibar, and this is placed by Ptolemy not far from its true position in 7° S. lat. But he added to this a bay of great extent as far as a promontory called Prasum (perhaps Cape Delgado), which he placed in 15A° 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 Boman general, whose itinerary was employed by Marinus. Taking, therefore, this parallel as the limit of knowledge to the south, while he retained that of Thule to the north, he assigned to the inhabited world a breadth of nearly 80°, instead of less than 60°, which it had occupied on the maps of Eratosthenes and Strabo,

It had been a fixed belief with all the Greek geographers from the earliest attempts at scientific geography not only that the length of the Inhabited World greatly exceeded its breadth, but that it was more than twice as great,—a wholly unfounded assumption, but to which their successors seem to have felt themselves bound to conform. Thus Marinus, while giving an undue extension to Africa towards the south, fell into a similar error, but to a far greater degree, in regard to the extension of Asia towards the east. Here also he really possessed a great advance in knowledge over all his predecessors, the increased trade with China for silk having led to an acquaintance, though of course of a very vague and general kind, with the vast regions in Central Asia that lay to the east of the Pamir range, which had formed the limit of the Asiatic nations previously known to the Greeks. But Marinus had learned that traders proceeding eastward from the Stone Tower—a station at the foot of this range—to Sera, the capital city of the Seres, occupied seven months on the journey, and from thence he arrived at the enormous result that the distance between the two points was not less than 36,200 stadia, or 3620 geographical miles. Ptolemy, while he justly points out the absurdity of this conclusion and the erroneous mode of computation on which it was founded, had no means of correcting it by any real authority, and hence reduced it summarily by one half. The effect of this was to place Sera, the easternmost point on his map of Asia, at a distance of 45 J° from the Stone Tower, which again he fixed, on the authority of itineraries cited by Marinus, at 24,000 stadia or 60° of longitude from the Euphrates, reckoning in both cases a degree of longitude as equivalent to 400 stadia, in accordance with his uniform system of allowing 500 stadia to 1° of latitude. Both distances were greatly in excess of the truth, independently of the error arising from this mistaken system of graduation. The distances west of the Euphrates were of course comparatively well known, nor did Ptolemy's calculation of the length of the Mediterranean differ very materially from those of previous Greek geographers, though still greatly exceeding the truth, after allowing for the permanent error of graduation. The effect of this last cause, it must be remembered, would unfortunately be cumulative, in consequence of the longitudes being computed from a fixed point in the west, instead of being reckoned east and west from Alexandria, which was undoubtedly the mode in which they were really calculated. The result of these combined causes of error was to lead him to assign no less than 180°, or 12 hours, of longitude to the interval between the meridian of the Fortunate Islands and that of Sera, which really amounts to about 130°.

But in thus estimating the length and breadth of the known world Ptolemy attached a very different sense to these terms from that which they had generally borne with preceding writers. All former Greek geographers, with the single exception of Hipparclms, had agreed in supposing the Inhabited World to be surrounded on all sides by sea, and to form in fact a vast island in the midst of a circumfluous ocean. This notion, which was probably derived originally from the Homeric fiction of an ocean stream, and was certainly not based upon direct observation, was nevertheless of course in accordance with the truth, great as was the misconception it involved of the extent and magnitude of the continents included within this assumed boundary. Hence it was unfortunate that Ptolemy should in this respect have gone back to the views of Hipparchus, and have assumed that the land extended indefinitely to the north in the case of Europe and Scythia, to the 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 Laud, as he calls it. But in the last case be 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. But unfortunately he had formed a totally false conception of the bearings of the coasts thus made known, and consequently of the position of the lands to which they belonged, and, instead of carrying the line of coast northwards from the Golden Chersonese (the Malay Peninsula) to China or the land of the Sinaj, 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 hypothesis that this land was continuous with the most southern part of Africa, so that the two enclosed one vast gulf, though a mere assumption, is stated by him as definitely as if it was based upon positive information ; and it was long received by mediaeval geographers as an unquestioned fact. This circumstance undoubtedly contributed to perpetuate the error of supposing that Africa could not be circumnavigated, in opposition to the more correct views of Strabo and other earlier geographers. On the other hand, there can be no doubt that the undue extension of Asia towards the east, so as to diminish by 50° of longitude the interval between that continent and the western coasts of Europe, had a material influence in fostering the belief of Columbus and others that it was possible to reach the Land of Spices (as the East Indian islands were then called) by direct navigation towards the west.

It is not surprising that Ptolemy should have fallen into considerable errors respecting the more distant quarters of the world ; but even in regard to the Mediterranean and its dependencies, as. well as the regions that surrounded them, with which he was in a certain sense well acquainted, the imperfection of his geographical knowledge is strikingly apparent. Here he had indeed some well-established data for his guidance, as far as latitudes were concerned. That of Massilia had been determined many years before by Pytheas within a few miles of its true position, and the latitude of Rome, as might be expected, was known with approximate accuracy. Those of Alexandria and Rhodes also were well known, having been the place of observation of distinguished astronomers, and the fortunate accident that the Island of Rhodes lay on the same parallel of latitude with the Straits of Gibraltar at the other end of the sea enabled him to connect the two by drawing the parallel direct from the one to the other. The importance attached to this line (36° N. lat.) by all preceding geographers has been already mentioned. Unfortunately Ptolemy, like his predecessors, supposed its course to lie almost uniformly through the open sea, wholly ignoring the great projection of the African coast towards. the north from Carthage to the neighbourhood of the straits. The erroneous position assigned to the former city has been already adverted to, and, being supposed to rest upon astronomical observation, doubtless determined that of all the north coast of Africa. The result was that he assigned to the width of the Mediterranean from Massilia to the opposite point of the African coast an extent of more than 11° of latitude, while it does not really exceed 63°.

At the same time he was still more at a loss in respect of longitudes, for which he had absolutely no trustworthy observations to guide him ; but he nevertheless managed to arrive at a result considerably nearer the truth than had been attained by 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 ; but unfortunately Ptolemy, in translating the conclusions thus arrived at into a scientific form, vitiated all his results by his erroneous system of graduation, and, while the calculation of Marinus gave a distance of 24,800 stadia as the length of the Mediterranean from the straits to the Gulf of Issus, this was converted by Ptolemy in preparing his tables to an interval of 62°, or just about 20° beyond the truth. Even after correcting the error due to his erroneous computation of 500 stadia to a degree, there remains an excess of nearly 500 geographical miles, which was doubtless owing to the exaggerated estimates of distances almost always made by navigators who had no real means of measuring them.

Another unfortunate error which disfigured the eastern portion of his map of the Mediterranean was the position assigned to Byzantium, which Ptolemy (misled in this instance by the authority of Hipparehus) placed in the same latitude with Massilia (43° 5'), thus carrying it up more than 2° above its true position. This had the inevitable effect of transferring the whole of the Euxine Sea— with the general form and dimensions of which he was fairly well acquainted—too far to the north by the same amount; but in addition to this he enormously exaggerated the extent of the Palus Mreotis (the Sea of Azotf), which he at the same time represented as having its direction from south to north, so that by the combined effect of these two errors he carried up its northern extremity (with the mouth of the Tanais and the city of that name) as high as 54° 30', or on the true parallel of the south shore of the Baltic. Yet, while he fell into this strange misconception with regard to the great river which was universally considered by the ancients as the boundary between Europe and Asia, he was the first writer of antiquity who showed a clear conception of the true relations between the Tanais and the Kha or Volga, which he correctly described as flowing into the Caspian Sea. With respect to this last also he was the first geographer after the time of Alexander to return to the correct view (already found in Herodotus) that it was an inland sea, without any communication with the Northern Ocean.

With regard to the north of Europe his views were still very vague and imperfect. He had indeed considerably more acquaintance with the British Islands than any previous geographer, and even showed a tolerably accurate knowledge of some portions of their shores. But his map was, in this instance, disfigured by two unfortunate errors,—the one, that he placed Ireland (which he calls Ivernia) altogether too far to the north, so that its southernmost portion was brought actually to a latitude beyond that of North Wales ; the other, which was probably connected with it, that the whole of Scotland is hoisted round, so as to bring its general extension into a direction from west to east, instead of from south to north, and place the northern extremity of the island on the same parallel with the promontory of Galloway. He appears to have been embarrassed in this part of his map by his having adopted the conclusion of Marinus—based upon what arguments we know not—that Thnle was situated in 63°, while at the same time he regarded it, in conformity with the received view of all earlier geographers, as the most northern of all known lands. In accordance with this same assumption Ptolemy supposed the northern coast of Germany, which he believed to be the southern shore of tin; Great Ocean, to have a general direction from west to east, while he placed it not very far from the true position of that of the Baltic, of the existence of which as an inland sea he was wholly ignorant, as well as of the vast peninsula of Scandinavia beyond it, and only inserted the name of Scandia as that of an island of inconsiderable dimensions. At the same time ho supposed the coast of Sarmatia from the Vistula eastwards to trend away to the north as far as the parallel of Thule ; nor did he conceive this as an actual limit, but believed the Unknown Land to extend indefinitely in this direction, as also to the north of Asiatic Scythia.

The enormous extent assigned by him to the latter region has been already adverted to ; but vague and erroneous as were his views concerning it, it is certain that they show a much greater approximation to the truth than those of earlier geographers, who possessed hardly a suspicion of the vast tracts in question, which stretch across Central Asia from the borders of Sarmatia to those of China. Ptolemy was also the first who had anything like a clear idea of the chain to which he gave the name of Imaus, and correctly regarded as having a direction across Scythia from south to north, so as to divide that great region into two distinct portions which he termed Scythia intra Imauni and Scythia extra Imaum, corresponding in some degree with those recognized in modern maps as Independent and Chinese Tartary. The Imaus of Ptolemy corresponds clearly to the range known in modern days as the Bolor or Pamir, which has only been fully explored in quite recent times. It was, however, enormously misplaced, being transferred to 140° E. long., or 80° east of Alexandria, the real interval between the two being little more than 40°.

It is in respect of the southern shores of Asia that Ptolemy's geography is especially faulty, and his errors are here the more unfortunate as they were associated with greatly increased knowledge in a general way of the regions in question. For more than a century before his time, indeed, the commercial relations between Alexandria, as the great emporium of the Roman empire, and India had assumed a far more important character than at any former period, and the natural consequence was a greatly increased geographical knowledge of the Indian peninsula. The little tract called the Periplus of the Erythreean Sea, about 80 A.D., contains sailing directions for merchants to the western ports of that country, from the mouth of the Indus to the coast of Malabar, and correctly indicates that the coast from Barygaza southwards had a general direction from north to south as far as the extremity of the peninsula (Cape Comorin). We are utterly ignorant of the reasons which induced Marinus, followed in this instance as in so many others by Ptolemy, to depart from this correct view, and, while giving to the coast of India, from the mouths of the Indus to those of the Ganges, an undue extension in longitude, to curtail its extension towards the south to such an amount as to place Cape Cory (the southernmost point of the peninsula) only 4° of latitude south of Barygaza, the real intervals being more than 800 geographical miles, or, according to Ptolemy's system of graduation, 16° of latitude ! This enormous error, which has the effect of distorting the whole appearance of the south coast of Asia, is associated with another equally extraordinary, but of an opposite tendency, in regard to the neighbouring island of Taprobane or Ceylon, the dimensions of which had been exaggerated by most of the earlier Greek geographers ; but to such an extent was this carried by Ptolemy as to extend it through not less than 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° to the south of the equator.

We have much less reason to be surprised at finding similar distortions in respect to the regions beyond the Ganges, concerning which he is our only ancient authority. During the interval which elapsed between the date of the Periplus and that of Marinus it is certain that some adventurous Greek mariners had not only crossed the great Gangetic Gulf and visited the land on the opposite side, to which they gave the name of the Golden Chersonese, but they had pushed their explorations considerably farther to the east, as far as Cattigara. It was not to be expected that these commercial ventures should have brought back any accurate geographical information, and accordingly we find the conception entertained by Ptolemy of these newly discovered regions to be very different from the reality. Not only had the distances, as was usually the case with ancient navigators in remote quarters, been greatly exaggerated, but the want of accurate observations of bearings was peculiarly unfortunate in a case where the real features of the coast and the adjoining islands were so intricate and exceptional. A glance at the map appended to the article MAP (vol. xv. Plate VII.) will at once show the entire discrepancy between the configuration of this part of Asia as conceived by Ptolemy and its true formation. Yet with the materials at his command we can hardly wonder at his not having arrived at a nearer approximation to the truth. The most unfortunate error was his idea that after passing the Great Gulf, which lay beyond the Golden Chersonese, the coast trended away to the south, instead of towards the north, and he thus placed Cattigara (which was probably one of the ports in the south of China) not less than 8^° south of the equator. It is probable that in this instance he was misled by his own theoretical conclusions, and carried this remotest part of the Asiatic continent so far to the south with the view of connecting it with his assumed eastward prolongation of that of Africa.

Notwithstanding this last theoretical assumption Ptolemy's map of Africa presents a marked improvement upon those of Eratosthenes and Strabo. But his knowledge of the west coast, which he conceived as having its direction nearly on a meridional line from north to south, was very imperfect, and his latitudes utterly erroneous. Even in regard to the Fortunate Islands, the position of which was so important to his system in connexion with his prime meridian, he was entirely misinformed as to their character and arrangement, and extended the group through a space of more than 5° of latitude, so as to bring down the most southerly of them to the real parallel of the Cape de Verd Islands.

In regard to the mathematical construction, or, to use the modern phrase, the projection of his maps, not only was Ptolemy greatly in advance of all his predecessors, but bis 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 indeed but 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 detective 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 special maps of different countries with the simple method employed by Marinus of drawing ^he parallels of latitude and meridians of longitude as straight lines, , ssuming in each ease the proportion between the two, as it really tood with respect to some one parallel towards the middle of the map, and neglecting the inclination 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's geographical work was devoted almost exclusively to the mathematical branch of his subject, and its peculiar arrangement, in which his results are presented in a tabular form, instead of being at once embodied in a map, was undoubtedly designed to enable the geographical student to construct his maps for himself, instead of depending upon those constructed ready to his hand. 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 condition to the present day. Unfortunately the specious appearance of the results thus presented to us has led to a very erroneous estimate of their accuracy, and it has been too often supposed 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 the same undoubting faith as was justly attached to his astronomical observations. It is only in quite recent times that his conclusions have been estimated at their just value, and the apparently scientific character of his work shown to be in most cases a specious edifice resting upon no adequate foundations.
There can be no doubt that the work of Ptolemy was from the time of its first publication accompanied with maps, which are regularly referred to in the eighth book. But how far those which are now extant represent the original series is a disputed point. In two of the most ancient MSS. it is expressly stated that the maps which accompany them are the work of one Agathodaamon of Alexandria, who " drew them according to the eight books of Chaudius 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 Agathodasmon. 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. —The above examination of the methods pursued by Ptolemy in framing his general map of the world, or according to the phrase universally employed by the ancients, the Inhabited World (_____), has already drawn attention to the principal extensions of geographical knowledge since the time of Strabo.
While anything like an accurate acquaintance was still confined to the limits of the Roman empire and the regions that immediately adjoined it, with the addition of the portions of Asia that had been long known to the Greeks, the geographical horizon had been greatly widened towards the east by commercial enterprise, and towards the south by the same cause, combined with expeditions of a military character, but which would appear to have been dictated by a spirit of discovery. Two expeditions of this kind had been carried out by Roman generals before the time of Marinus, which, starting from Fezzan, had penetrated the heart of the African continent due south as far as a tract called Agisymba, "which was inhabited by Ethiopians and swarmed with rhinoceroses." These statements point clearly to the expeditions having traversed the great desert and arrived at the Soudan or Negroland. But the actual position of Agisymba cannot be determined except by mere conjecture. The absurdly exaggerated view taken by Marinus has been already noticed ; but, even after his estimate had been reduced by Ptolemy by more than one-half, the position assigned by that author to Agisymba was doubtless far in excess towards the south. But, while this name was the only result that we know to have been derived from these memorable expeditions, Ptolemy found himself in possession of a considerable amount of information concerning the interior of northern Africa (from whence derived we know not), to which nothing similar is found in any earlier writer. Unfortunately this new information was of so crude and vague a character, and is presented to us in so embarrassing a form, as to perplex rather than assist the geographical student, and the statements of Ptolemy concerning the rivers Gir and Nigir, and the lakes and mountains with which they were connected, have exercised the ingenuity and baffled the sagacity of successive generations of geographers in modern times to interpret or explain them. It may safely be said that they present no resemblance to the real features of the country as known to us by modern explorations, and cannot be reconciled with them except by the most arbitrary conjectures.

It is otherwise in the case of the Nile. To discover the source of that river had been long an object of curiosity both among the Greeks and Romans, and an expedition sent out for that purpose by the emperor Nero had undoubtedly penetrated as far as the marshes of the White Nile ; but we are wholly ignorant of the sources from whence Ptolemy derived his information. But his statement that the mighty river derived its waters from the confluence of two streams, which took their rise in two lakes a little to the south of the equator, was undoubtedly a nearer approach to the truth than any of the theories concocted in modern times before the discovery in our own days of the two great lakes now known as the Victoria and Albert Nyanza. He at the same time notices the other arm of the river (the Blue Nile) under the name of the Astapus, which he correctly describes as rising in another lake. In connexion with this subject he introduces a range of mountains running from east to west, which he calls the Mountains of the Moon, and which have proved a sad stumbling-block to geographers in modern times, but may now be safely affirmed to represent the real fact of the existence of snow-covered mountains (Kilimanjaro and Kenia) in these equatorial regions.

Much the same remarks apply to Ptolemy's geography of Asia as to that of Africa. In this case also he had obtained, as we have already seen, a vague knowledge of extensive regions, wholly unknown to the earlier geographers, and resting to a certain extent on authentic information, though much exaggerated and misunderstood. But, while these informants had really brought home some definite statements concerning Serica or the Land of Silk, and its capital of Sera, there lay a vast region towards the north of the line of route leading to this far eastern land (supposed by Ptolemy to be nearly coincident with the parallel of 40°) of which apparently he knew nothing, but which he vaguely assumed to extend indefinitely northwards as far as the limits of the Unknown Land. The Jaxartes, which ever since the time of Alexander had been the boundary of Greek geography in this direction, still continued in that of Ptolemy to be the northern limit of all that was really known of Central Asia. Beyond that he places a mass of names of tribes, to which he could assign no definite locality, and mountain ranges which he could only place at haphazard. The character of his information concerning the south-east of Asia has been already adverted to. But, strangely as he misplaced Cattigara and the metropolis of Sina3 connected with it, there can be no doubt that we recognize in this name (variously written Thina? and Sinai) the now familiar name of China ; and it is important to observe that he places the land of the Sinse immediately south of that of the Seres, showing that he was aware of the connexion between the two, though the one was known only by land explorations and the other by maritime voyages.

In regard to the better known regions of the world, and especially those bordering on the Mediterranean, Ptolemy according to his own account followed for the most part the guidance of Marinus. The latter seems to have relied to a great extent on the work of Timosthenes (who flourished more than two centuries before) in respect to the coasts and maritime distances. Ptolemy, however, introduced many changes, some of which he has pointed out to us, though there are doubtless many others which we have no means of detecting. For the interior of the different countries the Roman roads and itineraries must have furnished him with a mass of valuable materials which had not been available to earlier geographers. But neither Marinus nor Ptolemy seems to have taken advantage of this last resource to the extent that we should have expected, and the tables of the Alexandrian geographer abound with mistakes—even in countries so well known as Gaul and Spain— which might easily have been obviated by a more judicious use of such Roman authorities.

Great as are undoubtedly the merits of Ptolemy's geographical work, it cannot be regarded as having any claim to be a complete or satisfactory treatise upon this vast subject. It was the work of an astronomer rather than a geographer, in the highest sense of the term. Not only did its plan exclude all description of the countries with which it dealt, their climate, natural productions, inhabitants, and peculiar features, all of which are included in the domain of the modern geographer, but even its physical geography strictly so called is treated in the most irregular and perfunctory manner. While Strabo was fully alive to the importance of the great rivers and mountain chains which (to use his own expressive phrase) "geographize " a country, Ptolemy deals with this part of his subject in so careless a manner as to be often worse than useless. Even in the case of a country so well known as Gaul the few notices that he gives of the great rivers that 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 Khine, though mentioning an obscure streamlet, otherwise unknown, because it happened to be the boundary between two Roman provinces.

The revival of the study of Ptolemy's work after the Middle Ages and the influence it exercised upon the progress of geography have been described in the article MAP (vol. xv. p. 520). His Geogmphia was printed for the first time ill a Latin translation, accompanied with maps, in 1478, and numerous other editions followed in the latter part of the 15th and earlier half of the 16th century, but the Greek text did not make its appearance till 1533, when it was published at Basel in 4to, edited by the celebrated Erasmus. All these early editions, however, swarm with textual errors, and are wholly worthless for critical purposes. The same may be said of the edition of Bertlus (Gr. and Lat., Leyden, 1618, typ. Elzevir), which was long the standard library edition of the work. It contains a new set of maps drawn by Mercator, as well as a fresh series (not intended to illustrate Ptolemy) by Ortelius, the Roman Itineraries, including the Tabula Peutingeriana, and much other miscellaneous matter. The first attempt at a really critical edition was made by Wilberg and Grashof (4to, Essen, 1842), but this unfortunately was never completed. The edition of Nobbe (3 vols. ISnio, Leipsic, 1843) presents the best Greek text of the whole work as yet available and has a useful index. But by far the best edition, so far as completed, is that published in Didot's Bibliotheca Classicvrum Graecorum (Paris, 1883), edited by Dr C. Muller, with a Latin translation and a copious commentary, geographical as well as critical. The first part, which is all that has yet appeared, contains only the first three books, without the Prolegomena, which will be anxiously expected by all students of Ptolemy. (E. H. B.)


Footnotes

3 De Morgan, in Smith's Dictionary of Greek and Iioman Biography, s.v. "Ptolemceus, Claudius."
4 KaipiKai, 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,
6 Aim., ed. Halma, i. 159.

1 Système de Politique Positive, iv. 173.

2 In this edition the Greek text and the French translation are given in parallel columns ; the latter, however, should not be read without reference to the former.
3 Delambrc begins his analysis of the Almagest thus—" L'Astronomie des Grecs est toute entière dans la Syntaxe mathématique de Ptolémée."
4 Narrien, An Historical Account of the Origin and Progress of Astronomy, Loudon, 1833.



The above article was written by two authors:

Ptolemy - Life; Astronomy
G. Johnston Allman, LL.B., F.R.S., Professor of Mathematics, Queen's College, Galway

Ptolemy - Geography
E. H. Bunbury, M.A., author of the History of Ancient Geography