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 Phaedo of Plato that Ptolemy devoted his life to astronomy and lived for forty years in the so-called IITepa TOD Kavwf30v, 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., 1 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 astronomiae, p. 177, or in the preface to Halma's edition of the Almagest, p. 61.
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 ( -r,unµa-ra ), 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 minutae primae " and " partes minutae secundae," whence our " minutes " ' Weidler and Halma give the ninth year; in the account of the eclipse of the moon in that year Ptolemy, however, does not say, as in other similar cases, he had observed, but it had been observed ( Almagest, iv. 9).
and " seconds " have arisen. It must not be supposed, however, that these sexagesimal divisions are due to Ptolemy; they must have been familiar to his predecessors, and were handed down from the Chaldaeans. 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 be 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 be calculated, and contains approximately lo p 32' 3". In like manner, the square on the chord of 90°, which is the side of the inscribed square, is twice the square on the radius; and the square on the chord of 120°, or the side of the equilateral triangle, is three times the square on the radius; these chords can thus be calculated approximately. Further, from the values of all these chords we can calculate at once the chords of the arcs which are their supplements.
This being laid down, we now proceed to give Ptolemy's exposition of the mode of obtaining his table of chords, which is a piece of 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 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 difference 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 formulae 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°, i 1° and three-fourths of t°, and lastly, the chords of 41°, 71°, 9°, tor, &c. - all those arcs, namely, as Ptolemy says, which being doubled are divisible by 3. Performing the calculations, he finds that the chord of 1 1° contains approximately 1 P 34' 55", and the chord of three-fourths of I° contains o p 47' 8". A table of chords of arcs increasing by I ° 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 r°; but, since this chord cannot be found geometrically from the chord of I °, 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 t°, which he does in the following manner chord 60' < 60 i.e. < 4 .'. chord I° < 4 chord 45'; chord 45' 45 3 3 again - chord < i.e. < 2, .'. chord I° > 3 chord 90'.
For brevity we use a modern notation. It has been shown that the chord of 45' is o p 47' 8" q.p., and the chord of 90' is 1 p 34' 15" q.p.; hence - it follows that approximately chord I° < I' 2' 50" 40'" and > I ' 2' 50".
| | | [MATHEMATICS Since these values agree as far as the seconds, Ptolemy takes 1 p 2' 50" as the approximate value of the chord of 0. The chord of I° being thus known, he finds the chord of one-half of a degree, the approximate value of which is o p 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 halfdegrees, from o° 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 halfdegree, in order that the chords of the intermediate arcs, which do not occur in the table, may be calculated, it being assumed that the increment of the chords of arcs within the table for each 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 (Alm. vi. ch. 7; i. 422, ed. Halma) Ptolemy solves a triangle in which the three sides are given by finding the segments of a side made by the perpendicular on it from the opposite vertex. It should be noticed also that the eleventh chapter of the first book of the 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 ( Alm. vi. ch. 7; i. 421, ed. Halma) takes 3 P 8' 30", 8 30 i.e. 3+60+3600=31416, as the value of the ratio of the circum ference to the diameter of a circle, and adds that, as had been shown by Archimedes, it lies between 3; and 3?-1. The foundation of spherical trigonometry is laid in chapter xi. on a few simple and useful lemmas. The starting-point is the wellknown theorem of plane geometry concerning the segments of the sines of a triangle made by a transversal: The segments of any side are in a ratio compounded of the ratios of the segments of the other two sides. This theorem, as well as that concerning the inscribed quadrilateral, was called after Ptolemy - naturally, indeed, since no reference to its source occurs in the Almagest. This 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 Tatter farther back and attributes it to Euclid, suggesting that it was given in his Porisms. 2 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 arc 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, but difficult to discover - from the theorem of Menelaus for a plane triangle the corresponding theorem for a spherical triangle: If the sides of a spherical triangle be cut by an arc of a great circle, the chords of the doubles of the segments 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 reader finds still more cumbrous, inasmuch as in each case it was necessary to express the relation in terms of chords - the equivalents of sines - only, cosines and tangents being of later invention. Such, then, was the trigonometry of the Greeks. Mathematics, indeed, has ever been, as it were, the handmaid of astronomy, and many important methods of the former arose 1 Ideler has examined the degree of accuracy of the numbers in these tables and finds that they are correct to five places of decimals. On the theorem of Menelaus and the rule of six quantities, see Chasles, Apercu historique sur l'origine et developpement des methodes en geometrie, note vi. p. 291. from the needs of the latter. Moreover, by the foundation of trigonometry, astronomy attained its final general constitution, in which calculations took the place of diagrams, as these latter had been at an earlier period substituted for mechanical apparatus in solving the ordinary problems. 3 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.4 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, what is the position of the oblique circle (the ecliptic), and the situation of the inhabited parts of the earth; that he will point out the differences of climates; that he will then pass on to the consideration of the motion of the sun and moon, without which one cannot have a just theory of the stars; lastly, that he will consider the sphere of the fixed stars and then the theory of the five stars called " planets." All these things - i.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 la y s 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, declination and longitude, and for the formation of a table of declinations to each degree of longitude. Delambre says he found both this and the table of chords very exact." In book ii., after some remarks on the situation of the 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 Comte, Systime de politique positive, iii. 3224 Captor, Vorlesungen fiber Geschichte der Mathematik, p. 356. | MATHEMATICS] | | 6 De Morgan, in Smith's Dictionary of Greek and Roman Biography, s.v. " Ptolemaeus, Claudius." 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 pole at Rhodes is 36°, a whole number, whereas at Alexandria he believed it to be 30° 58'; the second is that Hipparchus had made at Rhodes many observations; the third is that the climate of Rhodes holds the mean place of the seven climates subsequently described. Delambre suspects a fourth reason, which he thinks is the true one, that Ptolemy had taken his examples from the works of Hipparchus, who observed at Rhodes and had made these calculations for the place where he lived. 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 he 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 temporal' to equinoctial hours and vice versa, and of the nonagesimal point and the point of orientation of the ecliptic. In the following chapters of this book he determines the angles formed by the intersections of the ecliptic - first with the meridian, then with the horizon, and lastly with the vertical circle - and concludes by giving tables of the angles and arcs formed by the intersection of these circles, for the seven climates, from the parallel of Meroe (thirteen hours) to that of the mouth of the Borysthenes (sixteen hours). These tables, he adds, should be completed by the situation of the chief towns in all countries according to their latitudes and longitudes; this he promises to do in a 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 " (av5p1 oµou Kai 4uXaX18E2). He begins by telling us how Hipparchus was led to discover the precession of the equinoxes; he relates the observations by which Hipparchus verified the eccentricity of the solar orbit imperfectly known to his Chaldaean predecessors, and gives the hypothesis of the eccentric by which he explained the inequality of the sun's motion. Ptolemy concludes this book by giving a clear exposition of the circumstances on which the equation of time depends. 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. 2 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.' We find in the same page another principle, or rather practical injunction, that in investigations founded on observations where great delicacy is required we should select those made at considerable intervals of time in order that the errors arising from the imperfection which is inherent in all observations, even in those made with the greatest care, may be lessened by being distributed over a large number of years. In the same chapter we find also the principle laid down that the object of mathematicians ought to be to represent all the celestial phenomena by uniform and circular motions. This principle is stated by Ptolemy in the manner which is unfortunately too common with him - that is to say, he does not give the least indication whence he derived it. We know, however, from Simplicius, on the authority of Sosigenes, 4 that Plato is said to have proposed the following 1 Kacpucai, temporal or variable. These hours varied in length with the seasons; they were used in ancient times and arose from the division of the natural day (from sunrise to sunset) into twelve parts. Alm. ed. Halma, i. 159. Systeme de politique positive, iv. 173. 4 This Sosigenes, as Th. H. Martin has shown, was not the astronomer of that name who was a contemporary of Julius Caesar, but a Peripatetic philosopher who lived at the end of the 2nd century. 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 hypothesis of this kind," 5 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 book. As to the observations which should be employed for the investigation of the motion of the moon, Ptolemy tells us that lunar eclipses should be preferred, inasmuch as they give the moon's place without any error on the score of parallax. The first thing to be determined is the time of the moon's revolution; Hipparchus, by comparing the observations of the Chaldaeans with his own, discovered that the shortest period in which the lunar eclipses return in the same order was 226,007 days and i hour. In this period he finds 4267 lunations, 4573 restitutions of anomaly and 4612 tropical revolutions of the moon less 71 ° q.p.; this quantity (71°) is also wanting to complete the 345 revolutions which the sun makes in the same time with respect to the fixed stars. He concluded from this that the lunar month contains 29 days and 31' 50" 8"' 20" of a day, very nearly, or 29 days 12 hours 44' 3" 20'" These results are of the highest importance. In order to explain this inequality, or the equation of the centre, Ptolemy makes use of the hypothesis of an epicycle, which he prefers to that of the eccentric. The fifth book commences with the description of the astrolabe of Hipparchus, which Ptolemy made use of in following up the observations of that astronomer, and by means of which he made his most important discovery, that of the second inequality in the moon's motion, now known by the name of the " evection." In order to explain this inequality he supposed the moon to move on an epicycle, which was carried by an eccentric whose centre turned about the earth, in a direction contrary to that of the motion of the epicycle. This' is the first instance in which we find the two hypotheses of eccentric and epicycle combined. The fifth book treats also of the parallaxes of the sun and moon, and gives a description of an instrument - called later by Theon the " parallactic rods "- devised by Ptolemy for observing meridian altitudes with greater accuracy. The subject of parallaxes is continued in the sixth book of the Almagest, and the method of calculating eclipses is there given. The author says nothing in it which was not known before his time. Books vii., viii. treat of the fixed stars. Ptolemy verified the fixity of their relative positions and confirmed the observations of Hipparchus with regard to their motion in longitude, or the precession of the equinoxes. 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 which Hipparchus had drawn up 265 years before Ptolemy, whereas others, of whom Laplace is one, think that it is the work of Ptolemy himself. The probability is that in the main the catalogue is really that of Hipparchus altered to suit Ptolemy's own time, but that in making the changes which were necessary a wrong precession was assumed. This is Delambre's opinion; he says, " Whoever may have been the true author, the catalogue is unique, and does not suit the age when Ptolemy lived; by subtracting 2° 40' from all the longitudes it would suit the age of Hipparchus; this is all that is certain." 8 It has been remarked that Ptolemy, living at Alexandria, at which city the altitude of the pole is 5° less than at Rhodes, where Hipparchus observed, could have seen stars which are not visible at Rhodes; none of these stars, however, are in Ptolemy's catalogue. The eighth book contains, moreover, a description of the milky way and the manner 5 Brandis, Schol. in Aristot. edidit acad. reg. borussica (Berlin, 1836), p. 498. 6 Eivayuyi l ds ri 4acvo va, c. i. in Halma's edition of the works of Ptolemy, vol. iii. (" Introduction aux phenomenes celestes, traduite du grec de Geminus," p. 9), Paris, 1819. 1819. This has been noticed by Pliny, who says, " Multiformi haec (luna) ambage torsit ingenia contemplantium, et proximum ignorari maxime sidus indignantium " (N.H., ii. 9). s Delambre, Histoire de l'astronomie ancienne, ii. 264. | | | [MATHEMATICS of constructing a celestial globe; it also treats of the configuration of the stars, first with regard to the sun, moon and planets, and then with regard to the horizon, and likewise of the different aspects of the stars and of their rising, culmination and setting simultaneously with the sun. The remainder of the work is devoted to the planets. The ninth book commences with what concerns them all in general. The planets are much nearer to the earth than the fixed stars and more distant than the moon. Saturn is the most distant of all, then Jupiter and then Mars. These three planets are at a greater distance from the earth than the sun.' So far all astronomers are agreed. This is not the case, he says, with respect to the two remaining planets, Mercury and Venus, which the old astronomers placed between the sun and earth, whereas more recent writers' have placed them beyond the sun, because they were never seen on the sun.' He shows that this reasoning is not sound, for they might be nearer to us than the sun and not in the same plane, and consequently never seen on the sun. He decides in favour of the former opinion, which was indeed that of most mathematicians. The ground of the arrangement of the planets in order of distance was the relative length of their periodic times; the greater the circle, the greater, it was thought, would be the time required for its description. Hence we see the origin of the difficulty and the difference of opinion as to the arrangement of the sun, Mercury and Venus, since the times in which, as seen from the earth, they appear to complete the circuit of the zodiac are nearly the same - a year.4 Delambre thinks it strange that Ptolemy did not see that these contrary opinions could be reconciled by supposing that the two planets moved in epicycles about the sun; this would be stranger still, he adds, if it is true that this idea, which is older than Ptolemy, since it is referred to by Cicero,' had been that of the Egyptians.' It may be added, as strangest of all, that this doctrine was held by Theon of Smyrna, 7 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 owing to these difficulties that Hipparchus - who loved truth above all things, and who, moreover, had not received from his predecessors observations either so numerous or so precise as those that he has left - had succeeded, a s far as possible, in representing the motions of the sun and moon by circles, but had not even commenced the theory of the five planets. He was content, Ptolemy continues, to arrange the observations which had been made on them in a methodic order and to show thence that the phenomena did not agree with the hypotheses of mathematicians at that time. He showed that in fact each planet had two inequalities, which are different for each, that the retrogradations are also different, whilst other astronomers admitted only single inequality and the same retrogradation; he showed further that their motions cannot be explained by eccentrics nor by epicycles carried along concentrics, but that it was necessary to combine both hypotheses. After these preliminary notions he gives from Hipparchus the periodic motions of the five planets, together with the shortest times of restitutions, in which, moreover, he has made some slight corrections. He then gives tables of the mean motions in longitude and of anomaly of each of the five planets,8 ' This is true of their mean distances; but we know that Mars at opposition is nearer to us than the sun. 2 Eratosthenes, for example, as we learn from Theon of Smyrna. 3 Transits of Mercury and Venus over the sun's disk, therefore, had not been observed. This was known to Eudoxus. Sir George Cornewall Lewis (An Historical Survey of the Astronomy of the Ancients, p. 155), confusing the geocentric revolutions assigned by Eudoxus to these two planets with the heliocentric revolutions in the Copernican system, which are of course quite different, says that " the error with respect to Mercury and Venus is considerable "; this, however, is an error not of Eudoxus but of Cornewall Lewis, as Schiaparelli has remarked. ' " Hunc [solem] ut comites consequuntur Veneris alter, alter Mercurii cursus " (Somnium Scipionis, De rep. vi. 17). This hypothesis is alluded to by Pliny, N.H. ii. 17, and is more explicitly stated by Vitruvius, Arch. ix. 4. 6 Macrobius, Commentarius ex Cicerone in somnium Scipionis, i. 197 Theon (Smyrnaeus Platonicus), Liber de astronomic, ed. Th. H. Martin (Paris, 18 49), pp. 1 74, 2 94, 296. Martin thinks that Theon, the mathematician, four of whose observations are used by Ptolemy ( Alm. ii. 176, 1 93, 1 94, 1 95, 196, ed. Halma), is not the same as Theon of Smyrna, on the ground chiefly that the latter was not an observer. 8 Delambre compares these mean motions with those of our modern tables and finds them tolerably correct. By " motion in and shows how the motions in longitude of the planets can be represented in a general manner by means of the hypothesis of the eccentric combined with that of the epicycle. He next applies his theory to each planet and concludes the ninth book by the explanation of the various phenomena of the planet Mercury. In the tenth and eleventh books he treats, in like manner, of the various phenomena of the planets Venus, Mars, Jupiter and Saturn. Book xii. treats of the stationary and retrograde appearances of each of the planets and of the greatest elongations of Mercury and Venus. The author tells us that some mathematicians, and amongst them Apollonius of Perga, employed the hypothesis of the epicycle to explain the stations and retrogradations of the planets. Ptolemy goes into this theory, but does not change in the least the theorems of Apollonius; he only promises simpler and clearer demonstrations of them. Delambre remarks that those of Apollonius must have been very obscure, since, in order to make the demonstrations in the Almagest intelligible, he (Delambre) was obliged to recast them. This statement of Ptolemy is important, as it shows that the mathematical theory of the planetary motions was in a tolerably forward state long before his time. Finally, book xiii. treats of the motions of the planets in latitude, also of the inclinations of their orbits and of the magnitude of these inclinations. 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 MaOflµaTCKi 7 vuvracs, for it was in fact the mathematical form of the work which caused it to be preferred to all others which treated of the same science, but not by " the sure methods of geometry and calculation." Accordingly, it soon spread from Alexandria to all places where astronomy was cultivated; numerous copies were made of it, and it became the object of serious study on the part of both teachers and pupils. Amongst its numerous commentators may be mentioned Pappus and Theon of Alexandria in the 4th century and Proclus in the 5th. It was translated into Latin by Boetius, but this translation has not come down to us. The Syntaxis was translated into Arabic at Bagdad 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 Tobit ben Korra. The Almagest was translated from the Arabic into Latin by Gerard of Cremona. 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 (d. 1461) and completed by his pupil and successor in the professorship of astronomy in the university of Vienna, Regiomontanus. The earliest edition of this epitome is that of Venice (1496), 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 (1538). 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. Other works of Ptolemy, which we now proceed to notice very briefly, are as follow. (1) Fav airXavwv &crr pwv cal ovvaycwy, Ercvnpaauvv, On the Apparitions of the Fixed Stars and a Collection of Prognostics. It is a cal' ndar of a kind common amongst the Greeks under the name of 7raplun - ny,ua, 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 131 hours to 151 hours - that is, from the latitude of Syene to that of the middle of the Euxine. This work was printed by Petavius in his Uranologium (Paris, 1630), and by Halma in his edition of the works of Ptolemy, vol. iii. (Paris, 1819). (2) `TzroNcrEts Tan, ,r%avw / .,l p wv Y f TON of pavlwv KUKAWv 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 Kavcwv l3aacXECwv (London, 1620), and afterwards by Halma, vol. iv. (Paris, 1820). (3) Kavwv 13ariXECwv, 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 (cf. G. Syncellus, Chronogr. ed. Dind. i. 388 seq.) was printed by Scaliger, Calvisius, Petavius, Bainbridge and by Halma, longitude " must be understood the motion of the centre of the epicycle about the eccentric, and by " anomaly " the motion of the star on its epicycle. | GEOGRAPHY] | | vol. iii. (Paris, 1819). (4) `Apµovucwv (3e(3Xla y'. This Treatise on Music was published in Greek and Latin by Wallis at Oxford (1682). It was afterwards reprinted with Porphyry's commentary in the third volume of Wallis's works (Oxford, 16 99). (5) Terpa(t3 Xos auvraEcs, Tetrabiblon or Quadripartitum. This work is astrological, as is also the small collection of aphorisms, called Kapnros or Centiloquium, by which it is followed. It is doubtful whether these works are genuine, but the doubt merely arises from the feeling that they are unworthy of Ptolemy. They were both published in Greek and Latin by Camerarius (Nuremberg, 1535), and by Melanchthon (Basel, 1553). (6) De analemmate. The original of this work of Ptolemy is lost. It was translated from the Arabic and published by Commandine (Rome, 1562). The Analemma is the description of the sphere on a plane. We find in it the sections of the different circles, as the diurnal parallels, and everything which can facilitate the intelligence of gnomonics. This description is made by perpendiculars let fall on the plane; whence it has been called by the moderns " orthographic projection." (7) Planisphaerium, The Planisphere. The Greek text of this work also is lost, and we have only a Latin translation of it from the Arabic. The " planisphere " is a projection of the sphere on the equator, the eye being at the pole - in fact what is now called " stereographic " projection. The best edition of this work is that of Commandine (Venice, 1558). (8) Optics. This work is known to us only by imperfect manuscripts in' Paris and Oxford, which are Latin translations from the Arabic. 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 poor geometer. (G. J. A.) The publication of a new edition of Ptolemy's works under the title, Claudii Ptolemaei opera quae exstant omnia, was recently undertaken at Leipzig. The first volume (in two parts, 1898, 1903) contains the Greek text of the Almagest edited by J. L. Heiberg. Consult also J. E. Montucla, Histoire des mathematiques, i. 293 J. B. J. Delambre, Connaissance des temps (1816); and Histoire de l'astronomie .ancienne, vol. 2; J. J. A. Caussin, Nouvelles memoires de l'acad. des inscriptions, t. vi.; P. Tannery, Recherches sur l'histoire de l'astronomie ancienne, ch. vi.-xv.; Narrieu, History of Astronomy (1833); Fabricius, Bibliotheca graeca, ed. Harles, vol. 5; Halma's1813-1816edition of his Almagest (Greek with French translation); A. Berry, A Short History of Astronomy, pp. 62-73; British Museum Catalogue. Geography. Ptolemy is hardly less celebrated as a geographer than as an astronomer, and his Geographike syntaxis exercised as great an influence on geographical progress (especially during the period of the Classical Renaissance), as did his Almagest on astronomical. This exceptional position was largely due to its scientific form, which rendered it convenient and easy of reference; but, apart from this, it was really the most considerable attempt of the ancient world to place the study of geography on a scientific basis. The astronomer Hipparchus had indeed pointed out, three centuries before Ptolemy, that the only way to construct a trustworthy map of the inhabited world would be by observations of the latitude and longitude of all the principal points on its surface. But the materials for such a map were almost wholly wanting, and, though Hipparchus made some approach to a correct division of the known world into zones of latitude, " climates " or klimata, as he termed them, trustworthy observations of latitude were then very few, while the means of determining longitudes hardly existed. Hence probably it arose that no attempt was made to follow up the suggestion of Hipparchus until Marinus of Tyre, who lived shortly before Ptolemy, and whose work is known to us only through the latter. Marinus' scientific materials being inadequate, he contented 'himself mostly with determinations derived from itineraries and other rough methods, such as are still employed where more accurate means of determination are not available. The greater part of Marinus' treatise was occupied with the discussion of his authorities, and it is impossible, in the absence of the original work, to decide how far his results attained a scientific form. But Ptolemy himself considered them, on the whole, so satisfactory that he made his predecessor's work the basis of his own in regard to all the Mediterranean countries, that is, in regard to :almost all those regions of which he had definite knowledge. In the more remote regions of the world, Ptolemy availed himself of Marinus' information, but with reserve, and himself explains the reasons that induced him sometimes to depart from his predecessor's conclusions. It is unjust to term Ptolemy a plagiarist from Marinus, as he himself fully acknowledges 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, unique among those ancient geographical treatises which have survived, that constitutes one great merit of Ptolemy's work. At the same time it shows the increased knowledge of Asia and Africa acquired since Strabo and Pliny. 1. Mathematical Geography. - As an astronomer, Ptolemy was of course better qualified to explain the mathematical conditions of the earth and its relations to the celestial bodies than most preceding geographers. His general views had much in common with those of Eratosthenes and Strabo. Thus 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, dividing the earth into five zones, the relations of which with astronomical phenomena were of course clear to his mind as a matter of theory, though in regard to the regions bordering on the equator, as well as to those adjoining the polar circle, he could have had no confirmation of his conclusions from actual observation. He adopted also from Hipparchus the division of the equatorial circle into 360 parts ( degrees, as they were subsequently called, though the word does not occur in this sense in Ptolemy), and supposed other circles to be drawn through this, from the equator to the pole, to which he gave the name of meridians. He thus, like modern geographers, conceived the whole surface of the earth as covered with a network of parallels of latitude and meridians of longitude, terms which he himself was the first extant writer to employ in this technical sense. Within the network thus constructed it was his task to place the outline of the world, so far as known to him. But at the very outset of his attempt he fell into an error vitiating all his conclusions. Eratosthenes (276-196 p.c.) was the first who had attempted scientifically to determine the earth's circumference, and his result of 250,000 (or 252,000) stadia, i.e. 25,000 (25,200) geographical miles, was generally adopted by subsequent geographers, including Strabo. Poseidonius, however ( c. 135-50 B.C.), reduced this to 180,000, and the latter computation was inexplicably adopted by Marinus and Ptolemy. This error made every degree of latitude or longitude (measured at the equator) equal to only 500 stadia (50 geographical miles), instead of its true equivalent of 600 stadia. The mistake would have been somewhat neutralized had there existed a sufficient number of points of which the position was fixed by observation; but we learn from Ptolemy himself that such observations for latitude were very few, while the means of determining longitudes were almost wholly wanting.' Hence the positions laid down by him were, with few exceptions, the result of computations from itineraries and the statements of travellers, liable to much greater error in ancient times than at the present day, from the want of any accurate mode of observing bearings, of measuring time (by portable instruments), or of estimating distances at sea, except by the rough estimate of the time employed in sailing from point to point. Even the use of the log was unknown to the ancients. But, great as were the errors resulting from such imperfect means of calculation, they were increased by the permanent error arising from Ptolemy's system of graduation. Thus if he concluded (from itineraries) that two places were 5000 stadia distant, he would place them 10° apart, and thus in fact separate them by 6000 stadia. Another source of permanent error (though of less importance), which affected all his longitudes, arose from his prime meridian. Here also he followed Marinus, who, supposing that the Fortunate Islands (vaguely answering to our Canaries plus the Madeira group) lay farther west than any part of Europe or Africa, had taken the meridian through the (supposed) outermost of this group as his prime meridian, from whence he calculated his longitudes eastwards to the Indian Ocean. But as both Marinus and Ptolemy had no exact knowledge of the islands in question, the line thus assumed was purely imaginary, drawn through the supposed position of an island which they placed 22° (instead of 9° 20') west of the Sacred Promontory (i.e. Cape St Vincent, regarded by Marinus and Ptolemy, as by previous geographers, as the westernmost point of Europe). Hence all Ptolemy's longitudes, reckoned eastwards, were about 7° less than they would have been if really measured from the meridian of Ferro, which continued so long in use. This error was the more unfortunate as the longitude was really calculated, not from this imaginary line, but from Alexandria, westwards as well as eastwards (as Ptolemy himself has done in his eighth book), and afterwards reversed, so as to suit the supposed method of computation. 1 Hipparchus pointed out the mode of determining longitudes by observations of eclipses, but the instance to which he referred (of the celebrated-'eclipse before the battle of Arbela, which was also seen at Carthage) was a mere matter of popular observation, of no scientific value. Yet Ptolemy seems to have known of no other. | | | [ GEOGRAPHY The equator was in like manner placed by Ptolemy at a considerable distance from its true geographical position. The place of the equinoctial line was well known to him as a matter of theory, but as no observations could have been made in those regions he could only calculate its place from that of the tropic, which he supposed to pass through Syene. And as he here, as elsewhere, reckoned a degree of latitude as equivalent to 500 stadia, he inevitably made the interval between the tropic and the equator too small by one-sixth; and the place of the former being fixed by observation, he necessarily carried up the supposed place of the equator too high by more than 230 geographical miles. But as he had practically no geographical acquaintance with the equinoctial regions this error was of little importance. With Marinus and Ptolemy, as with preceding Greek geographers, the most important line for practical purposes was the parallel of 36° N., which, passing through the Straits of Gibraltar, Rhodes Island and the Gulf of Issus, and thus dividing the Mediterranean (as Dicaearchus and his successors usually regarded it) into two, was continued in theory along the chain of Mt Taurus till it joined the mountains north of India; thence to the Eastern Ocean it was regarded as constituting the dividing line of the inhabited world, along which the length of the latter must be measured. But so inaccurate were the observations and so imperfect the materials at command, even in regard to the best known regions, that Ptolemy, following Marinus, describes this parallel as passing through Caralis in Sardinia and Lilybaeum in Sicily, the one being really in 39° 12' lat., the other in 37° 50'. Still more strangely he places Carthage 1 ° 20'20' south of the dividing parallel, while it really lies nearly i ° north of it. The problem that had especially attracted the attention of geographers from Dicaearchus to Ptolemy was to determine the length and breadth of the inhabited world. This question had been fully discussed by Marinus, who had arrived at conclusions widely different from his predecessors. Towards the north, indeed, there was no great difference of opinion, the latitude of Thule being generally recognized as that of the highest northern land, and this was placed both by Marinus and Ptolemy in 63° N., not far beyond the true position of the Shetland Islands, which had come 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 outermost of the Fortunate Islands. But towards the south and east Marinus gave an enormous extension to Africa and Asia, beyond what had been known to or suspected by earlier geographers, and, though Ptolemy reduced Marinus' calculations, he retained an exaggerated estimate of their results. The additions thus made to the estimated dimensions of the known world were indeed in both directions based upon a real extension of knowledge, derived from recent information; but the original statements were so perverted by misinterpretation as to give results (in map-construction) differing widely from the truth. The southern limit of the world had been fixed by Eratosthenes and even by Strabo at the parallel which passed through the eastern extremity of Africa (Cape Guardafui), the Cinnamon Region (Somaliland) and the country of the Sembritae (Sennaar). This parallel, which would correspond nearly to that of 10° of true latitude, they supposed to be situated at a distance of 3400 stadia (340 geographical miles) from that of Meroe (the position of which was pretty accurately known) and 13,400 to the south of Alexandria; while they conceived it as passing eastward through Taprobane (Ceylon, often Ceylon plus Sumatra?), universally recognized as the southernmost land of Asia. Both these geographers were ignorant of the vast extension of Africa to the south of this line and even of the equator, and conceived it as trending away west from the Cinnamon Land and then north-west to the Straits of Gibraltar. Marinus had, however, learned from itineraries both by land and sea the fact of this extension, of which he had conceived so exaggerated an idea that even after Ptolemy had reduced it by more than half it was still much in excess of the truth. The eastern 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?), placed by Ptolemy not far from 7° S. To this he added a bay extending to Cape Prasum (Delgado?), which he placed in 15° S. At the same time he assumed the position in about the same parallel of a region called Agisymba, inhabited by Ethiopians and abounding in rhinoceroses, which was supposed to have been discovered by a Roman general, Julius Maternus, whose itinerary was employed by Marinus. Taking, therefore, this parallel as the limit of knowledge to the south, while he retained that of Thule to the north, Ptolemy assigned to the inhabited world a breadth of nearly 80°, instead of less than 60°, as in Eratosthenes and Strabo. It had been a common belief among Greek geographers, from the earliest attempts at scientific geography, not only that the length of the inhabited world greatly exceeded its breadth, but that it was more than twice as great, an unfounded assumption to which. their successors seem to have felt themselves bound to conform. Thus Marinus, while extending his Africa unduly southward, exaggerated Asia still more grossly eastward. Here also he really possessed a great advance in knowledge over all his predecessors, the silk trade with China having led to an acquaintance, though of a vague and general kind, with regions east of the Pamir and Tian Shan, the limits of Asia as previously known to the Greeks. Marinus had learned that traders proceeding eastward from the Stone Tower (near the Pamir?) to Sera, the capital of the Seres (inland China?), occupied seven months on the journey; thence he calculated that the distance between the two points was 36,200 stadia or 3620 geographical miles. Ptolemy, while he points out the erroneous mode of computation on which this conclusion was founded, could not correct it by any real authority, and hence reduced it summarily by one half. He therefore placed Sera (Singanfu?), the easternmost point on his map of Asia, 452° 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 (in this latitude) as equivalent to 400 stadia. Both distances were greatly in excess, independently of error arising from 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. This last, it must be remembered, would be cumulative, the longitudes being computed from a fixed point in the west, instead of being reckoned east and west from Alexandria, which was undoubtedly the mode in which they were really calculated. These causes of error combined to make Ptolemy allow 180° long., or 12 hours' interval, between the Fortunate Islands meridian and Sera (really about 130°). But in thus estimating the length and breadth of the known world, Ptolemy attached a very different sense to these terms from that which they had generally borne. Most earlier Greek geographers and " cosmographers " supposed the inhabited world to be surrounded on all sides by sea, and to form a vast island in the midst of a circumfluous ocean. This notion (perhaps derived from the Homeric " ocean stream," and certainly not based upon direct observation) was nevertheless in accordance with truth, great as was the misconception involved of the continents included. But Ptolemy in this respect went back to Hipparchus, and assumed that the land extended indefinitely north in the case of eastern Europe, east, south-east and north in that of Asia, and south, south-west and south-east in that of Copyright Statement
These files are public domain. Bibliography Information
Chisholm, Hugh, General Editor. Entry for 'Ptolemy (Mathematician)'. 1911 Encyclopedia Britanica. https://www.studylight.org/encyclopedias/eng/bri/p/ptolemy-mathematician.html. 1910.
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