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PTOLEMY (CLAUDIUS PTOLEMAEUS)

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PTOLEMY (See also:CLAUDIUS See also:PTOLEMAEUS)  , the celebrated mathematician, astronomer and geographer, was a native of See also:Egypt,' but there is an uncertainty as to the See also:place of his See also:birth . Some See also:ancient See also:manuscripts of his See also:works describe him as of See also:Pelusium, but See also:Theodorus Meliteniota, a See also:Greek writer on See also:astronomy of the 2 The See also:Ptolemies were not in antiquity distinguished by the ordinal See also:numbers affixed to their names by See also:modern scholars and represented according to. the usual See also:convention by See also:Roman'figures . This is merely done for our convenience . In the See also:case of the later Ptolemies different systems of notation prevail according as the problematic Eupatpr and Philopator Neos are reckoned in or not . See also:MATHEMATICS] 12th See also:century, says that he was See also:born at Ptolemais Hermii, a Grecian See also:city of the Thebaid . It is certain that he observed at See also:Alexandria during the reigns of See also:Hadrian and See also:Antoninus See also:Pius, and that he survived Antoninus . See also:Olympiodorus, a philosopher of the Neoplatonic school who lived in the reign of the See also:emperor Justinian, relates in his scholia on the See also:Phaedo of See also:Plato that See also:Ptolemy devoted his See also:life to astronomy and lived for See also:forty years in the so-called Hrepa rob' Kavm(3ou, probably elevated terraces of the See also:temple of See also:Serapis at See also:Canopus near Alexandria, where they raised pillars with the results of his astronomical discoveries engraved upon them . This statement is probably correct; we have indeed the See also:direct See also:evidence of Ptolemy himself that he made astronomical observations during a See also:long See also:series of years; his first recorded observation was made in the See also:eleventh See also:year of Hadrian, 127 A.D.,i and his last in the fourteenth year of Antoninus, 151 A.D . Ptolemy, moreover, says, " We make our observations in the parallel of Alexandria." St Isidore of See also:Seville asserts that he was of the royal See also:race of the Ptolemies, and even calls him See also:king of Alexandria; this assertion has been followed by others, but there is no ground for their See also:opinion . Indeed See also:Fabricius shows by numerous instances that the name Ptolemy was See also:common in Egypt . Weidler, from whom this is taken, also tells us that according to Arabian tradition Ptolemy lived to the See also:age of seventy-eight years; from the same source some description of his See also:personal See also: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 See also:preface to See also:Halma's edition of the Almagest, p .

61 . Mathematics . Ptolemy's See also:

work as a geographer is discussed below, and an See also:account of the discoveries in astronomy of See also:Hipparchus and Ptolemy is given in the See also:article ASTRONOMY: See also:History . Their contributions to pure mathematics, however, require to be noticed here . Of these the See also:chief is the See also:foundation of See also:trigonometry, See also:plane and spherical, including the formation of a table of chords, which served the same purpose as our table of sines . This See also:branch of mathematics was created by Hipparchus for the use of astronomers, and its exposition was given by Ptolemy in a See also:form so perfect that for 1400 years it was not surpassed . In this respect it may be compared with the . See also:doctrine as to the See also:motion of the heavenly bodies so well known as the Ptolemaic See also:system, which was See also:paramount for about the same See also:period of See also:time . There is, however, this difference, that, whereas the Ptolemaic system was then overthrown, the theorems of Hipparchus and Ptolemy, on the other See also:hand, will be, as See also:Delambre says, for ever the basis of trigonometry . The astronomical and trigonometrical systems are contained in the See also:great work of Ptolemy, 'H yaOrlµariKi7 o vraEts, or, as Fabricius after See also:Syncellus writes it, Meyak17 vuvra is rC7c avrpovoµias; and in like manner Suidas says ouros [Hro11.] eypailte See also:TOP dyav avrpovoµov ijroi vuvraEiv . The Syntaxis of Ptolemy was called '0 0yas avrpovoµos to distinguish it from another collection called '0 j. wphs avrpovoµos, also highly esteemed by the Alexandrian school, which contained some works of See also:Autolycus, See also:Euclid, See also:Aristarchus, See also:Theodosius of Tripolis, Hypsicles and See also:Menelaus . To designate the great work of Ptolemy the See also:Arabs used the superlative µeylvrn, from which, the article al being prefixed, the hybrid name Almagest, by which it is now universally known, is derived . We proceed now to consider the trigonometrical work of Hipparchus and Ptolemy .

In the ninth See also:

chapter of the first See also:book of the Almagest Ptolemy shows how to form a table of chords . He sup-poses the circumference divided into 36o equal parts (TSohiara), and then bisects each of these parts . Further, he divides the See also:diameter into 120 equal parts, and then for the subdivisions of these he employs the sexagesimal method as most convenient in practice, i.e. he divides each of the sixty parts of the See also:radius into sixty equal parts, and each of these parts he further subdivides into sixty equal parts . In the Latin See also:translation these subdivisions become " partes minutae primae " and " partes minutae secundae," whence our " minutes " Weidler and Halma give the ninth year; in the account of the See also:eclipse of the See also:moon in that year Ptolemy, however, does not say, as in other similar cases, he had observed, but it had been observed (Almagest, iv . 9).619 and " seconds " have arisen . It must not be supposed, however, that these sexagesimal divisions are due to Ptolemy; they must have been See also:familiar to his predecessors, and were handed down from the Chaldaeans . Nor did the formation of the table of chords originate with Ptolemy; indeed, See also:Theon of Alexandria, the See also:father of See also:Hypatia, who lived in the reign of Theodosius, in his commentary on the Almagest says expressly that Hipparchus had already given the doctrine of chords inscribed in a circle in twelve books, and that Menelaus had done the same in six books, but, he continues, every one must be astonished at the ease with which Ptolemy, by means of a few See also:simple theorems, has found their values; hence it is inferred that the method of calculation in the Almagest is Ptolemy's own . As starting-point the values of certain chords in terms of the diameter were already known, or could be easily found by means of the Elements of Euclid . Thus the See also:side of the hexagon, or the chord of 6o°, is equal to the radius, and therefore contains sixty parts . The side of the decagon, or the chord of 36°, is the greater segment of the radius cut in extreme and mean ratio, and therefore contains approximately 37" 4' 55" parts, of which the diameter contains 120 parts . Further, the square on the side of the See also:regular pentagon is equal to the sum of the squares on the sides of the regular hexagon and of the regular decagon, all being inscribed in the same circle (Eucl . XIII. io); the chord of 72° can therefore be calculated, and contains approximately 70P 32' 3" .

In like manner, the square on the chord of 90°, which is the side of the inscribed square, is twice the square on the radius; and the square on the chord of 120°, or the side of the equilateral triangle, is three times the square on the radius; these chords can thus be calculated approximately . Further, from the values of all these chords we can calculate at once the chords of the arcs which are their supplements . This being laid down, we now proceed to give Ptolemy's exposition of the mode of obtaining his table of chords, which is a piece of See also:

geometry of great elegance, and is indeed, as De See also:Morgan says, " one of the most beautiful in the Greek writers." He takes as basis and sets forth as a lemma the well-known theorem, which is called after him, concerning a See also:quadrilateral inscribed in a circle: The rectangle under the diagonals is equal to the sum of the rectangles under the opposite sides . By means of this theorem the chord of the sum or the difference of-two arcs whose chords are given can be easily found, for we have only to draw a diameter from the common vertex of the two arcs the chord of whose sum or difference is required, and See also:complete the quadrilateral; in one case a See also:diagonal, in the other one of the sides is a diameter of the circle . The relations thus obtained are See also:equivalent to the fundamental formulae of our trigonometry See also:sin (A+B) =sin A See also:cos B+cos A sin B, sin (A—B)=sin A cos B—cos A sin B, which can therefore be established in this simple way . Ptolemy then gives a geometrical construction for finding the chord of See also:half an arc from the chord of the arc itself . By means of the foregoing theorems, since we know the chords of 72° and of 60°, we can find the chord of 12°; we can then find the chords of 6°, 3°, I2° and three-fourths of 1°, and lastly, the chords of 41° 71°, 9°, toa°, &c.—all those arcs, namely, as Ptolemy says, which being doubled are divisible by 3 . Performing the calculations, he finds that the chord of II° contains approximately 1p 34' 55", and the chord of three-fourths of 1° contains op 47' 8" . A table of chords of arcs increasing by I a° can thus be formed; but this is not sufficient for Ptolemy's purpose, which was to See also:frame a table of chords increasing by half a degree . This could be effected if he knew the chord of one-half of 1°; but, since this chord cannot be found geometrically from the chord of II°, inasmuch as that would come to the trisection of an See also:angle, he proceeds to seek in the first place the chord of I°, which he finds approximately by means of a lemma of great elegance, due probably to See also:Apollonius . It is as follows: If two unequal chords be inscribed in a circle, the greater will be to the less in a less ratio than the arc described on the greater will be to the arc described on the less . Having proved this theorem, he proceeds to employ it in See also:order to find approximately the chord of I °, which he does in the following manner chord 6o' < 62 i.e .

< .. chord I° < 4 chord 45'; chord 45' 45 3' 3 again chord 6o' 60' i.e . < 2, chord I ° > 3 chord 90' . For brevity we use a modern notation . It has been shown that the chord of 45' is op 47' 8" q.p., and the chord of 90' is I' 34' 15" q.p.; hence it follows that approximately chord I° < 1p 2' 50" 40'" and > 1p 2' 50" . Since these values agree as far as the seconds, Ptolemy takes 1p 2' 50" as the approximate value of the chord of O . The chord of I° being thus known, he finds 'the chord of one-half of a degree, the approximate value of which is op 31' 25", and he is at once in a position to complete his table of chords for arcs increasing by half a degree . Ptolemy then gives his table of chords, which is arranged in three columns; in the first he has entered the arcs, increasing by half-degrees, from o° to 180°; in the second he gives the values of the chords of these arcs in parts of which the diameter contains 12o, the subdivisions being sexagesimal; and in the third he has inserted the thirtieth parts of the See also:

differences of these chords for each half-degree, in order that the chords of the intermediate arcs, which do not occur in the table, may be calculated, it being assumed that the increment of the chords of arcs within the table for each See also:interval of 3o' is proportional to the increment of the arc) . Trigonometry, we have seen, was created by Hipparchus for the use of astronomers . Now, since spherical trigonometry is directly applicable to astronomy, it is not surprising that its development was See also:prior to that of plane trigonometry . It is the subject-See also:matter of the eleventh chapter of the Almagest, whilst the See also:solution of plane triangles is not treated separately in that work . To resolve a plane triangle the Greeks supposed it to be inscribed in a circle; they must therefore have known the theorem—which is the basis of this branch of trigonometry: The sides of a triangle are proportional to the chords of the See also:double arcs which measure the angles opposite to those sides . In the case of a right-angled triangle this theorem, together with Eucl .

I . 32 and 47, gives the complete solution . Other triangles were resolved into right-angled triangles by See also:

drawing the perpendicular from a vertex on the opposite side . In one place (Alm. vi. ch . 7; i . 422, ed . Halma) Ptolemy solves a triangle in which the three sides are given by finding the segments of a side made by the perpendicular on it from the opposite vertex . It should be noticed also that the eleventh chapter of the first book of the See also:Alma gest contains incidentally some theorems and problems in plane trigonometry . The problems which are met with correspond to the following: See also:Divide a given arc into two parts so that the chords of the doubles of those arcs shall have a given ratio; the same problem for See also:external See also:section . Lastly, it may be mentioned that Ptolemy (Alm. vi. ch . 7; i . 421, ed .

Halma) takes 3P 8' 3o", i.e . 3-{ 6—0+3600=3.1416, as the value of the ratio of the circum- ference to the diameter of a circle, and adds that, as had been shown by See also:

Archimedes, it lies between 3; and 3',$ . The foundation of spherical trigonometry is laid in chapter xi. on a few simple and useful lemmas . The starting-point is the well-known theorem of plane geometry concerning the segments of the sines of a triangle made by a transversal: The segments of any side are in a ratio compounded of the ratios of the segments of the other two sides . This theorem, as well as that concerning the inscribed quadrilateral, was called after Ptolemy—naturally, indeed, since no reference to its source occurs in the Almagest . This See also:error was corrected by See also:Mersenne, who showed that it was known to Menelaus, an astronomer and geometer who lived in the reign of the emperor See also:Trajan . The theorem now bears the name of Menelaus, though most probably it came down from Hipparchus; See also:Chasles, indeed, thinks that Hipparchus deduced the See also:property of the spherical triangle from that of the plane triangle, but throws the origin of the latter farther back and attributes it to Euclid, suggesting that it was given in his Porisms.2 See also:Carnot made this theorem the basis of his theory of transversals in his See also:essay on that subject . It should be noticed that the theorem is not given in the Almagest in the See also:general manner stated above; Ptolemy considers two cases only of the theorem, and Theon, in his commentary on the Almagest, has added two more cases . The proofs, however, are general . Ptolemy then See also:lays down two lemmas: If the chord of an arc of a circle be cut in any ratio and a diameter be See also:drawn through the point of section, the diameter will cut the arc into two parts the chords of whose doubles are in the same ratio as the segments of the chord; and a similar theorem in the case when the chord is cut externally in any ratio . By means of these two lemmas Ptolemy deduces in an ingenious manner—easy to follow, but difficult to discover—from the theorem of Menelaus for a plane triangle the corresponding theorem for a spherical triangle: If the sides of a spherical triangle be cut by an arc of a great circle, the chords of the doubles of the segments ofany one side will be to each other in a ratio compounded of the ratios of the chords of the doubles of the segments of the other two sides . Here, too, the theorem is not stated generally; two cases only are considered, corresponding to the two cases given in plane .

Theon has added two cases . The proofs are general . By means of this theorem four of See also:

Napier's formulae for the solution of right-angled spherical triangles can be easily established . Ptolemy does not give them, but in each case when required applies the theorem of Menelaus for spherics directly . This greatly increases the length of his demonstrations, which the modern reader finds still more cumbrous, inasmuch as in each case it was necessary to See also:express the relation in terms of chords—the equivalents of sines—only, cosines and tangents being of later invention . Such, then, was the trigonometry of the Greeks . Mathematics, indeed, has ever been, as it were, the handmaid of astronomy, and many impo;•tant methods of the former arose 3Ideler has examined the degree of accuracy of the numbers in these tables and finds that they are correct to five places of decimals . 2 On the theorem of Menelaus and the See also:rule of six quantities, see Chasles,, Aperpu historigue sur l'or+gine et developpement See also:des methodes en geometrie, See also:note v1. p . 291.from the needs of the latter . Moreover, by the foundation of trigonometry, astronomy attained its final general constitution, in which calculations took the place of diagrams, as these latter had been at an earlier period substituted for See also:mechanical apparatus in solving the See also:ordinary problems.3 Further, we find in the application of trigonometry to astronomy frequent eke amples and even a systematic use of the method of approximations—the basis, in fact, of all application of mathematics to See also:practical questions . There was a disinclination on the See also:part of the Greek geometer to be satisfied with a See also:mere approximation, were it ever so See also:close; and the unscientific agrimensor shirked the labour involved in acquiring the knowledge which was indispensable for learning trigonometrical calculations . Thus the development of the calculus of approximations See also:fell to the See also:lot of the astronomer, who was both scientific and practical' We now proceed to See also:notice briefly the contents of the Almagest .

It is divided into thirteen books . The first book, which may be regarded as See also:

introductory to the whole work, opens with a See also:short preface, in which Ptolemy, after some observations on the distinction between theory and practice, gives See also:Aristotle's See also:division of the sciences and remarks on the certainty of mathematical knowledge, " inasmuch as the demonstrations in it proceed by the incontrovertible ways of See also:arithmetic and geometry." He concludes his preface with the statement that he will make use of the discoveries of his predecessors, and relate briefly all that has been sufficiently explained by the ancients, but that he will treat with more care and development whatever has not been well understood or fully treated . Ptolemy unfortunately does not always See also:bear this in mind, and it is sometimes difficult to distinguish what is due to him from that which he has borrowed from his predecessors . Ptolemy then, in the first chapter, presupposing some preliminary notions on the part of the reader, announces that he will treat in order—what is the relation of the See also:earth to the heavens, what is the position of the oblique circle (the See also:ecliptic), and the situation of the inhabited parts of the earth; that he will point out the differences of climates; that he will then pass on to the See also:consideration of the motion of the See also:sun and moon, without which one cannot have a just theory of the stars; lastly, that he will coneider the See also:sphere of the fixed stars and then the theory of the five stars called "See also:planets." All these things—i.e. the phenomena of the heavenly bodies—he says he will endeavour to explain in taking for principle that which is evident, real and certain, in resting everywhere on the surest observations and applying geometrical methods . He then enters on a See also:summary exposition of the general principles on which his Syntaxis is based, and adduces arguments to show that the See also:heaven is of a spherical form and that it moves after the manner of a sphere, that the earth also is of a form which is sensibly spherical, that the earth is in the centre of the heavens, that it is but a point in comparison with the distances of the stars, and that it has not any motion of translation . With respect to the revolution of the earth See also:round its See also:axis, ,which he says some have held, Ptolemy, while admitting that this supposition renders the explanation of the phenomena of the heavens much more simple, yet regards it as altogether ridiculous . Lastly, he lays down that there are two See also:principal and different motions in the heavens—one by which all the stars are carried from See also:east to See also:west uniformly about the poles of the See also:equator; the other, which is See also:peculiar to some of the stars, is in a contrary direction to the former motion and takes place round different poles . These preliminary notions, which are all older than Ptolemy, form the subjects of the second and following chapters . He next proceeds to the construction of his table of chords, of which we have given an account, and which is indispensable to practical astronomy . The employment of this table presupposes the evaluation of the obliquity of the ecliptic, the knowledge of which is indeed the foundation of all astronomical See also:science . Ptolemy in the next chapter indicates two means of determining this angle by observation, describes the See also:instruments he employed for that purpose, and finds the same value which had already been found by Eratosthenes and used by Hipparchus . This " is followed by spherical geometry and trigonometry enough for the determination of the connexion between the sun's right See also:ascension, See also:declination and See also:longitude, and for the formation of a table of declinations to each degree of longitude .

Delambre says he found both this and the table of chords very exact." In book ii., after some remarks on the situation of the habitable parts of the earth, Ptolemy proceeds to make deductions from the principles established in the preceding book, which he does by means of the theorem of Menelaus . The length of the longest See also:

day being given, he shows how to determine the arcs of the See also:horizon intercepted between the equator and the ecliptic—the See also:amplitude of the eastern point of the ecliptic at the See also:solstice—for different 8 See also:Comte, Systeme de politique See also:positive, iii . 324 . 4 Cantor, Vorlesungen fiber Geschichte der Mathematik, p . 356 . $ De Morgan, in See also:Smith's See also:Dictionary of Greek and Roman See also:Biography, s.v . See also:Ptolemaeus, See also:Claudius." degrees of obliquity of the sphere; hence he finds the height of the See also:pole and reciprocally . From the same data he shows how to find at what places and times the sun becomes See also:vertical and how to calculate the ratios of gnomons to their equinoctial and solstitial shadows at See also: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 See also: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 See also:climate of Rhodes holds the mean place of the seven climates subsequently described . Delambre suspects a See also:fourth See also: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 See also:

southern limit of the habitable See also:quarter of the earth . For each parallel or climate, which is determined by the length of the longest day, he gives the See also:latitude, a principal place on the parallel, and the lengths of the shadows of the See also:gnomon at the solstices and See also:equinox . In the next chapter he enters into particulars and inquires what are the arcs of the equator which See also:cross the horizon at the same time as given arcs of the ecliptic, or, which comes to the same thing, the time which a given arc of the ecliptic takes to cross the horizon of a given place . He arrives at a See also:formula for calculating ascensional differences and gives tables of ascensions arranged by 10° of longitude for the different climates from the equator to that where the longest day is seventeen See also:hours . He then shows the use of these tables in the investigation of the length of the day for a given climate, of the manner of reducing temporal' to equinoctial hours and See also:vice versa. and of the nonagesimal point and the point of See also:orientation of the ecliptic . In the following chapters of this book he determines the angles formed by the intersections of the ecliptic—first with the See also:meridian, then with the horizon, and lastly with the vertical circle—and concludes by giving tables of the angles and arcs formed by the intersection of these circles, for the seven climates, from the parallel of Meroe (thirteen hours) to that of the mouth of the Borysthenes (sixteen hours) . These tables, he adds, should be completed by the situation of the chief towns in all countries according to their latitudes and longitudes; this he promises to do in a See also:separate See also:treatise and has in fact done in his See also:Geography . Book iii. treats of the motion of the sun and of the length of the year . In order to understand the difficulties of this question Ptolemy says one should read the books of the ancients, and especially those of Hipparchus, whom he praises " as a See also:lover of labour and a lover of truth " (&v&pl ¢rXoaovw m 0µo6 sal cktX 046) . He begins by telling us how Hipparchus was led to discover the precession of the equinoxes; he relates the observations by which Hipparchus verified the eccentricity of the See also:solar See also:orbit imperfectly known to his Chaldaean predecessors, and gives the See also:hypothesis of the See also:eccentric by which he explained the inequality of the sun's motion . Ptolemy concludes this book by giving a clear exposition of the circumstances on which the See also:equation of time depends . Ptolemy, moreover, applies Apollonius's hypothesis of the See also:epicycle to explain the inequality of the sun's motion, and shows that it leads to the same results as the hypothesis of the eccentric .

He prefers the latter hypothesis as more simple, requiring only one and not two motions, and as equally See also:

fit to clear up the difficulties . In the second chapter there are some general remarks to which See also:attention should be directed . We find the principle laid down that for the explanation of phenomena one should adopt the simplest hypothesis that it is possible to establish, provided that it is not contradicted by the observations in any important respect ? This See also:fine principle, which is of universal application, may, we think—regard being paid to its place in the Almagest—be justly attributed to Hipparchus . It is the first See also:law of the " philosophia prima " of Comte.' We find in the same See also:page another principle, or rather practical See also:injunction, that in investigations founded on observations where great delicacy is required we should select those made at considerable intervals of time in order that the errors arising from the imperfection which is inherent in all observations, even in those made with the greatest care, may be lessened by being distributed over a large number of years . In the same chapter we find also the principle laid down that the See also:object of mathematicians ought to be to represent all the See also:celestial phenomena by See also:uniform and circular motions . This principle is stated by Ptolemy in the manner which is unfortunately too common with him—that is to say, he does not give the least indication whence he derived it . We know, however, from See also:Simplicius, on the authority of See also:Sosigenes,' that Plato is said to have proposed the following 1 Kac.pucai, temporal or variable . These hours varied in length with the seasons; they were used in ancient times and arose from the division of the natural day (from sunrise to sunset) into twelve parts . Alm. ed . Halma, i . 159 .

Systbme de politique positive, iv . 173 . ' This Sosigenes, as Th . H . See also:

Martin has shown, was not the astronomer of that name who was a contemporary of See also:Julius See also: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 " See also:Eudoxus of See also:Cnidus was the first of the Greeks to take in hand hypothesis of this See also: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 See also: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, See also: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 e 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 See also:score of See also:parallax . The first thing to be determined is the time of the moon's revolution; Hipparchus, by comparing the observations of the Chaldaeans with his own, discovered that the shortest period in which the lunar eclipses return in the same order was 126,007 days and 1 See also:hour . In this period he finds 4267 lunations, 4573 restitutions of See also:anomaly and 4612 tropical revolutions of the moon less 71t q.p.; this quantity (72°) 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 See also:month contains 29 days and 31' S0" 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 See also:

astrolabe of Hipparchus, which Ptolemy made use of in following up the observations of that astronomer, and by means of which he made his most important See also:discovery, that of the second inequality in the moon's motion, now known by the name of the " See also:evection." In order to explain this inequality he supposed the moon to move on an epicycle, which was carried by an eccentric whose centre turned about the earth in a direction contrary to that of the motion of the epicycle . This is the first instance in which we find the two hypotheses of eccentric and epicycle combined . The fifth book treats also of the parallaxes of the sun and moon, and gives a description of an See also:instrument—called later by Theon the ' parallactic rods "—devised by Ptolemy for observing meridian altitudes with greater accuracy . The subject of parallaxes is continued in the See also:sixth book of the Almagest, and the method of calculating eclipses is there given . The author says nothing in it which was not known before his time . Books vii., viii. treat of the fixed stars . Ptolemy verified the fixity of their relative positions and confirmed the observations of Hipparchus with regard to their motion in longitude, or the precession of the equinoxes . The seventh book concludes with the See also:catalogue of the stars of the See also:northern hemisphere, in which are entered their longitudes, latitudes and magnitudes, arranged according to their constellations; and the eighth book commences with a similar catalogue of the stars in the constellations of the southern hemisphere . This catalogue has been the subject of keen controversy amongst modern astronomers . Some, as See also:Flamsteed and See also:Lalande, maintain that it was the same catalogue which Hipparchus had drawn up 265 years before Ptolemy, whereas others, of whom See also:Laplace is one, think that it is the work of Ptolemy himself .

The See also:

probability is that in the See also:main the catalogue is really that of Hipparchus altered to suit Ptolemy's own time, but that in making the changes which were necessary a wrong precession was assumed . This is Delambre's opinion; he says, " Whoever may have been the true author, the catalogue is unique, and does not suit the age when Ptolemy lived; by subtracting 2° 40' from all the longitudes it would suit the age of Hipparchus; this is all that is certain."' It has been remarked that Ptolemy, living at Alexandria, at which city the See also:altitude of the pole is 5° less than at Rhodes, where Hipparchus observed, could have seen stars which are not visible at Rhodes; none of these stars, however, are in Ptolemy's catalogue . The eighth book contains, moreover, a description of the milky way and the manner See also:Brandis, Schol. in Aristot. edidit acad. reg. borussica (See also:Berlin, 1836), p . 498 . e Eieayuy"} cis ra 4acv6Eteva, c. i. in Halma's edition of the works of Ptolemy, vol. iii . (" Introduction aux phenomenes celestes, traduite du grec de Geminus," p . 9), See also:Paris, 1819 . 7 This has been noticed by See also:Pliny, who says, " Multiformi haec (See also:luna) ambage torsit ingenia contemplantium, et proximum ignorari maxime sidus indignantium " (N.H., ii . 9) . Delambre, Histoire de l'astronomie ancienne, ii . 264 . of constructing a celestial globe; it also treats of the configuration of the stars, first with regard to the sun, moon and planets, and then with regard to the horizon, and likewise of the different aspects of the stars and of their rising, See also:culmination and setting simultaneously with the sun .

The See also:

remainder of the work is devoted to the planets . The ninth book commences with what concerns them all in general . The planets are much nearer to the earth than the fixed stars and more distant than the moon . See also:Saturn is the most distant of all, then See also:Jupiter and then See also:Mars . These three planets are at a greater distance from the earth than the sun.' So far all astronomers are agreed . This is not the case, he says, with respect to the two remaining planets, See also:Mercury and See also:Venus, which the old astronomers placed between the sun and earth, whereas more See also:recent writers2 have placed them beyond the sun, because they were never seen on the sun.' He shows that this reasoning is not See also:sound, for they might be nearer to us than the sun and not in the same plane, and consequently never seen on the sun . He decides in favour of the former opinion, which was indeed that of most mathematicians . The ground of the arrangement of the planets in order of distance was the relative length of their periodic times; the greater the circle, the greater, it was thought, would be the time required for its description . Hence we see the origin of the difficulty and the difference of opinion as to the arrangement of the sun, Mercury and Venus, since the times in which, as seen from the earth, they appear to complete the See also:circuit of the See also:zodiac are nearly the same—a year.4 Delambre thinks it See also:strange that Ptolemy did not see that these contrary opinions could be reconciled by supposing that the two planets moved in epicycles about the sun; this would be stranger still, he adds, if it is true that this See also:idea, which is older than Ptolemy, since it is referred to by See also:Cicero,' had been that of the Egyptians.' It may be added, as strangest of all, that this doctrine was held by Theon of Srnyrna,7 who was a contemporary of Ptolemy or somewhat See also:senior to him . From this system to that of Tycho See also:Brahe there is, as Delambre observes, only a single step . We have seen that the problem which presented itself to the astronomers of the Alexandrian See also:epoch was the following: it was required to find such a system of equable circular motions as would represent the inequalities in the apparent motions of the sun, the moon and the planets . Ptolemy now takes up this question for the planets; he says that " this perfection is of the essence of celestial things, which admit of neither disorder nor inequality," that this planetary theory is one of extreme difficulty, and that no one had yet completely succeeded in it .

He adds that it was owing to these difficulties that Hipparchus—who loved truth above all things, and who, moreover, had not received from his predecessors observations either so numerous or so precise as those that he has See also:

left—had succeeded, as far as possible, in representing the motions of the sun and moon by circles, but had not even commenced the theory of the five planets . He was content, Ptolemy continues, to arrange the observations which had been made on them in a methodic order and to show thence that the phenomena did not agree with the hypotheses of mathematicians at that time . He shower that in fact each See also:planet had two inequalities, which are different for each, that the retrogradations are also different, whilst other astronomers admitted only single inequality and the same retrogradation; he showed further that their motions cannot be explained by eccentrics nor by epicycles carried along concentrics, but that it was necessary to combine both hypotheses . After these preliminary notions he gives from Hipparchus the periodic motions of the five planets, together with the shortest times of restitutions, in which, moreover, he has made some slight corrections . He then gives tables of the mean motions in longitude and of anomaly of each of the five planets ,6 i This is true of their mean distances; but we know that Mars at opposition is nearer to us than the sun . 2 Eratosthenes, for example, as we learn from Theon of See also:Smyrna . ' Transits of Mercury and Venus over the sun's disk, therefore, had not been observed . 4 This was known to Eudoxus . See also:Sir See also:George Cornewall See also:Lewis (An See also:Historical Survey of the Astronomy of the Ancients, p . 155), confusing the See also:geocentric revolutions assigned by Eudoxus to these two planets with the