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 is equal to the radius, and therefore contains 6o parts. The sides of the regular pentagon and decagon inscribed in a circle, which are the chords subtending arcs of and respectively, are found by means of propositions in Euclid's Book xiii. ; expressed in terms of the "parts" of the diameter, they are found to be (nearly) 70P 32' 3" and 37P respectively. (It should be noted that the ratio of the chord subtending any angle to the diameter of the circle is the same as the ratio of half the chord to the radius; hence, expressed on this system, the chord of any angle is equivalent to the trigonometrical sine of half the angle.) Now "Ptolemy's Theorem" enables us to deduce from the chords of two angles the chord of their sum or difference; the theorem is equivalent to the formulae Sin (A +B) = SinA Cos B+Cos ASinB. Another proposition which Ptolemy gives, shows how to deduce from the chord of any angle the chord of half the angle; it is equivalent to the formula Sin' 10=1(1 —Cos0). Yet another proposition gives the equivalent of As we know the lengths of (crd. and (crd. we deduce (crd. ; from this we deduce successively the chords of and and again from these (by "Ptolemy's Theorem") the chords of Ior etc.; we have there fore a table of chords going by gradations of Ir. It remains to find the chord of 0, and thence the chord of in order to obtain a table going by gradations of To find (crd. Ptolemy uses a clever method of "interpolation" based on a proposition already assumed as known by Aristarchus of Samos (fl. 28o B.c.) and equivalent to the trigonometrical formula Sin a /Sin33 < a/13 (where 1.7r> a>0). It follows from the proposition in question that (crd. : (crd. < i : 4 and (crd. : (crd.
: r, whence 4- (crd. (crd. (crd. ). Now Ptolemy has found (crd. to be op 47' 8" and (crd. ) to be IP 34' 15". Thus 4 of the former is it 2' 50" nearly (actually IP 2' 50i") and of the latter is IP 2' 50". Thus (crd. 0), being both greater and less than IP 2' 50", may be taken to be It' 2' 50" as nearly as possible. Ptolemy deduces that (crd. is very nearly op 31' 25"; and he is therefore in a position to complete his table going by gradations of 4 a degree. No wonder that De Morgan called this exposition "one of the most beautiful in the Greek writers." Ptolemy further uses the same method of proportional increase ("interpolation") to find the chords of anglqs containing a number of odd minutes between o' and 3o', inserting in the table opposite each chord, in a third column, of the excess of that chord over the one before. One particular result is worth quoting. Since (crd. is found to be IP 2' 50", the whole circumference =36o X (IP 2' 5o") nearly, and, the length of the diameter being i2oP it follows that it =3 X (I d-A-1 a) or which is equivalent to ir =3.14166.
Trigonometry, as 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.
Ptolemy obtains all that he wants in the shape of spherical trigonometry from the one fundamental proposition known as "Menelaus' Theorem" applied to the sphere, a theorem concerning the segments into which the sides of a spherical triangle are cut by a transversal. Menelaus gave the proof in his Sphaerica, Ptolemy gives it with two simple propositions on which it depends in the Syntaxis i. 13 ; the proof assumes as known the corre sponding proposition for a plane triangle. The theorem for the sphere, though called by the name of Menelaus, must certainly have been known to Hipparchus, while the theorem for the plane triangle at least was probably known to Euclid. When Ptolemy requires the equivalent of one of our formulae in the solution of spherical triangles, he does not quote such propositions but proves them afresh each time by means of Menelaus' theorem. These various applications include the equivalent of such formulae for a right-angled spherical triangle as (I) Sina= Sinc SinA, (2) Tana =Sinb TanA, (3) Cosc = Cosa Cosb, and (4) Tanb =Tans CosA.
We are, above all, indebted to Ptolemy for very full particulars of observations and investigations by Hipparchus as well as of the earlier observations which Hipparchus recorded, e.g., that of a lunar eclipse in 721 B.C. The work of Ptolemy is, in general, evidently based upon Hipparchus to a degree that makes it far from easy to identify Ptolemy's own contributions to the subject, except where he propounds a definite theory of the motions of the five planets, for which Hipparchus had only collected material in the shape of observations made before his time or by himself. The "Syntaxis."—The contents of the Syntaxis can here be only very briefly indicated. Book i. contains the indispensable preliminaries to the study of the Ptolemaic system, treating of the earth, its spherical shape and its position in the centre of the universe, the circular movements of the heavenly bodies, the propositions required for the preparation of the Table of Chords, this table itself, the angle of obliquity of the ecliptic and Ptolemy's own method of determining it, and, lastly, so much of spherical geometry and trigonometry as is necessary for the determination of the connection between a star's right ascension, declination, latitude and longitude, and for the formulation of a table of declinations to each degree of longitude. Book ii. contains matter similar to that of Autolycus' On the Moving Sphere, i.e., problems on the sphere, with special reference to the difference between various latitudes, the length of the longest day at any degree of latitude, and the like. Book iii. treats of the length of the year and the motion of the sun on the eccentric and epicycle hypothesis. Ptolemy observes that, to understand the difficulties of the question, we should read the treatises of the ancients, and especially those of Hipparchus, "that enthusiastic worker and lover of truth." He mentions here Hipparchus' dis covery of the precession of the equinoxes (a subject to which he returns in viii. 2). In chapter 1, Ptolemy also lays down some general principles which are worth quoting, namely, (I) that, in seeking to explain phenomena, we should adopt the simplest pos sible hypothesis, provided that it is not contradicted in any im portant respect by the observations; (2) that in investigations depending on observations in which great delicacy is required, we should select those made at long intervals of time in order that the error due to the imperfection of our instruments may be lessened by being distributed over a large number of years. Book iv. deals with the length of the months and the motions of the moon, and Book v. with the same subject continued, the con struct ion of the astrolabe, the diameters of the sun, the moon and the earth's shadow, the distance of the sun, and the dimensions of the sun, moon and earth. Book vi. is on the conjunctions and oppositions of the sun and moon, solar and lunar eclipses and the periods of their recurrence. Books vii. and viii. treat of the fixed stars, including precession. Book vii. ends with the catalogue (arranged under constellations) of the stars in the northern hemis phere, in which are entered the latitudes, longitudes and magni tudes; Book viii. begins with a similar catalogue for the southern hemisphere; the two catalogues include 1,022 separate stars as compared with the 85o or more included in Hipparchus' catalogue. Book viii. also describes the Milky Way and the construction of a celestial globe. Books ix. to xiii. are devoted to the planets. Book ix. begins with general remarks on the order of the planets, the difficulties in the way of framing a theory of the subject, the periods of revolution of the five planets, the different hypotheses that are possible ; then (ch. 7) Ptolemy passes to the separate case of Mercury. Book x. treats of Venus; Book xi. of Jupiter and Saturn; Book xii. deals with the stationary points and retro gradations of each of the planets, and with the greatest elonga tions of Mercury and Venus; Book xiii. considers the motions of the planets in latitude, the inclination of their orbits and its magnitude.