PTOLEMY into 120 equal parts, and then for the subdivisions of these he em- ploys the sexagesimal method as most convenient in practice, i.e., he divides each of the sixty parts of the radius into sixty equal parts, and each of these parts he further subdivides into sixty equal parts. In the Latin translation these subdivisions become " partes minute primae" and "partes minute secundse," whence our "minutes" 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 Chal- dseans. Nor did the formation of the table of chords originate with Ptolemy ; indeed, Theou 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 70? 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 in- scribed in a circle : The rectangle under the diagonals is equal to the sum of the rectangles under the opposite sides. By means of this theorem the chord of the sum or of the difference of two arcs whose chords are given can be easily found, for we have only to draw a diameter from the common vertex of the two arcs the chord of whose sum or difference is required, and complete the quadri- lateral ; 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, 1, and three-fourths of 1, and lastly, the chords of 4^, 7^, 9, 10^, &c., all those arcs, namely, as Ptolemy says, which being doubled are divisible by 3. Performing the calculations, he finds that the chord of 1J contains approximately IP 34' 55", and the chord of three-fourths of 1 contains OP 47' 8". A table of chords of arcs increasing by 1^ can thus be formed ; but this is not suffi- cient for Ptolemy's purpose, which was to frame a table of chords increasing by half a degree. This could be effected if he knew the chord of one-half of 1 ; but, since this chord cannot be found geometrically from the chord of 1^, inasmuch as that would come to the trisection of an angle, he proceeds to seek in the first place the chord of 1, which he finds approximately by means of a lemma of great elegance, due probably to Apollonius. It is as follows : If two unequal chords be inscribed in a circle, the greater will be to the less in a less ratio than the arc described on the greater will be to the arc described on the less. Having proved this theorem, he proceeds to employ it in order to find approximately the chord of 1, which he does in the following manner chord 60' 60 . 4 , . 4 45' 1 ' "* 3' ' Ch rd < 3 ch rd 45 ' ' again chord 90' 90 . -:^,i.e. t 60' ' 3 , , , 2 , , . rt , , .'.chord 1 >-= chord 90. 2' 3 chord 60' For brevity we use modern notation. It has been shown that the chord of 45' is OP 47' 8" q.p., and the chord of 90' is IP 34' 15" q.p. ; hence it follows that approximately chord 1 <1P 2' 50" 40'" and > IP 2' 50*. Since these values agree as far as the seconds, Ptolemy takes IP 2' 50" as the approximate value of the chord of 1. The chord of 1" 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 to 180 ; in the second he gives the values of the chords of these arcs in parts of which the diameter contains 120, the subdivisions being sexagesimal ; and in the third he has inserted the thirtieth parts of the differences of these chords for each half-degree, in order that the chords of the intermediate arcs, which do not occur in the table, may 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. 1 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 piano triangles is not treated separately in that work. To resolve a plane triangle the Greeks supposed it to be inscribed in a circle ; they must therefore have known the theorem which is the basis of this branch of trigonometry The sides of a triangle are proportional to the chords of the double arcs which measure the angles opposite to those sides. In the case of a right-angled triangle this theorem, together with Eucl. I. 32 and 47, gives the complete solution. Other triangles were resolved into right- angled triangles by drawing the perpendicular from a vertex on the opposite side. In one place (Aim., vi. c. 7 ; vol. i. p. 422, ed. Halma) Ptolemy solves a triangle in which the three sides arc 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 inci- dentally 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 sec- tion. Lastly, it may be mentioned that Ptolemy (Aim., vi. 7 ; vol. i. 8 30 p. 421, ed. Halma) takes SP 8' 30", i.e., 3 + -^ + ^^ =3'1416, as the value of the ratio of the circumference to the diameter of a circle, and adds that, as had been shown by Archimedes, it lies between 3f and 3f . The foundation of spherical trigonometry is laid in chapter xi. on a few simple and useful lemmas. The starting-point is the well- known theorem of plane geometry concerning the segments of the sides of a triangle made by a transversal : The segments of any side are in a ratio compounded of the ratios of the segments of the other two sides. This theorem, as well as that concerning the inscribed quadrilateral, was called after Ptolemy naturally, indeed, since no reference to its source occurs in the Almagest. This error was corrected by Mersenne, who showed that it was known to Menelaus, an astronomer and geometer who lived in the reign of the emperor Trajan. The theorem now bears the name of Menelaus, though most probably it came down from Hipparchus ; Chasles, indeed, thinks that Hipparchus deduced the property of the spheri- cal triangle from that of the plane triangle, but throws the origin of the latter further 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 for- mulae for the solution of right-angled spherical triangles can be easily i Ideler has examined the degree of accuracy of the numbers in these tables and finds that they are correct to five places of decimals. a On the theorem of Menelaus and the rule of six quantities, see Chasles, Aperyu Historique sur I'Origine et Developpement des Methods en Giometrie, note vi. p. 291.
