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78 PAÑCASIDDHĀNTIKĀ IV. 2 The ancients however expressed the sines in minutes-length or, more accurately, in minutes and seconds-lengths, the maximum sine called Trijyā (meaning ‘the sine of three signs’, i.e. 90°), occur- ring separately in the work to make up the ratio. This is the way in which they conceived the sine (meaning ‘bow-string’ from its Sanskrit equivalent śiñjinī, synonymous with jyā). In Fig. 1-b. A₃ E F₃ D is the circumference of the circle, centre B. A part of the circumference like ADF, A₁D F₁, etc. is called dhanus (literally, ‘bow’) or arc. Fig. IV. 1-b The straight lines ACF, A₁C₁F₁, etc. forming the ‘bow-strings’ of the respective ‘bows’ are the jyās or full sines. But in actual practice, the halves of the full sines AC, A₁C₁, etc. above are used with the name of ‘sines’, with respect to the half-bows or arcs, AD, A₁D₁, etc. Because the arcs AD etc. are as the angles ABD etc., the sines AC etc. are spoken of with respect to the angles ABD (= ABC) etc. also. Thus, AC is the sine of ∠ ABD or arc AD, A₁C₁ is the sine of ∠ A₁BD or arc A₁D₁ and so on. It is this connection of the sine with the arc that has given it the nature of a length, which is expressed in minutes and seconds on account of the connection of the arc with the angle at the centre. It may be mentioned here that CD, C₁D, C₂D etc., appearing like the arrows on the respective bow-strings, are called śara (meaning ‘arrow’). If A₃BD is a right angle, i.e. three signs, then, obviously, A₃B is the sign of this angle, i.e. it is the sine of three signs, and therefore called trijyā. Its length is clearly half the diameter A₃BF₃, i.e. the radius, equal to 120′. Now, let the angle ABD be equal to one sign, i.e. 30°. ABD = DBF = 30°. ∴ ∠ ABF = 60°. AB = BF, being radii. ∴ ∠ BAF = ∠ BFA = 60°. Thus ABF is an equilateral triangle, and AF = AB = 120′. ∴ AC = AF/2 = 60′. Thus sine 30° = 60′. Its karaṇī is its square, viz. (60′)² = 3600, the karaṇī of Meṣa as mentioned by the text. Then, let ∠ A₂BD be equal two signs, or 60°. A₂BD = DBF₂ = 60°. ∴ C₂ is a right angle. So the karaṇī of 2 signs = A₂C₂² = A₂B² − BC₂² = A₂B² − AC², (∵ △ A₂B C₂ ≡ △ BAC), = 120² − 60² = 10,800, A₂B being the radius. This also agrees with what the text says. (The square root of 10800 minutes, i.e. 103′ 55″, is the sine of 2 signs, which agrees with the value given in the table.)