पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

80 PAÑCASIDDHĀNTIKĀ IV. 5 I. The 4th karaṇī = ¼ [sin² 30° + (120 − sin 60°)²] = ¼ [60² + (120 − 103′ 55″.33)²] = ¼ [3600 + (16′ 4″.67)²] = ¼ (3600 + 258 97/144) = ¼ (3858 97/144) = 964 385/576 ∴ the fourth sine, i.e. sine 15° = √964 385/576 = 31′ 4″. II. The 4th karaṇī = 60 (120 − sin 60°) = 60′ (120′ − 103′ 55″.33) = 60′ × 16′ 4″.67 = 964 385/576 From this, sin 15° = √964 385/576 as before = 31′ 4″. We shall prove the first formula geometrically, and derive the second from the first. Fig. IV. 2 In Fig. 2, FBD = θ, and DF is its arc of which the sine wanted is DE. DE² = the wanted karaṇī (i.e. sine² θ). DFA = 2DF. ∴ DE = ½DA. ∴ sin² θ = DE² = ¼DA² = ¼(AC² + CD²). Now, ∵ AC = sine 2 arc FD = sin 2θ, AC² = sin² 2θ; and ∵ CD² = (BD − BC)² = (BD − AG)² = [120′ − sin (90° − 2θ)]², sin²θ = ¼ (AC² + CD²) = ¼ [sin² 2θ + {120′ − sin (90° − 2θ)}²] From this, sinθ = √sin²θ. From I, we can derive II thus: ¼ [sin² 2θ + {120′ − sin (90° − 2θ)}²] = ¼ {sin² 2θ + 120² + sin² (90° − 2θ) − 2 × 120 × sin (90° − 2θ)} = ¼ {120² + 120² − 2 × 120 × sin (90° − 2θ)} (∵ sin² 2θ + sin² (90° − 2θ) = sin² 2θ + Cos² 2θ = radius²) 2 × 120′ = ──────── {120′ − sin (90° − 2θ)} 4 = 60′ {120′ − sin (90° − 2θ)} Now, of TS, Thibaut alone proves the formula, while Sud. Dvivedi uses it. The form of the first formula given by them differs from that given by us, and is as follows:- sin²θ = (½ sin²2θ) + [½ {120′ − sin (90° − 2θ)}]² Though their formula is correct, it entails more work, and to give the formula in this form they have made many unwarranted changes in the already correct readings. We have made only one correc- tion, and that grammatical, for the sake of syntax, viz. dhanurdviguṇapadāt into dhanurdviguṇam padāt, which entails the occurrence of an extra syllable, which can be explained, as before, by rules of prosody. Or, let the reading be śeṣe tviṣṭe dhanurdvi-guṇam padāt projjhya etc. Now follow six verses giving the sines, computed by the author himself.