सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

107 radius is taken to be 120, and the corresponding H sines are called Laghu-jyās or simpler H sines used where great accuracy is not required. Śrīpati took the radius to be 3270 units in addition to 120 as did Brahmagupta. Muñjāla took 488′ and some others some other values also. Out of these 3438′ alone has a right significance (Vaṭēśwara took 3272) Bhāskara says under verses 1 to 5 under Jyotpatti- Vāsanā in the Golādhyāya that the Hindu astronomers got the values of the main H sines of 30°, 45°, 60°, 18° and 36° by inscribing regular polygons in a circle. They are called the pancha-jyakās or the fundamental H sines. From these the others were calculated according to the methods given by Bhaskara as follows. To start with, we have the fundamental formula H sin²θ + H cos²θ = R² (from fig-6) I In addition to this formula, Bhāskara gives another formula (verse 10, 11 Ibid) H Sin θ/2 = √(H sin² θ + H vers² θ) = √(½ R. H vers θ) II In the commentary under the above verses, he has given the method by which II was obtained (Ref. fig. 7) BM = H sin θ where AÔB = θ ; also AM = H vers θ and AB² = AM² + MB². Let N be the mid-point of AB. ½AB = AN = H sin θ/2 ∴ H sin θ/2 = ½AB = ½ √(AM² + MB²) == ½ √(H sin² θ + H vers² θ) which proves the first part of II. Again from the right—angled triangle ABC, AB² = AM · AC = H vers θ × 2R ∴ H sin θ/2 = ½AB = ½ √(2R H vers θ) = √((R H vers θ) / 2) which proves the second part.