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.