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

260 First half of the verse 40. To obtain the shadow S and K the Chāyakarṇa of any shadow (H sin z × 12) / (H cos z) = S and (R × 12) / (H cos z) = K. Comm. (Ref. fig. 44). (12. H sin z) / (H cos z) = 12 tan z = S. Also 12 / K = cos z = (H cos z) / R so that (12 R) / (H cos z) = K. The Hindu method of looking at this through the similarity of ΔS OM☉ and Ogn the gnomonic triangle. is as follows. ☉M is called Mahā-Śaṅku ie. H cos z ; ☉L is Dṛkjya or H sin z = OM. 12 / (H cos z) = S / (H sin z) so that S = 12 H sin z/H cos z. Also, On / O☉ = 12 / (H cos z) ie. K / R = 12 / (H cos z) ∴ K = (12 R) / (H cos z) . It will be noted that fig. 44 pertains to any vertical plane. Second half of Verse 40. The Dinārdha-Karṇa is equal to (R × k) / Hṛti where k is the Viṣuvat-Karṇa. Comm. The formula is derived through twice apply- ing the rule of three or what is the same, through the similarities of two sets of triangles From fig. 39, O₁A / Aa = Hṛti / Dinārdha-Śaṅku = k / 12 (a) and from fig. 44 On / O☉ = 12 / Dinārdha-Śaṅku = K / R (b) where K is the required Chāyākarṇa. Dividing (a) by (b) Hṛti / 12 = k / 12 × R / K ∴ K = kR / Hṛti