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

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above differences in the reverse order give the H versines.
The rest of the contents of the verses is simple, the proces-
ses being based on the ‘Rule of three’.
Verse 16. Rectification of the next H sine difference
known as Bhōgya—Khanda.
The difference of the previous and the following H
sine —differences being multiplied by the remaining degrees
and divided by 20, the result is subtracted from the arith-
metic mean of the previous and following H sine-differences
to give the rectified H sine —difference, in question.
Comm. This is a formula for interpolation which
agrees with the interpolation formula given by Ball in his
spherical astronomy on page 18 in the form y = yo + x/h
(y₁ — yo) + x (x — h) / 2h² (y₂ — 2y₁ + yo). This formula is
a re-statement of the formula enunciated by Brahmagupta
in his work Brahma Sphuta Siddhānta as well as Uttara-
Khandakhādya in the form “गतभोग्यखण्डकान्तरदलविकलवधात्
शतैर्नवभिराप्तैः, तद्युतिदलं युतोनं भोग्यादूनाधिकं भोग्यम्” wherein in
the place of ten-degree-interval, a fifteen-degree-interval
ie 900'-interval was taken. Rule of three is a linear for-
mula of interpolation, whereas the above is a quadratic
formula reflecting much credit on the mathematical genius
of Brahmagupta.
We shall now see how the formula is applied and what
mathematical significance it has. Suppose it is required to
find the H sine of 24° from the previous table of H sine —
differences given for intervals of 10° — from the table H
sin 10° = 21, H sine 20 = 41, H sine 30 = 60 where R is
taken to be 120. Now to find H sin 24°, we are asked to
rectify the next H sine —difference namely 19', where the
table is 21, 20, 19 etc. As a first approximation, applying
rule of three H sin 24° = 41 + ⁴/₁₀ × 19=48·6=48'—36"