Page:Indian mathematics, Kaye (1915).djvu/21

did Ptolemy, Pauliśa divides it into 120 parts; for as $\scriptstyle {\sin {\frac {a}{2}}={\frac {{\text{chord }}a}{2}}}$ this division of the radius enabled him to convert the table of chords into sines without numerical change. Āryabhata gives another measure for the radius (3438′) which enabled the sines to be expressed in a sort of circular measure.

We thus have three distinct stages:
(a) The chords of Ptolemy, or $\scriptstyle {{\text{ch'd }}\alpha }$, with r=60
(b) The Pauliśa sine or $\scriptstyle {\sin {\frac {\alpha }{2}}={\frac {{\text{ch'd }}\alpha }{2}}}$, with r=120
(c) The Aryabhata sine or $\scriptstyle {\sin {\frac {\alpha }{2}}={\frac {3}{\pi }}\cdot {\frac {{\text{ch'd }}\alpha }{2}}}$, with r=3438′
To obtain (c) the value of $\scriptstyle {\pi }$ actually used was $\scriptstyle {{\frac {600}{191}}(=3.14136)}$

Thus the earliest known record of the use of a sine function occurs in the Indian astronomical works of this period. At one time the invention of this function was attributed to el-Battâni [A.D. 877—919] and although we now know this to be incorrect we must acknowledge that the Arabs utilised the invention to a much more scientific end than did the Indians.

In some of the Indian works of this period an interpolation formula for the construction of the table of sines is given. It may be represented by $\scriptstyle {\Delta _{n+1}=\Delta _{n}-{\frac {{\text{Sin }}n.\alpha }{{\text{Sin }}\alpha }}}$ where $\scriptstyle {\Delta _{n}={\text{Sin }}n.\alpha -{\text{Sin }}(n-1)\alpha }$. This is given ostensibly for the formation of the table, but the table actually given cannot be obtained from the formula.

10. Āryabhata.—Tradition places Āryabhata (born A.D. 476) at the head of the Indian mathematicians and indeed he was the first to write formally on the subject.^[1]

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1. Although Āryabhata's Ganita, as first published by Kern, is generally accepted as authentic, there is an element of doubt in the matter.