Page:Indian mathematics, Kaye (1915).djvu/46

 (c) Āryabhata's alphabetic notation also had no place-value and differed from the Brāhmī notation in having the smaller elements on the left. It was, of course, written and read from left to right. It may be exhibited thus: $$\begin{array}{l c r r r r r r r r r r}\scriptstyle{\text{Letters}}&\scriptstyle{..}&\scriptstyle{k}&\scriptstyle{kh}&\scriptstyle{g}&\scriptstyle{gh}&\scriptstyle{\dot{n}}&\scriptstyle{c}&\scriptstyle{ch}&\scriptstyle{j}&\scriptstyle{jh}&\scriptstyle{\tilde{n}}\\\scriptstyle{\text{Values}}&\scriptstyle{..}&\scriptstyle{1}&\scriptstyle{2}&\scriptstyle{3}&\scriptstyle{4}&\scriptstyle{5}&\scriptstyle{6}&\scriptstyle{7}&\scriptstyle{8}&\scriptstyle{9}&\scriptstyle{10.}\\\end{array}$$ $$\begin{array}{l c r r r r r r r r r r}\scriptstyle{\text{Letters}}&\scriptstyle{..}&\scriptstyle{\overset{t}{.}}&\scriptstyle{\overset{t}{.}h}&\scriptstyle{\overset{d}{.}}&\scriptstyle{\overset{d}{.}h}&\scriptstyle{\overset{n}{.}}&\scriptstyle{t}&\scriptstyle{th}&\scriptstyle{d}&\scriptstyle{dh}&\scriptstyle{n}\\\scriptstyle{\text{Values}}&\scriptstyle{..}&\scriptstyle{11}&\scriptstyle{12}&\scriptstyle{13}&\scriptstyle{14}&\scriptstyle{15}&\scriptstyle{16}&\scriptstyle{17}&\scriptstyle{18}&\scriptstyle{19}&\scriptstyle{20.}\\\end{array}$$ $$\begin{array}{l c r r r r r r r r r r r r r}\scriptstyle{\text{Letters}}&\scriptstyle{..}&\scriptstyle{\overset{p}{.}}&\scriptstyle{ph}&\scriptstyle{b}&\scriptstyle{bh}&\scriptstyle{m}&\scriptstyle{y}&\scriptstyle{r}&\scriptstyle{l}&\scriptstyle{v}&\scriptstyle{\acute{s}}&\scriptstyle{sh}&\scriptstyle{s}&\scriptstyle{h}\\\scriptstyle{\text{Values}}&\scriptstyle{..}&\scriptstyle{21}&\scriptstyle{22}&\scriptstyle{23}&\scriptstyle{24}&\scriptstyle{25}&\scriptstyle{30}&\scriptstyle{40}&\scriptstyle{50}&\scriptstyle{60}&\scriptstyle{70}&\scriptstyle{80}&\scriptstyle{90}&\scriptstyle{100.}\\\end{array}$$ The vowels indicate multiplication by powers of one hundred. The first vowel a may be considered as equivalent to $\scriptstyle{100^0}$, the second vowel $$\scriptstyle{i=100^1}$$ and so on. The values of the vowels may therefore be shown thus: $$\begin{array}{l c r r r r r r r r r}\scriptstyle{\text{Vowels}}&\scriptstyle{..}&\scriptstyle{a}&\scriptstyle{i}&\scriptstyle{u}&\scriptstyle{\overset{r}{.}i}&\scriptstyle{\overset{l}{.}i}&\scriptstyle{e}&\scriptstyle{ai}&\scriptstyle{o}&\scriptstyle{au}\\\scriptstyle{\text{Values}}&\scriptstyle{..}&\scriptstyle{1}&\scriptstyle{10^2}&\scriptstyle{10^4}&\scriptstyle{10^6}&\scriptstyle{10^8}&\scriptstyle{10^{10}}&\scriptstyle{10^{12}}&\scriptstyle{10^{14}}&\scriptstyle{10^{16}}\\\end{array}$$ The following examples taken from Āryabhata's Gītikā illustrate the application of the system: The notation could thus be used for expressing large numbers in a sort of mnemonic form. The table of sines referred to in paragraph 9 above was expressed by Āryabhata in this notation which, by the way, he uses only for astronomical purposes. It did not come into ordinary use in India, but some centuries later it appears occasionally in a form modified by the place-value idea with the following values: