Page:Indian mathematics, Kaye (1915).djvu/32

 with diagonals 63 and 56, etc. He also introduces a proof of Ptolemy's theorem and in doing this follows Diophantus (iii, 19) in constructing from triangles (a, b, c) and $$\scriptstyle{(\alpha,~\beta,~\gamma,)}$$ new triangles $$\scriptstyle{(a\gamma,~b\gamma,~c\gamma,)}$$ and $$\scriptstyle{(\alpha c,~\beta c,~\gamma c,)}$$ and uses the actual examples given by Diophantus, namely (39, 52, 65) and (25, 60, 65). 18. An examination of the Greek mathematics of the period immediately anterior to the Indian period with which we are now dealing shows that geometrical knowledge was in a state of decay. After Pappus (c. A.D. 300) no geometrical work of much value was done. His successors were, apparently, not interested in the great achievements of the earlier Greeks and it is certain that they were often not even acquainted with many of their works. The high standard of the earlier treatises had ceased to attract, errors crept in, the style of exposition deteriorated and practical purposes predominated. The geometrical work of Brahmagupta is almost what one might expect to find in the period of decay in Alexandria. It contains one or two gems but it is not a scientific exposition of the subject and the material is obviously taken from western works.