Vol. Ill, No. 7.] J^otes on Indian Mathematics. 497 



-a rule he gives for the construction of right-angled triangles with 

 rational sides. This rule he gives twice over without knowing it; 

 firstin paragraph 35, section iv., chapter xii., and again in paragi^apli 

 38 of chapter xviii, (Colehrooke's edition). In the first case he 

 gives directions for the construction of ' half a rectangle ' and in the 

 second for an isosceles triangle (a douhle right-angled triangle). 

 The two rules are mathematically identical but worded very dif- 

 ferently. The only possible explanation of their occurrence 

 appears to be that Brahmacrupta took them from two different 

 works which he used for his compilation. The rule in question 

 is a generalisation of the two rules that Proclus attributes to 

 Pythagoras and to Plato ; and had always been a proposition par- 

 ticularly interesting to the Greeks,^ That Brahmagupta was the 

 original generaliser is altogether improbable ; no one familiar 

 with his mathematics could possibly conceive it unless, like 

 ijhasles, they had been misled by Colebrooke and others. As a 

 matter of fact the formula was given by Alkharki the translator 



of Diophantus. 



Next come the * indeterminate ' or Diophantine equations. 

 The connection between Brahmagupta and Diophantus was dis- 

 cussed by Colebrooke but not from the most enlightened point 

 of view. We now know that Diophantus lived prior to Brahma- 

 gupta ; that his favourite subject was indeterminate equations of 

 the second degree ; that parts of his works were lost ; tliat his 

 mathematical work was carried on by Hypatia atid others. 

 Brahmagupta gives us numbers of indeterminate equations; he 

 gives a method of solving quadratics the same as that employed 

 by Diophantus, while the other method he gives is practically the 

 same as that of Nonius ; he uses the sexagesimal notation and 

 many Greek mathematical terms, and it can be stated with perfect 

 accuracy that no section of mathematics is touched upon by him 

 that had not been dealt with by the Greeks.^ 



In the work quoted above it is intimated that M. ibn Musa, 

 ^nd through him the Arabians generally, derived their mathemati- 



cal knowledge from the Hindus. Gow* also states that *'in the 

 time of Al-Mansur (754 — 775) the Arabian commerce with India 

 had brought to the knowledge of Bagdad the Siddhanta. This 

 also was translated and the Arabs acquii^ed the numerical »ym- 

 i)ols.'* On what authority such statements are made I cannot say, 

 but I suspect they may be traced back to Colebrooke, who, how- 

 ever, is not quite so culpable in this matter as his commentators 

 Colebrooke states that M, ibn Ibrahim Alfarazi translated or 

 -adapted an Indian work on astronomy, and this work of Alfarazi 

 was known as Sind-hind or Hind-sind, "It signifies," he says, 



1 The conclaaions given in this section are based npon a much fuller 

 investigation than is here indicated : it is proposed to give, on another occa- 

 fiioTij a more detailed expoHition of the resnlts of this investigation. 



* History of Greek Mathematics. 



