ARITHMETIC GEOMETRY ALGEBRA. 313 



culation, the advantages of the decimal notation are placed 

 in a striking light. The work begins thus : " Having 

 bowed to the deity, whose head is like an elephant, whose 

 feet are adored by gods, who, when called to mind, restores 

 his votaries from embarrassment, and bestows happiness on 

 his worshippers, I propound this easy process of computa- 

 tion, delightful by its elegance, perspicuous with words, 

 concise, soft, and correct,!and pleasing to the learned." The 

 definitions are given in the form of an introduction, and are 

 followed by an invocation : " Salutation to Ganesa, re- 

 splendent as a blue and spotless lotus, and delighting in the 

 tremulous motion of the dark serpent which is continually 

 twining within his throat." The rules of arithmetic are 

 then delivered in verse, and addressed to Lilavati, a young 

 and charming female, who appears to be receiving the in- 

 structions of the author, and to whom the examples of the 

 rules are usually proposed as questions to be resolved. 



The arithmetic is followed by a treatise on geometry, in- 

 ferior in excellence certainly to the treatise on algebra, yet 

 well deserving of attention. We have here the celebrated 

 proposition, that the square on the hypotenuse of a right- 

 angled triangle is equal to the squares on the sides contain- 

 ing the right angle ; and other propositions which form part 

 of the system of modem geometry. There is one proposi- 

 tion remarkable, namely, that which discovers the area of a 

 triangle when its three sides are known. This does not 

 seem to have been known to the ancient Greek geometers. 



It is a most singular circumstance that, with such a body 

 of mathematical science as has descended from a very re- 

 mote period to the present time, there is almost an entire 

 want of all analysis or synthetic demonstration ; for this it 

 is not easy to assign a cause. Some learned men in Europe 

 have supposed, that the entire ignorance of the modern Hin- 

 doos of the demonstrations of their rules is a satisfactory 

 proof that they are not the inventors of the science ; or else 

 that the knowledge of the mathematics has declined so much 

 that they have no longer any idea of the fundamental prin- 

 ciples and the practical operations which they have been 

 taught by their ancestors. 



The algebra of the Hindoos comes next to be considered. 

 We have seen that the age in which Arya Bhatta lived was 

 probably not very different from that of Diophantus. It 



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