88 OBSERVATIONS on the 



to have been fct down is greater than y, the integer next greater 

 is placed in the table. Thus the fine 3°. 45' being, when accu- 

 rately expreffed in their way, 224'. 49", is put down 225'; and 

 fo of the reft. The numbers, therefore, in tliefe tables, are only 

 fo for exacl as never to differ more than half a ininute from tlie 

 truth, and this very limited degree of accuracy gives no doubt 

 to their trigonometry the appearance of an infant fcience : But 

 when, on the other hand, we consider the principles and rules of 

 their calculations, rather than the numbers adlually calculated, 

 we find the marks of a fcience in full vigovir and maturity : and 

 we will acknowledge, that the Hindoo mathematicians did not 

 fatisfy themfelves with the degree of accuracy above mentioned, 

 from any incapacity of attaining to greater exacflnefs. 



Their rules for conftrudling their tables of fines, may be re- 

 duced to two, VIZ. the one for finding the fine of the leaft arch 

 in the table, that of 3°. 45', and the other for finding the fines 

 of the mi;ltiples of that arch, its triple, quadruple, l^c. Both of 

 thefe Mr Davis has tranflated, judging very rightly, that it was 

 impoflible to give two more curious fpecimens of die geometri- 

 cal knowledge of the Hindoo philofophers ; the firfl is extracted 

 from a commentary on the Surya Siddhanta j the other from 

 the Surya Siddhanta itfelf. 



6. With refpedl to the firfl, the method proceeds by the con- 

 tinual bife(5lion of the arch of 30°, and correfpondent extrac- 

 tions of the fquare root, to find the fine and co-fine of the half, 

 the fourth part, the eighth part, and fo on, of that arch. The 

 rule, when the fine of an arch is given, to find that of half the 

 arch, is precifely the fame with our own : " The fine of an arch 

 being given, find the co-fine, and thence the verfed fine, of the 

 fame arch : then multiply half the radius into the verfed fine, and 

 the fquare root of the produd is the fme of half the given arch." 

 Now, as the fine of 30°, was well known to thofe mathemati- 

 cians to be half the radius, it was of confequence given : thence, 



by 



