172 REMARKS on ths 



As fome of the former rules, therefore, have ferved to fix 

 the time, fo does this, in fome meafure, to afcertain the place 

 of its invention. It is the limplification of a general rule, 

 adapted to the circumflances of the torrid zone, and fuggefted 

 to the aftronomers of Hindoftan by their peculiar fituation. It 

 implies the knowledge of the circles of the fphere, and of fphe- 

 rical trigonometry, and perhaps argues a greater progrefs in 

 mathematical reafoning, than a theorem that was perfedlly ac- 

 curate would have done. The firft geometers muft naturally 

 have dreaded nothing fo much as any abatement in the rigour 

 of their demonflrations, becaufe they would fee no limits to the 

 error and uncertainty, in which they might, by that means, be 

 involved. It was long before the mathematicians of Greece 

 underflood how to fet bounds to fuch errors, and to afcertain 

 their utmoft extent, whether on the fide of excefs or defedl ; 

 in this art, they appear to have received the firft lelTons fo late 

 as the age of Archimedes. 



47. The 



time, reckoned after the Indian manner, a; =. ^Jl.g^l {tan.O x — ; — h tan.O' X 



G 



+ &c.) 



6G' 



If z: 24°, then iaii. — -4452, and the firft term of this formula gives x 



372.957 X -"^ — — -^^ — , which is the fame with the rule of the Brahmins. 



G G 



For that rule, reduced into a formula, is 2x ~ — — ^( 1- + 1 3 



G \ 3 15 9 / 



fi25 2c65 



-^ , or a; = — - — 



G G 



g 



They have therefore conoputed the coefficient of with fufficient accuracy ; the 



G 



error produced by the omillion of the reft of the terms of the feries will not exceed 1', 



even at the tropics, but, beyond them, it increafes faft, and, in the latitude of 45?, 



would amount to 8'. 



