TRIGONOMETRr of the BRAHMINS. 



99 



of the two extreme arches, as the fine of the difference of the 

 arches to the fine of twice that difference." Tliis theorem is well 

 known in Europe ; it is juftly reckoned a very remarkable pro- 

 perty of the circle ; and it.ferves to fliew, that the numbers in a 

 table of fines conflitute a feries, in which every term is formed 

 exadly in the fame way, from the two preceding terms, viz. by 

 multiplying the lafl by a certain, conftant number, and fub- 

 trading the laft but one from the produdl. 



9. Now, it is worth remarking, that this property of the tabic 

 of fines, which has been fo long known in the Eaft, was not ob- 

 ferved by the mathematicians of Europe till about two hundred 

 years ago. The theorem, indeed, concerning the circle, from 

 which it is deduced, under one fliape or another, has been 

 known to them from an early period, and may be traced up to 

 the writings of Euclid, where a propofition nearly related to it 

 forms the 97th of the Data : " If a ftraight line be drawn with- 

 in a circle given in magnitude, cutting off a fegment containing 

 a given angle, and if the angle in the fegment be bifeded by a 

 ftraight fine produced till it meet the circumference ; the 

 ftraight fines, which contain the given angle, fhall both of them 

 together have a given ratio to tlie fl:raight line which bifeds the 

 angle." This is not precifely the fame with the theorem which 

 has been fhewn to be the foundation of the Hindoo rule, but 

 differs from it only by affirming a certain relation to hold 

 among the chords of arches, which the other affirms to hold 

 of their fines. It is given by Euclid as ufeful for the con- 

 ftrudlion of geometrical problems ; and trigonometry being 

 then unknown, he probably did not think of any other applica- 

 tion of it. But what may feem extraordinary is, that when, 

 about 400 years afterwards, Ptolemy, the aftronomer, con- 

 ftruded a fet of trigonometrical tables, he never confidered Eu- 

 clid's theorem, though he was probably not ignorant of it, as 

 ■having any connedion with the matter he had in hand. He, 



M 2 therefore, 



