98 OBSERVATIONS on the 



which, added to the third jyapinda, gives the fourth; and fo on 

 unto the twenty-fourth or laft." 



It is not immediately obvious on what geometrical principle 

 this rule is founded, but a flight change in the enunciation 

 will remove the difficulty. The remainder, it muft be ob- 

 ferved, from which the qviotient is always directed to be taken 

 away, is the difference between the two fines lafl computed; and 

 hence th^ rule may be exprefTed more generally : Divide any 

 line by 225, and fubtradl the quotient, or the integer neareft 

 the quotient, from the difference between that fine and the fine 

 next lefs ; the remainder is the difference between the fame fine 

 and the fine next greater ; and therefore if it be added to the 

 former, will give the latter. If then, (fig. 3. PI. I.), GA, GC, GE, 

 be three contiguous arches in the table, of which the dif- 

 ferences AC, CE, of confequence are equal, and of which 

 the fines are AB, CD, and EF, the rule, as laft ftated, gives us 



CD 



CD— AB , for the difference between CD and EF, and 



225' » 



therefore EF=zCD + CD— AB- ~=:2CD — — ~ AB, and al- 



foEF+AB = CD(2-^)=zCD(~). But 225 is the fme of 



the arch 3 ° . 45', and 449 of twice that arch, as already fhewn ; 

 and, therefore, according to this rule, if there be three arches, 

 of which the common difference is 3°. 45', the fme of the 

 mean arch will always have to the fum of the fines of the ex- 

 treme arches, a given ratio, that namely, which the fine of 

 3°. 45' has to the fine of twice 3°. 45', or of 7°. 30'; now, 

 this is a true propofition ; and therefore we are in polleffion of 

 the principle on which the Hindoo canon is conftru6led. 



8. The geometrical theorem, which is thus fhewn to be the 

 foundation of the trigonometry of Hindoflan, may alfo be more 

 generally enunciated. " If there be three arches in arithmetical 

 progrefBon, the fine of the middle arch is to the fum of the fines 



■ of 



