14 DISSERTATION SECOND. [part i, 



and of course nearer (o the curvilinear area, which was 

 always intermediate between them. In prosecuting this 

 sort of approximation, a result was at length found out, 

 which must have occasioned no less surprise than delight 

 to the mathematician who first encountered it. The result 

 I mean is, that, when the series of inscribed figures was 

 continually increased, by multiplying the number of the 

 sides, and diminishing their size, there was an assignable 

 rectilineal area, to which they continually approached, so 

 as to come nearer it than any difference that could be sup- 

 posed. The same limit would also be observed to belong 

 to the circumscribed figures, and therefore it could be no 

 other than the curvilineal area required. 



It appears to have been to Archimedes that a truth of 

 this sort first occurred, when he found that two-thirds of 

 the rectangle, under the ordinate and abscissa of a parabo- 

 la, was a limit always greater than the inscribed rectilineal 

 figure, and less than the circumscribed. In some other 

 curves, a similar conclusion was found, and Archimedes 

 contrived to show that it was impossible to suppose that 

 the area of the curve could differ from the said limit, with- 

 out admitting that the circumscribed figure might become 

 les«, or the inscribed figure greater than the curve itself. 

 The method of Exhaustions was the name given to the in- 

 direct demonstrations thus formed. Though few things 

 more ingenious than this method have been devised, and 

 though nothing could be more conclusive than the demon- 

 strations resulting from it, yet it laboured under two very 

 considerable defects. In the first place, the process by 

 which the demonstration was obtained was long and diffi- 

 cult ; and, in the second place, it was indirect, giving no 

 insight into the principle on which the investigation was 

 founded. Of consequence, it did not enable one to find 

 out similar demonstrations, nor increase oue's power of 



