[ 36 ] 



n 



m^-n m-f-n 



fore the curve ADB is to aX~ as i to "—^—y and therefore 



II m-; n 



equal to x aX""- O. E. D. 



If the ordinate BD be oblique to the bafe, the area, found 

 as above, rauft be diminiflied in the ratio of radius to the 

 fine of the angle made by the ordinate and bafe. 



This demonftration being admitted, the whole dodrine of 

 quadratures becomes a branch of prime and ultimaie ratios^ 

 and confequently of pure geometry. 



We are to obferve, that the reafon why the curves treated 

 of above are perfedly quadrable is, becaufe the redangles in- 

 fcribed in the curve are to the refpedive redangles infcribed 

 in the exterior fpace, ultimately, in a given ratio, whence the 

 curve will be to that fpace (Cor. Lem. 4. Prin.) and confe- 

 quently to the circumfcribed redangle, in a given ratio. But 

 this is not the cafe in the circle, which therefore is not quad- 

 table by this method, at lead in its prefent ftate. But though 

 the ratio of the redangles infcribed in a quadrant to their 

 correfponding redangles in the exterior fpace of a circumfcribed 

 fquare perpetually varies from the beginning of the quadrant 

 to the end, yet this variation is regular, beginning from th& 

 finite ratio of 2 to i, and conftantly approaching the infinite 

 ratio of i to nothing. The law of which approach may be 

 thus determined : 



If 



