160 Mijcellanea Curiofa* 



Ag ( ex. gr. upon AB and Ab ) as the 

 Area AHB to the Area AEB. The Demon- 

 Uration of which depends upon a General 

 Theorem, which I deduced very eafily from 

 Prop* 35- Newt. /?. 324. 



COROL I 



Let BG, bg, be infinitely fmall parts of 

 the Lines AG, Ag, and let bB be produced 

 to L\ I fay, that the Refinance upon BG 

 (which call e) is to the Refiftance upon bg 

 (which call E) as GL 2 : GB 2 . 



For e : E : : KHgb : FEgb that is, e : E : : 

 bg x bH : bg x bE (by the foregoing Lemma ) 

 therefore e : E : : bH : bE ^ that is, e : E : : 

 CM^ : BC : : CM 2 : BG 2 . But CM 2 : BG 2 : : 



BC 



GL 2 : GB 2 (becaufe of the fimilar Trian- 

 gles BMC, GLB.) Therefore e : E : : GL 2 ; 

 GB 2 . Q_: E : D. 



COROL. II. 



The Refinance upon the infinitely fmall 

 part GB, is = GL 3 . For if all the infinite- 

 GB 2 



ly fmall parts in the Line Ag (as bg) be fup- 

 pos'd equal, then the Refinance upon bg, 

 may be exprefs'd by bg, that is E~bg, and 

 fo E=GL. Therefore (by Cor. 1.) e : GL : : 

 GL* : GB 2 , whence e= GLK Q; E : D. 



GB 2 



COROL 



