OF A THEOREM, Sec. 331 



cond, may fo vary as to become equal to each other, or 

 to form an interfeclion. By the fame reafon, the fingle 

 root of the firft, and the remaining root of the fecond, 

 may each fo vary as to become equal, or to form ano- 

 ther interferon; and therefore a right line cuts a line 

 of the fecond order in two points. 



If a line of the firft order be compared with a line of 

 the n order, it is alfo evident that the fingle root of the 

 firft. line may in the fame manner be fo varied with each 

 of the n roots of the fecond line as to become equal ; 

 and therefore a right line may cut a line of the n order 

 in n points. 



Let a line of the m order be now compared with a 

 line of the order n; then as each fingle root of the firft: 

 line may become equal, in the fame manner, to every 

 root in the fecond, it therefore follows, that for every 

 unit in m there may be n interferons; and as there are 

 m units, there confequently will be mn interferons. 



The fame method may be applied to the determina- 

 tion of the points, lines, and furfaces, that arife from the 

 interferons of lines, furfaces, and folids j by consider- 

 ing that the number of times that p may be taken from 

 m, and q at the fame time from «, will be 



= m.m — 1 p, X n.n — 1 q 



1 . 2 . 3...P, X 1 .2.3...,q 



Aaz AVIL 



