03O DEMONSTRATION 



DEMONSTRATION 



OF A 



THEOREM 



Concerning the Intersections of Curves, 



By REUBEN BURROW, Esq. 



TN Stone's Mathematical Dictionary is the following 

 **• paragraph : " Two geometrical lines of any order, 

 " will cut one another in as many points as the number 

 " expreffes which is produced by the multiplication of 

 " the two numbers exprefling thofe orders." And 

 Mr. Braikonridge, in the Preface to his Exercitatio Geo- 

 metrica de Defcriptione Curvarum, fays, " Mr. George 

 " Campbell, now Clerk of the Stores at Woolwich, has 

 *' got a neat demonftration of the fame, which I hope 

 " he willpublim." As it does not appear that Mr. Camp- 

 bell ever publifhed any thing, except a paper on the 

 roots of equations, and a fmall treatife on the plagia- 

 rifms of Maclaurin, it is very probable his demonftra- 

 tion is loft, and therefore it may not be improper to 

 publifh the following. 



The.equation of a line of the firft order has one root, 

 or function of the abfcifs, for the ordinate; of the fecond 

 order, two; and fo on. 



In equations for two right lines, the roots may fovary 

 and accommodate themfelves to each other, that the 

 quantities exprefling the ordinates may be equal ; and 

 as there is only one cafe where this may happen, there- 

 fore two right lines can only interject, in one point. 



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

 the fecond, or an equation of oneroot with an equation 

 of twoj the root of the firft, and a fingle root of the fe- 

 cond, 



