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Having thus fiiewn, that this method has more latitude than 

 has been generally imagined, I fliall endeavour to evince the truth 

 of it in the cafe of a fexinomial ; thereby demonftrating its uni- 

 verfality, and fhewing hovir the furds, one by one, vanifli, not- 

 withftanding the enormous appearance of furd redangles at firft ; 

 for the number of furd redangles, arifing from the involution of 

 a multinomial furd quantity to the fquare, is always equal to the 

 fum of the natural numbers between o and the number of the 

 parts conneded by the figns + or — , and therefore rapidly en- 

 creafes. — Vide Saunderfon's Alg. Tom. 2, Sedion 422, (fub finem ;) 

 & Simpfon's Alg. Chapter on Combinations. 



Let there be propofed the equation, confifting of a rational and 

 five quadratic furds, x + K^a+ v'3 — >/ c—s/ d—,jf— o. 



After tranfpofition, the equation muft either have two parts 

 at one fide, and four at the other, in which cafe, after involution, 

 there will be feven furd redangles ; or one part at one fide, and 

 five at the other, from whence would arife, after involution, ten 

 furd redangles ; or finally, three parts at each fide of the equa- 

 tion, from whence, after involution, refult fix furd redangles, i. e. 

 three at each fide. Let us take the equation from whence the 

 leafl; number of furd redangles refult ; 



fquared x'-^a-\-b-V%x\/a-\-%i: ^'b-V 1 V ah = c-\- d +_/+ 2 -/cd 

 + 2 -/€/+ 2 ^/df^ 



Let 



