[ 326 ] 



they can be reduced to three and a rational, whence the fim- 

 plification is manifeft : The fame I have tried in a fcptinomial, 

 and the rcfult was agreeable to the former; though the third 

 involution, produced twenty-nine furd redangics bcfides ra- 

 tionals. 



The example propofed by Newton, where he mentions M. Fer- 

 mat's method, will readily prove that the method juft laid down 

 fhews the connedion of the parts of the equation much better 

 and is much more brief, the whole being carried on in one 

 fingle equation ; that example is an equation confifting of a 

 cubic furd, two quadratics and a rational, and which Sir Ifaac 

 has given us without the operation. Dodor Hales in his Analyfis, 

 98th Sedion, has given the operation where even the fubfti- 

 tuted letters rife to the cubic dimenfion. I fliall now fubjoin 

 the operation according to the method now propofed : 



•Jay — v/a' — ay — %a =-/ay'^ which cubed and coUeding fimilar 

 terms becomes —\^a^ + i ^a^—2ay \/ay—i'^a^—2ay ^a'^—ay -\- i2a^ 



•Jay—y'^ = ay^ ; affuming i^a-—ay ^l p and 13^^ + lay = q 

 by tranfpofition we have, 

 ps/ay \- 12a' s/ay—y^ = ay^ -\- 14^' ■\-q-</a^ —ay and fquaring, 



p^ay + i\/^ay— i^^ay+ 2^pay\^a- —ay = ay+ igta + qa — gay 



+ zSay + 2qay + 2Baq -/a^ — ay. 



whence 



