VOL. XVIII.] VHILOSOPHICAL TRANSACTIONS. 643 



operose calculation. But the author has no where given his method of inves- 

 tigation or demonstration, though it seemed most to require it ; especially as it 

 is erroneously printed in his book, which may easily mislead the unskilful. Now 

 the 5th power of a + e consists of these parts, a' -f 5 a* e + 10 a^ e^ + 10 a^ e^ 

 + 5 a e* + e' = a* + i' ; hence b = 5 a" e + ]0 a^ e^ + lOa" ^ + 5 ae*, re- 

 jecting e^ on account of its smallness ; therefore — = a' e -f- 2 a' e'* -f 2 a e' 

 -j- e* ; then adding -J- «* on each side, and extracting the square-root, it is 

 y^a* + ~= ^^ ^ a* + a^ e + 2 a^ e^ -\- 2 a e^ + e* = la" + a e + e^ ; then 

 subtracting -J- a^ from each side, and extracting the square-root again, we shall 



have v ■/ t'^'* + ^ T«^= ^ i^ a^ + a e -\- e^ = -^ a + e i and adding -i. a, it 



isa-f-e = -i-a -f V 's/ ^ a* + -J- a'' the 5th root of the power a* + b. 



But if it were a^ — b, by taking a greater than just, the rule would be, -J- a -|- 



/- 



This rule approximates wonderfully fast, so that there can scarcely be any 

 need of repetition. And while I was considering this matter, I fell upon a 

 general method of forms for any power whatever, which is elegant enough, and 

 which I cannot prevail on myself to conceal, since in the higher powers they at 

 least triple the known figures of the root. And these forms, both rational and 

 irrational, proceed as follows : 



a b 



^''7+1 = 4- a 4- / f a a -h 3^, or « + j^fp. 

 /7+b = ^ a -I- /i a « + 6^., or a + ^^j. 



2 «^ + i 6' 



/T^b = A a -1- y^aa + :^^, or a + — ^^. 



And so on for the other higher powers. But when a is assumed greater than 

 the true root, (which will be done with advantage whenever it is nearer the 

 truth than the next less integer number) then by a proper change in the signs, 

 the same expressions of the roots arise, viz. 



^a'^-b-^a + y^aa-^^, or a- ^J^^j; 



4 N 2 



