648 PHILOSOPHICAL TRANSACTIONS. [anNO 16Q4. 



figures. But to be content with fewer, we may write t b ± t e' instead of t b, 



or we may add or subtract , ^^ ; to or from the root before found. But 



the equation cannot be explained by any other root, because the power to be 



resolved, 350, exceeds the cube of '-3-' o"" -f^- 



Exam. 3. Let there be taken the equation z* — 80 z' + 1998 z^ — 14937 z 



+ 5000 = O, which Dr. Wallis finds, ch. 62 of his Algebra, in the resolution 



of a difficult arithmetical problem, the root of which he obtained very accurately 



by Vieta's method ; and which Mr. Raphson also uses as an example of his 



method, p. 25, 26. Now this equation is of such a form as admits of several 



positive roots ; and, what increases the difficulty, the coefficients are very great 



in respect of the given resolvend. But, that it may be the easier managed, let 



it be divided, and according to the known rules of pointing, let — z"* + 8 z' 



— 20z^+ 15z = 0,5, where z is -rV z in the proposed equation; and for a 



first supposition take a = 1 ; then + 2 — 5e — 2ee + 4e^ — e* — 0,5 = 0; 



, V-^ss + bt — 4. « V37 — 5 , 



that is, 14- = 5 e + 2 ee ; hence e = ^ = , and 



therefore z = 1,27. Hence it appears that 12,7 is a near root of the equation. 

 Secondly, supposing z = 12,7 ; then, according to the table of powers, 



b s t u 



— 26014,4641 — 8193,532 e — 967,74 ee — 50,8 e' — e* 

 + 163870,640 + 38709,60 e + 3048 ee + 80 e* 



— 322257,42 — 50749,2 e — I998 ee 

 + 189699,9 + 14937 e 



— 5000 



+ 298,6559 — 5296,1 32 e + 82,26 ee + 29,2 e^ — e* = O. 



Therefore — 298,6559 = — 5296,132 e, + 82,26 e e, the root of which is e z= 

 ^ss-VjT7^t ^ 2648.066 - v6m6s6A0602, ^ ^05644080331, less than 



just. But to correct it, y/J^J"/] = S^l = >OOOOOOg9ll7, and there- 



fore e corrected is ,05644179448. But if still more figures of the root be de- 

 sired, from e corrected let there be formed tue^ — t e* = 0,43105602423, and 



^ « - VjTs -Bt -tuei + te* 2648,066 - V6987685,67496557577 _ 



t °'" 82,26 ~ 



,05644179448074402 = e ; hence a -{• e = z the root is very accurately 



12,75644179448074402, the same as found by Dr. Wallis. 



Here it may be observed that repeating the calculus always triples the true 



