VOL. XVIII.] VHILOSOVHICAL TRANSACTIONS. Q4Q 



figures in the assumed a, which the first correction, or — ^ — =, quintuples, 



which operation is easily performed by logarithms. The other correction, after 

 the first, also doubles the true figures ; so that on the whole it makes the 

 assumed true figures seven-fold. But the first is commonly quite sufficient for 

 most arithmetical uses. And by what is here said about the figures rightly 

 assumed in the root, I mean that when a is distant from the true root not above 

 a 10th part, then the first figure is rightly assumed ; if within a 100th part, the 

 first two figures are right; if within a 1000th part, the first three figures are 

 right; which consequently treated according to our rule, the true figures soon 

 become nine. 



It remains now to add something concerning our rational formula, viz. 



s b 



— ^T^-^j which seems expeditious enough, and is not much inferior to the 



former, since it triples the given number of true figures. Now, forming 

 an equation from a ± e = z, as before, it will soon appear whether the 

 assumed a be greater or less than just, since se ought always to have a sign 

 contrary to that of the difference between the resolvend and its homo- 

 geneum produced from a. Then supposing that + b+se + t€e:=0, 

 the divisor is ss — tb when t and b have the same sign, but ss -^ bt 

 when they have different ones. But it seems most convenient for practice 



to write the theorem thus, e = ^, since then the business is performed 



s 

 by one multiplication and two divisions, which otherwise would require three 

 multiplications and one division. Thus, taking an example of this method 

 from the root of the foregoing equation, 12,7, &c. in which 

 298,6559 — 5296,1 32 e + 82,20 ee + 29,2 e' — e* = O, and therefore 

 + b — s + t + M 



b 1-1 , t b 



= e, that IS, make s : t :: b : — , or 



tb 



s 



5296,132) 298,6559 + 82,26 (4,63875 ; hence the divisor s — 



— = 5291,49325) 298,6559 (0,05644 1 ... = e, viz. five true figures 



added to the assumed root. But this formula cannot be corrected like the 

 irrational one ; so that if more figures of the root be desired, it is better to 

 repeat the calculation by making a new assumption ; and then the new quotient, 

 by tripling the number of true figures, will abundantly satisfy even the most 

 scrupulous calculator. 



VOL. III. 4 O 



