644 PHILOSOPHICAL TRANSACTIONS. [aNNO i6Q4. 



a b 



or a — 

 or a — 

 or a — 

 or a — 



3 a3 - I b' 

 ab 



4 a" _ 6* 



a_b 



5 «J - 4 6* 



a b . 



l/a^ — b = ^a+ y^^a a — -|-, ... ^ ^ . -,. 



^ ' I r « s 15 at 6 a* — 4 J 



And between these two limits always lies the true root, being rather nearer 

 to the irrational than to the rational formula ; but the quantity e, found by the 

 irrational formula, always errs in excess, as the quotient resulting from the 

 rational always errs in defect ; therefore, when it is + b, the irrational form 

 gives the root too great, and the rational one too little ; but the contrary when 

 it is — b. And this may be sufficient concerning the finding the roots of pure 

 powers ; which, however, for ordinary purposes, may be performed more easily, 

 and accurately enough, by means of logarithms ; but whenever the root is to 

 be extracted very accurately, and beyond the extent of the logarithms, recourse 

 must necessarily be had to such methods as these. Besides, as the invention 

 and contemplation of these formulas led me to a certain general rule for the 

 roots of affected equations, which I trust will be of good use to the students 

 in algebra and geometry, I was willing here to give some account of this disco- 

 very in as clear a manner as I can. 



Now having given in the Philosophical Transactions, N° 1 88 -|-, a very easy 

 and general construction of all affected equations, not exceeding the biquadratic, 

 from that time I have always had a great desire of performing the same in num- 

 bers. But soon after that Mr. Raphson seemed in a good measure to have satis- 



* All the above rational formulas may be reduced to one easy and general expression, which will 

 be easier both to remember and lo practise than any of the single individual ones ; and which ig 

 deduced in the following manner. First, the general theorem for all the above particular ones, and 



in the same form, is a + r— , which reduces to -"''."'" ^" "*" .^ . x a, for the near value of 



n— I' gna" + (n—l)i 



"«" + 8~ 4 



the root, by only reducing the quantity into a common denominator, and where n denotes the index 

 of the power whose root is to be extracted. Now, by putting N = a" + 6 the given number, of 

 which o" is the nearest complete power, a being its root; then bysubstitutiag N co a" for the difference 



b, the last general form above will become , '^ 1 ."*" ,"~ X\-, X n, for the approximated root. 

 ° (b + I) a» + (n — I) N '^'^ 



Or, in a proportion it is, as (n + 1) o" + (» — 1) N : (n + 1) N + (» — 1) <»" - " = r the root 

 sought ; that is, in words, as » + 1 times the nearest power, added to » — 1 times the given num- 

 ber, is to n +' 1 times the given number added to n — 1 times the nearest power, so is the near as- 

 sumed root to the root required. And this is the same general formula as that first demonstrated, by 

 a different way, in Dr. Mutton's Tracts, published in 1786, p. 49, 

 \ P. 376 of this vol. of these Abridgments. 



