4 PHILOSOPHICAL TRANSACTIONS. [anNO 1694-6. 



great; or perhaps i.OS ; and upon trial it will be found not too large ; for the 

 cube of '2.08 is but 8.998912. 



If this first step be not near enough, this cube subtracted from 9.0{)0()00, 

 leaves a new b = O.OOI088, which divided by 3 a-= 12.9796, gives 0.000084 — ; 

 which will be somewhat too great but not much. So that if to 2.08 we add 

 0.000084 — , the result 2.080084 will be too great, but 2.080083 will be more 

 too little. So that either of them, at the second step, gives the true root within 

 an unit in the sixth place of decimal parts. 



Hitherto I have pursued the method most affected by the ancients, in seeking 

 a square or cube, and the like of other powers, always less than the just value, 

 that it might be subtracted from the number proposed, leaving b a positive 

 remainder ; thereby avoiding negative numbers. But since the arithmetic of 

 negatives is now so well understood, it may in this be advisable, to take the 

 next greater, in case that be nearer to the true value, rather than the next less. 

 According to this notion, for the square root of 5, I would say, it is 2 +> 

 somewhat more than 2, and inquire, how much more ? but for the square root 

 of 8 I would say, it is 3 — , somewhat less than 3 ; and inquire, how much 

 less? taking, in both cases, that which is nearest to the just value. And what 

 is said of these is easily applicable to the higher powers. 



I shall omit that of the biquadrate, because here perhaps it may be thought 

 most advisable to extract the square root of the number proposed, and then the 

 square root of that root. But if we would do it at once, we are from n to 

 subtract a^, the greatest biquadrate contained in it, to find the remainder b = 

 4 A^ E + 6 A^ e" + 4 A E^ + E^ ; which remainder, if we divide by 4 a% the 

 quotient will certainly be too large for e ; if by 4 a^ + 6 a^ + 4 a + 1, it will 

 certainly be too little ; and we are to use our discretion in taking some inter- 

 mediate number. And if we chance not to hit on the nearest, the inconve- 

 nience will be only this, that our leap will not be so great as otherwise it might 

 be. Which will be rectified by another b at the next step. 



For the sursolid of five dimensions, we are, from n to subtract a\ the 

 greatest sursolid therein contained, to find the remainder b = 3 a^ e -|- 10 A' 

 E- -\- 10 a' e' + 5 a e* + e' ; which if we divide by 5 a\ the result will be 

 somewhat too great, if by 5 a* -|- 10 a^ + 10 a' -|- 5 a + 1, the result will cer- 

 tainly be less than the true e. The just value of e being somewhat between 

 these two ; where we are to use our discretion, wiiat intermediate number to 

 take. Which, according as it proves too great or too little, is to be rectified at 

 the next step. 



If, to direct us in the choice of such intermediate number, we should mul- 

 tiply rules or precepts for such choice, the trouble of observing them would be 



3 



