642 PHILOSOPHICAL TRANSACTIONS. [ANNO I694. 



this formula comes rather nearer the truth than the former, but errs in point 

 of excess, as the other in defect ; and it seems more commodious for practice, 

 since the repetition of the calculus is only the continual addition or subtraction 



of the quantity -— , as the small supplement e becomes known ; so that it may 



rather be written in the former case 



T « + V -f aa + ^-^~, and 4- a 4- Vt « « + ^-^— in the latter. 

 And by either form the figures already known, in extracting the root, are at 

 least tripled ; which must prove an acceptable compendium, and therefore I 

 congratulate the inventor upon it. But that the benetit of these rules may the 

 better appear, I shall add an example or two. 



Example 1. Required to find the side of the double cube, ov oi a aa -\- b 



= 2. Here a = 1, /; = 1, and — = ^ ; therefore o- + / ^'^ or 1,26 will 



be found near the true root. Now the cube of 1,26 is 2,000376, there-. 



fore 0,63 + */ 0,3969 — ^^g— , or 0,63 + V 0,3968005291005291 = 



1,259921049895 — , which exhibits tiie side of the double cube to 13 figures 

 with very little trouble, via. by only one division and extraction of the square 

 root ; whereas by the common way it is well known how laborious it must have 

 been. Hence this calculus may be continued at pleasure, by increasing the 



square by the addition of '— : which correction in this case brings only the 



increase of a unit in the 14th figure. 



Example 1. Required to find the side of a cube equal to the wine gallon, 

 containing 231 solid inches. The next less cube is 2l6, its root being 6, and 

 the remainder is 15 = Z) ; therefore the first approximation will be 3 + v^ 9 -)- ». 

 for the root : and since ^9,8333 ... is 3,1358 . . ., it appears that 6,1358 = 

 a -f- e. Now make 6,1358 = a, then its cube is 231,000853894712, and 



according to the rule 3,o679 + ^9,41201041 - ^'^^^^^^f^l = 



6,13579243966195897, which is the side of the given cube very exactly, 

 being true in the 18th figure, and only falling short in the 19th ; which calcu- 

 lation I performed in an hour's time. And this formula is deservedly preferred 

 before the rational one, which, on account of its large divisor, cannot be used 

 without much trouble, in comparison of the irrational one, as manifold experi- 

 ence has informed me. 



Now the rule for the root of the pure sursolid, or 5th power, is of a little 

 higher inquiry, and yet performs the business much more perfectly; for it at 

 least quintuples the given figures of the j-oot, and yet it requires not a very 



