COMPUTATION OF SQUARES ANB CUBES. 173 



"that his method lii&kes it unneceflhry to calculate them ; for On the compu- 

 by mere comparifon of p. 9 with p. 125, it will be feen that ^jj^^^^j^^^^**"^'^*;^ 

 we both ufe (he fame numbers. In this inftance, asjn that of 

 the fquares, the real difference between the two methods 

 confiils in the manner of finding thefe additional quantities. 

 H. G. mufl therefore mean, that he avoids the neceflity of 

 ^Calculating the table given in p. 8. Now in this I do not 

 'tlirnk that any advantage is gained. For if the firft differ- 

 'Hfences are to be found feparately, 1 have (hewn that they can 

 he more eafily found by the method which I have recora- 

 raended ; and much expedition will be obtained by making a 

 feparate table of them, which is impoffible if we work accord- 

 ing to the diredions of H. G. I may, with fome reafon, 

 complain, that H. G. has not made his example a fair parallel 

 to mine ; for I have fet down every figure which need be ufed, 

 while he has declined printing any of the operations whereby 

 he determined his firfl: differences. It is poffible, indeed, 

 from long pra6lice in arithmetical computations, that he may 

 be able to cafl his eye from one cube to the other, and fo , . 



find the firft differences, without copying them out on the 

 fide of his paper for fubtradion. I ftiall not dwell upon the 

 danger of miftake in fuch a procefs : the quickeft and mod 

 accurate eye could hardly avoid frequent errors, and the com- 

 mon arithmetician will find it attended by confiderable diffi- 

 culty. But granting that all this difficulty could be com- 

 pletely conquered, ftill it would give no fuperiority to the 

 method recommended by H. G. ; for it might be applied to mine 

 with equal, if not greater eafe and advantage. Thus, in the ex- 

 ample given in p. 125 ; if F were computed by the addition 

 of C and D, inftead of the fubtraftion of A from E ; if we 

 confider I =r F -1- G infiead of = H - E, M = I -f- K in. 

 ftead of z: L — H, &c, we (hould have the differences com- 

 puted according to my rules. We may remark likewife, in 

 this fliorter method of calculating, that, according to the ar- 

 rangement given by H. G. himfelf in p. 129., the ,e)'e mufl 

 neceffarily pafs over two intermediate lines of .figures in fub- 

 trafling one cube from the other, whereas the two. rows of 

 figures which I would have added to one another, may 

 always be placed contiguous. This certainly is an advan- 

 tage ; but at the fame time I only mention it to fhew the 



grounds 



