12 COMPUTATION OF SQUARES AND CUBES. 



Computationof fame time it may be remarked, Kiat this is an extreme cafe, 

 cubes. which can never occur but under particular circumfiances: 



for it only happens when tlie number to be fquared is lels than 

 a multiple of the higheft number in the table, and greater than 

 the fame multiple of the number next lefs than the higheft; 

 that is, for a table like Mr. Councer^s, it muft be lefs than 

 m. 12854-0, and greater than m . 128539 : under thefe circura^ 

 ilances, if a be greater than 1, the cafe will occur which is 

 the fubjed of the above caution. 



It remains for us to confider the method of finding the cubes 

 of numbers which exceed thofe given in the table. 



Upon the principle to which we referred before, that x"* x 

 y^ = ajl*", we may find the cube of a number which is a mul- 

 tiple of any one contained in the table, by firaple multiplica- 

 tion : for the cube of any number multiplied by 8, will give 

 the cube of double that number ; the cube multiplied by 29, 

 64, or 125, will give us the cube of 3, 4, or 5 times the 

 number : Thus, 



3368928641.271 = the cube of 14991 



26951429154168 = the cube of 29982. 



1045678375000 = the cube of 10150 



27 



731974862500© 

 2091356750000 



28233316125000 = the cube of 30450. 



But as the cubes increafe more rapidly even than the fquares, 

 it will flill be more necelfary in this cafe than in the former, to 

 efiablifli fome means c>f finding the cubes of thofp high numr 

 bers which are either prime or not exactly a fmall multiple of 

 a number contained in the table ; and by proceeding in a man- 

 ner fimilar to that which we ufed for the fquares, we eafily 

 efiablifli a general rule for this purpofe : For"^^ + «T = '^^-^^ 



a:^ -\-3an''-3an. X* -{-a^ ~na^ = n.x -\-a]^ -{-n .n^ ~ I .x^ -{-n. 



n-l . Sax^-i-l-n.a^zznxx+^^ + n-^j.n-^l r^-fSax* -f 

 l—n.a^. Determine^ therefore, n, x, and «, in the fame 



manner 



