IQ COMfUTATION or SQUARES AND CUBE3, 



Computation of In this manner we may find the fquares of all even numbers 

 cu^s! *^ which do not exceed the double of the table ; we may alfo find 



the fquares of numbers, to a greater extent, which are mul- 

 tiples of 3 : the multiples of 4 and 5 may be fquared likewife, 

 if we multiply by 16 and 25 ; but fquare numbers increafe 

 very rapidly, and confequently this method can be only ufed 

 (with advantage) for the fmall multiples of the tabular num- 

 bers. We muft, therefore, confider fome other method which 

 may be applicable to prime numbers and multiples of large 

 ones. 



The fquare of 2x -f I zi 4j-« -f- 4x -f I zr 2x^ -f- 4r -f 2 -f 



2j^ - 1 ZZ 2.7+71^ + 2x* - I, or 2.x -fFl' 4- x* - I. 

 If, therefore, we want to find the fquare of an odd number 

 which does not exceed the double of thofe which are fquared 

 in the table, we muft divide the next even number below it 

 by 2, and this half will be equivalent to x in the above equa- 

 tion ; the next number above this half will give us x-j-l* 

 Having thefe two fmaller numbers, we may find their fquares 

 by the table; and the fum of their fquares multiplied by 2, 

 will exceed the fquare required by 1. Thus, if it were re- 

 quired to find the fquare of 257079 = 2. 128539 -f 1 : Here 

 X = 128539, and X -f 1 = 128540; therefore, by Mr. Coun- 

 cer's table, the fquare of 128540 would be 16522531600 

 the fquare of 128539 would be 16522274521 



33044806121 

 2 



66089612242 

 Therefore the fquare of 257079 is 66089612241. 



Inftead of examining each particular cafe, when the number 

 to be fquared is more than the double of thofe in the table, it 

 will be beft to confider the theorem, from which a general rule 

 may be deduced, nx -\-a\^ ZT 7i*x* -}- 2anx -{- a^ zz wx* -|- 

 2a«x -\-na^'{-n*^n,x^-\'l—n,a^ZZ:n . x-J-a^ -|- n . n—i, x^ 

 -}- l-n.««— n.x-|-«\2-}-/i — 1.x*-}- l-w.a'. Hence we 

 TOuft multiply x« by n ~ 1 , to the produd addx-j-^V, multi- 

 ply the fum of thefe two quantities by w, and this laft produft 

 will exceed the fquare required by /»— i.a«. The only cau- 

 tion neceflary, is in dividing the given number by x; for as 



we 



