COMPUTATION OF SQUARES AND CUBES. 5 



lation cannot be effe&ed by a (ingle operation, I believe I IhaJI Computation of. 

 be joined by all perfons accuftomed to arithmetical computa- ^^^^^^f* *" 

 tion, in recommending the ufe of addition* in preference to 

 fublradtion, and of multiplication in preference to divifion. 

 Your correfpondent H. G. does not feem to have paid fuffi- 

 cient attention to thefe confiderations ; and I fear that he not 

 only would be foon wearied, if he were to work by the me- 

 thod which he recommends, but (what is of ftill more confer 

 quence) he could not depend for any continuance upon the 

 accuracy of his computations. 



The only method by which an extenfive table like Mr, 

 Councer's could have been calculated, muft have been by the 

 conftant addition of differences ; and the rules for this method 

 may be eafiiy deduced from the following confiderations : Let 

 x-\'3a, x-^2a, x-j-ff> &c. be any numbers in arithmetical pro- 

 greffion. Then by the binomial theorem. 



x+3al* = a:' + 6ax ^ 9a' 



x-\-2i\ 2 = X* 4- 4«x -{- 4a* 

 x-f.aV=:r*+2aa:4- a» 



x" ^=.x^ 



IJl Differences. 

 2ax -{- 9^* 



2ax -f- 3a» 



2ax 4- a« 



2d Differences^ 



2a* 



2o* 



It will be foreign from my prefent purpofe to enter into a 

 particular confideration of the manner in whjch thefe differ- 

 ences arife. It will be fufficient to remark, that the fecond 

 difference is conftant, and that its value will be always 2, 

 when we confider a asiz 1, or the progreffion x -f- 3a, x-f-2c, 

 &c. as a feries of common numbers differing from one an- 

 other by unity. Hence the fir ft differences of the fquares of 

 fuch numbers become themfelves an arithmetical progreflion, 

 of which the common difference is 2 ; and confequently, if 

 we know the firft difference between the fquares of any two 

 fuch numbers, we can find the difference between the fquares 

 of the two next. 



* If this opinion wanted any argument in its favour, I might 

 fupport it by the authority of that able mathematician Captain 

 Mendoza, who has calculated his new tabks in fuch a manner as tp 

 ^ake ajl the equations additional. 



What 



