334 Mr. J. W. L. Glaisher on Multiplication 



tors exceeds 20,000 the method loses its advantages, as the 

 la si -written formula requires three entries and a duplication. 



An ordinary multiplication table, or Pythagorean table, 

 giving the product ab for arguments a, b is of double entry, and 

 so could not be carried to any very considerable extent on account 

 of the great bulk of the table. Herwart ab Hohenburg's table 

 of 1610, referred to in §§ 9-12, extends to 1000x1000, as 

 also is the case with Crelle's JRechentafeln, which are in general 

 use ; but the Pythagorean table has never been carried beyond 

 this limit. The question of the reduction of the process of 

 multiplication to that of addition or subtraction is one that is 

 interesting both from a practical and historical point of view. 



§ 3. In the Philosophical Magazine, [IV.] vol. vii. pp. 431, 

 432 (1854), Sylvester gave the generalization of the quarter- 

 square formula for the product of n quantities in the following 

 form : — 



" Let #!, 2 , 3 , . . . 6 n be disjunctively equal to 1, 2, 3, ... n; 

 then 



(2.4.6... 2n)(ai a 2 . . . a n ) 



= (aQ i + a Q +a Qz + ... + a 07 ) n -?.(-a Q +a d2 + ... + a Qn ) n 



+ 2(-a 9l -a e2 + a e3 + ... + a 9n ) n + ... 



+ {-)\-a Q -a Q -...-a 9 jr (1) 



The first and last terms, the second and last but one, &c, are 

 identical and may be united, there being one term left over in 

 the middle if n be even ; viz. this becomes 



(4.6.8... 2n){a x a 2 ... a n ) = (a 9i + a 92 + a 93 . . . + a 9n ) n 



-t(-a 9i + a 92 + a 9z ... + a 9r ) n 



+ &c; (2) 



the last term being 



{-) m ^(- a e l -^9 2 '"-^e m + a e m +i + a em+2'" + a en) n 

 if n = 2m + l, and 



%{-) m Z(-a 9 -a 92 . . .-a 0m + a gm+1 + a gm+2 ... 4 a 9n ) n 



if n = 2m. This last expression is integral, notwithstanding the 

 factor J, since each term composing it occurs twice ; as, ex. gr., 

 when n = 2, the whole term is -J{(« — b) 2 + {b — a) 2 }. These 

 are the formulae given by Sylvester on p. 432. The equation 

 (1) is there proved by showing that the expression on the 

 right-hand side vanishes when a n = 0, and therefore when 

 ax = 0, a 2 = 0, &c, so that it =ka ± a 2 . . . a n , and determining 

 the numerical factor k by putting a x = a 2 — . . . = a n =l. 



§ 4. The formula? can, however, be proved in the manner 



