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



cations had been published, suggested the calculation of such 

 tables in two papers — ' Note on a Formula by aid of which, and 

 of a table of Single Entry, the continued product of any set of 

 Numbers .... may be effected by additions and subtractions 

 only, without the use of Logarithms ' (Philosophical Magazine, 

 [IV.] vol. vii. p. 430), and ' On Multiplication by aid of a 

 Table of Single Entry ' (Assurance Magazine, vol. iv. p. 236). 

 Both these papers were probably written together ; but 

 there is added to the former a postscript, in which reference 

 is made to Voisin and to Shortrede's manuscript. Professor 

 Sylvester gives a generalization of the formula for ab as the 

 difference of two squares, in which the product a x a 2 . . . a n is 

 expressed as the sum of nth powers of a l3 a 2 , . . . a n , connected 

 by additive or subtractive signs. For the product of three 

 quantities the formula is 



abc=-^{(a + b + cy — (a + b—c) z — (c + a — b) d — (b + c— a) 3 }. 



And at the end of the ( Philosophical-Magazine ' paper Pro- 

 fessor Sylvester has added some remarks on how a Table to 

 give triple products should be arranged. 



"At the end of a! memoir, " Sur divers points d' Analyse," 

 Laplace has given a section, / " Sur la Reduction des Fonctions 

 en Tables " (Journal de VEcole Poly technique, cah. xv. t. viii. 

 pp. 258-265, 1809), in which he has briefly discussed the 

 question of multiplication by a table of single entry. His 

 analysis leads him to the method of logarithms, quarter squares, 

 and also to the formula 



sin a sin b — \ {cos (a—b)— cos (a + b)}, 



by which multiplication can be performed by means of a table 

 of sines and cosines. On this he remarks: — i Cette maniere 

 ingenieuse de faire servir des tables de sinus a la multiplication 

 des nombres, fut imaginee et employee un siecle environ avant 

 l'invention des logarithmes.' 



" It is worth notice that the quarter-square formula is de- 

 duced at once from 



sin a sin b = \ {cos {a— b)— cos (a + 5)}, 



by expanding the trigonometrical functions and equating the 

 terms of two dimensions ; similarly from 



sin a sin b sin c = J {sin (a + c — b) + sin (a-f b — c) 



+ sin(& + (j — a) — sin(a + 5 + c)}, 



by equating the terms of three dimensions we obtain 



abc = £ i {(a + b + cy-&c.}, 



