ON MATHEMATICAL TABLES. 23 



nth. powers of a^, a^, . , . «„, connected by additive or subtraetive signs. For 

 the product of three quantities the formula is 



abc=^{{a+b + cf-(a + b-cy-(c + a-hf-{b + c-ayi 



And at the end of the ' Philosophical-Magazine ' paper, Prof. Sylvester has 

 added some remarks on how a table to give triple products should bo 

 arranged. 



At the end of a memoir, " Sur divers points d' Analyse," Laplace has given 

 a section " Sur la Eeduction des Fonctions en Tables " (Journal de I'Ecole 

 Polytechnique, 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 

 aual}-sis leads him to the method of logarithms, quarter squares, and also to the 

 formula siua sin&=r|{cos(rt— 6)— cos(a + 6)}, by which multiplication can 

 be performed by means of a table of sines and cosines. On this he remarks, 

 " Cette maniere ingenieuse de faire servir des tables de sinus a la multiplication 

 des nombres, fut imaginee et employe'e un siecle environ avaut I'iuveution 

 des logarithmes." 



It is worth notice that the quarter-square formula is deduced at once from 

 sin rt sin 6 =g { cos (« — i) — cos (a + b)},'by expanding the trigonometrical func- 

 tions and equating the terms of two dimensions ; similarly from sin a sin b 

 sin c = j{sin (« + c— 6) + sin(a + 6— c) + sin(6-|-c— a) — sin (a + 6 + c)}, by 

 equating the terms of three dimensions, we obtain abc=-^{(a-\-b-^cy — &c. }, 

 as written down above, and so on, the general law being easily seen. We 

 may remark that there is an important distinction between the trigonometrical 

 formulas and the algebraical deductions from them, viz. that by the latter to 

 multiply two factors we require a table of squares, to multiply three a table 

 of cubes, and so on ; {. e. each different number of factors requii'es a sepa- 

 rate table; while one and the same table of sines and cosines wiU serve to 

 multi])ly any number of factors. This latter property is shared by tables of 

 logarithms of numbers, the use of which is of course in every way preferable ; 

 still it is interesting to note the inferiority that theoretically attaches to the 

 algebraical compared with the trigonometrical formulo3. Other remarks on the 

 subject of multiplication by tables are to be found in § 3, art. 1. 



It is almost imnecessary to remark that a table of squares may be used 

 instead of one of quarter squares if the semisum and semidifference of the 

 numbers to be multiplied be taken as factors. Tables of squares and cubes 

 are described in the next section. 



*Voisin, 1817. Quarter squares of numbers from unity to 20,000. We 

 have taken the title from the introduction to Mr. Latjndt's ' Quarter Squares' 

 (1856). De Morgan also so describes the work. We have seen no copy; but 

 there is one in the Graves Library, although we were unable to find it : it 

 will be described from inspection in the supplement to this Eeport. 



Leslie, 1820. On pp. 249-250 there is a table of quarter squares of 

 numbers from 1 to 2000, reprinted from Voisijt, 1817, whose work Leslie 

 met with at Paris in 1819. There is also given, facing p. 208, a large folding 

 sheet, containing an enlarged multiplication table, exhibiting products from 

 11x11 to 99x9 9, the table being of triangular form. There are also, on 

 the same sheet, two smaller tables, the fii'st giving squares, cubes, square 

 roots (to seven places), cube roots (to six places), and reciprocals (to seven 

 places) of numbers from 1 to 100, and the second being a small m.ultiplication 

 table from 2 x 2 to 25 x 25. In the first edition (1817, pp. 240) the quarter- 

 square table does not appear ; and in the folding sheet (which follows the 



