by a Table of Single Entry. 333 



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

 seen. We may remark that there is an important distinction be- 

 tween the trigonometrical formulas and the algebraical deduc- 

 tions 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 ; i. e. each different number of factors requires a 

 separate table, while one and the same table of sines and co- 

 sines will serve to multiply 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 inter- 

 esting to note the inferiority that theoretically attaches to the 

 algebraical compared with the trigonometrical formulae." 



The object of this paper is to enter in some detail into the 

 matters briefly referred to in the above extract. 



§ 2. The method of quarter squares depends upon the for- 

 mula 



a b = L( a + h y-L( a - b y ; 



so that, with the aid of a table of quarter squares, in order to 

 multiply two numbers it is only necessary to enter the table 

 with their sum and difference as arguments and take the dif- 

 ference of the tabular results. The first table of quarter 

 squares was published by Voisin at Paris in 1817, and extends 

 to 20,000 ; and the largest that has appeared was published 

 by the late Mr. S. L. Laundy, and extends to 100,000. 

 General Shortrede's manuscript table, that extended to 200,000, 

 and so would enable five figures to be multiplied by five 

 figures, has not been printed. Ludolf, who in 1690 published 

 a table of squares to 100,000, explains in his introduction how 

 it can be applied to effect multiplications by means of the 

 above formula. The title of Voisin's work is Tables des mul- 

 tiplications, ou logarithmes des nombres entiers depuis 1 jusqiia 

 20,000, au moyen desquelles on peut multiplier tous les nom- 

 bres qui rfexcedent pas 20,000 par 20,000, et generalement 

 faire toutes les multiplications dont le produit n'excede pas 

 400,000,000 . . . , par Antoine Voisin .... By a logarithm is 

 here meant a quarter square, viz. Voisin calls a a root, and \a 2 

 its logarithm. If the sum of the two numbers to be multiplied 

 exceeds the limits of the table, but each of the numbers is in- 

 cluded in it, the multiplication is to be effected by means of 

 the formula 



ab = <L{\d l + \V-\(a-Vf\. 



Voisin is thus justified in stating that by means of his table, 

 any two numbers neither of which exceeds 20,000 may be 

 multiplied together ; but it is clear that if the sum of the fac- 



