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



function of y only, it is found that 



giving the method of logarithms ; and assuming 

 «3,=*(X+Y)-£(X-Y), 



it is found that solutions are : — 



and 



a = sin X, y 



utions are : — 



<i>(X + Y) = i(x+yy; 



= sinY, 0(X + Y)=-icos(X + Y). 



Laplace then shows that the values off(x,y) can be calculated 

 by means of a table of single entry, if the differential equation 

 obtained by eliminating c from the equation f(x, y) = c be of 

 the form Sdfa? + Tdy — 0, S being a function of x only, and T a 

 function of y only. 



It is clear that the formula 



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



does reduce multiplication to addition or subtraction by means 

 of a table of sines; and Laplace's remark that tables of sines 

 had actually been used in this manner for about a century 

 before the invention of logarithms led me to search for the 

 history of this curious method. 



§ 7. The method in question was called jwosthaphceresis, 

 often written in Greek letters 7rpoa6acf>aip€G-L^, and had its 

 origin in the solution of spherical triangles. A careful exa- 

 mination of the history of the method is given by Scheibel in 

 his Einleihing zur mathematisclien Bitch erkenntniss : siebentes 

 Stiick (Breslau, 1775), pp. 13-20 ; and there is also an account 

 in Kastner's Geschichte der Mathematik, t. i. (1796) pp. 566- 

 569, in Montucla's Histoire des Mathematiques , t. i. pp. 583-585 

 and 617-619, and in Klu geYsWorterbuc7i (1808), article Pros- 

 thaphceresis. The method consists in the use of the formula 



sin a sin b = J- {cos (a — b)— cos (« + £)}, 



by means of which the multiplication of two sines is reduced 

 to the addition or subtraction of two tabular results taken from 

 a table of sines ; and as such products occur in the solution 

 of spherical triangles, the method affords the solution of sphe- 

 rical triangles in certain cases by addition and subtraction 

 only. It seems to be due to Wittich, of Breslau, who was 

 assistant for a short time to Tycho Brahe*; and it was used by 



* Christmann, in his Tlieoria Lunce, states that the first inventor of the 

 method was Werner of Nuremberg, who employed it in a treatise De 

 Triangulis, which was never printed (Montacla, t. i. p. 584). 



