340 Mr, J. W. L. Glaisher on Multiplication 



cosines, or if a sine or cosine is a divisor, the process is more 

 troublesome — that these multiplications can be effected by the 

 formula, but that the requisite transformations are more la- 

 borious than the multiplication itself, which is purely mecha- 

 nical, while with the prosthaphaeresis method more care is re- 

 quired and, in addition, it is more difficult to obtain accuracy 

 in the result. These remarks seem to be obviously just; and it 

 is clear that the method could not be a good one for the 

 ordinary multiplication of numbers not given in the form of 

 sines of angles, as four entries of the tables would be neces- 

 sary, besides very troublesome interpolations. 



§ 8. Napier published his Canon Mirificus in 1614; and then 

 the prosthaphaeresis method was at once superseded by loga- 

 rithms. The latter process requires only three entries of the 

 table in order to multiply two numbers, and, even regarded 

 merely as a multiplication method, is greatly superior in every 

 respect to that of prosthaphaeresis, which requires four entries. 

 Before the invention of logarithms the object was to arrange 

 formulae, of which numerical values were required, as sums of 

 sines by prosthaphaeresis, so that they might admit of calculation 

 by addition or subtraction ; since the invention of logarithms 

 the object has always been to throw formulae into the form of 

 products. 



Regarded as processes for effecting multiplications, the 

 methods of (1) prosthaphaeresis, (2) logarithms, and (3) 

 quarter squares, may be compared as follows : — The first 

 theoretically solves the question, but is impracticable as a 

 general method. The second is the best method, if only a few 

 figures (viz. 6 or at the most 9 figures) of the product are 

 wanted : if n numbers are to be multiplied together, only n + 1 

 entries are required. The method of quarter squares is the 

 best if only two numbers are to be multiplied together, and 

 if all the figures of the product are wanted. Only two entries 

 are required ; and a table from 1 to 200,000 (which would 

 only occupy a moderate octavo volume) would give the pro- 

 duct of any two five-figure numbers by two entries and one 

 subtraction, no interpolations being necessary. A Pytha- 

 gorean table of this extent would be absolutely impossible, as 

 it would occupy 100 x 100, or 10,000 volumes similar to Crelle's 

 Rechentafeln, which in the new edition occupies one volume 

 folio (see § 10). The quarter-square method seems not to have 

 been much used, partly because it has never become generally 

 known, and partly because no table exceeding 100,000, and 

 therefore available for all five-figure numbers, has been pub- 

 lished. Such tables are also only suitable for the one purpose 

 of multiplication, while logarithms have a great variety of uses. 



