DEVELOPMENT OF MATHEMATICAL THOUGHT. 651 



degi'ee were arrived at. The forward or direct process 

 was easy enough, though even here assumptions or arbit- 

 rary rules were included which escaped notice for a long 

 time; l)ut the real labour of the analysts only began 

 with the inverse problem — viz., given any compound 

 (juantity, similar in structure to those directly produced 

 by multiplication of binomials, to find the factors or 

 binomials out of which it can be compounded. Now 

 it was found that as in the arithmetical process of 

 division, the invention of fractional quantities ; as in 

 that of extraction of roots, the irrational quantities 

 had to be introduced : so in the analysis of compound 

 algebraical expressions into binomial factors, a new 

 (quantity or algebraical conception presented itself. It 

 was easily seen that this analysis could be carried out 

 in every case only by the introduction of a new unit, 

 algebraically expressed by the square root of the nega- 

 tive unity. There was no difficulty in algebraically 

 indicating the new quantity as we indicate fractions 

 and irrational quantities ; the difficulty lay in its inter- 

 pretation as a number. Since the time of Descartes 

 geometrical representations of algebraical formulic had 

 become the custom, and it was therefore natural when 

 once the new, or so-called imaginary, unit was formally 

 admitted, that a geometrical meaning should be attached 

 to it. 



Out of the scattered beginnings of these researches i9 

 two definite problems gradually crystallised: the one, ""'*!"'*"''' ^'"i 

 a purely formal or mechanical one — ^■i/., the geo- i"""*^'*^'"'*- 

 metrical representation of the extended conception of 

 quantity, of the complex quantity ; the other, a logical 



