DEVELOPMENT OF MATHEMATICAL THOUGHT. 681 



foremost in having with unrivalled fertility propounded 

 theorems which were as difficult to prove as the 

 manner in which they had been arrived at was mysteri- 

 ous. The great analytical genius of Euler, who possessed 

 unequalled resources in the solution of single problems, 

 spent much time and power in unravelling the riddles 

 of Fermat. In the theory of equations the general 

 solution beyond the fourth degree battled the greatest 

 thinkers. The time had come when in both branches 

 a systematic study of the properties had to be at- 

 tempted. This was done for the theory of numbers by 

 Gauss, for that of equations by Abel. Every great 

 step in advance of this kind in mathematics is accom- 

 panied by, and dependent on, skilful abbreviations, and 

 an easy algorithm or mathematical language. An as- 

 semblage of elements held together by the simplest 

 operations or signs of arithmetic — namely, those of 

 addition and multiplication — is much easier to deal 

 with if it can be arranged with some regularity, and 

 accordingly methods were invented by which algebraical 

 [expressions or forms were made symmetrical and homo- 

 geneous ; ^ the latter property signifying that each term 



39. 

 Symmetry. 



Arithmetic,' gave an interesting 

 [theorem by wliich the number of 

 imaginary roots of an equation can 

 be determined ; he left no proof, 

 and the theorem was discussed by 

 jEuler and many other writers, till 

 at last Sylvester in 1866 found the 

 broof of it in a more general 

 theorem. In more recent times 

 Jacob Steiner published a great 

 lumber of theorems referring to 

 dgebraical curves (see Crelle's 

 ; Journal,' vol. xlvii. ) which have 

 been compared by Hesse with the 

 I' riddles of Fermat." Luigi Cre- 

 nona succeeded at last in proving 



them by a general sj^nthetical 

 method. 



1 The introduction of homogene- 

 ous expressions marks a great 

 formal advance in algebra and 

 analj'tical geometry. The first in- 

 stance of homogeneous co-ordinates 

 is to be found in Mobius's "Bary- 

 centric Calculus" (1826), in which 

 he defined the position of any point 

 in a plane by reference to three 

 fundamental points, considering 

 each point as the centre of gravity 

 of those points when weighted. 

 " The idea of co-ordinates appears 

 here for the first time in a new 



