DEVELOPMENT OF MATHEMATICAL THOUGHT. 725 



application it may lead to a widening of ideas, to an 

 enlargement of views, to a removing of artificial and con- 

 ventional barriers of thought. As I stated early in this 

 chapter, the attempts of Gauss to prove the fundamental 

 theorem of algebra, that every equation has a root, 

 suggested to him the necessity of introducing complex 

 numbers ; the development of the theory of congruences 

 and of residues — notably of the higher residues — con- 

 firmed this necessity. In the year 1831, in his memoir 

 on biquadratic residues, he annovmces it as a matter of 

 fundamental importance. In the earlier memoir he had 

 treated this extension of the field of higher arithmetic 

 .as possible, but had reserved the full exposition. And 

 before he redeemed this promise the necessity of doing 

 so had been proved by Abel and Jacobi, who had created 

 the theory of elliptic functions, showing that the concep- 

 tion of a periodic function (such as the circular or har- 

 monic function) could be usefully extended into that 

 theory, if a double period— a real and an imaginary 

 ■one — were introduced. A simplification similar to that 

 which this bold step led to in the symbolic represen- 

 tation of those higher transcendents, had been discovered 

 by Gauss to exist in the symbolical representation of 

 the theory of biquadratic residues which only by the 

 simultaneous use of the imaginary and the real unit 

 '" presented itself in its true simplicity and beauty." In 

 this theory it was necessary to introduce not only a posi- 

 tive and negative, but likewise a lateral system of count- 

 ing — i.e., to count not only in a line backwards and for- 

 wards, but also sideways in two directions, as Gauss 

 showed very plainly in the now familiar manner. At the 



