THE HISTORY OF MATHEMATICS 405 



The older theory of numbers in general dealt with integers and to 

 a less extent with fractions, square roots, etc., that is, numbers which 

 could be formed out of the integers by the ordinary operations of 

 arithmetic. Gauss opened the way to a more extended idea of the 

 meaning which might be attached to numbers by introducing those 

 which are the solutions of an ordinary algebraic equation of any degree, 

 whose coefficients are rational; such numbers are called algebraic. 

 They naturally introduced complex numbers and have properties such 

 as divisibility more extended than those which play a large part in our 

 ordinary number system. This soon demanded a theory of congruences 

 which, in their simplest form are numbers which, when divided by a 

 given factor (called the modulus), always leave the same remainder 

 (a residue^. The theory of forms was another development which 

 arose. Since Gauss' time the ideas have been greatly extended to many 

 other kinds of numbers in which special classes have special proper- 

 ties, and these classes are the main subjects of investigation. 



The extensive development of the theory of functions of a complex 

 variable is perhaps the most significant achievement of the last hun- 

 dred years. The imaginary was always arising in such questions as the 

 solutions of quadratic equations and in the new branches. The next 

 step was to give a geometric interpretation of a complex number by 

 showing that it could represent in a single symbol the two co-ordinates 

 of a point in a plane. A function of a complex variable was therefore 

 a function which could take values over an area as against one of a 

 real variable which could only take values along a line. The majority 

 of functions become infinite for one or more values of the variable and 

 these infinities play the major part in the development of the theory. 

 To Cauchy belongs the honor of starting the work in the third decade 

 of the nineteenth century, examining such questions as the possibility 

 of developing such functions in series, their integrals, and the actual 

 existence of functions of different kinds. Closely following him, 

 Weierstrass and Riemann developed Cauchy's ideas, the former by bas- 

 ing his arguments on a special form — a series of powers of the 

 variable — and the latter by using geometrical and even physical ideas 

 for progress. Their successors have merged these different modes of 

 development and have continued to investigate with success the repre- 

 sentation of a function under given conditions, and the limitations of 

 a given function. At the same time special functions, particularly those 

 known as elliptic, were being developed by Abel and Jacobi, and the 

 latter extended them in a very general manner. We have now many 

 groups of such functions, some of which have become of sufficient im- 

 portance to have whole treatises devoted to them. 



The study of functions of real variables during the past half cen- 

 tury has been largely due to the critical spirit which has compelled 



