482 MATHEMATICS 



and a theory of point aggregates. In the theory of point aggregates 

 the notion of limiting points gives rise to important classes of aggre- 

 gates as discrete, dense, everywhere dense, complete, perfect, con- 

 nected, etc., which are so important in the function theory. 



In the general theory two notions are especially important, 

 namely, the one to one correspondence of the elements of two ag- 

 gregates, and well-ordered aggregates. The first leads to cardinal 

 numbers and the idea of enumerable aggregates, the second to trans- 

 finite or ordinal numbers. 



Two striking results of Cantor's theory are these: the algebraic 

 and therefore the rational numbers, although everywhere dense, are 

 enumerable; and secondly, one-way and n-way space have the 

 same cardinal number. 



Cantor's theory has already found many applications, especially 

 in the function theory, where it is to-day an indispensable instrument 

 of research. 



Functions of Real Variables The Critical Movement 



One of the most conspicuous and distinctive features of mathe- 

 matical thought in the nineteenth century is its critical spirit. Be- 

 ginning with the calculus, it soon permeates all analysis, and toward 

 the close of the century it overhauls and recasts the foundation of 

 geometry and aspires to further conquests in mechanics and in the 

 immense domains of mathematical physics. 



Ushered in with Lagrange and Gauss just at the close of the 

 eighteenth century, the critical movement receives its first decisive 

 impulse from the teachings of Cauchy, who in particular introduces 

 our modern definition of limit and makes it the foundation of the 

 calculus. We must also mention in this connection Abel, Bolzano, 

 and Dirichlet. Especially Abel adopted the reform ideas of Cauchy 

 with enthusiasm, and made important contributions in infinite series. 



The figure, however, which towers above all others in this move- 

 ment, whose name has become an epithet of rigor, is Weierstrass. 

 Beginning at the very foundations, he creates an arithmetic of real 

 and complex numbers, assuming the theory of positive integers to be 

 given. The necessity of this is manifest when we recall that until 

 then the simplest properties of radicals and logarithms were utterly 

 devoid of a rigorous foundation; so, for example, 



V2 N/5=ViO log 2+log 5=log 10 



Characteristic of the pre-Weierstrassean era is the loose way in 

 which geometrical and other intuitional ideas were employed in 

 the demonstration of analytical theorems. Even Gauss is open to 

 this criticism. The mathematical world received a great shock 

 when Weierstrass showed them an example of a continuous function 



