736 



SCIENTIFIC THOUGHT. 



grade of infinity ; it has a higher, perhaps the second, 

 power. 1 



In all these, and in many similar investigations, a 

 conception has gradually emerged which was foreign to 

 older mathematics, but which plays a great and useful 

 part in modern mathematical thought. Older mathe- 

 matics, ever since the introduction of general arithmetic 

 or algebra, centred in the conception of equality and in 

 the solution of equations. Everything was reduced to 

 magnitude. But there are other relations besides those 

 of magnitude, of more or less. Often in practical pur- 

 suits, if we cannot find a counterpart or write down 

 an exact numerical equation, we can gain information 

 by a correspondence. This conception of correspondence 

 plays a great part in modern mathematics. It is the 

 fundamental notion in the science of order as dis- 

 tinguished from the science of magnitude. If older 

 mathematics were mostly dominated by the needs of 

 mensuration, modern mathematics are dominated by the 

 conception of order and arrangement. It may be that 

 this tendency of thought or direction of reasoning goes 

 hand in hand with the modern discovery in physics, 

 that the changes in nature depend not only or not so 

 much on the quantity of mass and energy as on their 

 distribution or arrangement. 



With these reflections we touch the limits of mathe- 



1 A summary of Prof. Cantor's 

 work is given by Prof. Schdnflies 

 in the ' Encyklop. Math. Wiss. , ' 

 vol. i. p. 184 sqq. The importance 

 of accurate definitions and distinc- 

 tions regarding the infinite and 

 the continuous is dwelt on and 



the different recent theories set 

 forth in a very lucid address to the 

 London Math. Society by Prof. 

 Hobson, " On the Infinite and In- 

 finitesimal in Mathematical Analy- 

 sis," November 1902. 



