THE HISTORY OF MATHEMATICS 411 



a proof can be constructed without a change of our axioms concerning 

 number or space. 



It will be seen from the sketchy remarks of the last few paragraphs 

 that at least one outstanding feature of pure mathematics during the 

 last century has been its emancipation from the trammels imposed by 

 any necessity for application to physical problems. It is, nevertheless, 

 necessary to say a few words about these applications, although the 

 major part of the story naturally comes under the history of physics. 

 Under the general term "applied mathematics," are included at least 

 three methods of study. In the first and simplest, we translate the 

 physical problems into symbols and deduce the consequences we desire 

 by mathematical methods. The work consists, therefore, of little more 

 than an argument on lines laid down by the mathematician. In the 

 second, a study of the formulae and relations which have arisen from 

 physical problems is made, without any special desire to apply them to 

 the phenomena: as indicated above, much of the pure mathematics 

 arose in this way, even before it was recognized that such study was a 

 quite legitimate intellectual exercise. In the third, the mathematical 

 processes used by the applied mathematician are studied in order to 

 find out their limitations, the extent of their validity, what extensions 

 they will admit, how more general methods may be obtained, the best 

 manner of treatment, and so on. This is not by any means an infertile 

 source of progress, as may be illustrated by Poincare's work on the 

 divergent series which are used to calculate the places of the moon 

 and planets. 



The most fundamental change in the attitude of applied mathe- 

 maticians has been in the recognition and working out of the conse- 

 quences of simple fundamental principles or laws. Foremost amongst 

 the latter is that known as the conservation of energy, brought into 

 prominence in the middle of the nineteenth century by the labors of 

 Helmholtz and Kelvin. It is now regarded as the chief invariant of the 

 universe and has been applied to every branch of physics. Owing to 

 the various forms which energy can take and to the fact that we are 

 practically compelled, in applying mathematics to a physical problem, 

 to deal only with some partial phase of it rather than with the whole, 

 we cannot always assume that the principle holds in a particular prob- 

 lem. But in the majority of such cases the energy which is lost or 

 changed from the particular form which we are considering is small 

 so that this loss may be neglected or allowed for. When the loss is 

 zero, the differential equations of the problem admit of an integral 

 which expresses this fact. Newtonian mechanics lead to two forms of 

 energy, kinetic (that due to motion) and potential (that due to posi- 

 tion) and the development of the mathematics of all material systems 

 has been mainly based on this separation. 



