94 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



He who attempts to adequately describe the full significance of the 

 service of Helmholtz to mathematical science must refer to each of 

 his two or three hundred published writings, for all of them, even 

 when mathematical language is not employed, excite the highest 

 interest of the mathematician by reason of their eminently acute log- 

 ical reasoning. But the difficulty of treating each research with full 

 justice becomes apparent when we consider that the historian must 

 follow his unparalleled investigations through all branches of natural 

 science. In short, physiologist, physicist, mathematician, philosopher, 

 and master alike of nature and art must he be who would regard so 

 great a thinker as Helmholtz not merely with wonder and amaze- 

 ment, but fully and intelligently. It was not the nature of his mind 

 to pursue mathematical investigations for their own sake, or to delight 

 in the discovery of purely abstract truths deduced from algebraic or 

 geometrical conceptions to find possible future use in the exact natural 

 sciences. On the contrary, he obtained his mathematical problems 

 direct from observation of nature, and this is certainly the only true 

 way, yet fruitful only in the hands of so great a master. His starting 

 point was the axiom that science, whose aim it is to apprehend nature, 

 must admit that she is capable of being understood, and for him that 

 meant no less than what his greatest pupil, Heinrich Hertz, has said: 

 a The necessary logical consequences of the inner conceptions of outer 

 phenomena should correspond with the necessary natural consequences 

 of the phenomena conceived of, which requires that the problems of 

 nature should be mathematically formulated." Thus in all his works 

 appears an inexhaustible richness of results full of interest from a 

 purely mathematical point of view, that nevertheless find a mechanical- 

 physical significance, and then lead to the discovery of profound and 

 general natural laws, which, when divested of their mathematical 

 exposition, have prepared the way not only in the natural sciences, but 

 also in the world at large, for essentially new conceptions of the pro- 

 cedure of natural events. He was interested in mathematical investi- 

 gations for their own sake only in treating of the axioms and founda- 

 tions of mathematical science. With this aim he made researches in 

 each of the three great branches of mathematics — geometry, arithme- 

 tic, and mechanics — which have marked radical advances both in 

 idiilosopfiy and in the whole development of mathematical physics. 

 Here also, in contrast with the methods of other distinguished mathe- 

 maticians engaged in the same or similar investigations, he continually 

 verified his deductions by references to observation and experience in 

 proceeding to attain the most abstract mathematical truths. 



The science of mathematics in its whole extent, as I shall here regard 

 it, deals with three independent primary conceptions, those of space, of 

 time, and of mass. The province of geometry embraces considerations 

 of space, that of arithmetic embraces time, and the relations of mass 

 to space and time constitute the subject-matter of mechanics and 

 mathematical physics. The view brought forward by Kant that space 



