324 A SHORT HISTORY OF SCIENCE 



the other, and moving on steadily in the direction it has deliberately 

 chosen. Helmholtz. 



Nature herself exhibits to us measurable and observable quan- 

 tities in definite mathematical dependence; the conception of a 

 function is suggested by all the processes of nature where we observe 

 natural phenomena varying according to distance or to time. Nearly 

 all the "known" functions have presented themselves in the attempt 

 to solve geometrical, mechanical, or physical problems. Merz. 



We have now reached a period of maturity in the evolution of 

 mathematical science beyond which any attempt to follow its 

 details would involve technical discussions outside the range of 

 this work. The present chapter will be devoted to a general 

 survey of modern tendencies in pure and applied mathematics, 

 in mechanics, in mathematical physics and in astronomy. The 

 most notable single fact in the centuries under discussion is the 

 increasing specialization resulting from the great expansion of 

 scientific knowledge. It is no longer possible for the individual 

 scholar to command the range at once of philosophy, mathe- 

 matics, physics, chemistry, and the natural sciences. It has even 

 become more and more difficult to have a general knowledge of 

 any one of these broad fields. 



MATHEMATICS AND MECHANICS IN THE EIGHTEENTH CENTURY. 

 The invention of the infinitesimal calculus by Newton and 

 Leibnitz was comparable in its relations and consequences with 

 the discovery of a new world by Columbus two centuries earlier. 

 As in that case the discovery was not an absolutely sudden one ; 

 other explorers had hoped or imagined, but only genius of that 

 highest order which we call inspired, gained the complete revela- 

 tion. The years next following the great discovery were natu- 

 rally a period of eager and wide-ranging exploration, of optimistic 

 self-confident pioneering. Such was the power of the new method, 

 that one might rashly hope no secret of nature could long resist 

 its attack. As circumnavigation of the globe was not long in 

 following the discovery of America, so the cycle of mathematical 

 knowledge might be completed. The parallel has failed. The 

 calculus grew out of the insistent grappling by mathematicians with 



