642 



SCIENTIFIC THOUGHT. 



figures ? What are the properties of these finite figures 

 as inferred from the properties of their infinitesimally 

 small parts ? The infinitesimal methods evidently corre- 

 sponded with the atomistic view of natural objects, 

 according to which the great variety of observable 

 phenomena, the endlessly complicated properties of 

 natural objects, could be reduced to a small number 

 of conceivable properties and relations of their smallest 

 parts, and could then be made intelUgible and calculable. 

 The general reader who is unacquainted with the 

 numberless problems and intricate operations of higher 

 mathematics can scarcely realise how in these few words 

 lie really hidden the great questions of all the modern 

 sciences of number and measurement ; the trained mathe- 

 matical student will recognise in a process of inversion 

 not only the rationale of such extensive doctrines as the 

 integral calculus, the calculus of variations, the doctrine 

 of series, the methods of approximation and interpolation, 

 but also the application of analysis to geometry, the 

 theory of curves of higher order, the solution of equations, 

 &c. All these various branches were diligently cultivated 

 by the great mathematicians of the eighteenth century, 

 mostly, however, with the object of solving definite 

 problems which were suggested by the applied sciences,^ 



1 In general it can be stated that 

 the impetus given to mathematical 

 research by the problems set by 

 the applied sciences has been im- 

 measurably greater than that which 

 can be traced to the abstract treat- 

 ment of any purely mathematical 

 subject. We have a good example 

 of this at the beginning of the 

 nineteenth century in the gi'eat 

 work of Laplace as summed up, 



for the most j)art, in the 'Me- 

 canique Celeste ' and the ' Theorie 

 des Probabilites,' which contain 

 the beginnings and the develop- 

 ment of a great number of purely 

 mathematical theories suggested 

 by problems in astronomy, physics, 

 aud statistics. On the other side 

 we have at the same time the so- 

 called "Combinational School " in 

 Germanj', whose members and 



