64 SCIENCE PROGRESS 



under which various solutions are valid. Some brilliant 

 generalisations and important results have been obtained by 

 this school, but these are derived rather as by-products of the 

 original investigation. The mathematician of the older school 

 starts with a definite problem, more or less idealised so as to 

 be susceptible of mathematical manipulation, and proceeds 

 to attack it by such methods as seem most promising. His 

 intuition guides him safely through difficulties which would 

 bring the pure mathematician to a halt. Both classes of work 

 are valuable and necessary, but it would be difficult to prove 

 that the latter, though the less showy, has not been the more 

 productive of results of importance. 



Darwin frequently found it necessary to resort to numerical 

 computation when further mathematical treatment became 

 impossible, and always preferred a quantitative to a qualitative 

 result. Although he disliked numerical calculation he would 

 never shrink from the most arduous computations if only there 

 seemed some possibility of throwing further light on his results. 

 He once told Lord Moulton that he detested arithmetic and 

 that his calculations were as tedious and painful to him as 

 they would have been to any other man, but that he realised 

 that they must be done and that it was impossible to train 

 any one else to do them. In tracing one of his periodic 

 orbits he casually remarks that he obtained for the purpose 

 seventy-five points, each of which involved three-quarters 

 of an hour's work. His results, which have been invaluable 

 in theories of cosmogony, fully justified the labour expended 

 in obtaining them and show such work to be equally necessary 

 with that of the newer school. Let due honour be given to 

 the men who do the necessary spade-work of science. Darwin 

 himself aptly described his methods as " the procedure of a 

 house-breaker who blows in a safe with dynamite instead of 

 picking the lock " ; and again, " to put at their lowest the 

 claims of this clumsy method, which may almost excite the 

 derision of the pure mathematician, it has served to throw 

 light on the celebrated generalisations of Hill and Poincare." 



The present volume forms the fifth and concluding volume 

 of Darwin's collected papers, the previous four volumes having 

 been published during the author's lifetime and containing 

 such of his papers as were published before 1910 as he wished 

 to see reproduced. Many scientific reports on various subjects 



