PRESIDENT S ADDRESS — SECTION A. 9 



the prelimiuaiy work has been done and the earliest attempts had 

 their turn and proved unsuccessful, it is best to let the question rest 

 in the mind and develop itself. Thus it would appear that one often 

 works hardest when one is doing nothing. 



The field of work to which his attention was first directed always 

 remained dear to him, but it is probably in the applications of analysis 

 to the solution of the differential equations of mathematical physics 

 that he found one of his favourite themes. To this point I shall return 

 later. 



I now look at Poincare, the great astronomer. And first we 

 notice that neither as a physicist nor as an astronomer did his work 

 lie in the laboratory or the observatory. The semdce which he 

 rendered these lay in the application of the methods of Analysis and 

 ■Geometry to the problems of Physics and Astronomy. His investiga- 

 tion of the form taken by a gravitating fluid mass in rotation led him 

 to most important theories on the separation of the earth and the 

 moon. Of this work Darwin, who had himself made important 

 discoveries in the same field, records that the memoir will always mark 

 an important epoch, not only in the history of Astronomy, but also in 

 that of the larger domain of General Dynamics. 



In his work on the stability of the solar system Poincare returned 

 to the problem treated by Laplace, and applied to it the new mathe- 

 matical instruments now available. His labours in this connexion 

 are to be found in his book, Les Melhodes Nouvelles de la Mecanique 

 Celeste. "What Newton's Principia did for Astronomy in the 17th 

 century, this work of Poincare's has done for the 20th century. We 

 are assured that all the advances likely to be made in the next 50 ye^rs 

 in Astronomy are certain to rest upon the foundation Poincare has 

 laid. 



Of the mathematician, the physicist, and the astronomer, we 

 have spoken. There still remains the philosopher. But to enter 

 upon a discussion of that part of his work which lies in the border- 

 ground between Philosophy and Mathematics is a task for which I 

 have neither the time nor the qualifications. The ordinary mathe- 

 matician meets this section of Poincare's work when he considers the 

 Principles of Mathematics in general ; and in particular the Principles 

 of Geometry. No student of Geometry can afford to neglect Poincare's 

 contribution to this modem development of Mathematics. It is a 

 pity that some other mathematical philosophers have not approached 

 the subject with the clear open vision of Poincare, and that they have 

 not placed their views before us in lucid language such as he delighted 

 to employ. 



I now turn to the subject which I have chosen for my address, 

 The Relation between Pure and Applied Mathematics. 



