THE HISTORY OF MATHEMATICS 407 



the eighteenth century: for the latter new foundations were laid by 

 Poincare some thirty years ago. 



Hydrodynamics has had less success in its applications. The pre- 

 diction of the tides, chiefly from the work of G. H. Darwin, and the 

 relative equilibrium of liquids in rotation by him and Poincare have 

 advanced in a satisfactory manner, but the knowledge of the motions 

 of bodies in actual fluids is still in an elementary condition, especially 

 when an attempt is made to apply it to under-sea and air craft. This, 

 of course, refers, not to the experimental side, but to developments 

 from the equations of motion. An interesting, but now almost 

 neglected subject is that of vortex rings in a perfect fluid, the main 

 features of which were given by Helmholtz and Lord Kelvin, giving rise 

 to a hypothesis, now abandoned, that the fundamental atoms of matter 

 consisted of such rings. The motions of our atmosphere have so far 

 defied attempts at explanation on any general plan. The theory of 

 sound, on the other hand, in the hands of Helmholtz and Lord Rayleigh, 

 has been well developed. The theory of elasticity is almost entirely a 

 creation of the present century and has found many applications. 



i 



Many of the ideas which are now fundamental in mathematics 

 have had their origin in an attempt to advance some particular branch. 

 Development has proceeded to a certain stage by means of known 

 methods and then stops, owing perhaps to mathematical difficulties or 

 to a failure of those methods to cast further light on it. A new method 

 of attack has then been evolved, showing new roads by which it may be 

 explored, after a time leading to openings which enable the investi- 

 gator to continue on the earlier lines. These new methods have then 

 been seen to be applicable to various other branches, thus forming 

 connecting links and shedding new light. Indeed, one not infrequently 

 meets with a statement that all mathematics can be based on some one 

 of these ideas. This may be true, but progress demands that the sub- 

 ject be cross cut in many ways: a new country may be opened out by 

 one great highway, but it is only well developed by several main roads 

 in different directions with numerous connecting branches. I shall 

 take up certain of these ideas and try to indicate briefly their bearing 

 on various mathematical topics. 



Some illustrations have already been given of the effect which a 

 critical examination of the logical processes used in mathematics has 

 produced. In geometry, the examination of Euclid's axioms has led to 

 the discovery of ideas of space other than those which were current in 

 earlier times. In analysis, algebras have been constructed in which 

 some of the familiar rules have been dropped or changed. The way 

 was thus opened to the examination of the foundations on which mathe- 

 matics rests. Here the work gets close to a consideration of the mode 



