588 POPULAR LECTURES AND ADDRESSES. 



for the furrow in any particular direction through 

 any particular place. It gives this and it cannot 

 give anything but this, for any plane isoperimetri- 

 cal problem whatever, or for any isoperimctrical 

 problem on a given curved surface of any kind. 



Beautiful, simple, and clear as isoperimetrics is 

 in geometry, its greatest interest, to my mind, is 

 in its dynamical applications. The great theorem 

 of least action, somewhat mystically and vaguely 

 propounded by Maupertuis, was magnificently 

 developed by Lagrange and Hamilton, and by 

 them demonstrated to be not only true throughout 

 the whole material world, but also a sufficient 

 foundation for the whole of dynamical science. 



It would require nearly another hour if I were 

 to explain to you fully this grand generalisation 

 for any number of bodies moving freely, such as 

 the planets and satellites of the solar system, or 

 any number of bodies connected by cords, links, or 

 mutual pressures between hard surfaces, as in a 

 spinning-wheel, or lathe and treadle, or a steam- 

 engine, or a crane, or a machine of any kind ; but 

 even if it were convenient to you to remain here an 



