1893.] on Isoperimetrical Problems. 117 



which the land is of different value in different parts. Take a 

 square foot of white paper and divide it into 144 square inches to 

 represent square miles, your forty inches of endless thread repre- 

 senting a forty miles wall to enclose the area you are to acquire. 

 Write on each square the value of that particular square mile of 

 land, and place your endless thread upon the paper, stretched round 

 a large number of smooth pins stuck through the paper into a draw- 

 ing-board below it, so as to enclose as much value as you can, judging 

 first roughly by eye and then correcting according to the sum of 

 the values of complete squares and proportional values of parts of 

 squares enclosed by it. In a very short time you will find with 

 practical accuracy the proper shape of the wall to enclose the 

 greatest value of the land that can be enclosed by forty miles of wall. 

 When you have done this you will understand exactly the subject of 

 the calculus of variations, and those of you who are mathematical 

 students may be inclined to read Lagrange, Woodhouse, and other 

 modern writers on the subject. The problem of Horatius Codes, 

 when not only the different values of the land in different places 

 but also the different speed of the plough according to the nature 

 of the ground through which the furrow is cut are taken into 

 consideration, though more complex and difficult, is still quite 

 practicable by the ordinary graphic method of trial and error. The 

 analytical method of the calculus of variations, of which I have told 

 you the result, gives simply the proper curvature for the furrow in 

 any particular direction through any particular place. It gives this 

 and it cannot give anything but this, for any plane isoperimetrical 

 problem whatever, or for any isoperimetrical 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 pro- 

 pounded 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 hour longer, I fear that 

 two hours of pure mathematics and dynamics might be too fatiguing. 

 I must, therefore, perforce limit myself to the two-dimensional, but 

 otherwise wholly comprehensive, problems of Dido and Horatius 

 Codes. Going back to the simpler included case of the railway of 

 minimum cost between two towns, the dynamical analogue is this : — 

 For price per unit length of the line substitute the velocity of a 



