March 29, 1894] 



NA TURE 



517 



which would be very valuable to the draughtsman if he were 

 required to produce an exceedingly accurate drawing of the 

 required curve. The curvature of the curve at any point is con- 

 vex towards the side on which the price per unit length of line 

 is less, and is numerically equal to the rate per mile perpen- 

 dicular to the line at which the Neperian logarithm of the price 

 per unit length of the line varies. This statement would give 

 the radius of curvature in fraction of a mile. If we wish to 

 have it in yards we must take the rate per yard at which the 

 Neperian logarithm of the price per unit length of the line 

 varies. I commend the Neperian logarithm of price in pounds, 

 shillings and pence to our Honorary Secretary, to whom no 

 doubt tt will present a perfectly clear idea; but less powerful 

 men would prefer to reckon the price in pence, or in pounds and 

 decimals of a pound. In every possible case of its subject the 

 "calculus of variations " gives a theorem of curvature less 

 simple in all other cases than in that very simple case of the 

 railway line of minimum first cost, but always interpretable and 

 intelligible according to the same principles. 



Thus in Dido's problem we find by the calculus of variations 

 that the curvature of the enclosing line varies in simple propor- 

 tion to the value of the land at the places through which it 

 passes ; and the curvature at any one place is determined by 

 the condition that the whole length of the ox-hide just com- 

 pletes the enclosure. 



The problem of Horatius Codes combines the railway 

 problem with that of Dido. In it the curvature of the boundary 

 is the sum of two parts ; one, as in the railway, equal to the 

 rate of variation perpendicular to the line, of the Neperian 

 logarithm of the cost in time per yard of the furrow (instead of 

 cost in money per yard of the railway) ; the other varying pro- 

 portionally to the value of the land as in Dido's problem, but 

 now divided by the cost per yard of the line, which is constant 

 in Dido's case. The first of these parts, added to the ratio of 

 the money-value per square yard of the land to the money-cost 

 per lineal yard of the boundary (a wall suppose), is the curvature 

 of the boundary when the problem is simply to make the most 

 you can of a grant of as much land as you please to take pro- 

 vided you build a proper and sufficient stone wall round it at 

 your own expense. This problem, unless wall-building is so 

 costly that no part of the offered land will pay for the wall 

 round it, has clearly a determinate finite solution if the offered 

 land is an oasis surrounded by valueless desert It has also a 

 determinate finite solution even though the land be nowhere 

 valueless, if the wall is sufficiently more and more expensive 

 at greater and greater distances from some place where there are 

 quarries, or habitations for the builders. 



The simplified case of this problem, in which all equal areas 

 of the land are equally valuable, is identical with the old well- 

 known Cambridge dynamical plane problem of finding the 

 motion of a particle relatively to a line of reference revolving 

 uniformly in a plane : to which belongs that considerable part 

 of the " Lunar Theory " in which any possible motion of the 

 moon is calculated on the supposition that the centre of gravity 

 of the earth and moon moves uniformly in a circle round the 

 sun, and that the motions of the earth and moon are exactly 

 in this plane. The rule for curvature which I have given 

 you expresses in words the essence of the calculation, and 

 suggests a graphic method for finding solutions by which not 

 uninteresting approximations ^ to the cusped and looped orbits 

 of G. F. Hill- and Poincare" can be obtained without dis- 

 proportionately great labour. 



In the dynamical problem, the angular velocity of the revolving 

 line of reference is numerically equal to half the value of the 

 land per square yard ; and the relative velocity of the moving 

 particle is numerically equal to the cost of the wall per lineal 

 yard in the land question. 



But now as to the proper theorem of curvature for each case ; 

 both Dido and Horatius Codes no doubt felt it instinctively and 

 were guided by it, though they could not put it into words, still 

 less prove it by the "calculus of variations." It was useless 

 knowledge to the bees, and, therefore, they did not know it ; 

 because they had only to do with straight lines. But as you are 

 not bees I advise you all, even though you have no interest in 

 acquiring as much property as you can enclose by a wall of 



1 Kelvin, " On Graphic Solution of Dynamical Problems." /'////. Mag. 

 1892 (2nd half-year). 



- Hill, " Researches in the Lunar Theory," Part 3. " National Academy 

 of Sciences," 1887. 



3 " M6thodes Nouvelles de la M^canique Celeste," p. 109 (1S92). 



given length, to try Dido's problem for yourselves, simplifying 

 it, however, by doing away with the rugged coast line for part 

 of your boundary, and completing the enclosure by the wall 

 itself. Take forty inches of thin soft black thread with its ends 

 knotted together and let it represent the wall ; lay it down on a 

 large sheet of white paper and try to enclose the greatest area 

 with it you can. You will feel that you must stretch it in a 

 circle to do this, and then, perhaps, you will like to read 

 Pappus (Liber V. Theorema II. Propositio II.) to find mathe- 

 matical demonstration that you have judged rightly for the case 

 of all equal areas ot the enclosed land equally valuable. Next 

 try a case in 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 representing 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 drawing-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 parti- 

 cular 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 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 hour longer, I fear that two hours of pure mathematics 

 and dynamics might be too fatiguing. I must, therefore, per- 

 force 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 point 

 moving in a plane under the influence of a given conservative 

 system of forces, that is to say, such a system that when material 

 particles not mutually influencing one another are projected 

 from one and the same point in different directions, but with 

 equal velocities, the subsequent velocity of each is calculable 

 from its position at any instant, and all have equal velocities in 

 travelling through the same place whatever may be their 

 directions. The theorem of curvature, of which I told you in 

 connection with the railway engineering problem, is now simply 

 the well-known elementary law of relatioii between curvature 

 and centrifugal force of the motion of a particle. 



The motion of a particle in a plane is, as Liouville has proved, 

 a case to which every possible problem of dynamics involving 

 just two freedoms to move can be reduced. But to bring you 

 to see clearly its relation to isoperimetrics, I must tell you of 

 another admirable theorem of Liouville's, reducing to a still 

 simpler case the most general dynamics of two-freedoms motion. 

 Though not all mathematical experts, I am sure you can all per- 



NO. 1274, VOL. 49J 



