

1893.] on Isoperimetrical Problems. 115 



the minuter the accuracy to be attained the greater the labour, of 

 course. You must not imagine that I suggest, as a thing of practical 

 engineering, the attainment of minute accuracy in the solution of a 

 problem thus arbitrarily proposed ; but it is interesting to know that 

 there is no limit to the accuracy to which this ideal problem may be 

 worked out by the methods which are actually used every day by 

 engineers in their calculations and drawings. 



The modern method of the " calculus of variations," brought into 

 the perfect and beautiful analytical form in which we now have it by 

 Lagrange, gives for this particular problem a theorem which would 

 be very valuable to the draughtsman if he were required to produce 

 an exceedingly accurate drawing of the required curve. The curva- 

 ture of the curve at any point is convex towards the side on which 

 the price per unit length of line is less, and is numerically equal to the 

 rate per mile perpendicular to the line at which the Neperian loga- 

 rithm 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 it 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 ca-es than in that 

 very simple case of the railway line of minimum first cost, but 

 always interpretable and intelligible according to the same prin- 

 ciples. 



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

 that the curvature of the enclosing line varies in simple proportion 

 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 completes 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 proportionally 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 



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