114 The Bight Hon. Lord Kelvin [May 12, 



and gives results in his Book V., which consists of fifty-five theorems 

 and fifty- seven propositions on the areas of various plane figures 

 having equal circumferences. In this Book, written originally 

 in Greek, we find (Theorem IX. Proposition X.) the expression 

 " isoperimetrical figures," which is, so far as I know, the first use of 

 the adjective "isoperimetrical" in geometry; and we may, I believe, 

 justly regard Pappus as the originator, for mathematics, of isoperi- 

 metrical problems, the designation technically given in the nineteenth 

 century* to that large province of mathematical and engineering 

 science in which different figures having equal circumferences, or 

 different paths between two given points, or between some two points 

 on two given curves, or on one given curve, are compared in con- 

 nection with definite questions of greatest efficiency and smallest 

 cost. 



In the modern engineering of railways an isoperimetrical problem 

 of continual recurrence is the laying out of a line between two towns 

 along which a railway may be made at the smallest prime cost. If 

 this were to be done irrespectively of all other considerations, the 

 requisite datum for its solution would be simply the cost per yard of 

 making the railway in any part of the country between the two towns. 

 Practically the solution would be found in the engineers' drawing 

 office by laying down two or three trial lines to begin with, and 

 calculating the cost of each, and choosing the one of which the cost is 

 least. In practice various other considerations than very slight 

 differences in the cost of construction will decide the ultimate choice 

 of the exact line to be taken, but if the problem were put before a 

 capable engineer to find very exactly the line of minimum total cost, 

 with an absolutely definite statement of the cost per yard in every 

 part of the country, he or his draughtsmen would know perfectly 

 how to find the solution. Having found something near the true 

 line by a few rough trials they would try small deviations from the 

 rough approximation, and calculate differences of cost for different 

 lines differing very little from one another. From their drawings 

 and calculations they would judge by eye which way they must 

 deviate from the best line already found to find one still better. At 

 last they would find two lines for which their calculation shows no 

 difference of cost. Either of these might be chosen ; or, according to 

 judgment, a line midway between them, or somewhere between them, 

 or even not between them but near to one of them, might be chosen, as 

 the best approximation to the exact solution of the mathematical 

 problem which they care to take the labour of trying for. But it is 

 clear that if the price per yard of the line were accurately given 

 (however determined or assumed) there would be an absolutely 

 definite solution of the problem, and we can easily understand that the 

 skill available in a good engineer's drawing-office would suffice to find 

 the solution with any degree of accuracy that might be prescribed ; 



* Example, Woodhouse's 'Isoperimetrical Problems/ Cambridge, 1810. 



