5i6 



NATURE 



[March 29, 1894 



ward from any given point A of the coast, the inland bounding line 

 must at its far end cut the coast line perpendicularly. Here, 

 then, to complete our solution, we have a very curious and in- 

 teresting, but not at all easy, geometrical question to answer : — 

 What must be the radius of a circular arc A D C, of given 

 length, and in what direction must it leave the point A, in order 

 that it may cut a given curve ABC perpendicularly at some un- 

 known point c? I do not believe Dido could have passed an ex- 

 amination on the subject, but no doubt she gave a very good 

 practical solution, and better than she would have found if she had 



just mathematics enough to make her fancy the boundary ought 

 to be a circle. No doubt she gave it different curvature in different 

 parts to bring in as much as possible of the more valuable parts of 

 the land offered to her, even though difference of curvature in 

 different parts would cause the total area enclosed to be less than 

 it would be with a circular boundary of the same length. 



The Roman reward to Horatius Codes brings in quite a new 

 idea, now well known in the general subject of isoperimetrics : 

 the greater or less speed attainable according to the nature of the 

 country through which the line travelled over passes. If it had 

 been equally easy to plough the furrow in all parts of the area 

 offered for enclosure, and if the value of the land per acre was 

 equal throughout, Codes would certainly have ploughed as nearly 

 in a circle as he could, and would only have deviated from a 

 single circular path if he found that he had misjudged its proper 

 curvature. Thus, he might find that he had begun on too large 

 a circle, and, in order to get back to the starting-point and com- 

 plete the enclosure before nightfall, he must deviate from it on 

 the concave side ; or he would deviate from it on the other side 

 if he found that he had begun on too small a circle, and that he 

 had still time to spare for a wider sweep. But, in reality, he 

 must also have considered the character of the ground he had to 

 plough through, which cannot but have been very unequal in 

 different parts, and he would naturally vary the curvature of 

 his path to avoid places where his ploughing must be very slow, 

 and to choose those where it would be most rapid. 



He must also have had, as Dido had, to consider the different 

 value of the land in different parts, and thus he had a very com- 

 plex problem to practically solve. He had to be guided both 

 by the value of the land to be enclosed and the speed at which 

 he could plough according to the path chosen ; and he had a 

 very brain-trying task to judge what line he must follow to get 

 the largest value of land enclosed before night. 



These two very ancient stories, whether severe critics will 

 call them mythical or allow them to be historic, are never- 

 theless full of scientific interest. Each of them expresses a 

 perfectly definite case of the great isoperimetrical problem to 

 which the whole of dynamics is reduced by the modern mathe- 

 matical methods of Euler, Lagrange, Hamilton and Liouville 

 (Liouville's Journal, 1840 1850). In Dido's and Horatius Codes' 

 problems, we find perfect illustrations of all the fundamental 

 principles and details of the generalised treatment of dynamics 

 which we have learned from these great mathematicians of the 

 eighteenth and nineteenth centuries. 



NO. J 274, VOL. 49] 



Nine hundred years after the time of Horatius Codes we find,, 

 in the fifth Book of the collected Mathematical and Physical 

 Papers of Pappus of Alexandria, still another idea belonging to 

 isoperimetrics — the economy of valuable material used for 

 building a wall ; which, however, is virtually the same as the 

 time per yard of furrow in Codes' ploughing. In this new case 

 the economist is not a clever princess, nor a patriot soldier, 

 but a humble bee who is praised in the introduction to the book 

 not only for his admirable obedience to the Authorities pf his 

 Republic, for the neat and tidy manner in which he collects 

 honey, and for his prudent thoughtfulness in arranging for its 

 storage and preservation for future use, but also for his know- 

 ledge of the geometrical truth that a "hexagon can enclose 

 more honey than a square or a triangle with equal quantities of 

 building material in the walls," and for his choosing on this 

 account the hexagonal form for his cells. Pappus, concluding 

 his introduction with the remark that bees only know as mucb 

 of geometry as is practically useful to them, proceeds to apply 

 what he calls his own superior human intelligence to investiga- 

 tion of useless knowledge, 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 isopa-i- 

 metrical problems, the designation technically given in the nine- 

 teenth 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 connection 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 calculat- 

 ing 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 draughts- 

 men 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 judg- 

 ment, 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 ; 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 m„inute 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 



5 Example, Woodhouse's "Isoperimetrical Problems," Cambridge, iSio. 



