1893.] on Isoperimetrical Problems. 113 



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 complete the enclosure 

 before night-fall, 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 complex 

 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 nevertheless 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 mathematical 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. 



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 intro- 

 duction to the book not only for his admirable obedience to the 

 Authorities of 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 much of geometry as is 

 practically useful to them, proceeds to apply what he calls his own 

 superior human intelligence to investigation of useless knowledge " 



Vol. XIV. (No. 87.) ! 5 ' 



