ISOPERIMETRICAL PROBLEMS. 577 



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 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 

 knowledge 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, 

 VOL. II P P 



