390 
PAPPUS OF ALEXANDRIA 
ambrosia in this form, they do not think it proper to pour it 
carelessly on ground or wood or any other ugly and irregular 
material; but, first collecting the sweets of the most beautiful 
flowers which grow on the earth, they make from them, for 
the reception of the honey, the vessels which we call honey 
combs, (with cells) all equal, similar and contiguous to one 
another, and hexagonal in form. And that they have con 
trived this by virtue of a certain geometrical forethought we 
may infer in this way. They would necessarily think that 
the figures must be such as to be contiguous to one another, 
that is to say, to have their sides common, in order that no 
foreign matter could enter the interstices between them and 
so defile the purity of their produce. Now only three recti 
lineal figures would satisfy the condition, I mean regular 
figures which are equilateral and equiangular; for the bees 
would have none of the figures which are not uniform. . . . 
There being then three figures capable by themselves of 
exactly filling up the space about the same point, the bees by 
reason of their instinctive wisdom chose for the construction 
of the honeycomb the figure which has the most angles, 
because they conceived that it would contain more honey than 
either of the two others. 
‘ Bees, then, know just this fact which is of service to them 
selves, that the hexagon is greater than the square and the 
triangle and will hold more honey for the same expenditure of 
material used in constructing the different figures. We, how 
ever, claiming as we do a greater share in wisdom than bees, 
will investigate a problem of still wider extent, namely that, 
of all equilateral and equiangular plane figures having an 
equal perimeter, that which has the greater number of angles 
is always greater, and the greatest plane figure of all those 
which have a perimeter equal to that of the polygons is the 
circle.’ 
Book V then is devoted to what we may call iso perimetry, 
including in the term not only the comparison of the areas of 
different plane figures with the same perimeter, but that of the 
contents of different solid figures with equal surfaces. 
Section (1). Isoperimetry after Zenodorus. 
The first section of the Book relating to plane figures 
(chaps. 1-10, pp. 308-34) evidently followed very closely 
the exposition of Zenodorus nepl icro per poor a-xyp-drcou (see 
pp. 207-13, above); but before passing to solid figures Pappus 
inserts the proposition that of all circular segments having