A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE COLLECTION. BOOKS IV, V 389
390 PAPPUS OF ALEXANDRIA
Book V. Preface on the Sagacity of Bees.
It is characteristic of the great Greek mathematicians that,
whenever they were free from the restraint of the technical
language of mathematics, as when for instance they had occasion
to write a preface, they were able to write in language of
the highest literary quality, comparable with that of the
philosophers, historians, and poets. We have only to recall
the introductions to Archimedes's treatises and the prefaces
to the different Books of Apollonius's Conies. Heron, though
severely practical, is no exception when he has any general
explanation, historical or other, to give. We have now to
note a like case in Pappus, namely the preface to Book V of
the Collection. The editor, Hultsch, draws attention to the
elegance and purity of the language and the careful writing
the latter is illustrated by the studied avoidance of hiatus. 1
The subject is one which a writer of taste and imagination
would naturally find attractive, namely the practical intelligence
shown by bees in
selecting the hexagonal form for the
cells in the honeycomb. Pappus does not disappoint us ; the
passage is as attractive as the subject, and deserves to be
reproduced.
'
It is of course to men that God has given the best and
most perfect notion of wisdom in general and of mathematical
science in particular, but a partial share in these things he
allotted to some of the unreasoning animals as well. To men,
as being endowed with reason, he vouchsafed that they should
do everything in the light of reason and demonstration, but to
the other animals, while denying them reason, he granted
that each of them should, by virtue of a certain natural
instinct, obtain just so much as is needful to support life.
This instinct may be observed to exist in very many other
species of living creatures, but most of all in bees. In the first
place their orderliness and their submission to the queens who
rule in their state are truly admirable, but much more admirable
still is their emulation, the cleanliness they observe in the
gathering of honey, and the forethought and housewifely care
they devote to its custody. Presumably because they know
themselves to be entrusted with the task of bringing from
the gods to the accomplished portion of mankind a share of
1
Pappus, vol. iii, p. 1233.
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 honeycombs,
(with cells) all equal, similar and contiguous to one
another, and hexagonal in form. And that they have contrived
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 rectilineal
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 themselves,
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, however,
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 isoperimetry',
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). lsoperimetry 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 wepl la-ouerpcou crxv yLOLTQ&v (see
pp. 207-13, above) ; but before passing to solid figures Pappus
inserts the proposition that of all circular segments having