the coming emergence by shortening the road, what must not
the adult do, who has so short a time to live and who is in so great a
hurry to leave the hateful darkness? He above any other should be a
judge of short cuts. To go from the murky heart of the tree to the
sun-steeped bark, why does he not follow a straight line? It is the
shortest way.

Yes, for the compasses, but not perhaps for the sapper. The length
traversed is not the only factor of the work accomplished, of the
total activity expended. We must take into account the resistance
overcome, a resistance which varies according to the depth of the more
or less hard strata and according to the method of attacking the woody
fibres, which are either broken across or divided lengthwise. Under
these conditions, whose value remains to be determined, can there be a
curve involving a minimum of mechanical labour in cutting through the
wood?

I was already trying to discover how the resistance may vary according
to depth and direction; I was working out my differentials and my
minimum integrals, when a very simple idea overturned my slippery
scaffolding. The calculation of variations has nothing to do with the
matter. The animal is not the moving body of the mathematicians, the
particle of matter guided in its trajectory solely by the motive
forces and the resistance of the medium traversed; it bears within
itself conditions which control the others. The adult insect does not
even enjoy the larva's privileges; it cannot bend freely in all
directions. Under its harness it is almost a stiff cylinder. To
simplify the explanation, we may liken the insect to a section of an
inflexible straight line.

Let us return to the Sirex, reduced by abstraction to its axis. The
metamorphosis is effected not far from the centre of the trunk. The
insect lies lengthwise in the tree with its head up, very rarely with
its head down. It must reach the outside as quickly as possible. The
section of an inflexible straight line that represents it nibbles away
a little wood in front of it and obtains a shallow cavity wide enough
to allow of a very slight turn towards the outside. An infinitesimal
advance is made; a second follows, the result of a similar cavity and
a similar turn in the same direction. In short, each change of
position is accompanied by the tiny deviation permitted by the slight
excess of width of the hole; and this deviation invariably points the
same way. Imagine a magneti