Does the trajectory imply the minimum of work? Yes, under the conditions of the insect's existence. If the larva had taken the precaution to place itself in a different direction when preparing for the nymphosis, to turn its head towards the nearest point of the bark instead of turning it lengthwise with the trunk, obviously the adult would escape more easily: he would merely have to gnaw straight in front of him in order to pass through the minimum thickness. But reasons of convenience whereof the grub is the sole judge, reasons dictated perhaps by weight, cause the vertical to precede the horizontal position. In order to pass from the former to the latter, the insect veers round by describing the arc of a circle. When this turn has been effected, the distance is completed in a straight line.
Let us consider the Sirex at his starting-point. His stiffness of necessity compels him to turn gradually. Here the insect can do nothing of its own initiative; everything is mechanically determined. But, being free to pivot on its axis and to attack the wood on either side of the sheath, it has the option of attempting this reversal in a host of different ways, by a series of connected arcs, not in the same plane. Nothing prevents it from describing winding curves by revolving upon itself: spirals, loops constantly changing their direction, in fact, the complicated route of a creature that has lost its way. It might wander in a tortuous maze, making fresh attempts here, there and everywhere, groping for ever so long without succeeding.
But it does not grope and it succeeds very well. Its gallery is still contained within one plane, the first condition of the minimum of labour. Moreover, of the different vertical planes that can pass through the eccentric starting-point, one, the plane which passes through the axis of the tree, corresponds on the one side with the minimum of resistance to be overcome and on the other with the maximum. Nothing prevents the Sirex from tracing his path in any one of the multitude of planes on which the path would possess an intermediate value between the shortest and the longest. The insect refuses them all and constantly adopts the one which passes through the axis, choosing, of course, the side that entails the shortest path. In brief, the Sirex' gallery is contained in a plane pointing towards the axis of the tree and the starting-point; and of the two portions of this plane the channel passes through the less extensive. Under the conditions, therefore, imposed upon him by his stiffness the hermit of the poplar-tree releases himself with the minimum of mechanical labour.