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 magnetic needle swung out of its position and tending to return to it while moving with a uniform speed through a resisting medium in which a sheath of a diameter slightly greater than the needle's opens bit by bit. The Sirex behaves more or less in the same fashion. His magnetic pole is the light outside. He makes for that direction by imperceptible deviations as his tooth digs. The problem of the Sirex is now solved. The trajectory is composed of equal elements, with an invariable angle between them; it is the curve whose tangents, divided by infinitely small distances, retain the same inclination between each one and the next; the curve, in a word, with a constant angle of contingence. This characteristic betrays the circumference of the circle. It remains to discover whether the facts confirm the logical argument. I take accurate tracings of a score of galleries, selecting those whose length best lends itself to the test of the compasses. Well, logic agrees with reality: over lengths which sometimes exceed four inches, the track of the compasses is identical with that of the insect. The most pronounced deviations do not exceed the small variations which we must reasonably expect in a problem of a physical nature, a problem incompatible with the absolute accuracy of abstract truths. The Sirex' exit-gallery then is a wide arc of a circle whose lower extremity is connected with the corridor of the larva and whose upper extremity is prolonged in a straight line which ends at the surface with a perpendicular or slightly oblique incidence. The wide connecting arc enables the insect to tack about. When, starting from a position parallel with the axis of the tree, the Sirex has passed gradually to a transversal position, he completes his course in a straight line, which is the shortest road.