mathematicians; and, looking upon it thus, we should be disposed to regard the form of motion which involves least effort as being chosen out of all possible forms, much in the same way as a man who has to perform a journey or to do a certain piece of work inquires how the journey or piece of work can be reduced to a minimum of trouble or expense. But the fact of the "principle of least action" being mathematically deducible from the principles of motion would seem to prove that there is in reality no choice in the matter, but that least action is as necessary a truth as is that of the least distance between two points on a sphere being that which is traced by the great circle joining them.
Just consider this question of two points on a sphere. As a matter of geometry it is easy to show that the shortest path between them is that given by the great circle, and this principle is now well recognized in navigation. But change the problem from geometry to dynamics, by supposing a particle to move on the surface of a smooth sphere under the action of a force tending to the center, as that exerted by an elastic string in a state of tension; then it is equally easy to prove that this particle, when started in any direction, will describe a great circle—that is, its motion will be such that the distance traversed by it in passing from its point of departure to any point in its path will be the shortest distance between those points. It might be said that the particle chose the easiest path, but in reality there was no choice, nothing but necessity; in other words, the dynamical minimum stands on the same footing as the geometrical.
In truth, the question of minimum comes under our notice very frequently and very curiously in nature. The path of a ray of reflected light may be determined upon the principle that it is the shortest possible; and this is not the only case in which the law of minimum is illustrated by optics. But take a very different case, that of the cells made by the bee. It is well known that the bee is a wonderful geometer. The cells consist of hexagonal prisms closed at the ends with three tiles having exactly the angles which with a given amount of material will make the cells most capacious, or with a given capacity will use the smallest amount of material. This has been long known, and has given rise to much speculation as to the manner in which the bee is guided to so remarkable a result. I am not aware that any satisfactory solution has yet been proposed; but the intellectual conception of the problem is much simplified if we bear in mind that the transverse section is the nearest form possible to a circle, and the form of the end of the cell the nearest possible to a sphere; so that it may be said that the instinct of making circular prismatic cells with spherical ends, and then clearing away unnecessary wax, is all the instinct which the bee requires. Let the reader observe that this is said, not with a view to depreciate the bee's architectural skill, but only for the
