GOD AND NATURE. 37 



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 mat- 

 ter of geometry it is easy to show that the shortest path between 

 them is that given by the great circle, and this princij)le 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 geomet- 

 rical. 



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 con- 

 ception 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 spheri- 

 cal 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 



