production of the Prismatic Structure of Basalt. 127 



the surface must become divided — the first and last having their 

 sides, singly or in pairs, respectively normal to three axes in 

 the plane of the figure crossing each other at angles of 60°, 

 and in the case of the square to two such axes crossing each 

 other at 90°. What, then, in nature determines her choice of 

 the complicated form of six sides in place of the simpler ones of 

 three and four sides respectively ? The reason is to be found in 

 the " principle of least action," which, as a universal law, governs 

 every operation of unbalanced force, the end to be obtained (to 

 restore equilibrium) being effected in such a manner as to require 

 the minimum expenditure of work. Now it can easily be shown 

 that the smallest amount of work necessary to split up a given 

 surface by contractile tensional forces within it will be that which 

 divides it into hexagons rather than into equilateral triangles or 

 squares, all being of equal area, and that whether the divisional 

 figures be large or small in proportion to the surface. 



In a mass contracting by cooling every particle tends to cause 

 every other to approach nearer to it ; and if we take any two 

 distant points a and b at random in such a body, all the parti- 

 cles round each point tend to approach it with forces propor- 

 tional to some function of their respective distances from a and 

 b. In a line drawn from a to b there are therefore equal and 

 opposite tensional forces ; and when their amount exceeds the 

 tensional resistance of the material, breach of continuity must 

 occur somewhere across the line a b ; and as every particle along 

 that line is urged by equal and opposite forces through paths 

 dependent upon the elasticity and coefficient of contraction of 

 the material and proportional to its distance from a and b re- 

 spectively, so the point of discontinuity must occur at the middle 

 point between a and b ; and if the line have determinate width 

 and breadth, i. e. be a small prism of the material between a and 

 b, its fracture must occur at its mid length and normal to its 

 axis. We are not concerned at present with determining the 

 distance between a and b for a given material under the condi- 

 tions of cooling at which breach of continuity must occur; but 

 we will presently return to it. Let us apply the above reason- 

 ing to the rigid film forming the surface of the cooling mass of 

 basalt already supposed. As every particle tends to draw every 

 other particle towards it, the surface of the film, when breach of 

 continuity occurs, must split up into figures of equal area ; for the 

 tensional pull in all directions round points in the surface such as 

 a and b is proportional to the number of particles, and to their 

 distances round each, or to equal areas round each. Now the rend- 

 ing forces, or those tending to produce splitting up, are propor- 

 tional to those areas ; and the resisting forces of the material 

 are proportional to the perimeters separating these areas — that 



