26. Structure of the Corporeal World. Mechanicism. -- (1) Corporeal substances consist of configurations of simple bodies. In accordance with earlier scientific notions, Plato admits the existence of four elementary bodies, water, air, fire, and earth; all of which, however, he reduces to regular geometrical figures: the regular tetrahedron is the fundamental form of fire, the regular octahedron that of air, the regular icosahedron that of water, the regular cube that of earth. The plane surfaces which form the sides of these four regular solids have, as generating forms, triangles; and these triangles realize the most perfect proportions: the right-angled scalene triangle for fire, air, and water; the right-angled isosceles triangle for earth.^{1} Thus Plato seeks for the reason of the world's beauty in what he regards as the deepest and ultimate elements of its constitution.

It is important to bear in mind that these surfaces are only sections of space and do not form the boundaries of any material mass. Suppress these geometrical forms and you obtain as a residue, not a formless substratum, but the on, i.e., void or vacuum: the elements of nature are not irreducible bodies, but irreducible surfaces -- a conception that harmonizes with the Platonic notion of matter.

Natural bodies are compounds of simple bodies. The phenomena of substantial change, of increase and decrease, are the outcome of a simple change in the disposition of the primary forms. Since water, air, and fire have the same scalene triangle as their source, a new arrangement of the polyhedric surfaces is all that is needed to bring about an interchange of water, air, and fire among themselves.^{2} The earth, on the contrary, having as base the isosceles triangle, which cannot be reduced to the scalene, may doubtless be mixed with the other elements, but cannot be changed into them, nor vice versa. In like manner, increase and diminution result from the union and separation of surfaces respectively.^{3} What determines these phenomena of change, of growth and decay? Motion.

(2) Motion is extrinsic to the elementary bodies: it comes from the world-soul. This latter in fact surrounds the whole world of sense (25) and exerts a mechanical pressure on all the bodies within it. As these are of unequal dimensions owing to their different shapes and different degrees of cohesiveness; and as on the other hand their plane surfaces give rise to projecting angles or corners by which they pierce one another in their never-ending motions: these many-sided figures cleave to one another and arrange themselves in ever-varying forms.

Thus, we find in Plato the two fundamental theses of Mechanicism (7).^{4} The originality -- and weak point -- of his presentation of it is his geometrical conception of the simple body. For, the real bodies in Nature around us are something very different from a mere collection of empty figures: Platonism has no justification to offer for its unwarranted transition from an empty circumscription of space to a positive, circumscribed content.^{5}

{1} The equilateral triangle is the hasis of the regular tetrahedron, octahedron, and icosahedron; for, it may be resolved into six right-angled scalene triangles (the hypotenuse of each of which is double the smallest side). On the other hand, the square -- which is the base of the cube -- is resolvable into right-angled isosceles triangles.

{2} For example, the unit of water (an icosahedron, i.e. with twenty sides) can be transformed into two units of air (octahedrons) together with one unit of fire (tetrahedron), 8 + 8 + 4 = 20.

{3} ZELLER, op. cit., II., i., pp. 789 sqq.

{4} We must, however, make this reserve: elementary bodies of the same kind move, in virtue of an internal tendency (weight), towards a place of their own, the latter differing for each kind.

{5} The universe is geocentric and spherical (which conflicts with the angular shape of the polyhedron); it includes the earth and seven concentric spheres surrounding the earth. The stars are beings endowed with life and intelligence, more perfect the nearer they are to the world-soul; the rotation of each around an axis is the index and correlative of its power of conscious self-reflection. Thus Plato descends gradually to the psychology of man and the animal kingdom. So the entire universe becomes a vast collection of living things, each one endowed with a soul of its own, -- a fact which does not seem to hinder the whole collection as such from being itself a perfect zôon, or living thing.