Timaeus — Plato

The construction of surfaces and solids. one, they will form one large mass of another kind. So much for their passage into one another. I have now to speak of their several kinds, and show out of what combinations of numbers each of them was formed. The first will be the simplest and smallest construction, and its element is that triangle which has its hypothenuse twice the lesser side.

When two such triangles are joined at the diagonal, and this is repeated three times, and the triangles rest their diagonals and shorter sides on the same point as a centre, a single equilateral triangle is formed out of six triangles ; and four equilateral triangles, if put together, make out of every three plane angles one sohd angle, being 55 that which is nearest to the most obtuse of plane angles; and out of the combination of these four angles arises the first solid form which distributes into equal and similar parts the whole circle in which it is inscribed.

The second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed. And the third body is made up of 120 triangular elements, forming twelve solid angles, each of them included in five plane equilateral triangles, having altogether twenty bases, each of which is an equilateral triangle.

The one element [that is, the triangle which has its hypothenuse twice the lesser side] having generated these figures, generated no more ; but the isosceles triangle produced the fourth elementary figure, which is compounded of four such triangles, joining their right angles in a centre, and forming one equilateral quadrangle.

Six of these united form eight solid angles, each of which is made by the combination of three plane right angles ; the figure of the body thus composed is a cube, having six plane quadrangular equilateral bases. There was yet a fifth combination which God used in the delineation of the universe.

Now, he who, duly reflecting on all this, enquires whether the worlds are to be regarded as indefinite or definite in number, will be of opinion that the notion of their indefiniteness is characteristic of a sadly indefinite and ignorant mind. He, however, who raises the question whether they are to be truly regarded as one or five, takes up a more reasonable

475 Timaeus. TiMAEUS. The first and simplest solid, the pyramid, has four equilateral triangular surfaces, each formed by the union of six rectangular scalene triangles.

The second species, the octahedron, has eight such surfaces, and the third, the icosahedron, twenty.

The fourth, the cube, has six square surfaces, each formed of four rectangular isosceles triangles.

There is also a fifth species. Although there are five elementary solids, there is but one world.

p. 476

The figures of earth, air, fire, water

Tiviaeus. TiMAEUS.

We have now to assign to the four elements their respective forms, — to earth the cube, to water the icosahedron, to air the octahedron, to fire the pyramid.

Individual particles cannot be seen : masses of each kind are visible.

position. Arguing from probabilities, I am of opinion that they are one ; another, regarding the question from another point of view, will be of another mind. But, leaving this enquiry, let us proceed to distribute the elementary forms, which have now been created in idea, among the four elements.

To earth, then, let us assign the cubical form ; for earth is the most immovable of the four and the most plastic of all bodies, and that which has the most stable bases must of necessity be of such a nature.

Now, of the triangles which we assumed at first, that which has two equal sides is by nature more firmly based than that which has unequal sides ; and of the compound figures which are formed out of either, the plane equilateral quadrangle has necessarily a more stable basis than the equilateral triangle, both in the whole and in the parts.

Wherefore, in assigning this figure to earth, we 56 adhere to probability ; and to water we assign that one of the remaining forms which is the least movable ; and the most movable of them to fire ; and to air that which is intermediate. Also we assign the smallest body to fire, and the greatest to water, and the intermediate in size to air ; and, again, the acutest body to fire, and the next in acuteness to air, and the third to water.

Of all these elements, that which has the fewest bases must necessarily be the most movable, for it must be the acutest and most penetrating in every way, and also the lightest as being composed of the smallest number of similar particles : and the second body has similar properties in a second degree, and the third body in the third degree.

Let it be agreed, then, both according to strict reason and according to probability, that the pyramid is the solid which is the original element and seed of fire ; and let us assign the element whi'ch was next in the order of generation to air, and the third to water.

We must imagine all these to be so small that no single particle of any of the four kinds is seen by us on account of their smallness : but when many of them are collected together their aggregates are seen. And the ratios of their numbers, motions, and other properties, everywhere God, as far as necessity allowed or gave consent, has exactly perfected, and harmonized in due proportion.