Macrobius' Commentarii in somnium Scipionis and Plato´s Cosmology: The Timaeus

 

Leaf from Commentarii in Somnium Scipionis: Diagram of Five Celestial and Five Earthly Zones

from The Walters Art Musem, License details

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The citation below comes from Plato's Cosmology: The Timaeus

by S. Marc Cohen, Professor Emeritus, Department of Philosophy, University of Washington

The coming to be of the elements

The four elements are “the most excellent four bodies that can come into being” (53e). But how do they come into being? What are they made of? Plato’s answer is that they are all made of triangles, and constructed in such a way as to explain how the transmutation of elements is possible.

Overview

Each kind of matter (earth, air, fire, water) is made up of particles (“primary bodies”). Each particle is a regular geometrical solid. There are four kinds of particles, one for each of the four kinds of matter. Each particle is composed of elementary right triangles. The particles are like the molecules of the theory; the triangles are its atoms.

The argument that all bodies are ultimately composed of elementary right triangles is given at 53c-d: all bodies are 3-dimensional (“have depth”) and hence are bounded by surfaces. Every surface bounded by straight lines is divisible into triangles. Every triangle is divisible into right triangles. Every right triangle is either isosceles (with two 45° angles) or scalene. So all bodies can be constructed out of isosceles and scalene right triangles.

The details

1. The two atomic triangles

   Plato notes (54a1) that there is only one kind of isosceles right triangle--namely, the 45°/45°/90° triangle--whereas there are “infinitely many” kinds of scalene. But of these, he tells us, “we posit one as the most excellent” (54a7), one “whose longer side squared is always triple its shorter side” (54b5-6). Plato describes the same scalene triangle, equivalently, as “one whose hypotenuse is twice the length of its shorter side” (54d6-7). (The angles of this triangle are thus 30°/60°/90°.)

   I’ll call the 30°/60°/90° triangles “a triangles” and the 45°/45°/90° triangles “b triangles.”

   a triangle (scalene, 30°/60°/90°)		b triangle (isosceles, 45°/45°/90°)

2. Construction of “faces” of particles out of the atomic triangles

   • Each face is either an equilateral triangle (t) or a square (s).

   • Equilateral triangles (t’s) are made of a triangles.

   • Squares (s’s) are made out of b triangles.

   • Plato’s description at 54e and 55b tells us that each t is made of 6 a’s, and each s is made of 4 b’s. (See diagrams, RAGP 640.) But 57c-d makes clear that he envisages other ways of constructing these faces out of primitive a’s and b’s.

3. Construction of solid particles out of the faces

   The construction of the particles is described at 54d-55c. The particles are identified with the four elements at 55d-56b. Click on the names of the elements to see a diagram of a particle of that element:
   1. Fire: a particle of fire is a tetrahedron (4-sided solid), made of 4 t’s consisting of 24 a’s altogether.

   2. Air: a particle of air is an octahedron (8-sided solid), made of 8 t’s consisting of 48 a’s altogether.

   3. Water: a particle of water is an icosahedron (20-sided solid), made of 20 t’s consisting of 120 a’s altogether.

   4. Earth: a particle of earth is a cube (6-sided solid), made of 6 s’s consisting of 24 b’s altogether.

4. Transformation of elements (described at 56c-57c)

   Inter-elemental transformations are among fire, air, and water only. Earth cannot be transformed into any of the others (54c, 56d).

   Transformations can be described at the level of equilateral triangles (that are the faces of the three solids). Since a fire molecule has 4 faces (one F is made up of 4 t), an air molecule 8 (one A is made up of 8 t), and a water molecule 20 (one W is made up of 20 t), any of the following transformations (for example) are possible.  (Each transformation is represented by an equation on the left; its geometrical basis is shown by the equation on the right.):

   1 A = 2 F	8 t = 2 × 4 t
   1 W  = 5 F	20 t = 5 × 4 t
   2 W  = 5 A	2 × 20 t = 5 × 8 t
   1 W  = 2 A + 1 F	20 t = (2 × 8 t) + 4 t
   1 W  = 3 F + 1 A	20 t = (3 × 4 t) + 8 t

5. Larger and smaller particles

   Since equilateral triangles can be constructed out of a’s (and squares out of b’s) in more than one way, it is possible to have “molecules” of each of the elements that have different numbers of atomic triangles (a’s and b’s). These might be considered “isotopes” of the basic molecules described by Plato (with each t made of 6 a’s, and each s made of 4 b’s).

   An equilateral triangle can also be constructed out of 2, or 8, or 18, a’s (and so on, ad infinitum).

   A square can also be constructed out of 2, or 8, or 16, b’s (and so on, ad infinitum).

   This means that one “normal” particle of earth (6 s = 24 b) can be transformed into 2 of the smaller “isotopes” of earth (6 s = 12 b)

   Similarly, 4 “normal” particles of water (containing 120 a’s each) can combine to form one huge particle of one of the larger “isotopes” of water (20 sides of 24 a’s each, for 480 a’s altogether)