“Let no one who cannot think geometrically enter”

In 387 B.C.E. Plato founded his philosophical school.

Plato, through Timaeus, says that the creator made the world soul out of

various ingredients, and formed it into a long strip.

The strip was then marked out into intervals.

First [the creator] took one portion from the whole

(1 unit)

 and next a portion double the first (2 unit)

 a third portion half again as much as the second

(3 unit)

 the fourth portion double the second (4 unit)

 the fifth three times the third (9 unit)

 the sixth eight times the first (8 unit)

 and the seventh 27 times the first(27 unit)

They give the seven integers; 1, 2, 3, 4, 8, 9, 27.

These contain the monad, source of all numbers,

the first even and first odd,

and their squares and cubes.

These seven numbers can be arranged as two progressions

Monad 1 Point   First even and odd 2 3 Line   Squares 4 9 Plane   Cubes 8 27 Solid  This is called Plato's Lambda, because it is shaped like the Greek letter lambda.

Plato deduces the need for the four elements in

Timaeus, 31B-32C

. it is necessary that nature should be visible and tangible ...

. . . But it is impossible for two things to cohere without the intervention of a third ...

... [and] the most beautiful analogy is when in three numbers,the middle is to the last as the first to the middle, . . . they become the same as to relation to each other.

But if the universe were to have no depth, one medium would suffice to bind all the natures it contains.

But . . . the world should be a solid, and solids are never harmonized by one, but always by two mediums.

Hence water and air are in the middle of fire and earth,

fabricating them in the same ratio to each other; so that

fire might be to air as air is to water and that

water is to earth.

fire/air = air/water = water/earth

Thus the ratio is constant between successive elements, giving a geometric progression.

Plato associates four of the Platonic Solid with the four elements. He writes,

We must proceed to distribute the figures we have just described between fire, earth, water, and air. . .

Let us assign the cube to earth, for it is the most immobile of the four bodies

and most retentive of shape

the least mobile of the remaining figures (icosahedron) to water

the most mobile (tetrahedron) to fire

the intermediate (octahedron) to air

"There still remained a fifth construction,

which the god used for embroidering the

constellations on the whole heaven."

Plato’s World 

A forms (circularity)

Immaterial B math objects (circle)

--------------------------------- C----------------------------- physical obj. (wheel)

materialD

images (pict. of wheel)E

 

ACCE

ABBC

CDDE

In the Pythagorean concept of the music of the spheres, the interval between the earth

and the sphere of the fixed stars was considered to be a diapason

--the most perfect harmonic interval.

From the sphere of the earth to the sphere of the moon; one tone;

from the sphere of the moon to that of Mercury, one half-tone;

from Mercury to Venus, one-half; from Venus to the sun, one and one-half tones;

from the sun to Mars, one tone; from Mars to Jupiter, one-half tone;

from Jupiter to Saturn, one-half tone; from Saturn to the fixed stars, one-half tone.

The sum of these intervals equals the six whole tones of the octave.