72.1 Adrastus treatment of geometrical proportion:

"He says the geometrical proportion is the only proportion in the full and proper sense and the primary one, because all the others require it, but it does not require them. The first ratio is equality (1/1), the element of all other ratios and of the proportion they yield.

He then derives a whole series of geometrical proportions from the proportion with equal terms, (1, 1, 1) according to the following law:

Give three terms in continued proportion, if you take three other terms formed of these, one equal to the first, another composed of the first and the second, and another composed of the first and twice the second and the third, these new terms will be in continued proportion.

In this manner, from the proportion with equal terms arises the double proportion, and from the triple, and so on, as follows. Take the equal proportion with the smallest possible terms, 1, 1, 1. Then take three terms according to the above rule:

1, 1 + 1 = 2, 1 + 2 + 1 = 4.

This is the double proportion (see pg 39). 1, 2, 4..etc. Now take 1, 2, 4 and proceed in the same way:

1, 2 + 1 = 3, 1 + 4 + 4 = 9.

This is the triple proportion (see pg 39). 1, 3, 9...etc. By continuing the process we obtain:

1, 1, 1
1, 2, 4
1, 3, 9
1, 4, 16
1, 5, 25
1, 6, 36
1, 7, 49
1, 8, 64
1, 9, 81
1, 10, 100

72.2.1 Concerning 10, 100,

72.2.2

72.3.1 Concerning the whole series,

72.3.2

Also note the last column 1, 4, 9, 16.. etc is the atomic shells proportioned to the rule of single squares.

The number in the third column are squares, those in the second column are the roots of these squares. The underlying notion seems to be that any number (represented by a line) has, in itself, and without the aid of any other factor, the power of multiplying itself or generating its own square by advancing as far as its own length into the second dimension....so the root number is the first power, the corresponding line is more commonly applied to the square, in which this potency of the root is developed and deployed. Hence, the square is the second power. The square contains the power that can be further deployed when the square advances into the third dimension and produces the cube, or third power. If we now continue Adrastus' geometrical proportions, we shall next reach the cube. Taking the double and triple proportions we have:

1, 2, 4, 8 and 1, 3, 9, 27

72.4

These are the two series that Plato takes later (35B) as the basis for the harmony of the World-Soul. Both series emanates from unity, in which all the powers concerned are conceived as gathered up. The series proceed through the first even, and the first odd, number to their squares and cubes.....

Nichomachus...repeats that this is the only proportion in the most proper sense and give the same examples: " the numbers proceeding from unity according to the double proportion":

1, 2, 4, 8, 16, 32, 64...

..and the triple proportion:

1, 3, 9, 27, 81, 243...

72.5.1

..and so on with the quadruple proportion, etc. He points out that the terms of these proportions have the properties that Plato mentions and later speaks of "the Platonic theorem, that the plane numbers are held together by one mean, the solids by two standing in proportion: for between two consecutive squares will be found only one mean preserving the geometrical proportion... and between two consecutive cubes only two". This is true of all proportions of the above pattern: e.g.

Root 2

72.6.1

Square 4 (2^2)

72.7.1

Solid 32

72.8.1

72.9 Cube 512 (8^3)

72.9.1

Platos Cosmology
888.4 P697co

"The special points of this pattern are:

All the plane numbers are squares; there are no oblongs.

Also such progressions cannot be continued to four and more terms without introducing fractions . Platos main point is emphasized in the concluding sentence: the worlds body, consisting of neither less nor more than 4 primary bodies, whose quantities are limited and linked in the most perfect proportion, is in unity and concord with itself and hence will not suffer dissolution from any internal disharmony of its parts. The bond is simply geometrical proportion",

72.10

"It is not a question of mechanical forces holding the world together"....

72.12 Summary. Transition to the World Soul 

The summary lists certain perfections for which the body of the universe is indebted to divine providence; then we learn that its axial rotation is due to its soul, which extends from its center to its circumference.

All this, then, was devised by the ever-existant god for the god who was one day to be. (B) According to this plan he(he?) made it smooth and uniform, everywhere equidistant from its centre, a body whole and complete compounded of complete bodies. And in the midst thereof he set a soul and extended it throughout the whole, and also wrapped its body round with soul on the outside; and so he established one world, round and revolving in a circle",

(Note: ellipse, egg or spiral?),

"solitary but able by virtue of its excellence to keep itself company, needing no other acquaintance or friend but sufficient unto itself. For all these reasons the world he brought into being was a blessed god.

72.13 Soul Prior to Body (34B-C)

The world soul, though prior to dignity to the body, is coeval with it; both are everlasting.

Now although this soul comes later in the account we are (C) now attempting, the god did not make it younger than the body; for when he united them he would not have allowed the elder to be ruled by the younger. Human nature partakes largely of the casual (not causal) and random, which becomes apparent in our speech; but the god made soul elder than body and prior in birth and excellence, to be the body's mistress and ruler.

72.14 Composition of the World Soul (35A)

The Demiurge (God) now compounds the world soul from certain kinds in intermediate Existence, Sameness and Difference. To understand the meaning of Platos symbolism here, it is necessary to have read the Sophist.

(35). The things of which he constructed soul and the manner of its composition were as follows: First, between the indivisible Existence, which remains always in the same state, and the divisible Existence that becomes in bodies, he compounded a third form of Existence out of both. Second, in the case of Sameness and that of Difference, he also on the same principle fashioned a compound intermediate between that kind of them which is indivisible and the kind that is divisible in bodies. Then, third, taking the three, he blended them all into one form, compelling the nature of Difference, hard though it was to mingle, into union with Sameness, and mixing them together with Existence.

72.15 Its Division into Harmonic Interval (35B-36B)

Timaeus speaks of the triple compound as if it were a strip of pliable material. It will presently be split lengthwise and bent round into circles; but the Demiurge first marks it off into divisions corresponding to the intervals of a musical scale. (B) Having thus made a unity of the three",

(Correspondences, Unity = 2, Fuller).

"he divided this whole into a suitable number of parts, each part being a blend of Sameness, Difference and Existence. And here is how he began the division:

Next he proceeded to fill up the double and the triple intervals, (36) cutting off further parts from the mixture and placing them between the terms, so that within each interval there were two means, the one (harmonic)", exceeding the one extreme and being exceeded by the other by the same fraction of the extremes, the other (arithmetical)",

72.16.1

"exceeding the one extreme by the same number whereby it was exceeded by the other.

These links produced intervals 3/2 and 4/3 and 9/8 (the tone) within the original intervals. (B) And he went on to fill all the intervals of 4/3 (fourths) with the interval 9/8 (the tone), leaving over in each a fraction. This remaining interval of the fraction had its terms in the numerical proportion 256 : 243 (semitone approx). By this time the mixture from which he was cutting off these proportions was all spent.

The final step, taken in the sentence that follows, is to fill in every tetrachord with two intervals of a tone (9/8) and a remainder (256/243) approximately equal to a semitone. This process applied throughout the remaining tetrachords, completes the entire range of notes from 1 to 27".

72.16.3

Platos Timaeus
By John Warrington
888.4 P697tiw

(After the author describes Proportions of the World Combination, from Plato's Timaeus, we read)...

"Now all these proportions are combined harmonically according to numbers, which proportions were scientifically divided according to a scale which reveals the elements and the means of the soul's combination. Now seeing that the earlier is more powerful in power and time than the later, the deity did not rank the soul after the substance of the body, but made it older by taking the first of unities, 384. Knowing this first, we can easily reckon the double (square) and the triple (cube); and all the terms together, with the compliments and eights, there must be 36 divisions, and the total amounts to 114, 695.

72.17.1

72.19 Diagram. Table of Tone Numbers 

l = limma semitone, ap. = apotome

Pythagorean Sourcebook and Library

(1)-(36) is table sequence for location purposes and division of series. C sharp to C'' sharp is

The Harmonic System of 15 chords in the Diatonic Mode 

© Copyright. Robert Grace. 1999.

72.25 The Theory of Discontinuous Groups 

"One of its simplest theorems, for example, proves that the symmetry elements of crystals can be grouped in 32",

72.26.1

"different ways, and no other"... and that "there are just 230 different ways of distributing identical objects of arbitrary shape regularly in space,..."

Rodin's I Ching, Paragraph 4 

Music of the Spheres
By Guy Murchie
523.M938

...Jose Arguelles explains... "a fundamental algorithm of multiplication or addition corresponding to succession and spontaneity or their inverse aspects and can be implied in their terms. The two properties which present this form":

The Golden Section: A/B = B(A+B)

The Greek "Harmonic" Means: A/C = (A-B) / (B-C)

72.28.1

Charles Henry
By Jose Arguelles
612. H3965za, pg. 112

The Golden Section:

1. Generated by formula 1 + Sq.rt5 / 2 = 1.618 or Phi.

2. "Multiplied" by itself, gives 2.617924, in fact,

3. itself plus 1, so that this 3rd number is

4. the sum of the preceding two, i.e.,

5. (Phi x Phi, Phi x Phi x Phi etc.).

72.30.1

A simpler version is:

The Fibonacci Series using whole numbers:

1, 1, 2, 3, 5, 8, 13, 21 etc. which have exactly the same "additive" properties as the Phi progression.

Number Symbolism
By C. Butler
133. 3359 B976

Phi = 1 + Sq.rt5 / 2 = 1.618. Its chief property is that when:

multiplied by itself produces 2.617924, itself (1.618) plus 1, so the third number (2.617924) is the sum of the first two numbers.

The exact procedure, using the Fibonacci Series whole numbers produce the exact same additive properties.

© Copyright. Robert Grace. 1999