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The proportion perfectly achieves this goal. For, when three numbers, three masses or three forces whatever, the mean is to the last what the first is to the mean and to the first what the last is to the mean, and if the mean becomes the first and the last, and the first and the last become the mean, it necessarily happens that all is the same, and that all being is in the same relation, all is one as before.

Plato - Timaeus

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Sometimes it helps to read alternative translations.

The quote is from section 32, (part?) VII (p97 of the book 109 of the pdf) in this copy of Plato's Timaeus translated by Richard Dacre Archer-Hind, 1888 - the first English edition.

"The best of bonds is that which makes itself and those which it binds as complete a unity as possible ; and the nature of proportion is to accomplish this most perfectly. For when of any three numbers, whether expressing three or two dimensions, one is a mean term, so that as the first is to the middle, so is the middle to the last ; and conversely as the last is to the middle, so is the middle to the first ; then since the middle becomes first and last, and the last and the first both become middle, of necessity all will come to be the same, and being the same with one another all will be a unity."

Also, it is from section 32 in this 1937 book (page 44 - 28 in the pdf) by Francis MacDonald Cornford (which I think has a better translation as well as commentary by the author).

(31 B) Now that which comes to be must be bodily, and so visible and tangible; and nothing can be visible without fire, or tangible without something solid, and nothing is solid without earth. Hence the god, when he began to put together the body of the universe, set about making it of fire and earth. But two things alone cannot be satisfactorily united (31 C) without a third; for there must be some bond between them drawing them together. And of all bonds the best is that which makes itself and the terms it connects a unity in the fullest sense; and it is of the nature of a continued geometrical proportion to effect this most perfectly. For whenever, of (32) three numbers, the middle one between any two that are either solids (cubes?) or squares is such that, as the first is to it, so is it to the last, and conversely as the last is to the middle, so is the middle to the first, then since the middle becomes first and last, and again the last and first become middle, in that way all will necessarily come to play the same part towards one another, and by so doing they will all make a unity."

According to Professor A. E. Taylor in his "Commentary", "The formula for the physics and physiology of the dialogue is that it is an attempt to graft Empedoclean biology on the stock of Pythagorean mathematics" (p18). He continues in his distinction of what Plato himself thought and what the character Timaeus presents,

"It is in fact the main thesis of the present interpretation that the teaching of Timaeus can be shown to be in detail exactly what we should expect in a fifth-century Italian Pythagorean who was also a medical man, that it is, in fact, a deliberate attempt to amalgamate Pythagorean religion and mathematics with Empedoclean biology"

As for Timaeus's meaning, consider Thomas Taylor's translation of Proclus in "Proclus on the Timaeus of Plato"

32a "For when either in three numbers, or masses, or powers, as is the middle to the first so is the last to the middle; and again, as is the last to the middle, so is the middle to the first; then the middle becoming both first and last, and the last and the first becoming both of them middles, it will thus happen that all of them will necessarily be the same. But becoming the same with each other, they will be one."

In the first place, it is requisite to explain what is here said mathematically; and in the next place, physically, as being that which is especially proposed to be effected. For it is not proper to separate the discussion from its appropriate theory. There are therefore some who think that Plato in these words defines the geometric middle, and among other things which they assert, they say that the geometric middle is properly exclusive of all the others analogy; but that the others may be justly called middles. Nicomachus also is of this opinion, and he speaks rightly. For geometric proportion is properly analogy; but it is requisite to call the others middle, as Plato also says further on in the generation of the soul. But the others are improperly called analogies. To others, however, these appear not to have apprehended the meaning of Plato properly.

and T. Taylor continues...

Plato clearly assumes the geometric middle. For it is the peculiarity of this proportion, that the first has the same ratio to the middle that the middle has to the third term. As, however, there are three middles, the  arithmetic, the geometric, and the harmonic, and these being such as we have shown them to be, Plato very properly assumes these three subjects, numbers, masses, and powers. For the arithmetical middle is in numbers; the geometrical is in a greater degree conversant with continued [than with discrete] quantity; and the harmonical middle is in powers. For it is conversant with sharp and flat sounds. And after this manner you may speak, distinguishing the middles according to their predominance.

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In this section Plato is describing a bit of geometry. He is describing a ratio using the geometric mean as it exists for certain lengths "first," "mean," and "last," which we will call "f," "m," and "l," respectively. To understand the idea focus on this part of the quotation:

"when... the mean is to the last what the first is to the mean and to the first what the last is to the mean, and if the mean becomes the first and the last, and the first and the last become the mean, it necessarily happens that all is the same..."

This becomes a "when x is true, if y then z is true" statement:

When f/m = m/l is true

if m/m = sqrt(fl)/m = m/sqrt(fl), then m/m = sqrt(fl)/c = m/sqrt(fl) = 1 is true

To give an example:

let f = 4, m = 6, l = 9,

when 4/6 = 6/9 is true

if 6/6 = sqrt(4*9)/4 = 6/sqrt(4*9), then 46/6 = sqrt(4*9)/6 = 6/sqrt(4*9) = 1

sqrt(4*9) = sqrt(36) = 6

if 6/6 = 6/6 = 6/6, then 6/6 = 6/6 = 6/6 = 1

The limited view of this quotation in the context of this question is more of a mathematical question than a philosophical one, as the quote is describing just a basic property of geometry. However, in the Timaeus dialog this quotation is part of a description of the (classical) elements. They are described as being composed at the smallest level of basic geometric shapes and Timaeus is relaying the mathematical beauty of their geometry. This concept is a central idea in much of the history of metaphysics. It is very often said that he is describing the golden ratio in this quotation, but that is contentious: see this post. In the answer Will Jagy mentions the example from his translation for the numbers 2, 4, and 8. Put into our formula it becomes:

When 2/4 = 4/8

if 4/4 = sqrt(2*8)/4 = 4/sqrt(2*8) then 4/4 = sqrt(2*8) = 4/sqrt(2*8) = 1

sqrt(2*8) = sqrt(16) = 4

if 4/4 = 4/4 = 4/4 then 4/4 = 4/4 = 4/4 = 1

Other examples are (3, 6, 12), (5, 10, 20), (4, 4, 4) etc. Any time the first and the mean and the mean and the last have the same ratio (f/m = m/l) this will be true. In a more formal sense, any three consecutive terms in a geometric progression will satisfy this property. For a final statement bringing this back to ontology... hopefully it goes without saying that modern physics and chemistry have proven that the classical elements are not truly elements and that Plato's geometric description of their atoms is incorrect.

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