Sources: Mainly Fulke's The Philosopher's Game.
Since, as mentioned, there were a number of variants of the game, this article sets out the commonalities. It is not intended to be taken as a full description on its own; rather, it should be considered the common first section of the Rhythmomachy reconstructions I will be writing.
Each player has 24 men, eight round, eight triangular, and eight square. Each man has a number on it; the rationale behind the numbers is discussed below. Rhythmomachy is unusual in that it is an asymmetrical game -- although each player has the same number of pieces, the numbers written on those pieces differ widely. I usually think of one side as "even", and the other "odd", but the reality is more complex than that. The two sides have contrasting colors, usually but not always black and white. I usually play black as odd and white as even out of habit, but that is a semi-arbitrary choice, based on the illustrations in Fulke. On average, the odd numbers are significantly higher than the even numbers. The men are usually flat, and double-sided, with the same number in the opposing color on the opposite side. For my personal sets, I paint the edges of the men with the color of the side they start on, to make them easier to sort for setup.
The first rank of rounds can be thought of as the "seeds" for the rest; they are the odd or even numbers less than ten. That is, the evens have 2, 4, 6, and 8; the odds have 3, 5, 7, and 9. Since each rank has four men, each of those flows from a particular seed. (Thus, for the first rank of squares on the even side, one man is based on 2, one on 4, and so on.)
The second rank of rounds are the first rank of rounds squared; thus, on the even side, they are 4, 16, 36, and 64, and on the odd are 9, 25, 49, and 81.
The first rank of triangles are the two rounds added together; thus, on the even side, they are 6 (2 plus 4), 20, 42, and 72, and on the odd are 12, 30, 56, and 90.
The second rank of triangles are the seed plus 1, squared. Thus, on the even side they are 9 (2 plus 1 squared), 25, 49, and 81, and on the odd are 16, 36, 64, and 100.
The first rank of squares are the two triangles added together. Thus, on the even side they are 15 (6 plus 9), 45, 91, and 153, and on the odd are 28, 66, 120, and 190.
The second rank of squares are twice the seed, plus one, squared. Thus, on the even side they are 25 ((2*2)+1, squared), 81, 169, and 289, and on the odd are 49, 121, 225, and 361.
Summarizing all of that into a neat table, we get:
|1st Rounds (x)||2||4||6||8||3||5||7||9|
|2nd Rounds (x2)||4||16||36||64||9||25||49||81|
|1st Triangles (x + x2)||6||20||42||72||12||30||56||90|
|2nd Triangles ((x + 1)2)||9||25||49||81||16||36||64||100|
|1st Squares (x + x2 + (x + 1)2)||15||45||91||153||28||66||120||190|
|2nd Squares ((2x + 1)2)||25||81||169||289||49||121||225||361|
The initial arrangement of the men varies from version to version, but typical arrangements can be seen in this Word 97 document and this GIF image (which was taken from Fulke). The ranks on which the men started varied significantly; the layout within those ranks tended to be the same, but there are versions with significantly different layouts.
The Kings are generally treated specially: in most variants, they have enhanced powers of movement, and capturing the opponent's King is at least part of the objective of the game. Usually, the King can be captured intact or piecemeal, capturing its component pieces out of the pyramid. The details vary from version to version, however.
While each version has a couple of degenerate captures (eg, surrounding a piece completely), capture is generally done mathematically. The details of this vary greatly, but in general you capture opposing pieces by creating mathematical relationships between those pieces and your own. For example, if two of your men are positioned by an enemy man, and they add up to his value, they capture him. Note that capture frequently does not require actually jumping into the enemy's space, as Chess does; you can often capture simply by setting the position up.
In most versions, captured men can then be subverted; when you capture the man, you flip him over (so that your color shows), and re-enter him on your side. This is essential in some cases, since most numbers exist only on one side or the other.
In general, all of these proportions are ways of defining a relationship between three numbers. In all of these examples, I will talk about three numbers A, B, and C, where A is the smallest number and C the largest. I will give examples for each, as well.
In other words, the numbers are rising in a simple progression, adding the same number each time. 5 plus 20 is 25; 25 plus 20 is 45; so 5, 25, and 45 form an arithmetic proportion. Another way to think of it is that there is a number X, such that (A + X) = B, and (B + X) = C.
There are 49 arithmetic proportions available among the numbers in Rhythmomachy, according to Fulke.
So for example, 3, 12, and 72 are in geometric proportion, because (3 x 6) = 12, and (12 x 6) = 72. 2, 4, and 8 are in geometric proportion, because (2 x 2) = 4, and (4 x 2) = 8. A more sophisticated example would be 4, 6, and 9, which are in geometric proportion because (4 x 1.5) = 6, and (6 x 1.5) = 9. (Yes, this is a period example; they understood fractions perfectly fine.)
There are 27 geometric proportions available in Rhythmomachy, according to Fulke.
So for example, 3, 4, and 6 are in harmonic proportion, because (6 / 3) = 2, and ((6 - 4) / (4 - 3)) = 2 as well. Or 25, 45, and 225, because (225 / 25) = 9, and ((225 - 45) / (45 - 25)) = 9 also.
Personally, I find harmonic proportions incredibly hard to work out in my head, although I understand them mathematically. There are 17 harmonic proportions available in Rhythmomachy, according to Fulke.