What did Aristotle mean by $(A+C):C::(B+D):D$ and $(A+C):(B+D)::C:D$ ?

Question

_{Source first encountered: p 482-484, Introducing Philosophy for Canadians: A Text with Integrated Readings (2011 1 ed).
Primary Source: Bekker Number 1131 B, around Lines 11-13, Nicomachean Ethics by Aristotle.
Translation 1: (2012) by Joe Sachs. The following cites a footnote of the translator.}

¹¹¹ To start with, A and B were people, while C and D were (say) sums of money. Alternating this proportion is illegitimate mathematically, since there is no ratio between a person and a dollar, but is perfectly intelligible in the situation to which the mathematical language is being applied. It says that person A is to (deserves) C dollars as (on the same grounds that) person B is to (deserves) D dollars. “So too is the whole to the whole” is an elliptical way of saying that composing the alternate ratios [(A+C):C::(B+D):D] $ (\color{green}{1. \dfrac{A + C}{C} = \dfrac{B + D}{D} })$ and re-alternating [(A+C):(B+D)::C:D] $ (\color{green}{2. \dfrac{A + C}{B + D} = \dfrac{C}{D} })$ brings back the original ratio of the two people A and B. That is, A with C dollars in his pocket and B with D dollars in his, maintain the same relation as do their relative merits, and neither has been unjustly enriched at the expense of the other. Proofs that these transformations of a proportion preserve proportionality in the results may be found in Euclid’s Elements, Bk. V, Props. 16 and 18. [...]

1. Please correct me if my rewrites (in green) are wrong.

2. What did Aristotle (and Joe Sachs) intend? 1 cannot be algebraically rechanged to 2.

Answer 1

What Aristotle uses in the cited passage ; Nic.Eth., 1131b, are some simple properties of proportions:

if $\dfrac A B = \dfrac C D$ [as the first term is to the second, so is the third to the fourth],

then $\dfrac A C = \dfrac B D$ [as the first is to the third, so is the second to the fourth].

Thus, $BC = AD$ and so : $AB + BC = AB + AD$, from which :

$(A+C)B = A(B+D)$.

Finally :

$$\dfrac A B = \dfrac {A+C}{B+D}$$ [as the first is to the second, so is the sum of the first and third to the sum of the second and fourth.].

In discussing distributive justice, that requires that equal persons receive equal shares, Aristotle uses proportions for an analogy :

[ 1131a ] Justice is therefore a sort of proportion; for proportion is not a property of numerical quantity only, but of quantity in general, proportion being equality of ratios, and involving four terms at least. [...] Thus the just also involves four terms at least, and the ratio between the first pair of terms is the same as that between the second pair. For the two lines representing the persons and shares are similarly divided: Now this is the combination effected by a distribution of shares, and the combination is a just one, if persons and shares are added together in this way. The principle of Distributive Justice, therefore, is the conjunction of the first term of a proportion with the third and of the second with the fourth; and the just in this sense is a mean between two extremes that are disproportionate, since the proportionate is a mean, and the just is the proportionate.

The gist of the analogy seems to be that what is just share lies between too large a share and too small a one, i.e. must be "proportionate".