In Euclid's day, the modern notion of real number did not exist; Euclid did not believe that the length of a line segment was a quantity measurable by number. But he did think it made sense to talk about the ratio of two lengths. In fact, he devotes Book V of his Elements to the study of such ratios, using the so-called Eudoxian theory of proportions. Here's how it works.

Let $w$ and $x$ be two magnitudes of the same kind (for instance two length), and let $y$ and $z$ be two magnitudes of the same kind (for instance two areas). Then according to Euclid, the ratio of $w$ to $x$ is said to be equal to the ratio of $y$ to $z$ if for all positive integers $m$ and $n$, if $nw$ is greater, equal, or less than $mx$, then $ny$ is greater, equal, or less than $mz$, respectively. Or to put it in more modern language, $w/x = y/z$ if the same rational numbers $m/n$ are less than both, the same rational numbers are equal to both, and the same rational numbers are greater than both.

In other words, a ratio is defined by the classes of rational numbers which are less than, equal to, and greater than it. If you've studied real analysis; this should look familiar to you: it is how the real number system is constructed using Dedekind cuts! In fact, Dedekind took the Eudoxian theory of proportions in Euclid's book V as the inspiration for his Dedekind cut construction. So to sum up, while Euclid wouldn't have thought of them as numbers, his notion of "ratios" corresponds to our notion of "positive real numbers".

Now with that background, I would like to try to prove using Euclid's system that multiplication of real numbers is commutative. First let me explain how the product of two ratios is defined. (Euclid uses the product in a few propositions including this one.) We say that the product of $w/x$ and $y/z$ is equal to $u/v$ if there exist magnitudes $r,s,$ and $t$ such that $w/x = r/s$, $y/z = s/t$, and $r/t = u/v$.

So in order to prove the commutativity of multiplication, we would need to prove the following:

Suppose that $b/c = e/f$ and $a/b = f/g$. Then $a/c =e/g$.

If we could prove that, then that would mean that the product of $b/c$ and $a/b$ is equal to the product of $a/b$ and $b/c$, which would immediately imply the commutativity of multiplication in general.

So how would I go about proving that? Euclid's Book V contains a lot of theorems about ratios that are potentially relevant, but I'm not sure how to proceed.

EDIT: I just posted a follow-up question on the distributive property.


1 Answer 1


From your link:

V.23. Perturbed ratios ex aequali. If u : v = y : z and v : w = x : y, then u : w = x : z.


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