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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 lengths), and let $y$ and $z$ be two magnitudes of the same kind (for instance two areas). Then [according to Euclid,][2] 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 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" basically corresponds to our notion of "positive real numbers".

Now with that background, my question is about the multiplication of real numbers. Here is how Euclid defines the product of ratios: 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 $u/v = r/t$. (This is well-defined by Euclid's proposition V.22) But this is not the standard way that multiplication is defined in the Dedekund cut construction of the real numbers, where you form a new cut by taking the products of the rational numbers in the two cuts that you're multiplying.

So my question is, how can we prove than Euclid's definition of real number multiplication is equal to the Dedekind cut definition? If I'm not mistaken, the problem basically reduces to proving the following:

For all positive integers $l$, $m$, and $n$ and all magnitudes $x$, $y$, and $z$:

If $l/m < x/y$ and $m/n < y/z$ then $l / n < x/ z$
If $l/m = x/y$ and $m/n = y/z$ then $l / n = x/ z$
If $l/m > x/y$ and $m/n > y/z$ then $l / n > x/ z$

And that in turn is equivalent to the following:

For all positive integers $l$, $m$, and $n$ and all magnitudes $x$, $y$, and $z$:

If $ly < mx$ and $mz < ny$ then $lz < nx$
If $ly = mx$ and $mz = ny$ then $lz = nx$
If $ly > mx$ and $mz > ny$ then $lz > nx$

So does anyone have any idea how to go about proving that? For reference, addition of magnitudes is associative and commutative, and magnitudes also obey the following properties:

V.1. Multiplication by numbers distributes over addition of magnitudes. $m(x_1 + x_2 + ... + x_n) = m x_1 + m x_2 + ... + m x_n$

V.2. Multiplication by magnitudes distributes over addition of numbers. $(m + n)x = mx + nx$

V.3. An associativity of multiplication. $m(nx) = (mn)x$

V.5. Multiplication by numbers distributes over subtraction of magnitudes. $m(x – y) = mx – my$

V.6. Uses multiplication by magnitudes distributes over subtraction of numbers. $(m – n)x = mx – nx$

EDIT: Out of the three statements I wanted to prove, I just realized that Euclid proved statement 2 in his proposition V.22. So now I just need to prove statements 1 and 3.

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  • $\begingroup$ Have you considered asking this on HSM SE ? There may be more users able to answer your question :) $\endgroup$ – BusyAnt Jul 11 '16 at 8:36
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    $\begingroup$ @BusyAnt I've never heard of that site before. In any case, this is less of a history question and more of a math problem. The history is quite clear: Euclid simply did not arrive at Dedekind's definition of real number multiplication. But the question is, could be have arrived at it using his system, even though he didn't actually do so. $\endgroup$ – Keshav Srinivasan Jul 11 '16 at 8:53
  • $\begingroup$ Extremely interesting question! Wish I could upvote it more than once .. $\endgroup$ – Han de Bruijn Sep 14 '16 at 14:08
  • $\begingroup$ Dedekind took the Eudoxian theory of proportions in Euclid's Book V as the inspiration for his Dedekind cut construction. Is there a reference for this or is it just your theory? $\endgroup$ – Han de Bruijn Sep 14 '16 at 14:12
  • $\begingroup$ 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. Euclid was right, of course. What is needed for a quantity to be measurable in a physical sense is a number plus some unit. Resulting in a calculus of ratios: length/unit = number. $\endgroup$ – Han de Bruijn Sep 14 '16 at 14:23
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Here's a proof of the first statement; the third statement can be proven similarly. If $ly<mx$ and $mz<ny$, then multiplying the first inequality by $n$ and the second by $l$, we find $lny<mnx$ and $lmz<lny$, which combine to give $lmz<mnx$. Dividing by $m$ now gives $lz<nx$.

Here we are using the fact that if $x$ and $y$ are magnitudes and $n$ is a positive integer, then $x<y$ iff $nx<ny$. I am not entirely familiar with how Euclid reasoned about these things, but this fact should be true in his system. Here's one way you might prove it, which I think should be valid in this context. If $x<y$, then there is a magnitude $z$ such that $y=x+z$, and then $ny=nx+nz$, so $nx<ny$. Similarly, $x=y$ implies $nx=ny$ and $x>y$ implies $nx>ny$, and so (assuming any two magnitudes are comparable) it follows that if $nx<ny$, we must have $x<y$.

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  • $\begingroup$ Hmm, does this mean that Euclid's system is far more transparent than Dedekind's? Then why not employ the former instead of the latter for defining the reals? $\endgroup$ – Han de Bruijn Sep 12 '16 at 19:45

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