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Preliminary note: With "Euclid" I don't mean a person but the mathematicians of the Euclidean period of which Euclid (if he had been one person) was a representative.


I imagine that Euclid could have thought about relating lengths and numbers in a bijective way even though they were considered completely different things. For rational lengths, it would have worked because for each rational length there is a number (which for Euclid was necessarily rational) and vice versa.

But unfortunately there do exist constructible lengths which provably are not rational (e.g. the length of the diagonal of the unit square) and for which Euclid hadn't got a (rational) number to assign to. So this may have been the main reason that he abondonded the plan (if he had it) to relate lengths and numbers in a bijective way: there were not enough numbers.

But there may have been another (admittedly speculative) reason: For the assignment to work, one has to pick an arbitrary line segment and assign the number 1 (the unit) to its length. Did possibly Euclid dislike the arbitrariness of assigning the "primordial" unit (from the realm of platonic entities, from which all numbers are built) to a "random" line segment (from the realm of "earthly" entities)?

But note, that he doesn't define (in Definition VII.1) the unit, but a unit. So this might not have been the reason. But the question arises why he didn't define the unit, which seems much more intuitive. (How would he have distinguished between different units?)

Alternatively, he could have started with two "primordial" points – named 0 and 1 – from which all other points and lengths can be built with straightedge and compass, and assign the number 1 to the length of the distinguished "primordial" line segment $\overline{01}$ – but that's even more speculative.

I admit that I might completely misunderstand Euclid's way of thinking. Any hint in which respect I do so would be welcome.


To my defense: Euclid's Definition VII.1 is really rather obscure:

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A unit is that by virtue of which each of the things that exist is called one.
source

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You state the main reason correctly. At the time of Euclid, the only known numbers were rational numbers, and it was discovered that you cannot measure segments constructed in geometry with rational numbers. Therefore, numbers were abandoned in geometry.

Instead Euclid (or his predecessors) developed a highly sophisticated theory of proportions, which can be shown to be equivalent to our theory of real numbers. So they could talk about length and area of a circle, for example. It took two thusand years after Euclid to develop a satisfactory theory of real numbers. So Euclid just could not "assign numbers to lengths".

I recommend a very nice book which discusses these things: R. Hartshorne, Companion to Euclid, AMS, 1970.

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  • $\begingroup$ How can the theory of proportions be shown to be equivalent to our theory of real numbers if the former did not even define addition of proportions? Not to mention that proportions can only be applied to constructed segments, which, given Euclid's tools, would give far less than real numbers. And still far less even if we throw in neusis and mechanical curves. $\endgroup$ – Conifold Oct 28 '18 at 23:35
  • $\begingroup$ Proportions can be defined for any segments, commensurable or not, this is the whole point. Addition of segments can be defined in a very simple way, geometrically. For the details, see the book I mentioned. Or look in Euclid, book V. Especially Proposition 4. $\endgroup$ – Alexandre Eremenko Oct 29 '18 at 14:14
  • $\begingroup$ @Conifold: It is a simple exercise that Proposition V.4 is equivalent to Dedekind's definition of a real number. But of course it is much more complicated than that of Dedekind. $\endgroup$ – Alexandre Eremenko Oct 29 '18 at 14:18
  • $\begingroup$ Addition of segments, not of their ratios, you'd need to fix a "unit" segment to reduce one to the other. That made no sense to Greeks, not even to Archimedes, and does not appear until Descartes. Dedekind constructs real numbers as cuts, this hypostatic abstraction is his big leap that nothing Greek comes close to. That he repurposes Eudoxian trick of V.4 (proving equality of ratios for pre-constructed magnitudes) pales in comparison. I think what you mean by "equivalent" is "theory of ratios can be translated into a fragment of real analysis with some parallels in proofs". $\endgroup$ – Conifold Oct 29 '18 at 22:31

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