Did Euclid really prove the existence of irrational numbers?

Question

In Proposition 10.10 of Euclid's Elements, Euclid tries to construct a line segment which is incommensurable with a given line segment. (Two line segments are incommensurable if there exists no common line segment that both are integer multiples of, or equivalently, if the ratio of their lengths does not equal a ratio of natural numbers.) Here's what he says:

Let A be the assigned straight line. It is required to find two straight lines incommensurable, the one in length only, and the other in square also, with A. Set out two numbers B and C which do not have to one another the ratio which a square number has to a square number, that is, which are not similar plane numbers, and let it be contrived that B is to C as the square on A is to the square on D, for we have learned how to do this. Therefore the square on A is commensurable with the square on D. And, since B does not have to C the ratio which a square number has to a square number, therefore neither has the square on A to the square on D the ratio which a square number has to a square number, therefore A is incommensurable in length with D.

This proof relies on Euclid's Proposition 10.9, which states in part that "squares which do not have to one another the ratio which a square number has to a square number also do not have their sides commensurable in length either". So what Euclid does is he chooses two natural numbers B and C such that the ratio of B to C is not equal to a ratio of square numbers. And then he constructs two squares whose areas are in the ratio of B to C. And finally he uses Proposition 10.9 to show that the sides of the two squares are incommensurable.

But my question is, where does Euclid get the fact that there exists two natural numbers B and C such that the ratio of B to C is not equal to a ratio of square numbers? Euclid just says "Set out two numbers B and C which do not have to one another the ratio which a square number has to a square number, that is, which are not similar plane numbers".

For those who don't know, two natural numbers $m$ and $n$ are called similar plane numbers if there exist natural numbers $p$, $q$, $r$, and $s$ such that $m=pq$, $n=rs$, and the ratio of $p$ to $q$ is equal to the ratio of $r$ to $s$, or equivalently the ratio of $p$ to $r$ is equal to the ratio of $q$ to $s$. Now using Euclid's Proposition 8.18 and Proposition 8.11, it's easy to prove that the ratio of similar plane numbers is equal to the ratio of square numbers. But did Euclid ever prove the converse of that statement, i.e. that if two numbers are not similar plane numbers then their ratio is not equal to the ratio of square numbers?

Because if Euclid didn't prove that, then he didn't really do the hard number theory work needed to prove that irrational numbers exist.

EDIT: I just found out that Euclid stated the result I want him to prove in a Lemma after Proposition 10.9:

It has been proved in the arithmetical books that similar plane numbers have to one another the ratio which a square number has to a square number, and that, if two numbers have to one another the ratio which a square number has to a square number, then they are similar plane numbers.

The translator says that the justification of this is "VIII.26 and converse". Here is what Proposition 8.26:

Similar plane numbers have to one another the ratio which a square number has to a square number.

But where does Euclid prove the converse, namely that "if two numbers have to one another the ratio which a square number has to a square number, then they are similar plane numbers"? That's the hard thing to prove. Euclid claims "it has been proved in the arithmetical books", i.e. in Books 7-9, but I can't seem to find such a proof.

Answer 1

Regarding similar plane numbers :

Definition VII.21 : Similar plane numbers are those which have their sides proportional.

Example : The numbers $18$ and $8$ are similar plane numbers. When $18$ is interpreted as a plane number with sides $6$ and $3$, and $8$ has sides $4$ and $2$, then the sides are proportional.

I.e. : $\dfrac 2 3 = \dfrac 4 6$.

Prop.X.9 amounts to proving that :

Line segments which produce a square whose area is an integer, but not a square number, are incommensurable with the unit length.

Euclid states : "Set out two numbers B and C which do not have to one another the ratio which a square number has to a square number, that is, which are not similar plane numbers".

We can consider a rectangle with sides $n$ and $1$, where $n$ is a number whatever that is not a square (like: $2,3,5,\ldots$); by Prop.II.14 we can build the square with the same area.

Now we have to compare the square B with area $n$ and the unit square C and proceed by contradiction (in the way that Aristotle's proof is managed) assuming that they are "produced" by lines A and D commensurable in length, i.e. whose values are $p$ and $q$, such that :

$n : 1 = p^2 : q^2$

and assume as usual that $p$ and $q$ are relatively prime.

Thus, from Prop.VII.27 also $p^2$ and $q^2$ are relatively prime.

Thus, by Prop.VII.21, they are the least of the numbers in that ratio.

But also $n$ and $1$ are so : relatively prime and the least of the numbers in that ratio.

Thus :

$n=p^2$ and $1=q^2$

from which we have to conclude that $n$ is a square, contrary to assumption.

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For incommensurable magnitudes, see :

Def.X.1. Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.

In Proposition X.2 the so-called euclidean algorithm : ἀνθυφαίρεσις (anthyphairesis, reciprocal subtraction) is used to prove that :

If, when the less of two unequal magnitudes is continually subtracted in turn from the greater that which is left never measures the one before it, then the two magnitudes are incommensurable.

See the comment for a geometrical example, and see Heath's edition, Vol.III, page 19 for the application to the side-diagonal case.

In Prop.X.5 he proves that :

Commensurable magnitudes have to one another the ratio which a number has to a number

followed by its converse : Prop.X.6.

In Prop.X.7 Euclide proves the contrapositive of X.6 :

Incommensurable magnitudes do not have to one another the ratio which a number has to a number

followed by X.8 : the contrapositive of X.5.

Thus, applying X.5 to the result regarding so-called Aristotelian proof, it gives us another example of incommensurable magnitudes.

The Aristotelian proof has been introduced into Elements as X.117, but it is considered an interpolation : maybe the "interpolator" decided to supplement Elements with this "obvious" result...

For details, see Salomon Ofman's works.

For a full-lenght book on the history of incommensurability in Greek mathematics, you can see :

• Wilbur Richard Knorr, The Evolution of the Euclidean Elements : A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry (1973).