Euclid's Elements, Book X, Proposition 19

Proposition 19

Lemma.

Since it has been proved that straight lines commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line is commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in square also.

But, if any straight line is commensurable in square with a given rational straight line, then, if it is also commensurable in length with it, in this case it is also called rational and commensurable with it both in length and in square, but, if again any straight line, being commensurable in square with a given rational straight line, is incommensurable in length with it, in this case it is also called rational but commensurable in square only.

Proposition 19

The rectangle contained by rational straight lines commensurable in length is rational.

Let the rectangle AC be contained by the rational straight lines AB and BC commensurable in length.
I say that AC is rational.

Describe the square AD on AB. Then AD is rational. I.46
X.Def.4

java applet or image And, since AB is commensurable in length with BC, while AB equals BD, therefore BD is commensurable in length with BC.

And BD is to BC as DA is to AC. VI.1

Therefore DA is commensurable with AC. X.11

But DA is rational, therefore AC is also rational. X.Def.4

Therefore, the rectangle contained by rational straight lines commensurable in length is rational.

Q.E.D.

This is the first proposition that deals with rational lines and rational squares. As required by definitions X.Def.I.3 and X.Def.I.3, there is some assigned straight line to act as a standard to which other lines and squares are compared for rationality. That line is usually not mentioned in the propositions.

In this proposition, it is assumed that both sides of the rectangle AB and BC are rational lines. That means these lines are commensurable in square to the standard line, that is, their squares are commensurable with the standard square. It is also assumed that AB and BC are commensurable with each other. Therefore the rectangle AC is commensurable with the square on AB, but that's commensurable with the standard square, so the rectangle AC is too.

The proposition is used several times starting with X.25. The lemma is used in X.23.

The next proposition is a converse of this one, but the language obscures that from notice.

Book X Introduction - Proposition X.18 - Proposition X.20.

Proposition 19
The rectangle contained by rational straight lines commensurable in length is rational.
Let the rectangle AC be contained by the rational straight lines AB and BC commensurable in length. I say that AC is rational.
Describe the square AD on AB. Then AD is rational.		I.46 X.Def.4
	And, since AB is commensurable in length with BC, while AB equals BD, therefore BD is commensurable in length with BC.
	And BD is to BC as DA is to AC.	VI.1
	Therefore DA is commensurable with AC.	X.11
But DA is rational, therefore AC is also rational.		X.Def.4
Therefore, the rectangle contained by rational straight lines commensurable in length is rational.
Q.E.D.