Commentary on Proposition X.9¶
Back to Proposition X.9.
A scholium to this proposition (Schol. x. No. 62) says categorically that the theorem proved in it was the discovery of Theaetetus.
If $a$, $b$ be straight lines, and
$$a : b = m : n,$$
where $m$, $n$ are numbers, then
$$a^2 : b^2 = m^2 : n^2;$$
and conversely.
This inference, which looks so easy when thus symbolically expressed, was by no means so easy for Euclid owing to the fact that $a$, $b$ are straight lines, and $m$, $n$ numbers. He has to pass from $a : b$ to $a^2 : b^2$ by means of VI. 20, Por. through the duplicate ratio ; the square on $a$ is to the square on $b$ in the duplicate ratio of the corresponding sides $a$, $b$. On the other hand, $m$, $n$ being numbers, it is VIII. 11 which has to be used to show that $m^2 : n^2$ is the ratio duplicate of $m : n$.
Then, in order to establish his result, Euclid assumes that, if two ratios are equal, the ratios which are their duplicates are also equal. This is nowhere proved in Euclid, but it is an easy inference from V. 22, as shown in my note on VI. 22.
The converse has to be established in the same careful way, and Euclid assumes that ratios the duplicates of which are equal are themselves equal. This is much more troublesome to prove than the converse ; for proofs I refer to the same note on VI. 22.
The second part of the theorem, deduced by reductio ad absurdum from the first, requires no remark.
In the Greek text there is an addition to the Porism which Heiberg brackets as superfluous and not in Euclid’s manner. It consists (1) of a sort of proof, or rather explanation, of the Porism and (2) of a statement and explanation to the effect that straight lines incommensurable in length are not necessarily incommensurable in square also, and that straight lines incommensurable in square are, on the other hand, always incommensurable in length also.
The Lemma gives expressions for two numbers which have to one another the ratio of a square number to a square number. Similar plane numbers are of the form $pm \cdot pn$ and $qm \cdot qn$, or $mnp^2$ and $mnq^2$, the ratio of which is of course the ratio of $p^2$ to $q^2$.
The converse theorem that, if two numbers have to one another the ratio of a square number to a square number, the numbers are similar plane numbers is not, as a matter of fact, proved in the arithmetical Books. It is the converse of VIII. 26 and is used in IX. 10. Heron gave it (see note on
27 above).
Heiberg however gives strong reason for supposing the Lemma to be an interpolation. It has reference to the next proposition, X. 10, and, as we shall see, there are so many objections to X. 10 that it can hardly be accepted as genuine. Moreover there is no reason why, in the Lemma itself, numbers which are not similar plane numbers should be brought in as they are.