Commentary on Proposition X.10¶
Back to Proposition X.10.
It would appear as though this proposition was intended to supply a justification for the statement in X. Def. 3 that it is proved that there are an infinite number of straight lines ($a$) incommensurable in length only, or commensurable in square only, and ($b$) incommensurable in square, with any given straight line.
But in truth the proposition could well be dispensed with ; and the positive objections to its genuineness are considerable.
In the first place, it depends on the following proposition, X. 11 ; for the last step concludes that, since
$$a^2 : y^2 = a : x,$$
and $a$, $x$ are incommensurable in length, therefore $a^2$, $y^2$ are incommensurable. But Euclid never commits the irregularity of proving a theorem by means of a later one. Gregory sought to get over the difficulty by putting X. 10 after
11 ; but of course, if the order were so inverted, the Lemma would still be in the wrong place.
Further, the expression ἐμάθομεν γάρ, “for we have learnt (how to do this),” is not in Euclid’s manner and betrays the hand of a learner (though the same expression is found in the Sectio Canonis of Euclid, where the reference is to the Elements).
Lastly the manuscript P has the number 10, in the first hand, at the top of X. 11, from which it may perhaps be concluded that X. 10 had at first no number.
It seems best therefore to reject as spurious both the Lemma and X. 10.
The argument of X. 10 is simple. If $a$ be a given straight line and $m$, $n$ numbers which have not to one another the ratio of square to square, take $x$ such that
$$a^2 : x^2 = m : n, \qquad \text{[X. 6, Por.]}$$
whence $a$, $x$ are incommensurable in length. [X. 9]
Then take $y$ a mean proportional between $a$, $x$, whence
$$a^2 : y^2 = a : x \qquad \text{[V. Def. 9]}$$
$$[= \sqrt{m} : \sqrt{n}],$$
and $x$ is incommensurable in length only, while $y$ is incommensurable in square as well as in length, with $a$.