Commentary on Proposition X.73¶
Back to Proposition X.73.
Euclid now passes to the irrational straight lines which are the difference and not, as before, the sum of two straight lines. Apotome (“portion cut off”) accordingly takes the place of binomial and the other terms follow mutatis mutandis. The first hexad of propositions (73 to 78) exhibit the six irrational straight lines which are really the result of extracting the square root of the six irrationals in the later propositions 85 to 90 (or, strictly speaking, of finding the sides of squares equal to the rectangles formed by each of those six irrational straight lines respectively with a rational straight line). Thus, just as in the corresponding propositions about the irrational straight lines formed by addition, the further removed irrationals, so to speak, come first.
We shall denote the apotome etc. by $(x - y)$, which is formed by subtracting a certain lesser straight line $y$ from a greater $x$. In X. 79 and later propositions $y$ is called by Euclid the annex (ἡ προσαρμόζουσα), being the straight line which, when added to the apotome or other irrational formed by subtraction, makes up the greater $x$.
The methods of proof are exactly the same as in the preceding propositions about the irrational straight lines formed by addition.
In this proposition $x, y$ are rational straight lines commensurable in square only, and we have to prove that $(x - y)$, the apotome, is irrational.
$x \sim y$, so that $x \smile y$: therefore, since
$$x : y = x^2 : xy,$$
$$x^2 \smile xy.$$
But $x^2 \frown (x^2 + y^2)$, and $xy \frown 2xy$; therefore
$$x^2 + y^2 \smile 2xy,$$
whence
$$(x - y)^2 \smile (x^2 + y^2).$$
But $(x^2 + y^2)$ is rational, therefore $(x - y)^2$, and consequently $(x - y)$, is irrational.
The apotome $(x - y)$ is of the form $\rho \sim \sqrt k . \rho$, just as the binomial straight line is of the form $\rho + \sqrt k . \rho$.