Commentary on Proposition X.111

Back to Proposition X.111.

This proposition proves the equivalent of the fact that $\sqrt{x} + \sqrt{y}$ cannot be equal to $\sqrt{x'} - \sqrt{y'}$, and $x + \sqrt{y}$ cannot be equal to $x' - \sqrt{y'}$.

We should prove these results by squaring the respective expressions; and Euclid’s procedure corresponds to this exactly.

He has to prove that

$$\rho + \sqrt{k} \cdot \rho \text{ cannot be equal to } \rho' - \sqrt{\lambda} \cdot \rho'.$$

For, if possible, let this be so.

Take the straight lines $\frac{(\rho + \sqrt{k} \cdot \rho)^2}{\sigma}$, $\frac{(\rho' - \sqrt{\lambda} \cdot \rho')^2}{\sigma}$; these must be equal, and therefore

$$\frac{\rho^2}{\sigma} (1 + k + 2\sqrt{k}) = \frac{\rho'^2}{\sigma} (1 + \lambda - 2\sqrt{\lambda}) \dots\dots\dots\dots\dots(1).$$

Now $\frac{\rho^2}{\sigma} (1 + k)$, $\frac{\rho'^2}{\sigma} (1 + \lambda)$ are rational and $\mathbin{\Cap}$; therefore

$$\left\{ \frac{\rho'^2}{\sigma} (1 + \lambda) - \frac{\rho^2}{\sigma} (1 + k) \right\} \mathbin{\Cap} \frac{\rho'^2}{\sigma} (1 + \lambda)$$

$$\smile \frac{\rho'^2}{\sigma} \cdot 2\sqrt{\lambda}.$$

And, since both sides are rational, it follows that

$$\left\{ \frac{\rho'^2}{\sigma} (1 + \lambda) - \frac{\rho^2}{\sigma} (1 + k) \right\} - \frac{\rho'^2}{\sigma} \cdot 2\sqrt{\lambda} \text{ is an \textit{apotome}}.$$

But, by (1), this expression is equal to $\frac{\rho^2}{\sigma} \cdot 2\sqrt{k}$, which is rational.

Hence an apotome, which is irrational, is also rational: which is impossible.

This proposition is the connecting link which enables Euclid to prove that all the compound irrationals with positive signs above discussed are different from all the corresponding compound irrationals with negative signs, while the two sets are all different from one another and from the medial straight line. The recapitulation following makes this clear.

The apotome and the irrational straight lines following it are neither the same with the medial straight line nor with one another.

For the square on a medial straight line, if applied to a rational straight line, produces as breadth a straight line rational and incommensurable in length with that to which it is applied, [x. 22] while the square on an apotome, if applied to a rational straight line, produces as breadth a first apotome, [x. 97] the square on a first apotome of a medial straight line, if applied to a rational straight line, produces as breadth a second apotome, [x. 98] the square on a second apotome of a medial straight line, if applied to a rational straight line, produces as breadth a third apotome, [x. 99] the square on a minor straight line, if applied to a rational straight line, produces as breadth a fourth apotome, [x. 100] the square on the straight line which produces with a rational area a medial whole, if applied to a rational straight line, produces as breadth a fifth apotome, [x. 101] and the square on the straight line which produces with a medial area a medial whole, if applied to a rational straight line, produces as breadth a sixth apotome. [x. 102]

Since then the said breadths differ from the first and from one another, from the first because it is rational, and from one another since they are not the same in order,
it is clear that the irrational straight lines themselves also differ from one another.
And, since the apotome has been proved not to be the same as the binomial straight line, [x. 111] but, if applied to a rational straight line, the straight lines
following the apotome produce, as breadths, each according to its own order, apotomes, and those following the binomial straight line themselves also, according to their order, produce the binomials as breadths,

therefore those following the apotome are different, and those following the binomial straight line are different, so that there are, in order, thirteen irrational straight lines in all,

Medial, Binomial, First bimedial, Second bimedial, Major, “Side” of a rational plus a medial area, “Side” of the sum of two medial areas, Apotome, First apotome of a medial straight line, Second apotome of a medial straight line, Minor, Producing with a rational area a medial whole, Producing with a medial area a medial whole.