Commentary on Proposition X.11

Back to Proposition X.11.

I shall henceforth, for the sake of brevity, use symbols for the terms “commensurable (with)” and “incommensurable (with)” according to the varieties described in X. Deff. 1—4. The symbols are taken from Lorenz and seem convenient.

Commensurable and commensurable with, in relation to areas, and commensurable in length and commensurable in length with, in relation to straight lines, will be denoted by $\frown$.

Commensurable in square only or commensurable in square only with (terms applicable only to straight lines) will be denoted by $\sim$.

Incommensurable (with), of areas, and incommensurable (with), of straight lines will be denoted by $\smile$.

Incommensurable in square (with) (a term applicable to straight lines only) will be denoted by $\sim\hspace{-0.5em}-$.

Suppose $a$, $b$, $c$, $d$ to be four magnitudes such that

$$a : b = c : d.$$

Then (1), if $a \frown b$,

$$a : b = m : n, \quad \text{where } m, n \text{ are integers, [x. 5]}$$

whence

$$c : d = m : n,$$

and therefore

$$c \frown d. \quad \text{[x. 6]}$$

  1. If $a \smile b$,

$$a : b \neq m : n, \quad \text{[x. 7]}$$

so that

$$c : d \neq m : n,$$

whence

$$c \smile d. \quad \text{[x. 8]}$$