Commentary on Proposition VIII.20

Back to Proposition VIII.20.

[Folio 0382 | p. 376]

This proposition is the converse of VIII. 18. If $a, c, b$ are in geometrical progression, $a, b$ are “similar plane numbers.”

Let $\alpha : \beta$ be the ratio $a : c$ (and therefore also the ratio $c : b$) in its lowest terms.

Then [VII. 20]

$$\begin{aligned} a &= m\alpha, & c &= m\beta, && \text{where } m \text{ is some integer,} \\ c &= n\alpha, & b &= n\beta, && \text{where } n \text{ is some integer.} \end{aligned}$$

[Folio 0383 | p. 377]

Thus $a$, $b$ are both products of two factors, i.e. plane. Again,

$$\alpha : \beta = a : c = c : b$$

$$= m : n. \qquad \text{[VII. 18]}$$

Therefore, alternately,

$$\alpha : m = \beta : n, \qquad \text{[VII. 13]}$$

and hence $m\alpha$, $n\beta$ are similar plane numbers. [Our notation makes the second part still more obvious, for $c = m\beta = n\alpha$.]