Commentary on Proposition IX.2

Back to Proposition IX.2.

[Folio 0391 | p. 385]

If $ab$ is a square number, $a$, $b$ are similar plane numbers. (The converse of IX. 1.) For

$$a : b = a^2 : ab. \hfill \text{[VII. 17]}$$

And $a^2$, $ab$ being square numbers, and therefore similar plane numbers, they have one mean proportional. [VIII. 18]

Therefore $a$, $b$ also have one mean proportional. [VIII. 8]

whence $a$, $b$ are similar plane numbers. [VIII. 20]