Commentary on Proposition VIII.11

Back to Proposition VIII.11.

[Folio 0369 | p. 363]

According to Nicomachus the theorems in this proposition and the next, that two squares have one geometric mean, and two cubes two geometric means, between them are Platonic. Cf. Timaeus 32 A sqq. and the note thereon, p. 294 above.

[Folio 0370 | p. 364]

$a^2$, $b^2$ being two squares, it is only necessary to form the product $ab$ and to prove that

$$a^2,\quad ab,\quad b^2$$

are in geometrical progression. Euclid proves that

$$a^2 : ab = ab : b^2$$

by means of VII. 17, 18, as usual.

In assuming that, since $a^2$ is to $b^2$ in the duplicate ratio of $a^2$ to $ab$, $a^2$ is to $b^2$ in the duplicate ratio of $a$ to $b$, Euclid assumes that ratios which are the duplicates of equal ratios are equal. This, an obvious inference from V. 22, can be inferred just as easily for numbers from VII. 14.