Commentary on Proposition VIII.18

Back to Proposition VIII.18.

[Folio 0378 | p. 372]

If $ab$, $cd$ be “similar plane numbers,” i.e. products of factors such that

$$a : b = c : d,$$

there is one mean proportional between $ab$ and $cd$; and $ab$ is to $cd$ in the duplicate ratio of $a$ to $c$ or of $b$ to $d$.

Form the product $bc$ (or $ad$, which is equal to it, by VII. 19).

Then

$$\left.\begin{array}{r} ab,\ bc \\ = ad \end{array}\right\},\ cd$$

is a series of terms in geometrical progression.

For

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

Therefore

$$a : c = b : d. \qquad \text{[VII. 13]}$$

Therefore

$$ab : bc = bc : cd. \qquad \text{[VII. 17 and 16]}$$

Thus $bc$ (or $ad$) is a geometric mean between $ab$, $cd$.

And $ab$ is to $cd$ in the duplicate ratio of $ab$ to $bc$ or of $bc$ to $cd$, that is, of $a$ to $c$ or of $b$ to $d$.