Commentary on Proposition VIII.19

Back to Proposition VIII.19.

[Folio 0381 | p. 375]

In other words, if $a : b : c = d : e : f$, then there are two geometric means between $abc$, $def$; and $abc$ is to $def$ in the triplicate ratio of $a$ to $d$, or $b$ to $e$, or $c$ to $f$.

Euclid first takes the plane numbers $ab$, $de$ (leaving out $c$, $f$) and forms the product $bd$. Thus, as in VIII. 18,

$$\left.\begin{array}{r} ab,\ bd \\ = ea \end{array}\right\},\ de$$

are three terms in geometrical progression in the ratio of $a$ to $d$, or of $b$ to $e$.

He next forms the products of $c$, $f$ respectively into the mean $bd$.

Then

$$abc,\ cbd,\ fbd,\ def$$

are in geometrical progression in the ratio of $a$ to $d$ etc.

For

$$\left.\begin{aligned} abc : cbd &= ab : bd = a : d \\ bd : fbd &= c : f \\ fbd : def &= bd : de = b : e \end{aligned}\right\} \qquad \text{[VII. 17]}$$

And

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

The ratio of $abc$ to $def$ is the ratio triplicate of that of $abc$ to $cbd$, i.e. of that of $a$ to $d$ etc.