Commentary on Proposition VIII.21

Back to Proposition VIII.21.

[Folio 0384 | p. 378]

The converse of VIII. 19. If $a, c, d, b$ are in geometrical progression, $a, b$ are “similar solid numbers.”

Let $\alpha, \beta, \gamma$ be the least numbers in the ratio of $a, c, d$ (and therefore also of $c, d, b$). [VII. 33 or VIII. 2]

Therefore $\alpha, \gamma$ are prime to one another. [VIII. 3]

They are also “similar plane numbers.” [VIII. 20]

Let

$$\alpha = mn, \quad \gamma = pq,$$

where

$$m : n = p : q.$$

[Folio 0385 | p. 379]

Then, by the proof of VIII. 20,

$$\alpha : \beta = m : p = n : q.$$

Now, ex aequali,

$$a : d = \alpha : \gamma, \qquad \text{[VII. 14]}$$

and, since $\alpha, \gamma$ are prime to one another,

$$a = r\alpha, \quad d = r\gamma, \quad \text{where } r \text{ is an integer.}$$

But $\alpha = mn$: therefore $a = rmn$, and therefore $a$ is “solid.” Again, ex aequali,

$$c : b = \alpha : \gamma,$$

and therefore

$$c = s\alpha, \quad b = s\gamma, \quad \text{where } s \text{ is an integer.}$$

Thus $b = spq$, and $b$ is therefore “solid.” Now

$$\begin{aligned} \alpha : \beta = a : c &= r\alpha : s\alpha \\ &= r : s. \end{aligned} \qquad \text{[VII. 18]}$$

And, from above,

$$\alpha : \beta = m : p = n : q.$$

Therefore

$$r : s = m : p = n : q,$$

and hence $a, b$ are similar solid numbers.