Commentary on Proposition V.20¶
Back to Proposition V.20.
Though, as already remarked, Euclid has not yet given us any definition of compounded ratios, Props. 20—23 contain an important part of the theory of such ratios. The term “compounded ratio” is not used, but the propositions connect themselves with the definitions of ex aequali in its two forms, the ordinary form defined in Def. 17 and that called perturbed proportion in Def. 18. The compounded ratios dealt with in these propositions are those compounded of successive ratios in which the consequent of one is the antecedent of the next, or the antecedent of one is the consequent of the next.
Prop. 22 states the fundamental proposition about the ratio ex aequali in its ordinary form, to the effect that,
$$\begin{aligned} \text{if} \quad & a \text{ is to } b \text{ as } d \text{ is to } e, \\ \text{and} \quad & b \text{ is to } c \text{ as } e \text{ is to } f, \\ \text{then} \quad & a \text{ is to } c \text{ as } d \text{ is to } f, \end{aligned}$$
with the extension to any number of such ratios; Prop. 23 gives the corresponding theorem for the case of perturbed proportion, namely that,
$$\begin{aligned} \text{if} \quad & a \text{ is to } b \text{ as } e \text{ is to } f, \\ \text{and} \quad & b \text{ is to } c \text{ as } d \text{ is to } e, \\ \text{then} \quad & a \text{ is to } c \text{ as } d \text{ is to } f. \end{aligned}$$
Each depends on a preliminary proposition, Prop. 22 on Prop. 20 and Prop. 23 on Prop. 21. The course of the proof will be made most clear by using the algebraic notation.
The preliminary Prop. 20 asserts that,
$$\begin{aligned} \text{if} \quad & a : b = d : e, \\ \text{and} \quad & b : c = e : f, \end{aligned}$$
$$\text{then, according as } a > = < c, \quad d > = < f.$$
For, according as $a$ is greater than, equal to, or less than $c$, the ratio $a : b$ is greater than, equal to, or less than the ratio $c : b$, $[\text{V. } 8 \text{ or V. } 7]$
$$\begin{aligned} \text{or (since} \quad & d : e = a : b, \\ \text{and} \quad & c : b = f : e) \end{aligned}$$
the ratio $d : e$ is greater than, equal to, or less than the ratio $f : e$,
$$[\text{by aid of V. } 13 \text{ and V. } 11]$$
and therefore $d$ is greater than, equal to, or less than $f$. $[\text{V. } 10 \text{ or V. } 9]$
It is next proved in Prop. 22 that, by V. 4, the given proportions can be transformed into
$$\begin{aligned} & ma : nb = md : ne, \\ \text{and} \quad & nb : pc = ne : pf, \end{aligned}$$
whence, by V. 20,
$$\begin{aligned} \text{according as} \quad & ma \text{ is greater than, equal to, or less than } pc, \\ & md \text{ is greater than, equal to, or less than } pf, \end{aligned}$$
so that, by Def. 5,
$$a : c = d : f.$$
Prop. 23 depends on Prop. 21 in the same way as Prop. 22 on Prop. 20, but the transformation of the ratios in Prop. 23 is to the following: (1)
$$ma : mb = ne : nf$$
(by a double application of V. 15 and by V. 11), (2)
$$mb : nc = md : ne$$
(by V. 4, or equivalent steps), and Prop. 21 is then used.
Simson makes the proof of Prop. 20 slightly more explicit, but the main difference from the text is in the addition of the two other cases which Euclid dismisses with “Similarly we can prove.” These cases are:
“Secondly, let A be equal to C; then shall D be equal to F. Because A and C are equal to one another, A is to B as C is to B. $[\text{V. } 7]$ But A is to B as D is to E, and C is to B as F is to E, wherefore D is to E as F to E; $[\text{V. } 11]$ and therefore D is equal to F. $[\text{V. } 9]$ Next, let A be less than C; then shall D be less than F. For C is greater than A, and, as was shown in the first case,
and, in like manner,
therefore F is greater than D, by the first case; and therefore D is less than F.”