Commentary on Proposition V.4

Back to Proposition V.4.

This proposition shows that, if $a$, $b$, $c$, $d$ are proportionals, then

$$ma \text{ is to } nb \text{ as } mc \text{ is to } nd;$$

and the proof is as follows:

Take $pma$, $pmc$ any equimultiples of $ma$, $mc$, and $qnb$, $qnd$ any equimultiples of $nb$, $nd$.

Since $a : b = c : d$, it follows [V. Def. 5] that,

$$\text{according as } pma \gtreqqless qnb, \ pmc \gtreqqless qnd.$$

But the $p$- and $q$-equimultiples are any equimultiples; therefore [V. Def. 5]

$$ma : nb = mc : nd.$$

It will be observed that Euclid’s phrase for taking any equimultiples of A, C and any other equimultiples of B, D is “let there be taken equimultiples E, F of A, C, and G, H other, chance, equimultiples of B, D,” E, F being called ἰσάκις πολλαπλάσια simply, and G, H ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια. And similarly, when any equimultiples (K, L) of E, F come to be taken, and any other equimultiples (M, N) of G, H. But later on Euclid uses the same phrases about the new equimultiples with reference to the original magnitudes, reciting that “there have been taken, of A, C, equimultiples K, L and of B, D, other, chance, equimultiples M, N”; whereas M, N are not any equimultiples whatever of B, D, but are any equimultiples of the particular multiples (G, H) which have been taken of B, D respectively, though these latter have been taken at random. Simson would, in the first place, add ἃ ἔτυχεν in the passages where any equimultiples E, F are taken of A, C and any equimultiples K, L are taken of E, F, because the words are “wholly necessary” and, in the second place, would leave them out where M, N are called ἄλλα, ἃ ἔτυχεν, ἰσάκις πολλαπλάσια of B, D, because it is not true that of B, D have been taken “any equimultiples whatever (ἃ ἔτυχε), M, N.” Simson adds: “And it is strange that neither Mr Briggs, who did right to leave out these words in one place of Prop. 13 of this book, nor Dr Gregory, who changed them into the word ‘some’ in three places, and left them out in a fourth of that same Prop. 13, did not also leave them out in this place of Prop. 4 and in the second of the two places where they occur in Prop. 17 of this book, in neither of which they can stand consistent with truth: And in none of all these places, even in those which they corrected in their Latin translation, have they cancelled the words ἃ ἔτυχε in the Greek text, as they ought to have done. The same words ἃ ἔτυχε are found in four places of Prop. 11 of this book, in the first and last of which they are necessary, but in the second and third, though they are true, they are quite superfluous; as they likewise are in the second of the two places in which they are found in the 12th prop. and in the like places of Prop. 22, 23 of this book; but are wanting in the last place of Prop. 23, as also in Prop. 25, Book XI.”

As will be seen, Simson’s emendations amount to alterations of the text so considerable as to suggest doubt whether we should be justified in making them in the absence of MS. authority. The phrase “equimultiples of A, C and other, chance, equimultiples of B, D” recurs so constantly as to suggest that it was for Euclid a quasi-stereotyped phrase, and that it is equally genuine wherever it occurs. Is it then absolutely necessary to insert $\ddot{\alpha} \text{ } \ἔ\tau\upsilon\chi\epsilon$ in places where it does not occur, and to leave it out in the places where Simson holds it to be wrong? I think the text can be defended as it stands. In the first place to say “take equimultiples of A, C” is a fair enough way of saying take any equimultiples whatever of A, C. The other difficulty is greater, but may, I think, be only due to the adoption of any whatever as the translation of $\ddot{\alpha} \text{ } \ἔ\tau\upsilon\chi\epsilon$. As a matter of fact, the words only mean chance equimultiples, equimultiples which are the result of random selection. Is it not justifiable to describe the product of two chance numbers, numbers selected at random, as being a “chance number,” since it is the result of two random selections? I think so, and I have translated $\ddot{\alpha} \text{ } \ἔ\tau\upsilon\chi\epsilon$ accordingly as implying, in the case in question, “other equimultiples whatever they may happen to be.”

To this proposition Theon added the following:

“Since then it was proved that, if K is in excess of M, L is also in excess of N, if it is equal, (the other is) equal, and if less, less, it is clear also that, if M is in excess of K, N is also in excess of L, if it is equal, (the other is) equal, and if less, less; and for this reason,

$$\text{as } G \text{ is to } E, \text{ so also is } H \text{ to } F.$$

From this it is manifest that, if four magnitudes be proportional, they will also be proportional inversely.”

Simson rightly pointed out that the demonstration of what Theon intended to prove, viz. that, if E, G, F, H be proportionals, they are proportional inversely, i.e. G is to E as H is to F, does not in the least depend upon this 4th proposition or the proof of it; for, when it is said that, “if K exceeds M, L also exceeds N etc.,” this is not proved from the fact that E, G, F, H are proportionals (which is the conclusion of Prop. 4), but from the fact that A, B, C, D are proportionals.

The proposition that, if A, B, C, D are proportionals, they are also proportionals inversely is not given by Euclid, but Simson supplies the proof in his Prop. B. The fact is really obvious at once from the 5th definition of Book V. (cf. p. 127 above), and Euclid probably omitted the proposition as unnecessary.

Simson added, in place of Theon’s corollary, the following:

“Likewise, if the first has the same ratio to the second which the third has to the fourth, then also any equimultiples whatever of the first and third have the same ratio to the second and fourth: And, in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and fourth.”

The proof, of course, follows exactly the method of Euclid’s proposition itself, with the only difference that, instead of one of the two pairs of equimultiples, the magnitudes themselves are taken. In other words, the conclusion that

$$ma \text{ is to } nb \text{ as } mc \text{ is to } nd$$

is equally true when either $m$ or $n$ is equal to unity.

As De Morgan says, Simson’s corollary is only necessary to those who will not admit $M$ into the list $M, 2M, 3M$ etc.; the exclusion is grammatical and nothing else. The same may be said of Simson’s Prop. A to the effect that, “If the first of four magnitudes has to the second the same ratio which the third has to the fourth : then, if the first be greater than the second, the third is also greater than the fourth ; and if equal, equal ; if less, less.” This is needless to those who believe once A to be a proper component of the list of multiples, in spite of multus signifying many.