Commentary on Proposition V.24

Back to Proposition V.24.

[Folio 0190 | p. 184]

Algebraically, if

$$a : c = d : f\text{,}$$

and

$$b : c = e : f\text{,}$$

then

$$(a + b) : c = (d + e) : f.$$

This proposition is of the same character as those which precede the propositions relating to compounded ratios; but it could not be placed earlier than it is because v. 22 is used in the proof of it.

Inverting the second proportion to

$$c : b = f : e\text{,}$$

it follows, by v. 22, that

$$a : b = d : e\text{,}$$

whence, by v. 18,

$$(a + b) : b = (d + e) : e\text{,}$$

and from this and the second of the two given proportions we obtain, by a fresh application of v. 22,

$$(a + b) : c = (d + e) : f.$$

The first use of v. 22 is important as showing that the opposite process to compounding ratios, or what we should now call division of one ratio by another, does not require any new and separate propositions.

Aristotle tacitly uses v. 24 in combination with v. 11 and v. 16, $Meteorologica$ III. 5, 376 a 22—26.

Simson adds two corollaries, one of which (Cor. 2) notes the extension to any number of magnitudes.

“The proposition holds true of two ranks of magnitudes whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude; as is manifest.”

Simson’s Cor. 1 states the corresponding proposition to the above with separando taking the place of componendo, viz., that corresponding to the algebraical form

$$(a - b) : c = (d - e) : f.$$

“Cor. 1. If the same hypothesis be made as in the proposition, the excess of the first and fifth shall be to the second as the excess of the third and sixth to the fourth. The demonstration of this is the same with that of the proposition if division be used instead of composition.” That is, we use v. 17 instead of v. 18, and conclude that

$$(a - b) : b = (d - e) : e.$$