Theorems 257–283
The propositions of this section and of the next are given in Euclid's words, with "magnitude" read as "bound." Euclid proves them of magnitudes of any kind, by taking equimultiples. Here each is proved from what a bound is, something determined by ratios of numbers, and so each proof is proper to arithmetic.
Three of Euclid's definitions for magnitudes are needed (the fourth, fifth and seventh definitions of Elements V).
For bounds ξ and η, the ratio of ξ to η is the bound ξ/η of Theorem 247. The next three theorems show that Euclid's tests come to sameness and order of these quotients.
Theorem 257 (Euclid V, Def. 4, of bounds). For any bounds ξ, η there is a number n with n • η > ξ.
Proof: ξ/η is a bound, so it has an upper ratio X (Definition 43), and there is a number n greater than X (Theorem 208). Then n > ξ/η, so n • η > ξ (Theorem 239).
Theorem 258 (the Bridge). Let ξ, η be bounds and m, n numbers. Then m • ξ exceeds, equals, or falls short of n • η according as the ratio n/m is less than, the same as, or greater than ξ/η.
Proof: Multiply or divide by m and by η (Theorems 239 and 247).
Theorem 259 (Euclid V, Def. 5). ξ, η, ζ, υ are in the same ratio in Euclid's sense if and only if ξ/η = ζ/υ.
Proof: By the Bridge, Euclid's condition says: every ratio n/m is less than, the same as, or greater than ξ/η exactly as it is with respect to ζ/υ. If ξ/η = ζ/υ this holds. If they differ, say ξ/η < ζ/υ, there is a ratio n/m between them (Theorem 253), and for it the condition fails.
Theorem 260 (Euclid V, Def. 7). ξ has to η a greater ratio than ζ to υ in Euclid's sense if and only if ξ/η > ζ/υ.
Proof: Euclid's condition says: for some m, n, m • ξ > n • η and m • ζ ≦ n • υ. By the Bridge this is ξ/η > n/m ≧ ζ/υ. Conversely, if ξ/η > ζ/υ, take n/m between them (Theorem 253).
What this shows. Eudoxus fixes the ratio of ξ to η by which ratios of numbers fall below it and which above. Those are the lower and upper ratios of the bound ξ/η. So Euclid's fifth definition is the sorting of Definition 43, said of two magnitudes before their ratio is taken as one thing. The relative maximum and minimum of Definition 43 are Eudoxus's definition with the ratio itself named.
By Theorem 259, "ξ is to η as ζ to υ" is ξ/η = ζ/υ. When this holds, write λ for the common quotient, so that ξ = λ • η and ζ = λ • υ.
Theorem 261 (Euclid V.1). If there be any number of magnitudes whatever which are, respectively, equimultiples of any magnitudes equal in multitude, then, whatever multiple one of the magnitudes is of one, that multiple also will all be of all.
m • ξ + m • η = m • (ξ + η). Theorem 238.
Theorem 262 (Euclid V.2). If a first magnitude be the same multiple of a second that a third is of a fourth, and a fifth also be the same multiple of the second that a sixth is of the fourth, the sum of the first and fifth will also be the same multiple of the second that the sum of the third and sixth is of the fourth.
m • ξ + n • ξ = (m + n) • ξ. Theorems 236 and 238.
Theorem 263 (Euclid V.3). If a first magnitude be the same multiple of a second that a third is of a fourth, and if equimultiples be taken of the first and third, then also ex aequali the magnitudes taken will be equimultiples respectively, the one of the second and the other of the fourth.
m • (n • ξ) = (m • n) • ξ. Theorem 237.
Theorem 264 (Euclid V.4). If a first magnitude have to a second the same ratio as a third to a fourth, any equimultiples whatever of the first and third will also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.
(p • ξ)/(q • η) = (p/q) • (ξ/η) = (p/q) • (ζ/υ) = (p • ζ)/(q • υ).
Theorem 265 (Euclid V.5). If a magnitude be the same multiple of a magnitude that a part subtracted is of a part subtracted, the remainder will also be the same multiple of the remainder that the whole is of the whole.
For ξ > η: m • ξ − m • η = m • (ξ − η). Theorems 234 and 238.
Theorem 266 (Euclid V.6). If two magnitudes be equimultiples of two magnitudes, and any magnitudes subtracted from them be equimultiples of the same, the remainders also are either equal to the same or equimultiples of them.
For m > n: m • ξ − n • ξ = (m − n) • ξ. As Theorem 265.
Theorem 267 (Euclid V.7). Equal magnitudes have to the same the same ratio, as also has the same to equal magnitudes.
If ξ = η then ξ/ζ = η/ζ and ζ/ξ = ζ/η. Theorem 247.
Porism. If magnitudes be proportional, they will also be proportional inversely: if ξ/η = ζ/υ then η/ξ = υ/ζ.
Theorem 268 (Euclid V.8). Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.
If ξ > η then ξ/ζ > η/ζ and ζ/η > ζ/ξ. Theorem 239.
Theorem 269 (Euclid V.9). Magnitudes which have the same ratio to the same are equal to one another; and magnitudes to which the same has the same ratio are equal.
If ξ/ζ = η/ζ, multiply by ζ. If ζ/ξ = ζ/η, invert first.
Theorem 270 (Euclid V.10). Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.
The converse of Theorem 268, since the three cases exclude one another (Theorem 217).
Theorem 271 (Euclid V.11). Ratios which are the same with the same ratio are also the same with one another.
By Theorem 259 this is sameness of bounds.
Theorem 272 (Euclid V.12). If any number of magnitudes be proportional, as one of the antecedents is to one of the consequents, so will all the antecedents be to all the consequents.
ξ + ζ = λ • η + λ • υ = λ • (η + υ). Theorem 238.
Theorem 273 (Euclid V.13). If a first magnitude have to a second the same ratio as a third to a fourth, and the third have to the fourth a greater ratio than a fifth has to a sixth, the first will also have to the second a greater ratio than the fifth to the sixth.
By Theorems 259 and 260 this is: if α = β and β > γ then α > γ.
Theorem 274 (Euclid V.14). If a first magnitude have to a second the same ratio as a third has to a fourth, and the first be greater than the third, the second will also be greater than the fourth; and, if equal, equal; and if less, less.
ξ = λ • η and ζ = λ • υ. Then ξ is greater than, equal to, or less than ζ as η is with respect to υ (Theorem 239).
Theorem 275 (Euclid V.15). Parts have the same ratio as the same multiples of them taken in corresponding order.
(m • ξ)/(m • η) = ξ/η.
Theorem 276 (Euclid V.16). If four magnitudes be proportional, they will also be proportional alternately.
ξ/η = ζ/υ if and only if ξ • υ = η • ζ, if and only if ξ/ζ = η/υ.
Euclid needs all four magnitudes of one kind here (a line cannot have a ratio to an area). Bounds are all of one kind, so the condition always holds.
Theorem 277 (Euclid V.17). If magnitudes be proportional componendo, they will also be proportional separando.
For ξ > η: (ξ − η)/η = ξ/η − 1. So if ξ/η = ζ/υ then (ξ − η)/η = (ζ − υ)/υ.
Theorem 278 (Euclid V.18). If magnitudes be proportional separando, they will also be proportional componendo.
(ξ + η)/η = ξ/η + 1.
Theorem 279 (Euclid V.19). If, as a whole is to a whole, so is a part subtracted to a part subtracted, the remainder will also be to the remainder as whole to whole.
Let ξ/η = α/β with α < ξ and β < η. Then ξ − α = λ • η − λ • β = λ • (η − β).
Theorem 280 (Euclid V.20 and V.22). If there be any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, they will also be in the same ratio ex aequali.
If ξ/η = δ/ε and η/ζ = ε/φ, then ξ/ζ = (ξ/η) • (η/ζ) = (δ/ε) • (ε/φ) = δ/φ. Theorem 280 is the same for greater, equal and less, by Theorem 274.
Theorem 281 (Euclid V.21 and V.23). If there be three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, they will also be in the same ratio ex aequali.
If ξ/η = ε/φ and η/ζ = δ/ε, then ξ/ζ = (ε/φ) • (δ/ε) = δ/φ, by Theorem 236.
Theorem 282 (Euclid V.24). If a first magnitude have to a second the same ratio as a third has to a fourth, and also a fifth have to the second the same ratio as a sixth to the fourth, the first and fifth added together will have to the second the same ratio as the third and sixth have to the fourth.
ξ/η + ε/η = (ξ + ε)/η. Theorem 238.
Theorem 283 (Euclid V.25). If four magnitudes be proportional, the greatest and the least are greater than the other two.
Let ξ/η = ζ/υ with ξ the greatest and υ the least. Then λ > 1 and η > υ. And (ξ + υ) exceeds (η + ζ) by (λ − 1) • η − (λ − 1) • υ = (λ − 1) • (η − υ).