Definition 58 · Theorems 284–297
Definition 58 (after Euclid X, Def. 1). Two bounds ξ and η are commensurable when there is a bound μ and numbers m, n with ξ = m • μ and η = n • μ. Such a μ is a common part of ξ and η. Bounds that have no common part are incommensurable.
Euclid says: "Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure." His "measure" is the common part. A bound is the thing itself.
Theorem 284 (Euclid X.1). Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.
Proof (Euclid's, of bounds): Let ξ > η. Some multiple n • η exceeds ξ (Theorem 257). From ξ subtract more than its half, and from n • η subtract η, which is not more than its half. What is left of ξ is less than what is left of n • η. Repeat. When one η is left of n • η, what is left of ξ is less than η.
This is the root of every later argument by exhaustion, and the first place where "as small as one pleases" is proved and not assumed.
Theorem 285 (Euclid X.2). If, when the less of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable.
Proof (Euclid's, of bounds): If ξ, η were commensurable, some μ would be a common part. As in Theorem 41, μ is a common part of every remainder. Each remainder is less than half of the one two before it, so some remainder is less than μ (Theorem 284). So μ would go a whole number of times into something less than itself: which is impossible.
Theorem 286 (Euclid X.3). Given two commensurable magnitudes, to find their greatest common measure.
Proof: As Theorem 42. The subtraction ends, since otherwise the magnitudes would be incommensurable (Theorem 285). The last remainder is a common part, and every common part is a part of it.
Theorem 287 (Euclid X.5). Commensurable magnitudes have to one another the ratio which a number has to a number.
If ξ = m • μ and η = n • μ, then ξ/η = m/n.
Theorem 288 (Euclid X.6). If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable.
If ξ/η = m/n, let μ be the m-th part of ξ (Theorem 247). Then ξ = m • μ and η = n • μ.
Theorem 289 (Euclid X.7). Incommensurable magnitudes have not to one another the ratio which a number has to a number.
Theorem 290 (Euclid X.8). If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.
These are Theorem 288 and Theorem 287 turned round.
In our terms. ξ and η are commensurable exactly when ξ/η is a rational bound (Definition 54), and incommensurable exactly when it is irrational. A bound is rational when it is commensurable with 1.
Theorem 291 (Euclid X.9). The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number will also have their sides commensurable in length. But the squares on straight lines incommensurable in length have not to one another the ratio which a square number has to a square number; and squares which have not to one another the ratio which a square number has to a square number will not have their sides commensurable in length either.
Read of bounds, with ξ • ξ for "the square on ξ":
Proof: If ξ/η = m/n, then (ξ • ξ)/(η • η) = (m • m)/(n • n). Conversely, if (ξ • ξ)/(η • η) = (m • m)/(n • n), then ξ/η and m/n have the same square, so they are the same (Theorem 255). The other two parts are these turned round.
Theorem 292 (after Theaetetus). Let p, q be numbers prime to one another. The side of p/q, that is, the bound whose square is p/q, is rational if and only if p and q are both square numbers.
Proof: If the side is a ratio, let it be a/b in least numbers. Then p/q = (a • a)/(b • b). The squares on a and b are prime to one another (Theorem 66), so they are the least in their ratio (Theorem 60), and so are p, q; so p is the square on a and q the square on b (Theorem 59). The converse is plain.
So the side of a number that is not square is never a ratio. Theorem 256 is the case of 2. This is the theorem Plato gives to Theaetetus (Theaetetus 147d to 148b).
Theorem 293 (Euclid X.11). If four magnitudes be proportional, and the first be commensurable with the second, the third will also be commensurable with the fourth; and, if the first be incommensurable with the second, the third will also be incommensurable with the fourth.
ξ/η = ζ/υ, and one is rational when the other is.
Theorem 294 (Euclid X.12). Magnitudes commensurable with the same magnitude are commensurable with one another also.
ξ/η = (ξ/ζ)/(η/ζ), and the quotient of rational bounds is rational (Theorems 248 to 251).
Theorem 295 (Euclid X.13). If two magnitudes be commensurable, and the one of them be incommensurable with any magnitude, the remaining one will also be incommensurable with the same.
Otherwise Theorem 294 would make the first two both commensurable with it.
Theorem 296 (Euclid X.15). If two commensurable magnitudes be added together, the whole will also be commensurable with each of them; and, if the whole be commensurable with one of them, the original magnitudes will also be commensurable.
If ξ = m • μ and η = n • μ, then ξ + η = (m + n) • μ. If ξ + η = p • μ and ξ = m • μ, then η = (p − m) • μ.
Theorem 297 (Euclid X.16). If two incommensurable magnitudes be added together, the whole will also be incommensurable with each of them; and, if the whole be incommensurable with one of them, the original magnitudes will also be incommensurable.
Theorem 296 turned round.