Incommensurable

A proof and a sonnet are two forms that share no common measure. What follows sets one down inside the other so that each stays wholly itself — fourteen lines that scan and rhyme, and a proof that holds step for step — and then gives an honest accounting of every place the two pull against each other. The seam is the subject.

I · The sonnet

1. Suppose the root of two could be confined
2. to one exact proportion, a to b,
3. with every common measure left behind,
4. the two coprime — of shared divisors, free.
5. Then square: so a-squared must be two b-squared,
6. which makes that square an even count, and more—
7. for odd times odd stays odd, as we declared;
8. thus a is even, two times c, therefore
9. four c-squared equals two b-squared; divide,
10. and two c-squared is b-squared, clean and whole;
11. so b, like a, is even — so implied,
12. an even root beneath an even goal.
13. But a and b were stripped of common ground—
14. yet both are even. So no such root is found.

II · The proof it makes

The claim is one of the oldest in mathematics: no fraction, however large its terms, has a square equal to 2. The sonnet argues it by contradiction. Stripped to symbols, the same argument runs:

1. Suppose √2 is rational. Then √2 = a/b for positive integers a, b, and we may take the fraction in lowest terms, so that gcd(a, b) = 1. (lines 1–4)
2. Square both sides: 2 = a²/b², hence a² = 2b². (line 5)
3. So a² is even. The square of an odd number is odd, so a cannot be odd; therefore a is even. Write a = 2c. (lines 6–8)
4. Substitute: (2c)² = 2b², that is 4c² = 2b²; divide by two to get b² = 2c². (lines 9–10)
5. So b² is even, and by the same step b is even. (lines 11–12)
6. But then 2 divides both a and b, so gcd(a, b) ≥ 2 — contradicting gcd(a, b) = 1. (lines 13–14)
7. The assumption is impossible. Therefore √2 is irrational. ∎

III · Line by line

A reading of the poem against the proof — what each line is required to carry, and where you can watch the form press on it.

Line	What it carries	Where the form shows
1	Assume √2 is rational.	confined stands in for expressible as a finite ratio — a poet’s verb doing a logician’s work.
2	√2 = a/b.	Clean: exact proportion is just ratio of two whole numbers.
3	Reduce to lowest terms.	common measure is the ancient name for a common divisor; it does double duty, naming the method and the theme.
4	gcd(a, b) = 1.	A near-restatement of line 3, kept because the rhyme wants a fourth line. The proof needs the fact once; the quatrain needs it twice.
5	a² = 2b².	Then square: opens on two stresses, not an iamb — the most audible metrical seam, and it falls exactly where the math is densest.
6	a² is even.	and more— carries no logic at all. It is there to reach the rhyme and hand the sense across the line break.
7	The parity lemma: an odd square is odd.	The line states the lemma’s ground but not its application; as we declared claims an authority the verse has not fully earned.
8	a is even; a = 2c.	therefore is stranded at the line’s end for the rhyme; its force only completes in line 9.
9	4c² = 2b²; divide by two.	divide is both the instruction — halve both sides — and the hinge of the turn to come.
10	b² = 2c².	clean and whole is ornament; whole winks at whole number without adding a step.
11	b is even.	so implied invokes line 7 again rather than re-proving it — an economy the formal column has to repay.
12	(restates b even).	The least load-bearing line in the poem: pure restatement, to fill the quatrain. root and goal are figures for b and b².
13	Recall gcd(a, b) = 1.	common ground reprises common measure; the dash enacts the volta.
14	Both even ⇒ contradiction ⇒ no such fraction exists.	root turns twice at once: the square root, and the root of a fraction that cannot be.

IV · The seams

Two lines do no logical work. Line 6’s and more— is filler that exists to reach a rhyme; line 12 is pure restatement. They are here because a Shakespearean sonnet wants three quatrains and a couplet, and the proof’s skeleton has fewer than twelve joints — so the form has slack, and slack fills with ornament. I have left them rather than disguise them. The apparatus is the point.

The real strain is the parity lemma: an even square has an even root. The poem leans on it twice — lines 7 and 11 — and proves it neither time. That is not a flaw to be hidden but a division of labour. The verse can name the lemma; only the prose beside it can discharge it: an odd number is 2k + 1, and (2k + 1)² = 2(2k² + 2k) + 1 is odd, so an even square cannot come from an odd root. Where the sonnet says as we declared and so implied, read it as a pointer to §II. The two columns are one argument; neither is complete alone.

Metrically, line 5 is the rough one. Then square: is a spondee where the meter wants an iamb, and a-squared / b-squared drag stress onto syllables an iamb would hurry past. I could smooth it. I have not — because the place where the mathematics is densest is exactly the place the meter breaks, and that correspondence is worth more than a clean foot.

What the poem does not do is lie. No line asserts anything false about the numbers. Every shortfall here is a shortfall of completeness or economy — a step compressed, a line padded — never a false step. That is the one line the form is not allowed to cross: it may strain, but it may not deceive.

V · Why the translation is possible at all

A proof by contradiction has a shape: assume, build, collide, withdraw. A Shakespearean sonnet has a shape too: three quatrains that build, a couplet that turns. Here the couplet is the collision — the reductio’s contradiction and the sonnet’s volta are the same event, arriving on the dash in line 13. The two forms can be made to coincide because both are machines for setting something up in order to overturn it. They are not as incommensurable as they look.

And the title is not decoration. The irrationality of √2 is the discovery that the diagonal of a square has no common measure with its side: the first magnitudes known to be incommensurable — sharing, literally, no common measure. The result is ancient. It is the territory of Euclid’s Elements, Book X; and Aristotle, in the Prior Analytics, alludes to a reductio of just this shape — one that proves the diagonal incommensurable by forcing odd numbers to come out equal to even. The parity collision in the couplet above is, in spirit, that same odd-and-even contradiction, some twenty-three centuries on, wearing a rhyme.

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