Euclid proves it without moving anything

The rearrangement proof is the one everybody meets, and it is over in a glance: two squares of the same size, each holding four copies of one triangle, and the leftovers must therefore be equal. It is complete, it is honest, and it has one property that its admirers rarely mention — it works by moving pieces.

Euclid’s proof does not move anything, and this essay is about why that is a different kind of argument rather than a stylistic preference.

What is being proved, and what is being proved instead #

The theorem is the same. The claim being established on the way is different, and that is where the interest is.

The rearrangement proof establishes that $a^2 + b^2$ and $c^2$ are equal as totals — the same collection of pieces adds up to both. Euclid establishes something more specific: the square on the hypotenuse divides into exactly two rectangles, and each rectangle equals one of the leg squares individually.

That is a stronger statement, and it is the statement that gets used. The rectangle beneath a leg has area $c \cdot p$, where $p$ is the piece of the hypotenuse cut off by the altitude, so $a^2 = c \cdot p$ — which is the geometric mean relation, the source of the similar-triangle proof, and the thing that makes the altitude worth drawing in the first place.

The move that does the work #

Euclid’s tool is one fact about area, and everything rests on it: a triangle on a fixed base with its apex anywhere on a fixed line parallel to that base has a fixed area.

The base does not change, the height does not change, so the area does not. Sliding the apex along that parallel changes the shape entirely and changes nothing about the size. The same statement for parallelograms is Elements I.35, and it is usually called a shear.

That is the whole engine. A shear is the one deformation available that changes shape while preserving area exactly, and Euclid’s proof consists of choosing what to shear.

The reason a shear is available at all is that area, unlike shape, depends on only two numbers — a base and a height — and a shear leaves both alone. Everything else about the figure is free to change. That is a very cheap invariant, and cheap invariants are the ones that prove things, in the same way that reducing a city to a graph keeps exactly the property the question was about and discards everything else.

The argument, in three sights #

Consider the square on one leg, and the triangle formed by two of its corners and the far end of the hypotenuse.

The apex of that triangle lies on the line containing the square’s opposite side. Both are perpendicular to the shared leg, so they are parallel, so the triangle has the same base and the same height as the square. Its area is half the square’s.

Now the same triangle, turned.

Rotate a quarter-turn about the vertex the leg and the hypotenuse share. One of the triangle’s sides was a side of the leg-square and becomes a side of the hypotenuse-square; the other was the hypotenuse and becomes the other side of the big square. The angle between them is unchanged, being a right angle plus the triangle’s own angle at that corner in both positions. So the two triangles are congruent — the same triangle, looked at from a different side.

And the turned triangle sits in the big square with its base on one of its sides and its apex on the altitude’s line, which is parallel to that side. So its area is half the rectangle.

Half the square. Same triangle. Half the rectangle. Therefore square equals rectangle, and running the argument on the other leg finishes the theorem.

It is worth noticing what the proof never does. It never computes an area. Not one number appears anywhere in it: the square’s area is never called $a^2$, the rectangle’s is never called $c \cdot p$, and the equality is established between two regions neither of which has been measured. That is the characteristic move of Greek geometry, where quantities are compared rather than evaluated, and it is the reason the Elements can prove things about lengths that turn out not to be numbers at all.

A modern reader trained on coordinates will find this the strangest part, and it is the part that ages best. An argument that never evaluates cannot be broken by a quantity that resists evaluation.

Why it is harder to hold #

Two proofs, one theorem, and one of them is much easier to carry away. It is worth being precise about why, because the reason is not that Euclid’s is longer — it is four sights against the rearrangement’s two.

The rearrangement proof shows a conservation: pieces are moved and nothing is created or destroyed, which is a physical intuition every reader already has. Euclid’s shows an equality between things that never touch. The square and the rectangle are in different parts of the picture, they have different shapes, and no piece of one ever occupies the other. The bridge between them is a triangle that is drawn twice, in two places, and the reader has to hold both copies at once and believe they are the same.

That is a genuinely harder cognitive task, and it is the price of a genuinely stronger conclusion. Nothing is being smuggled: the rearrangement proof does not establish the rectangle-by-rectangle version, and Euclid’s does.

There is also an ancient reason for the difficulty. Euclid could not use the rearrangement proof in the form given today, because moving figures around raises a question his axioms were built to avoid — whether a shape is unchanged by being carried through space. Book I’s fourth common notion, “things which coincide with one another are equal”, is doing that work, and it was already regarded as the weakest of them. The shear proof needs no motion at all, only parallels, which is why it is the one in the Elements.

What it costs #

Nothing, computationally, and that is worth saying because the two proofs differ in what they hand to a calculation.

The rearrangement proof hands over an identity. Euclid’s hands over the two relations $a^2 = c\,p$ and $b^2 = c\,q$, with $p + q = c$, and those are the working equations of a great deal of elementary geometry: they give the altitude as $h = \sqrt{pq}$, they give the similar-triangle proof directly, and they are how the theorem generalises to the relation between a chord and the diameter through its foot.

The altitude relations are also what make the theorem recursive. Each of the two smaller triangles the altitude creates is similar to the original — same angles, smaller — so the same construction can be run inside either of them, and again inside the four that produces, forever. The rearrangement proof has no such structure: it is a single arrangement that either works or does not.

That is a general difference between the two styles and it decides which one generalises. An argument built from a relation can be iterated; an argument built from a tiling can only be exhibited. It is the reason the similar-triangle route reaches the ratio results and the trigonometric identities while the dissection route reaches a great many beautiful dissections and stops.

They also give the fastest hand computation of a square root that predates any algorithm — the geometric mean construction. To find $\sqrt{k}$, lay a segment of length $k$ next to one of length $1$, draw the semicircle on their sum as diameter, and erect a perpendicular at the join. Its height is $\sqrt{k}$, because the angle in a semicircle is a right angle and $h^2 = pq = k \cdot 1$. That construction is Euclid’s relation used backwards, and it is not available from the rearrangement.

Where it needs a condition #

The proof uses parallels three times — to place the apex of the first triangle, to place the apex of the turned one, and to know the altitude is parallel to the square’s sides. Parallels are exactly what the fifth postulate governs, so the proof depends on it and cannot survive without it.

That dependence is not an accident of the argument. The theorem itself fails on a curved surface: on a sphere, a triangle with two right angles at its base has $a^2 + b^2 \ne c^2$ by an amount that grows with area, and on a hyperbolic surface the inequality runs the other way. The right-angle relation is a statement about flatness, and every proof of it must therefore use flatness somewhere. In the rearrangement proof it hides in the assumption that four copies of a triangle and a square tile a square; here it is in the parallels, where it is easier to see.

There is one more condition and it is easy to miss: the altitude has to land inside the hypotenuse. It does, for a right-angled triangle, because the other two angles are acute — but the same construction on an obtuse triangle puts the foot outside the segment, the two rectangles stop being a partition, and what comes out instead is the law of cosines with a term that does not vanish.

The history, and the sacrificed ox #

Elements I.47 is the proof’s home, and Book I is arranged so that this is where it can first be stated: it needs the theory of parallels, the congruence criteria, and the parallelogram results, and it arrives immediately after all three. The converse follows as I.48, which is what makes the theorem usable for construction rather than only for measurement.

The proof is sometimes called the windmill, or the bride’s chair, after what the completed figure resembles. The names are medieval and the resemblance is faint.

Whether it is Euclid’s own is unknown, and the attribution to Pythagoras is worse than unknown — no text within five centuries of him credits him with it, and the relation was in use in Babylon a thousand years before he was born, on tablets listing triples far too large to have been found by trial. Plimpton 322 lists fifteen of them, including $(12709, 13500, 18541)$, which is not a number anybody stumbles into.

What is fair to say is that the proof is Greek and the relation is not, and the distinction is the point of this whole site: the relation is a fact that can be discovered by measuring, and a proof is a different kind of object, which is why a picture that establishes something and a picture that illustrates it are not the same picture.

What the picture cannot show #

The four sights are four still frames of an argument whose whole content is a correspondence — the triangle in position one and the triangle in position two are the same triangle. A figure cannot show sameness. It can show two triangles and rely on the reader to accept a claim of congruence, which the generator checks and the picture does not.

That is the exact opposite of the rearrangement proof’s situation, where the picture carries the whole argument and the reader needs nothing else. Here the picture is a scaffold and the words are load-bearing, which is an honest thing for a figure-first site to admit about one of its own figures.

The shear itself is also not drawn. What is drawn is the triangle before and after the turn; the sliding of an apex along a parallel — the step that makes the areas equal — happens between the frames, and a static figure has nowhere to put it. A reader who has not seen a shear animated has to supply it, and that is the largest gap in the essay.

The ladder from here #

Rungs on this anchor above this one: the converse, and rope-stretching as construction rather than measurement. The similar-triangle proof, which is the same two relations arrived at without any squares. Triples on a circle, and the rational parametrisation. The tree of triples, where every primitive triple appears exactly once. The law of cosines as the case where a term survives. De Gua’s theorem and the version in $n$ dimensions. The failure on curved surfaces. Distance as a definition rather than a theorem, and the $p$-norms. Incommensurability, which is the crisis this relation produced. And the exponent question, drawn as a surface.

Below it, and already written: the dissection.

Two proofs, two things proved #

The standing lesson is not that one proof is better. It is that two proofs of the same statement can establish different amounts, and the amount is not visible in the statement.

The rearrangement gives an equality of totals and gives it instantly. Euclid gives an equality of parts, at the cost of an argument that has to be held rather than seen, and the parts are what later mathematics actually consumes: the geometric mean, the altitude relation, the construction of square roots, the extension to chords of a circle. A reader who knows only the dissection knows the theorem. A reader who knows I.47 knows the theorem and the machinery.

That gap is the argument for climbing a ladder at all. Each rung is a distinct thing proved, not the same thing said again at a different volume, and the test of whether a rung is worth writing is whether it leaves something behind that the rung below it does not.

It is also a warning about proof-without-words as a genre, which this site is otherwise enthusiastic about. A picture that establishes a theorem instantly is a picture that has chosen the weakest sufficient statement, because that is the one a glance can hold. The stronger statement usually needs the slower argument. The gnomon picture shows that odd numbers sum to squares and says nothing about why; the harmonic bars show a total climbing and cannot show that it climbs forever. In each case the picture is honest and the picture is not the whole of it, and the honest thing for a figure-first collection to do is say which of its figures are proofs and which are scaffolding.

This one is scaffolding, and it is the better argument.