Pinned between two sequences

The rings make the disc’s area obvious and leave a gap: the stack is a staircase at every finite count and a triangle at none of them, so the argument is complete only once a limit has been defined. That definition arrived about two thousand years after the picture.

What Archimedes had instead was a method that proves the same result with no limit anywhere in it, and the way it works is worth understanding because it is a different kind of argument — one that never produces the answer and only refutes the alternatives.

Refuting instead of computing #

The dissection says: here is the area, look. Exhaustion says: suppose the area were anything other than this, and watch the supposition fail.

To show a disc has the area of a triangle with base the circumference and height the radius, Archimedes argues twice.

Suppose the disc were larger than the triangle. Then the excess is some definite amount. Inscribe a square, then an octagon, then a sixteen-sided polygon, each time doubling; at every doubling the polygon takes up more than half of what the disc still has over it, so after enough steps the remaining excess is smaller than the amount supposed. But every inscribed polygon has area less than the triangle’s — its perimeter is shorter than the circumference and its apothem is less than the radius — so the disc, which is now within the supposed excess of an inscribed polygon, is less than the triangle plus nothing. Contradiction.

Suppose the disc were smaller. The same argument with circumscribed polygons, closing from the other side.

Neither is more than one, so it is equal.

Nothing in that argument exhibits the area. A modern reader notices immediately that the two halves are a limit in disguise — and the disguise matters, because the argument is valid without any theory of limits, and no such theory existed.

The step nobody skips #

The claim that at every doubling the polygon takes up more than half of what is left is the whole engine, and it deserves its own sentence because it is where the argument’s rigour lives.

Doubling the side count adds a triangle on every existing side, with its apex on the circle. That triangle is more than half of the circular segment it sits inside — the segment is contained in the rectangle whose base is the chord and whose height is the segment’s height, and the triangle is exactly half that rectangle. So each doubling removes more than half the gap.

A quantity that loses more than half at each step passes below any bound, which is a statement about whole numbers rather than about limits: after $k$ steps the gap is under $2^{-k}$ times what it started at, and a bound is a number, so some $k$ works. Euclid states this as a proposition in Book X and Archimedes cites it.

That is the whole of what a modern limit replaces. The limit is more convenient and buys enormously more; the exhaustion argument needs nothing beyond the Archimedean property of magnitudes, and it is what makes the result a theorem in a setting with no real numbers in it.

The recursion, which is where the number came from #

The method proves the area equals the triangle’s. Computing $\pi$ is the second use, and it is a recursion with two means in it and no trigonometry anywhere.

Write $a_n$ for the circumscribed perimeter divided by the diameter and $b_n$ for the inscribed one. Then

$$a_{2n} = \frac{2 a_n b_n}{a_n + b_n}, \qquad b_{2n} = \sqrt{a_{2n}\, b_n}.$$

A harmonic mean and a geometric mean. Starting from the hexagon, where $b_6 = 3$ exactly and $a_6 = 2\sqrt3$, six doublings reach the 384-gon and the figure’s table.

The figures compute the recursion and then check it, by comparing each row against $n\sin(\pi/n)$ and $n\tan(\pi/n)$. That is the point of doing it this way rather than evaluating the closed forms: the recursion is what Archimedes had, the trigonometric expressions are the modern gloss, and the agreement is a measurement of the gloss rather than a use of it.

At ninety-six sides the bracket is $3.14103 < \pi < 3.14271$, and Archimedes states $3\tfrac{10}{71} < \pi < 3\tfrac17$ — his fractions are a little outside the true bracket, because he replaced each square root by a rational bound in the safe direction at every step. Every approximation he made was made in the direction that keeps the inequality true, which is the discipline the method requires and the reason the final statement is a theorem rather than an estimate. It is also the discipline that interval arithmetic formalises today: carry a lower and an upper bound through every operation, round each outward, and whatever comes out is guaranteed to contain the answer. Archimedes did it by hand on six square roots.

Why the hexagon, and not the square #

Starting from the hexagon looks like a choice and is not much of one, and the reason is worth a paragraph because it is the only place in the computation where an exact value is available for free.

A regular hexagon inscribed in a circle has side exactly equal to the radius — the six triangles it splits into are equilateral. So its perimeter is exactly six radii, three diameters, and $b_6 = 3$ with no square root at all. The circumscribed hexagon needs one: its side is $2r/\sqrt3$, so $a_6 = 2\sqrt3$.

Starting from the square gives $b_4 = 2\sqrt2$ and $a_4 = 4$, which is also clean, and either works. What the hexagon buys is that it is closer to the circle to begin with — its bracket is $3$ to $3.464$ against the square’s $2.828$ to $4$ — so every doubling from it is worth a little more. Six doublings from the hexagon reach the same accuracy as about seven from the square.

The deeper reason the hexagon is the natural start is that three is the one exact perimeter available, and in a computation where every subsequent step introduces a square root that has to be bounded by hand, beginning with an exact number matters.

What else the same method settles #

Exhaustion is not a technique for circles. Archimedes used it for nearly everything he proved about curved figures, and three of the results are worth naming because they show the range.

The parabolic segment. The area cut off by a chord is four-thirds of the inscribed triangle, proved by filling the gaps with smaller triangles whose total is a geometric series of ratio a quarter. That is the argument the dissected square is about, done without a limit — Archimedes proves the total cannot be more and cannot be less.

The sphere. Its surface is four times its great circle and its volume two-thirds of the circumscribing cylinder’s, which Archimedes regarded as his best result and asked to have carved on his tomb. The proof brackets the sphere between inscribed and circumscribed solids of revolution built from polygons, which is the circle argument turned about an axis.

The spiral. The area swept in one turn of the curve $r = a\theta$ is a third of the circle it ends on, by bracketing with sectors. That one has no dissection at all and is pure exhaustion.

The common shape is worth stating: in each case a curved region is trapped between two families of straight-sided ones, each family is computable, and the two close. What varies is the ingenuity of the trapping, which is where all the difficulty is — the logical apparatus is the same three lines every time.

How good it is, and how good that was #

The 96-gon gives two correct decimal places. That is worth setting against what it cost: six doublings, each needing a square root extracted by hand on numbers of several digits, in a numeral system without positional notation.

The convergence is quadratic in the sense that matters. The gap $a_n - b_n$ falls by a factor of about four at each doubling, so each step buys a little over half a decimal digit. Reaching ten digits this way needs about thirty-three doublings and a polygon of some fifty thousand million sides, which is why nobody did.

Ludolph van Ceulen spent much of his life on exactly this recursion and reached thirty-five places by 1610, with a polygon of $2^{62}$ sides. The number was carved on his tombstone. Within a century the method was obsolete: the arctangent series and their relatives converge geometrically — at a fixed proportion a term rather than at a rate tied to a side count — and Machin reached a hundred places in 1706 with far less arithmetic.

So the exhaustion method was the best available for nineteen centuries and was superseded by something easier as soon as infinite series became usable. That ordering — the rigorous method first, the convenient one much later — is the reverse of the usual pattern, and it is the same ordering the rings themselves have: the picture is old and the licence to believe it is new.

The same bracket elsewhere #

Pinning a quantity between two sequences is a manoeuvre rather than a fact about circles, and it turns up in several places here.

A needle dropped on a floor is the opposite extreme in the same collection — a method for $\pi$ with no bracket at all, whose error shrinks like one over the square root of the number of throws and which after a thousand needles is lucky to have one decimal place right.

Two-sided brackets prove existence; one-sided bounds prove absence. Oresme’s blocks bound from below and the conclusion is divergence; Archimedes brackets from both and the conclusion is a value. The difference is not rigour but what is available: a quantity known to be above every bound needs no upper one.

The construction that is nearest is rectangles under a curve, where left-endpoint and right-endpoint sums bracket the integral for a monotone function and the gap is the last rectangle. That is the same trap in one dimension, and the modern definition of an integral takes it as a definition: the integral is the common value when the two closings agree, which is Archimedes’ conclusion promoted to a definition.

The two quantities the bracket is about #

There is an ambiguity in the phrase the circle is between the two polygons that Archimedes is careful about and a modern reader usually is not, and it matters because the two readings prove different things.

Perimeters. The inscribed polygon’s perimeter is less than the circumference and the circumscribed one’s is greater. The first is a fact about a convex curve containing a polygon — a shorter path between the same points; the second needs the circumscribed polygon to be convex and to contain the circle, and Archimedes states it as a postulate rather than proving it, because it is a statement about arc length and he has no definition of arc length to prove it from.

Areas. The inscribed polygon’s area is less and the circumscribed one’s greater, and both of those are immediate from containment.

The table on this page is about perimeters, which is the one the $\pi$ computation needs. The area version gives a bracket too — $\tfrac{n}{2}\sin(2\pi/n)$ below and $n\tan(\pi/n)$ above — and it converges at the same rate. The two brackets are different sequences closing on different numbers, and that they close on numbers related by a factor of the radius is the theorem the dissection draws rather than an obvious fact.

That an assumption about arc length has to be postulated is worth carrying. A length along a curve is a harder object than an area inside one, and the postulate Archimedes writes down — that a convex curve enclosing another is longer — is exactly the thing a definition of arc length has to deliver and that no dissection supplies.

A proof by contradiction has no picture #

Both polygons are drawn at one side count. The argument is about all of them, and the table reports six. That every doubling tightens the bracket is checked at each drawn row and argued in prose for the rest — the argument being the more-than-half claim, which no drawing at a fixed count contains.

The contradiction is not drawn. Archimedes’ proof supposes the area is greater, derives an absurdity, supposes it is less, derives another. Neither supposition is a picture; what is drawn is the arrangement the suppositions are made about, which is the whole of what a figure can do for a proof by contradiction.

And the rational bounds are not drawn either. The table gives the exact bracket at each row, computed in floating point and checked. Archimedes’ own numbers involve replacing each square root by a fraction, and the discipline of always rounding in the safe direction is the part of his computation that carries the rigour — and it is arithmetic bookkeeping with no geometry to show.

Still open: the same slicing, one dimension up #

The rings settle a disc. Slicing a solid the same way settles a sphere, and the step that makes it work is a comparison rather than a dissection: at every height a hemisphere’s cross-section has the area of the matching slice of a cylinder with a cone removed. That principle — equal slices, equal volume — is Cavalieri’s, it is where the sphere’s two formulas come from, and it gives a false answer the moment the slices are taken in the wrong direction. That is the slice that has to match.

What the method could not do, and why #

Exhaustion proves equalities and it does not discover them. That limitation is structural and is the reason the method, for all its rigour, did not become a calculus.

Every argument above begins suppose the area is not this particular value. The value has to be known in advance, from somewhere else, and Archimedes’ Method — a treatise lost until 1906, when Heiberg found it under a prayer book — says where he got them. He weighed the figures. Slicing a solid into strips and balancing them on a lever against strips of a known solid gives the answer immediately and is, as he says himself, no proof at all; the exhaustion argument is then written to establish what the balancing suggested.

So the working practice was exactly the pairing this page is about: a mechanical or dissective argument to find the answer, and an exhaustion argument to prove it. Both were his and he regarded only the second as mathematics.

What a limit adds is that the same apparatus finds and proves at once. A Riemann sum is a construction whose value is the answer, not a test of a value proposed beforehand — which is why the same slicing with a limit written down is a method and exhaustion is a verification.

That difference is the reason the interval was two thousand years. The missing piece was never cleverness; it was a definition that let a sequence’s limit be the object under discussion rather than a candidate for it.

Proving and showing #

The habit is to keep two things separate that are easily run together.

The dissection shows and does not prove; the exhaustion proves and does not show. A modern account of the disc’s area does both with one argument, because the limit supplies the rigour the picture lacks and the picture supplies the reason the limit converges — and it is easy to forget that the combination took two thousand years to become available.

The test worth applying to any argument by picture is whether a two-sided bracket exists for it. If the picture’s claim can be trapped between two computable sequences, the picture is an illustration of a theorem. If only one side is available, the picture is evidence for something weaker than it appears to be saying — and if neither is, it is not an argument at all.