The spiral that measures its own circle

A curve that divides any angle found an instrument in a curve: Hippias’s quadratrix, drawn by two uniform motions, turns heights into angles and so divides any angle in any ratio, and its foot sits at $2/\pi$ of the side, a length that squares the circle. It mentioned, in passing, that the quadratrix was not the only such curve. The other famous one is older in its full treatment and more surprising in how it works.

Archimedes wrote On Spirals around 225 BC, crediting the idea to his friend Conon of Samos. The curve is traced by a point that moves steadily outward along a ray while the ray turns steadily about its end. After one full turn the point has travelled out a fixed distance. The distance from the centre is then proportional to the angle turned:

$$r = a\theta.$$

That alone makes it an angle divider, exactly as the quadratrix is. But the spiral’s more remarkable property is carried not by its points but by its tangent, and it gives a way of turning the curved length of a circle into a straight one.

In the figure, the spiral completes one turn and ends at $P$, at distance $1$ from the centre $O$. The tangent to the spiral at $P$ is drawn, and it crosses the line through $O$ perpendicular to $OP$ at a point $T$. The length $OT$ comes out at $6.2832$: $2\pi$ times $OP$, the circumference of the circle of radius $OP$. The spiral has laid the circle’s circumference out along a straight line.

Distance and angle are one quantity #

The curve is easiest to understand by watching it being drawn.

At an eighth of the turn the point is an eighth of the way out; at three eighths, three eighths of the way; at the end of the turn, all the way out to the circle. Each point’s distance from the centre and its angle round the centre are the same number in two units. The figure checks, at every drawn instant, that the distance equals the fraction of the turn completed.

This is the property the quadratrix had, with a different pairing: there, height was proportional to angle; here, distance from the centre is. Nothing about the curve needs a coordinate system: it is defined entirely by two rates, one of turning and one of moving out, and every property below follows from their fixed ratio. The spiral’s version is more natural in one respect. The quadratrix is drawn inside a square and covers only angles up to a right angle. The spiral covers a whole turn, and continued, as many turns as desired.

Dividing an angle by dividing a length #

Given the spiral, dividing any angle in any ratio is a matter of dividing a length.

Draw the angle with one arm along the spiral’s starting direction. Its other arm meets the spiral at a point $P$, and the distance $OP$ is proportional to the angle. Divide $OP$ into three equal parts — the straightedge buys nothing and its neighbours show that compass and straightedge divide a length into any number of equal parts. Draw circles about $O$ through the division points. Each circle meets the spiral at a point whose distance from $O$ is one or two thirds of $OP$, and so whose angle is one or two thirds of the given angle. The rays to those points trisect the angle.

The same works for any ratio a length can be cut in, and for angles beyond a right angle, which the quadratrix cannot reach — the figure trisects an angle of $240°$. The trisection problem that no compass and straightedge can solve in general is here reduced to a step they can do, with the spiral supplying the translation between length and angle.

The tangent that unrolls the circle #

The spiral’s own proof of its rectifying property uses one fact about tangents to curves given by an angle-to-distance rule.

For a curve described by its distance $r$ from $O$ as a function of the angle $\theta$, the tangent at a point meets the line through $O$ perpendicular to the radius at a distance

$$OT = \frac{r^2}{\,dr/d\theta\,}$$

from $O$ — the polar subtangent. For the spiral, $r = a\theta$ and $dr/d\theta = a$, so

$$OT = \frac{(a\theta)^2}{a} = a\theta \cdot \theta = r\theta.$$

And $r\theta$ is the length of the arc of the circle of radius $r$ through the angle $\theta$. At the end of one full turn, $\theta = 2\pi$, and $OT = 2\pi r$ — the whole circumference.

The relation holds at every point, not only at the end of a turn. After half a turn the point is at distance $\tfrac12$, and the tangent cuts off a length $\pi \cdot \tfrac12 = 1.5708$: the half-circumference of the circle through it. The figures compute the tangent from the curve’s direction at the point, intersect it with the perpendicular, and check that the length is the arc to within a billionth.

Archimedes proved the full-turn case as Propositions 18 to 20 of On Spirals, without derivatives, by an argument that approaches the tangent from both sides and shows that any other line through $P$ either cuts the spiral or misses the required length. His proofs of the preliminary propositions use verging constructions — the sliding-mark step of the mark that changes what is reachable — which is one reason the spiral belongs in the same family of instruments as the marked ruler.

Why the tangent knows the arc #

The formula hides a picture that makes the result almost obvious. Follow the point for a moment longer, as the ray turns through a tiny angle $d\theta$. The point moves outward by $a\,d\theta$, because the outward speed is fixed. It also moves sideways, round the centre, by $r\,d\theta$, because a point at distance $r$ turning through $d\theta$ sweeps an arc of that length. Its motion is the combination of the two, and the tangent points along that combination.

So the tangent makes a small right triangle with the radius: its side along the radius is $a\,d\theta$ and its side across the radius is $r\,d\theta$. The large triangle $OPT$ — right-angled at $O$, with one side along the radius and one across it — is similar to the small one, because the tangent is the hypotenuse of both. Similar triangles have proportional sides:

$$\frac{OT}{OP} = \frac{r\,d\theta}{a\,d\theta} = \frac{r}{a} = \theta.$$

So $OT = \theta \cdot OP = r\theta$, the arc of radius $r$ through angle $\theta$. The tangent’s slope records the ratio of sideways to outward motion, and for this curve that ratio is the angle itself. At the end of a full turn the angle is $2\pi$, and the tangent triangle is $2\pi$ times as tall as it is wide.

Squaring the circle from the tangent #

Once a straight segment equal to the circumference is in hand, the circle can be squared with compass and straightedge alone.

The area of a circle of radius $r$ is half its circumference times its radius — the observation a circle unrolled made by cutting the disc into rings and laying them out as a triangle of height $r$ and base $2\pi r$. So a rectangle with sides $OT/2 = \pi r$ and $r$ has the circle’s area. Converting a rectangle into a square of equal area is a classical compass-and-straightedge step: the square’s side is the geometric mean of the rectangle’s sides, found with a semicircle.

The spiral therefore squares the circle, and it does so more directly than the quadratrix, which produced $2/\pi$ and needed a proportion to invert it. What the compass could not supply was a straight length equal to a curved one. The spiral supplies it through its tangent, and the rest is Euclid.

A third of the circle, by exhaustion #

Archimedes found the area inside the first turn too, and the argument is a model of the method later called exhaustion.

Cut the full turn into $n$ equal angles. In each, the spiral’s distance from $O$ runs from $a\theta_k$ to $a\theta_{k+1}$, so the region inside the spiral in that angle contains a circular sector of the smaller radius and is contained in one of the larger. The sectors’ areas are $\tfrac12 r^2\,\Delta\theta$, and because the radii grow linearly, their sums are sums of squares: $\sum k^2$, which Archimedes knew in closed form. The inscribed and circumscribed totals differ by exactly one sector of the full circle, $\pi/n$ when the circle has radius $1$, so the gap shrinks to nothing, and the only value both can approach is

$$\frac{1}{3}\,\pi r^2.$$

The first turn encloses exactly a third of the circle through its end. The computation is the sum $\tfrac12\int r^2\,d\theta = \tfrac12 a^2 \int_0^{2\pi} \theta^2\,d\theta$ in modern notation — the polar area formula of an area measured by walking round it — and Archimedes carried it out eighteen centuries before that notation, with sums of squares and a double bound.

The catch: the tangent is not a compass step #

Every construction above depends on having the spiral drawn. That is the same objection Sporus raised against the quadratrix: to draw the curve, one must already coordinate a turning motion with an outward one at the exact right rates, which presupposes the very ratio between straight and curved that the curve is used to find.

The spiral adds a second objection, which the quadratrix does not face. The rectification needs the tangent at $P$. Drawing a tangent to a circle is a compass-and-straightedge step; drawing a tangent to the spiral is not. There is no finite sequence of circles and lines that produces it from the spiral’s points alone, because the tangent’s direction depends on the rate at which the curve is turning outward — a derivative — and a derivative is a limit. Ancient geometers could characterise the tangent, as Archimedes did, but not construct it from the curve by the permitted operations.

Pappus of Alexandria, writing about 340 AD, sorted problems by the instruments they need: plane problems, solved with lines and circles; solid problems, which need conic sections; and linear problems, which need other curves, among them the quadratrix and the spiral. He held that a problem should be solved with the simplest instrument that can do it, and regarded solving a plane or solid problem with a spiral as a fault of method. By his classification the rectification of the circle is irreducibly linear, and later mathematics — the cube that will not double and Lindemann’s proof that $\pi$ is transcendental — showed that he was right: no conic, and no finite algebraic construction, reaches $\pi$.

The same objection, twice #

The quadratrix was criticised on two counts. Sporus complained that drawing it presupposes the ratio it is meant to find, since the two motions must finish together; and that its foot, the point on the base that carries $2/\pi$, is not actually traced by the motions at all — at the final instant the radius and the falling line coincide, and the crossing that defines the foot is a limit, not a crossing.

The spiral answers the first complaint no better: it too needs a turning motion and an outward motion in a fixed ratio. And its answer to the second is to move the difficulty rather than remove it. The spiral’s end point $P$ is traced honestly, but the length that matters, $OT$, is carried by the tangent, and a tangent is a limit of secants. Both curves deliver $\pi$ only through a limit — the quadratrix at its foot, the spiral at its tangent — and that is no coincidence. $\pi$ is transcendental, so no finite sequence of algebraic operations reaches it; any construction that does must contain an infinite process somewhere, and the only question is where it is hidden.

The comparison is instructive about what a construction is. The price of a construction counted the steps of compass-and-straightedge constructions and asked for the fewest; a construction with a transcendental curve has a step that no count can price, because it is not a finite operation at all.

Other spirals, other lengths #

The spiral whose distance grows in proportion to its angle is not the only spiral with a useful tangent. In the equiangular or logarithmic spiral, $r = e^{b\theta}$, the distance grows in proportion to itself, and the tangent makes the same angle with the radius at every point — the property that shows up in shells and in the growth patterns of plants, and the curve that the rectangle that eats itself approximates by quarter-circles when a golden rectangle is cut into squares over and over. Where Archimedes’ spiral adds a fixed distance for each turn, the equiangular spiral multiplies by a fixed factor, so it looks the same at every scale.

Evangelista Torricelli found in 1645 that this spiral, although it winds round its centre infinitely many times as it approaches it, has a finite length from any point in to the centre: the length is the distance from the centre divided by the cosine of the constant angle. That was one of the first rectifications of a curve in the seventeenth century, when finding a curve’s length was believed by many to be impossible in general. The two spirals divide the labour between them: Archimedes’ spiral rectifies circles, and the equiangular spiral is rectified by a formula in its own terms.

What the figures can and cannot show #

The curves are computed, not drawn by motion. Each point of the spiral is placed by its formula; the figures check the proportionality of distance and angle at every drawn instant, and check each claimed division and each claimed length against the angle or arc it is about.

The tangent is computed from the derivative. The figures find the tangent’s direction from the rate of change of the curve’s position, which is exactly the step no compass supplies. They verify Archimedes’ relation; they do not construct the tangent by the ancient means, because there is none.

The area is bracketed at one fineness. Twelve sectors pin the area between $0.92$ and $1.18$; the claim that $\pi/3$ is the only number between every such pair is the argument, and the figure shows one stage of it.

Still open: what a curve can reach #

Adding a curve to the instruments enlarges the set of reachable lengths, and for algebraic curves there is a measure of how much: two instruments with one reach found that conics reach exactly the numbers whose degree is built from twos and threes. For transcendental curves like the spiral and the quadratrix, no such measure exists. The spiral reaches $\pi$, and every length obtained from $\pi$ and the constructible numbers by the field operations; it reaches the division of every angle in every rational ratio, and so the sines and cosines of every rational multiple of any given angle.

What else it reaches is not characterised. Whether, for instance, the spiral combined with compass and straightedge can construct $e$, or $\sqrt[3]{2}$, or the side of a regular heptagon from a given unit — the second two are algebraic, and reachable by conics, but not obviously by this curve, since the spiral turns lengths into angles and back without solving cubic equations — is not settled by any general theorem. The theory of constructions with transcendental instruments has no analogue of the degree that settles the algebraic case, and each such question has to be answered, if at all, by an argument of its own.

A curve that carries a ratio #

The habit worth keeping is to ask what an instrument encodes.

A compass encodes equal distances; a straightedge encodes collinearity; a marked ruler encodes a fixed length that can slide. The spiral encodes the ratio between a straight motion and a turning one, and every one of its powers — dividing angles, rectifying the circle, squaring it, enclosing a third of it — is that ratio read off in a different way. The spiral does not compute $\pi$; it contains it, built in by the rule that draws it, and each construction merely points to where it already is.