A curve that divides any angle

Every instrument in this sequence has been measured by the numbers it can reach. Compass and straightedge reach the numbers built from the rationals by square roots. A marked ruler and a conic drawn at will reach cube roots as well, and so trisect angles; a sliding mark reaches at least one quintic. Every one of those reaches is a set of algebraic numbers — roots of polynomials with whole-number coefficients — because each operation solves a polynomial equation.

The oldest curve drawn to solve a construction problem goes further, and does it by abandoning equations altogether. It is traced by motion, and the motion has time in it, and time does not care about polynomials. Hippias of Elis described it around 420 BC to divide an angle into three; Dinostratus, a century later, used it to square the circle, which is why it is called the quadratrix.

Two motions and their crossing #

Take a square with side 1 and a corner $O$ at the origin. A radius pivots on $O$, starting upright and turning at a steady rate until it lies along the base. At the same time a horizontal line starts along the top of the square and falls at a steady rate until it too lies along the base. Both motions begin together and end together.

At any instant the radius and the line cross at one point, and the point traces a curve. Because both motions are uniform and simultaneous, when the line has fallen to height $y$, the radius has turned to angle $y \times 90°$. So the curve’s point at height $y$ lies on the ray at angle $\tfrac{\pi}{2} y$, and its coordinates are

$$\left( y \cot\!\left(\tfrac{\pi y}{2}\right),\ y \right).$$

The figure checks, at each of the drawn instants, that the crossing point lies on the radius it should. The curve starts at the top corner and bends down to meet the base — and where it meets the base is the interesting part.

The curve is not a circle, not a conic and not the graph of any polynomial. Its equation involves a cotangent, and the cotangent is where the circle enters: the curve records, for each height, the horizontal distance at which a ray of the matching angle reaches that height. It is a table of the tangent function drawn as a shape, and a table of the tangent function is exactly what a compass cannot draw.

Heights become angles #

The property that makes the quadratrix an instrument is that it converts one kind of quantity into another. Heights are lengths, and lengths can be divided in any ratio with compass and straightedge: to cut a segment into $n$ equal parts, draw any other ray from one end, mark off $n$ equal steps along it with the compass, and join and draw parallels. Angles cannot be divided like that — dividing an angle by three is impossible with those tools. But on the quadratrix, angle is proportional to height, so dividing the height divides the angle.

The procedure is short. Draw the angle’s arm from $O$; it meets the curve at some point $P$, whose height $h$ is proportional to the angle. Divide $h$ into three equal parts. From each division point draw a horizontal line to the curve. The rays from $O$ through those points of the curve divide the angle into three equal parts, because their heights are $h/3$ and $2h/3$ and angles on the curve are proportional to heights.

Nothing in that procedure depends on three. Any ratio a segment can be cut in, the angle can be cut in.

So the quadratrix divides every angle into any number of equal parts. A marked ruler trisects; a conic trisects; the quadratrix quinquisects, septisects and undecisects with equal ease. It does not solve one more kind of equation. It sidesteps equations.

The one step the compass contributes #

The quadratrix does the conversion between heights and angles; the compass and straightedge do the dividing, and it is worth seeing that they can. To cut a segment $AB$ into $n$ equal parts, draw any second ray from $A$, and step the compass along it $n$ times, from $A$ to points $C_1, C_2, \dots, C_n$ at equal spacing. Join $C_n$ to $B$ and draw parallels to $C_n B$ through $C_1, \dots, C_{n-1}$. They cut $AB$ at equal intervals, by similar triangles. Drawing a parallel is itself a compass-and-straightedge step, so the whole division is classical.

That is the division of a segment, and it works for any $n$ and indeed for any ratio of two constructible lengths. The corresponding division of an angle does not exist — there is no analogue of drawing parallels that cuts an arc into equal parts — and the obstruction is algebraic: cutting an angle into three needs the root of a cubic, into five the root of a quintic, and so on. The quadratrix’s contribution is to transport the problem from angles, where division is hard, to lengths, where it is easy, and back.

One angle, trisected three ways #

It is worth putting the quadratrix beside the two trisections that came before it, because they reach the same answer for completely different reasons.

Archimedes’ marked ruler trisects by sliding a segment of fixed length until its ends lie on a line and a circle; the position it stops at solves the cubic $4c^3 - 3c = \cos\theta$ for $c = \cos(\theta/3)$. A conic trisects by meeting a circle at the same root of the same cubic. Both are finite operations that happen to be able to solve a cubic, and both fail at the quintic that dividing an angle into five requires.

The quadratrix trisects without solving the cubic at all. It never computes $\cos(\theta/3)$ from $\cos\theta$; it reads the angle as a height, divides the height, and reads the result back as an angle. The cubic that made trisection hard is simply not on its route. That is why it does five, seven and eleven as easily as three — none of those equations is on its route either — and why its reach cannot be described by degrees. A marked ruler is a machine for solving one family of equations; the quadratrix is a machine for measuring angles, and a machine that measures angles has no reason to find one division harder than another.

Squaring the circle from the foot #

The curve meets the base of the square at a definite point. As the height $y$ goes to zero, $y \cot(\pi y/2)$ tends to $2/\pi$, because $\tan(\pi y/2) \approx \pi y/2$ for small $y$. So the foot of the quadratrix is at distance $2/\pi$ from the corner, about $0.6366$ of the side.

That single length is enough to square the circle — to construct a square with the same area as a given circle — which compass and straightedge cannot do, because it needs the length $\sqrt\pi$ and $\pi$ is not algebraic.

The theorem Dinostratus proved, in the terms of his time, is that the quarter-arc of the circle is to the side of the square as the side is to the foot of the quadratrix: $\tfrac{\pi}{2} : 1 = 1 : \tfrac{2}{\pi}$. Given the foot, the fourth proportional of $\tfrac{2}{\pi}$, $1$ and $1$ is a compass-and-straightedge construction, and it produces a segment of length $\tfrac{\pi}{2}$. Doubling it gives $\pi$. A rectangle of sides $\pi$ and $1$ has the area of a circle of radius 1, and a rectangle can be turned into a square of the same area by taking the geometric mean of its sides — another classical construction. So the circle is squared, exactly, once the foot of the quadratrix is given.

Sporus’s objection #

The last four words carry the whole difficulty, and they were pointed out in antiquity. Pappus of Alexandria, writing around AD 320, records the objections of Sporus of Nicaea to the quadratrix, and they are exact.

The first objection is that the construction presupposes what it is meant to find. To make the radius and the line finish together, the ratio of their speeds must be set in advance — the radius sweeps a quarter-arc of length $\pi/2$ in the time the line falls a distance 1 — and setting that ratio already requires knowing $\pi$.

The second objection is sharper. The foot of the curve, the one point needed for squaring the circle, is the one point the motions do not produce. At the final instant the line and the radius both lie along the base, and they coincide rather than cross: every point of the base is on both. The curve’s points are defined only at positive heights, and they approach $2/\pi$ without reaching it. The figure measures the approach: the gap is $(\pi/6)\,y^2$ to high accuracy, and it is positive at every height drawn, down to a millionth of the side, where it is about $5 \times 10^{-13}$. The foot is a limit, and constructing a limit is not what the classical rules allow.

Sporus’s point is not pedantry. It is the precise place where the quadratrix leaves the world of finite constructions. Every operation of the compass, the straightedge, the marked ruler and the conic finishes after finitely many steps and produces an exact point. Even the marked ruler, which slides until a condition is met, stops at a definite position that the condition determines, and that position solves a polynomial equation. The quadratrix produces its useful point only at the end of a continuous process, as a limit — and it is exactly the step to a limit that takes it beyond the algebraic numbers.

What each instrument reaches #

The regular polygons make the comparison concrete.

Gauss’s list comes from the degree of $\cos(2\pi/n)$: a polygon is constructible with compass and straightedge exactly when that degree is a power of 2, which happens when $n$ is a power of 2 times distinct Fermat primes. A trisector adds degrees divisible by 3, and so the primes of the form $2^a 3^b + 1$ — Pierpont primes — and 7, 13 and 19 join the list. The 11-gon and the 23-gon remain out of reach, because 11 and 23 are not Pierpont primes; the eleven-sided polygon needed the sliding mark’s quintic.

The quadratrix draws every regular polygon, because a regular $n$-gon needs only the division of a full turn into $n$ equal angles, and the quadratrix divides any angle into any number of parts. It does not reach one more algebraic number at a time. It reaches all the angles at once, and with them $\pi$, which is transcendental — the root of no polynomial with whole-number coefficients, as Lindemann proved in 1882. No instrument that works by solving polynomial equations can produce it; an instrument that works by uniform motion can, at the price of a limit.

Other curves that do the same #

The quadratrix is not the only such curve. The Archimedean spiral, traced by a point moving outward along a ray at a steady rate while the ray turns at a steady rate, has the same property in polar form: distance from the centre is proportional to angle, so dividing a distance divides an angle, and Archimedes used it to rectify the circle — to construct a straight segment equal to its circumference. The cycloid and the curves of Nicomedes and Diocles belong to the same tradition of curves generated by motion and pressed into service for problems the straightedge and compass could not solve.

What they share is that each couples two motions whose ratio encodes $\pi$ or an angle, and each yields its result either through a point the motions do produce, as in dividing an angle, or through a limit, as in squaring the circle. The Greek classification that grew out of Pappus’s discussion — planar problems for the straightedge and compass, solid problems for conics, and linear problems for these curves — is, read with modern eyes, the distinction between quadratic, cubic and transcendental numbers.

What the figures take for granted #

The curve is drawn, not constructed. Every figure computes the quadratrix from its formula, $x = y\cot(\pi y/2)$, which uses $\pi$ — exactly the knowledge Sporus said the construction needed. The figures show what the curve does once it exists; how it could be produced, with what precision and by what apparatus, is the objection, not the drawing.

The division checks are numerical. Each division of an angle is checked by computing the angle of the ray through the curve and comparing it with the intended fraction, to nine decimal places. The exact statement — angle proportional to height — is the definition of the curve, and the check confirms that the formula implements it.

The foot is drawn as a point, and it is not one of the curve’s points. In the first and third figures the foot of the quadratrix is marked on the base at $2/\pi$ as though the curve reached it. The curve as defined by the motions does not; the mark is the limit, placed where the formula sends it, which is the step Sporus refused.

The polygon table is a classification, applied. The rows for compass and straightedge and for the trisector are computed from the arithmetic criteria — Fermat primes and Pierpont primes — rather than by constructing each polygon; the row for the quadratrix is the observation that it divides every angle.

Still open: which curves are instruments #

The classical question the quadratrix raises — what can be constructed with a given curve as an additional tool — has sharp answers for conics and for the marked ruler at the cubic level, and much less beyond. For transcendental curves the question changes character: the quadratrix reaches every angle and $\pi$, but which other transcendental numbers it reaches in finitely many steps, combined with compass and straightedge, is not characterised. It reaches $\pi$, and the cosines and sines of every rational multiple of it; beyond those, no invariant of the kind that degree provides for algebraic numbers is available to say what else it reaches or cannot.

There is a modern reformulation of Sporus’s objection too. Computable analysis asks which real numbers can be produced to any desired precision by a finite procedure — and $2/\pi$, like $\pi$, is computable, so the limit the quadratrix needs can be approached as closely as wanted by a finite process. The classical rules demanded an exact point in finitely many steps; the computable ones demand arbitrary precision in finitely many steps. Between those two standards, which is the right model of “construction”, is a question about definitions rather than about curves, and it is still argued.

Leaving the equations behind #

Every earlier instrument reached further than the one before by solving a harder equation — quadratic, cubic, quintic. The quadratrix reaches further by solving none. It links a length to an angle through time, and time is uniform, and the link it makes is exactly proportional. That one proportionality divides every angle and, through a limit that its critics were right to point at, squares the circle. The ancient geometers who rejected it as a construction were not being fussy; they had found the line between the finite operations whose reach is algebraic and the continuous ones whose reach is not, two thousand years before anyone could say what an algebraic number was.

That line is also the organising principle of everything that came before in this sequence. Each instrument was characterised by the degrees of the equations it solves — two for the compass, three for the conics and the trisector, five somewhere in the reach of the sliding mark — and the degree was always the obstruction. The quadratrix has no degree. It is the instrument for which the question “which numbers can it reach?” can no longer be answered by counting, and that is why, for all its power, it has no theory to match the tidy ones below it.