The curve that does not ask where it started

Release four beads from four different heights on the same frictionless curve and they reach the bottom at four different times. That is what a curve does. There is one shape for which it is not true.

The four times are $1.003205$ seconds each. They are not approximately equal and they are not equal for small releases only; the highest bead starts at the cusp of the arch, travels $5.7$ times as far as the lowest, and arrives with it.

What the condition actually is #

The property has nothing to do with the shape as a shape. It is a statement about two quantities: the distance travelled along the curve from the bottom, written $s$, and the height above the bottom, written $y$.

A bead released at arc length $s_0$ arrives with speed $v = \sqrt{2g(y_0 - y)}$ at every point, by conservation of energy alone. The time is

$$T = \int_0^{s_0} \frac{ds}{\sqrt{2g\,(y(s_0) - y(s))}}$$

and the question is what function $y(s)$ makes that integral independent of $s_0$.

Suppose $y = s^2/8a$ for some constant $a$. Then $y(s_0) - y(s) = (s_0^2 - s^2)/8a$, and

$$T = \int_0^{s_0}\frac{ds}{\sqrt{g(s_0^2 - s^2)/4a}} = 2\sqrt{\frac{a}{g}}\int_0^{s_0}\frac{ds}{\sqrt{s_0^2 - s^2}} = \pi\sqrt{\frac{a}{g}}.$$

The $s_0$ cancels. It cancels because the extra distance a higher release has to travel is exactly paid for by the extra speed the extra height buys, and “exactly” is arranged by the square.

There is a shorter way to say the same thing. Differentiate $y = s^2/8a$ and the component of gravity along the curve is $-mg\,dy/ds = -(mg/4a)\,s$: a restoring force proportional to the displacement along the curve. That is simple harmonic motion in the variable $s$, and a harmonic oscillator’s period does not know its amplitude. The isochronism is not a coincidence about a curve; it is harmonic motion, arranged geometrically instead of by taking a limit.

The curve that satisfies $y = s^2/8a$ turns out to be the path of a point on the rim of a rolling wheel of radius $a$. That fact is a consequence and the physics never uses it.

What the circle does instead #

The comparison worth making is with the curve a string and a bob give for nothing.

A bead on a circular arc of radius $R$ has $y = R(1 - \cos(s/R))$, which expands to $s^2/2R - s^4/24R^3 + \dots$. The first term is the harmonic one — comparing it with $s^2/8a$ shows a circle imitates a cycloid of radius $R/4$ near the bottom — and the second term is the whole of the problem.

The circle’s rise is second order in the amplitude, which is the reason a pendulum keeps passable time in spite of not being isochronous. At nine degrees the error is $0.15$ per cent; at ninety it is $18$ per cent. Halving the swing quarters the error, so a clock that keeps its amplitude constant is nearly as good as one that does not need to, and keeping the amplitude constant is what an escapement is for.

That is worth stating plainly, because it decides the history. The problem is real, and it is small, and the practical answer turned out to be to hold the amplitude still rather than to remove the dependence.

The mechanism, and why the evolute matters #

Constraining a bob to a cycloid is not obviously possible. A track would work and would rub. Huygens found a way to do it with a string.

The construction rests on a fact about the curve that is not obvious and is not general: the evolute of a cycloid is another cycloid of the same size. The evolute is the curve traced by the centres of curvature, which is also the curve a taut string unwinds from. So a string unwinding from cycloidal cheeks traces a cycloid, and no other curve does this to itself.

The figure draws the path point by point from the wrapping construction, and compares it with the cycloid it is supposed to be. The two agree to five parts in a thousand million million, which is the precision of the arithmetic rather than of the geometry.

The physical content is the shortening. A plain pendulum has a fixed length, so wide swings take longer; the cheeks shorten the effective length as the swing widens, by exactly the amount that cancels the effect. The correction is applied by the suspension rather than by the bob.

What the cheeks are cancelling is the circular pendulum’s amplitude dependence, and the shape of that dependence is what makes the problem interesting: the rise is slow, and then it is not slow at all. A quarter of a per cent at ten degrees is tolerable in a clock; four per cent at forty-five is not, and the curve between them is the whole reason a clockmaker cares about keeping the amplitude fixed.

The instructive failure #

Huygens’ clocks with cheeks kept worse time than his clocks without them, and the reason is a good example of a correct calculation being the wrong thing to build.

A string wrapping against a metal surface rubs, and the friction is at the suspension, where a plain pendulum has almost none. A real string has bending stiffness, so it does not lie flat on the cheek; it stands off, by an amount that depends on the tension and therefore on the amplitude, which introduces an amplitude dependence of its own with no reason to be small. And the cheeks must be made to a shape rather than to a length, so an error in their profile is an error at every amplitude at once.

A suspension that rubs is a suspension with a lower quality factor, and the amplitude then falls faster between impulses — which moves the pendulum along that same curve during every swing. So friction does not merely waste energy here; it converts a constant-amplitude clock into a varying-amplitude one, and the isochronism the cycloid was supposed to guarantee is lost to the thing the cycloid introduced.

The error removed is second order in the amplitude. The errors introduced are first order in the contact, and there was no reason to expect them to be smaller.

There is a further difficulty that is easy to miss and is arithmetic rather than engineering. The cheeks correct a pendulum swinging in a plane, and a real clock’s bob does not stay in a plane: a small sideways component turns the swing into a slow ellipse, and on an elliptical path the string touches one cheek differently from the other. The correction is exact for the motion it was designed for and is not even defined for the motion the clock actually has.

And the whole apparatus is aimed at an error that the marine problem does not have in the form Huygens supposed. A ship’s motion does not merely change the amplitude; it changes the direction of down, at a frequency close enough to the pendulum’s own that the two interact. A clock that is perfectly isochronous about the wrong vertical is not a clock. That is why the eventual answer at sea was a balance wheel and a spring — an oscillator with no gravity in it at all — rather than a better pendulum. What replaced the cheeks is the arrangement every pendulum clock has used since: a plain suspension, a small amplitude, and an escapement that returns exactly the energy lost so that the amplitude does not drift.

The mathematics was never at fault, and it is still exact. What the episode measures is the difference between removing a term from an equation and removing an error from an instrument.

Solving it backwards, which is where integral equations began #

The problem so far has been: here is a curve, what is the descent time. The interesting version is the other one — here is a descent time, what is the curve — and it was asked by Abel in 1823.

Write the shape as $s(y)$, the arc length reached at height $y$. Then the time from release at height $h$ is

$$T(h) = \int_0^{h} \frac{1}{\sqrt{2g(h-y)}}\,\frac{ds}{dy}\,dy,$$

which is a relation between two functions in which the unknown, $ds/dy$, sits inside an integral. That is an integral equation, and it was the first one anybody solved. Abel’s answer inverts it: for a prescribed $T(h)$ the required $ds/dy$ is recovered by a second integral of the same kind, so the map from curves to timing laws is invertible and each law has exactly one curve.

Feeding it $T(h) =$ constant returns $ds/dy \propto 1/\sqrt{y}$, hence $s \propto \sqrt{y}$, hence $y \propto s^2$ — and the cycloid. So the curve is not found by trying shapes and testing them; it is computed from the requirement, and the requirement determines it uniquely.

The same inversion answers questions nobody asks about clocks. Prescribe a period that rises with amplitude in a chosen way and there is a curve for it; prescribe one that falls and there is a curve for that too. A circular arc’s law is one particular member of that family, and there is nothing privileged about it except that a string produces it for free.

What the inversion needs is that the descent time depends only on the release height and not on anything else about the release, which is where the frictionless, single-degree-of-freedom idealisation enters. It is exact for a bead on a wire and false for a ball on a track.

The same curve, a different question #

The cycloid answers a second question that has nothing to do with equal times, and the coincidence is worth pressing on.

Johann Bernoulli’s problem of 1696 was: between two points, which path gets a bead from one to the other in the least time? Not the straight line, which is the shortest — the answer is the cycloid, which dives steeply at the start to buy speed before it needs distance.

The numbers here are $0.84972$ seconds for the cycloid against $1.08708$ for the straight line, with a circular arc and a parabola between them at $0.87452$ and $0.87551$. The margin over the good-but-wrong curves is small — three per cent — and the margin over the straight line is not.

Two quite different questions, one answer. The connection is that both are statements about the relation between arc length and height, and it is the same relation: making $y$ proportional to $s^2$ is what equalises the times, and it is also what balances the extra distance against the extra speed in the way a minimum requires.

This is one of the earliest problems solved by asking which path makes a quantity stationary rather than by writing down forces, and the method that grew out of it is what the least-action ladder is about. Bernoulli’s own solution used an optical analogy — a ray in a medium whose speed increases with depth bends exactly as the fastest path does — which is Fermat’s principle doing mechanics.

The same correction, in things that are not pendulums #

The amplitude dependence being removed here is not a fact about pendulums. It is what happens to any oscillator whose restoring force stops being proportional to the displacement, and the cycloid is the one case where the geometry can be chosen to prevent it.

The same correction turns up in everything with a minimum. Three quite different potentials agreeing near the bottom give the same frequency for small oscillations and different frequencies for large ones — so the amplitude dependence is not a peculiarity of pendulums but the generic behaviour, and the harmonic case is the exception that has none.

A diatomic molecule’s bond is a spring only near the bottom of its well; higher vibrational levels sit closer together, and the spacing between them is a direct measurement of the fourth-order term that the pendulum’s $s^4$ correction is the mechanical version of. A perfect spring’s heat capacity does not depend on its stiffness precisely because a perfect spring is harmonic; a real one’s does, because it is not.

The same term decides whether a mechanical resonator is any use as a frequency reference. Drive a slightly anharmonic oscillator hard and its resonant frequency shifts with the drive amplitude, which turns an amplitude fluctuation into a frequency fluctuation and puts a floor under how stable the reference can be. Every high-quality oscillator, from a quartz crystal to the trapped ion whose transition defines a second, is operated at an amplitude small enough that the fourth-order term is below the required stability — which is Huygens’ second solution, applied everywhere, rather than his first.

The challenge, and the lion’s claw #

The brachistochrone was not a problem anybody had; it was a challenge, issued to humiliate, and the way it was answered founded a branch of mathematics.

Johann Bernoulli published it in June 1696, addressed to “the sharpest mathematicians of the whole world”, with six months allowed. Leibniz asked for longer so that foreign mathematicians might have time, and the deadline was moved to Easter.

Newton received it in January 1697, at the end of a day at the Royal Mint, where he was then Warden and occupied with recoinage rather than with mechanics. He worked on it that evening, had it by four the next morning, and sent the solution to the Royal Society without his name on it. Bernoulli, reading the anonymous paper, is said to have remarked that he recognised the lion by his claw.

Five solutions appeared: Newton’s, Leibniz’s, l’Hôpital’s, and one from each of the two Bernoulli brothers. The two brothers’ are the interesting pair, and their difference is the point of telling the story.

Johann’s is beautiful and special. He noticed that a light ray in a medium whose speed increases with depth bends according to Snell’s law at every level, and that the path of least time for light is therefore the path of least time for the bead — so the problem could be transferred wholesale into optics and solved there. It is an elegant argument and it works because this particular problem happens to be about least time.

Jakob’s is laborious and general. He varied the curve directly: perturb it a little, work out what the change in the total time is, and require it to vanish. That method knows nothing about optics, and it applies to any quantity of the form “an integral along a path” — least time, least length, least action, anything.

It was Jakob’s that mattered. Euler took it up in the 1740s and turned it into a systematic procedure, and Lagrange completed it; the result is the calculus of variations, and the entire modern statement of mechanics is written in it. The elegant argument solved the problem and the clumsy one founded the subject.

The brothers, who had set and answered the challenge in part to score off one another, spent the following decade quarrelling about priority in public, which is the least useful part of the episode and the reason most people have heard of it.

The tooth that had to be a cycloid, and then did not #

Huygens’ cheeks were an exactly correct construction that lost to a cruder alternative because the crude one tolerated the errors a workshop makes. The same story played out over the same curve in a different piece of machinery, and the ending is the same.

A pair of gears has to turn at a constant ratio. That is not automatic: if the tooth profiles are wrong the ratio fluctuates within each mesh, and the resulting vibration is what a badly made gear train sounds like. The condition for constancy is a geometric one — the common normal at the point of contact must always pass through the same point on the line of centres — and it is a genuine constraint on the shape of a tooth.

Cycloids satisfy it. Make the part of the tooth above the pitch circle an epicycloid and the part below a hypocycloid, both generated by rolling circles, and the ratio is exactly constant. That construction was worked out in the seventeenth century, contemporaneously with the cheeks, and it is why clock and watch wheels have the tooth shapes they do to this day.

Euler proposed the alternative. Take a circle, wrap a string round it, and trace the path of the string’s end as it unwinds: that curve is the involute of the circle, and a tooth cut to it also satisfies the condition exactly. It is the same unwinding construction the cheeks use, applied to a circle rather than to a cycloid.

Both are exact, and the involute won everything except horology, for two reasons that have nothing to do with the geometry being right.

It tolerates a wrong centre distance. Move two cycloidal gears a fraction of a millimetre apart and the constant-ratio condition fails, because the rolling circles no longer match. Move two involute gears apart and the ratio is unchanged — the contact point slides along a different part of the profile and the normal still passes through the pitch point. Bearings wear, shafts deflect, and housings are made to a tolerance, so a gear that does not care where its neighbour is has an enormous advantage over one that does.

And it can be cut with a straight-edged tool. The involute’s conjugate profile is a straight-sided rack, so one cutter of one shape generates correct teeth for every tooth count, by rolling. Cycloidal teeth need a different cutter for every combination.

The pattern is exactly the cheeks’ pattern. An exact solution that assumes the machine is built precisely, against an equally exact solution that keeps working when it is not — and the second wins, in a workshop, every time.

What the pictures cannot show #

Every curve here is frictionless, and friction is not a small correction to a race. A bead on a real track loses energy at a rate that depends on the normal force, which is largest where the curve is most sharply bent — precisely the steep start the cycloid relies on. The fastest real descent is not a cycloid, and how far it departs depends on the coefficient rather than on the geometry.

The bead is a point. A rolling ball has some of its energy in rotation, which slows every path by the same factor only if it rolls without slipping everywhere, and the steep start is exactly where it does not.

Rotation takes something out of a descent, and the amount is fixed by the mass distribution. A rolling body arrives later than a sliding one by a factor that depends on how far its mass sits from its axis — the same factor at every point of the path only if the body’s shape does not change — which is why a cycloidal track designed for a sliding bead is not the fastest track for a rolling ball.

The tautochrone property is exact and the constraint is not. Any real cheek has a profile with a tolerance on it, and the isochronism degrades in proportion. What is exact is a statement about a curve; what is built is a statement about a machined surface.

And the whole comparison assumes gravity is uniform. It is the assumption that makes $y$ a potential proportional to height, and it holds to a part in ten million over the height of a clock case. Over the height of a mountain it does not, and the tautochrone for an inverse-square field is a different curve entirely.

Every statement in this essay about the circular arc’s failure is a statement about one curve: the fractional error in replacing $\sin\theta$ by $\theta$. That error is third order in the angle, so it is invisible for small swings and unavoidable for large ones, and the entire seventeenth-century project of cycloidal cheeks was an attempt to remove a third-order term by mechanical means.

The ladder from here #

Later rungs on this anchor: the escapement, which is the practical answer that replaced the cheeks and is a control problem rather than a geometry one; Kater’s reversible pendulum, which measures $g$ to five figures without anyone measuring a length; the spherical pendulum, where the swing is allowed a second dimension and the path stops closing; and the pendulum in a lift, where the direction the curve should be built around is no longer vertical.

The neighbouring ladders are the pendulum and its small lie, which is where the amplitude dependence is first met, the period that depends on the swing, which computes it exactly, and every minimum is a parabola, which is the statement this whole essay is a geometrical construction of.