The vessel whose level falls evenly

The bottle that pours at one rate however full it is made water leave a vessel at a steady rate by moving the point at which the pressure is fixed: a sealed bottle with a tube through its stopper holds its outlet at the head of the tube’s tip whatever the level, so the flow does not slow as the bottle empties. That is a way to make the flow uniform. A water clock wants something slightly different, and older: a level that falls uniformly, so that a scale on the vessel’s wall reads the time. The two are not the same, because a level falling uniformly is a statement about the vessel’s shape as well as about its outflow, and the oldest solution to the problem is a statement about shape alone.

Why a cylinder will not do #

The pressure that only knows depth set out the rule: the pressure at the floor of a vessel of still water is the weight of the column above it, per unit area, whatever the vessel’s shape. A hole in the floor converts that pressure into speed. Water leaves at about $\sqrt{2gh}$, the speed it would have reached falling freely through the depth $h$ — Evangelista Torricelli’s result of 1643 — multiplied by a coefficient of about 0.61 that accounts for the jet narrowing just after it leaves a sharp-edged hole. The outflow is therefore proportional to the square root of the depth. Halve the depth and the flow falls by about thirty per cent.

In a cylinder the surface area is the same at every height, so the level falls at a rate proportional to the outflow, which is to say to the square root of the depth. That rate falls as the vessel empties, and the level traces a parabola in time: fast at the start, slow at the end. The first figure has the numbers. A cylinder holding twenty-five litres to a depth of thirty-five centimetres, draining through a hole eight-tenths of a millimetre across, takes sixteen hours to empty, and its level has fallen halfway in under a third of that. A cylinder makes a clock whose hours are long at the bottom of the vessel and short at the top.

That clock can still be read, if its marks are placed where the level actually is at each hour rather than spaced evenly. The second figure draws them. On the cylinder they crowd towards the floor, the first hour’s fall six times the second-last’s, and the last few hours can hardly be told apart. A clock-maker who wanted the marks evenly spaced, so that one ruler served, had to change the vessel.

The quartic bowl #

The level falls at a rate equal to the outflow divided by the area of the water’s surface. To keep that ratio constant while the outflow grows as $\sqrt h$, the area must grow as $\sqrt h$ too, and since the area of a circle grows as the square of its radius, the radius must grow as the fourth root of the height:

$$\frac{dh}{dt} = -\frac{C_d\,a\sqrt{2gh}}{\pi r(h)^2} = \text{constant} \quad\Longrightarrow\quad r \propto h^{1/4}.$$

The vessel this describes is a bowl whose wall is a quartic curve, $h \propto r^4$. Near the floor it is very narrow and flares quickly; near the rim its wall is almost vertical, because the fourth root changes slowly at large heights. In the second figure its hour marks are evenly spaced from top to bottom, and in the first figure its level falls in a straight line. The calculation behind the figures integrates the level from the outflow law step by step, never from the formula; the bowl’s level falls at one rate to within a per cent until the last centimetre, where the hole’s finite size begins to matter.

The same problem in another form is one that the curve that does not ask where it started solved for a pendulum: find a shape that makes a time independent of something it would otherwise depend on. Christiaan Huygens found the cycloid, along which a sliding bead reaches the bottom in the same time from any starting height. The quartic bowl is the water clock’s cycloid. It makes the level’s rate independent of the depth, by the same strategy of letting the shape do the compensating that a mechanism would otherwise have to do.

One rule for every outflow law #

The quartic bowl is one case of a rule that covers every outlet. Suppose the outflow grows as some power of the depth, $Q \propto h^p$. A level that falls evenly needs a surface area proportional to the outflow at every depth, so the area must grow as $h^p$ and the radius as $h^{p/2}$. Torricelli’s hole has $p = \tfrac12$ and wants a radius growing as the fourth root of the height. A long, fine tube, in which viscosity sets the flow, has $p = 1$ and wants a radius growing as the square root. And an outlet whose flow does not depend on the depth at all, $p = 0$, wants a radius that does not change: a plain cylinder.

That last case is not hypothetical. The hourglass that keeps time found that sand leaves an opening at a rate independent of how much sand is above it, because the walls of the glass carry the weight and the grains at the opening feel almost none of it. For sand the outflow exponent is zero, and so an hourglass needs no shaping at all: any bulb, cylindrical or pear-shaped, empties its sand at a steady rate, and the glass-blower can give it whatever form is pleasing. A water clock of the same shape would be useless. The difference between the two oldest kinds of timer is the exponent in their outflow laws, and the shape a vessel must have is that exponent written in glass or stone.

The rule also says why the even-falling bowl empties sooner than a cylinder of the same volume and height. The bowl keeps most of its water near the top, where the head is greatest and the flow fastest, and has little near the floor, where a cylinder spends its slow last hours. In the first figure the bowl is empty in 12.9 hours against the cylinder’s 16.0. An even scale and a short run come together here, from the same choice of shape.

What the Egyptian clock-makers did #

The oldest surviving water clock was found at Karnak and carries the name of Amenhotep III, about 1400 BC. It is an outflow clock of alabaster: water ran out through a small hole near its base, and the time was read from the level against marks on its inner wall. Its sides are not vertical. The vessel is a truncated cone, wider at the top, with walls sloping outward at an angle of the order of seventy degrees from the horizontal, and the slope is there to make the level fall more evenly than it would in a cylinder.

A cone is the shape that can be turned on a wheel or cut with straight tools, and it is a fair compromise. The third figure compares the rate the level falls at each depth. In a cylinder it changes by a factor of three or more over the vessel’s height. In a cone matched to the quartic bowl at its rim and near its floor it varies by about a quarter, fastest about a third of the way down, where the cone is narrower than the bowl would be. The fourth figure turns those rates into clock errors.

With evenly spaced marks the cylinder runs more than three hours fast in the middle of a sixteen-hour run. The cone is never more than about forty minutes out, and the quartic bowl, which the even scale suits, keeps within five minutes, the residue coming from the last centimetre, where the outflow law is no longer exact. The sloping sides of the Karnak vessel buy most of the improvement a perfect shape would.

The Karnak clock was not marked with one even scale. Its inner wall carries twelve columns of marks, one for each month, because the Egyptian hour was a twelfth of the night, and nights are longer in winter than in summer. A clock that measured unequal hours needed a different scale every month, and the marks in each column are spaced to suit the vessel’s own draining as well as the season. The ancient clock-maker’s problem was therefore harder than the one the figures solve: not a uniform hour, but a scale that turned a falling level into hours that changed length through the year.

A clock that keeps the outflow steady instead #

The other way to an even scale changes the outflow rather than the vessel. If water is fed into a receiving vessel from a supply held at a constant head, it arrives at a constant rate, and the level in a cylindrical receiver rises evenly. Ctesibius of Alexandria built clocks on this principle in the third century BC: a supply tank was kept brim-full by an overflow, so its head never changed, and the water it delivered raised a float in the receiver, which drove a pointer or a figure along a scale. The principle is the same one the bottle that pours at one rate used, arrived at two thousand years earlier: pin the head and the flow follows.

The two strategies divide the work differently. The outflow clock with a shaped vessel needs no moving parts and no supply, only a vessel cut to the right curve. The inflow clock needs a steady supply and an overflow, but its receiver can be any shape that is easy to make, and its float can turn gears. Medieval and Islamic clock-makers developed the second into astronomical clocks with automata; the shaped outflow vessel never needed more than a potter.

When viscosity drains the vessel #

The quartic bowl rests on Torricelli’s law, which describes water accelerating through a hole whose walls play no part. A long, narrow outlet is different. If the water must pass along a fine tube, the friction of its walls dominates, the flow follows the law the fourth power in a pipe derived for viscous flow, and the flow is proportional to the pressure — to the depth itself — rather than to its square root. The even-falling vessel then needs its area proportional to the depth, so its radius grows as the square root of the height: a much gentler flare near the floor than the quartic bowl’s.

Most outlets lie between, paying a little of each loss, and the last figure computes the ideal shape for three of them. The shape depends on which loss dominates, and that is decided by the outlet’s Reynolds number, the ratio of the water’s inertia to its viscosity in the jet. A hole eight-tenths of a millimetre across in a thin floor runs at a Reynolds number of more than a thousand, viscosity takes five per cent of the head, and the quartic bowl is right. A tube a third of a millimetre wide and three centimetres long gives most of the head to viscosity, and the ideal vessel is close to the square-root shape. But such a tube passes water so slowly that the twenty-five-litre vessel would take nearly two weeks to drain. A clock that marks hours in a vessel of a useful size works in the inertial regime, where the quartic bowl is the shape, and that is what the Karnak potter’s cone approximates.

Reading the level #

A clock is only as good as the reading of its scale, and a water level is harder to read than it looks. Water climbs a few millimetres up a clean wall, by the capillary rise that how high water will climb computed, so the level against a mark is a curved meniscus rather than a line, and in a narrow part of a vessel the curve can be a sizeable fraction of an hour’s fall. The quartic bowl is narrowest exactly where its marks are closest to the outlet, which is one practical reason a real clock is never run to the last centimetre. Ancient outflow clocks typically marked the hours by small drilled holes or notches on the inner wall, read from above as the falling water uncovered them, which turns the problem into telling whether a hole is wet — easier by lamplight than judging a line.

How far the argument reaches #

The figures treat the discharge coefficient as fixed at 0.61, the value for a sharp-edged hole at moderate speeds. It changes as the head falls and the jet through the hole slows, by tens of per cent at the lowest heads, which bends the last part of every curve. They treat the water as pure and the hole as clean; a real clock’s hole silts and furs with lime, which slows it over months, and the ancient clocks are thought to have been recalibrated against the stars. Temperature changes water’s viscosity by about two per cent per degree, which matters only in the viscous regime; in the inertial regime a clock is nearly indifferent to it. And evaporation from the open surface, a few millimetres a day in a warm room, is negligible over one night’s run but not over a week’s.

What the figures cannot show #

The figures show vessels as perfect surfaces of revolution, and a real clock’s interior is not. Its hole is not exactly round, its floor not exactly flat, and the small departures from the ideal shape near the outlet, where the vessel is narrowest, matter most. They cannot show the scales the ancient clocks actually carried, or how they were made. Whether the marks on the Karnak vessel were computed, copied from an earlier clock, or set by filling and timing against the stars is not known with certainty. Nor can the figures show how the clock was read: at night, by lamplight, by a priest looking down into a dark vessel at a dark waterline against carved notches, which is a source of error larger than any the figures compute.

Still open: how the ancient scales were made #

Surviving outflow clocks from Egypt and from the Greek and Roman worlds differ in their slopes, and their scales differ in how well they match their vessels. Some appear to have been calibrated empirically, by filling them and marking the level against an independent time, and others seem to follow a rule. How much the clock-makers understood about why sloping sides helped, whether any of them knew a curved wall would do better, and how the twelve monthly scales of the Karnak clock were laid out, are questions argued from measurements of the few vessels that survive, of which most are fragments. Experimental reconstructions, filled with water and timed, have checked how well particular vessels keep time. They have not settled how the knowledge was arrived at.

The habit worth carrying away is to ask which of two quantities a design must hold fixed. Water leaves a hole in a vessel’s floor at a rate proportional to the square root of the depth, so a cylinder’s level falls fast at first — sixteen hours to empty, half its depth gone in a third of the time, an evenly marked scale 202 minutes fast — while a bowl whose radius grows as the fourth root of the height lowers its level at one rate, 27.0 mm/h at every depth; a straight-sided cone, the shape of the oldest surviving water clock, cuts the cylinder’s error to 41 minutes, and if viscosity rather than inertia governs the outlet the right shape becomes r ∝ h^(1/2). A clock can keep time by holding its flow steady or by shaping its vessel to the flow it has, and the oldest one did the second.