The search for a steady beat

The first oscillator people used was the turning Earth. One turn brings the sun back to the same place in the sky, and that is a day. Within the day, the shadow of an upright stick, called a gnomon, swings round as the sun crosses the sky. It is shortest at midday, when the sun is highest, and longer towards morning and evening. Mark where the shadow falls and you have a sundial.

For most of history the hours stretched with the seasons. Daylight was split into twelve equal parts, whatever its length, so a summer hour was longer than a winter one. These are called seasonal hours. In Damascus, a daytime seasonal hour lasts about 72 of our minutes at midsummer and about 50 at midwinter.

Five times a day, Muslims pray at times set by the sun. The midday prayer begins once the sun has passed its highest point. In most schools of law, the afternoon prayer begins when a stick's shadow has grown beyond its midday length by the length of the stick itself; the Ḥanafī school waits until it has grown by twice that. The sunset prayer follows sunset, while nightfall and the first light of dawn fix the other two.

Working these moments out in advance, for every day of the year, grew into a science of its own, ʿilm al-mīqāt. A mosque timekeeper, the muwaqqit, is first recorded in thirteenth-century Egypt. His job was to work out the times of prayer, using sundials, astrolabes and tables of the sun's motion.

Seasonal hours made a poor unit of measurement, since an hour in June was not the same length as an hour in December. An equal hour, one twenty-fourth of a whole day and night, stays the same all year, and a sundial can show it if its gnomon is set parallel to the Earth's axis. The sky seems to turn around that axis at fifteen degrees an hour, so a gnomon lying along it sees the sun circle it steadily, at that same rate, in every season.

In 1371–72 Ibn al-Shāṭir, head muwaqqit of the Umayyad Mosque in Damascus, built a marble sundial about two metres by one for one of its minarets. Its main gnomon points at the celestial pole, the point the whole sky turns around. Fragments survive, making it the oldest known sundial built this way, a design long credited to later Europeans.

A water clock measures time by a flow instead of a shadow, so it works at night and under cloud. Some of the oldest, from Egypt more than 3,000 years ago, are stone vessels with a small hole near the bottom and marks inside. As the water drips out, its falling level shows the hours.

The trouble is pressure. Water presses harder with depth, so a full vessel pushes water out faster than a nearly empty one, and the level falls quickly at first and slowly later. The Egyptian vessels had sloping sides to make up for it. A better cure is to feed the clock from a tank topped up to one constant level, with any extra spilling away. The depth above the hole never changes, so the pressure stays the same and the water drips out at a steady rate.

In 1206 al-Jazarī, chief engineer at the Artuqid court in Diyarbakır, finished his Book of Knowledge of Ingenious Mechanical Devices. It sets out about fifty machines with instructions for building them, several of them water clocks. The best known is the elephant clock. Hidden in the elephant's body is a tank of water, and floating in it is a bowl with a small hole in its base. Water seeps in, and after thirty minutes the bowl fills and sinks.

As it sinks, it pulls strings that release a ball, which drops into a serpent's mouth and tips the serpent forward. That movement strikes a drum, raises a figure's hand to show whether a half or a full hour has passed, and hauls the sunken bowl back to the surface. A steady trickle of water has become a cycle that repeats every half hour, and the figures announce each one.

The first all-mechanical clocks appeared in thirteenth-century Europe. A weight hanging from a cord turns a train of gears as it falls, and left alone it would simply unwind in a rush. What holds it back is the escapement. A swinging bar catches the teeth of an escape wheel, and each swing lets just one tooth slip past before the next is caught. The falling weight supplies the energy, the swinging bar sets the pace, and the gears count the swings.

The gears also divide the count into hours and minutes, at ratios fixed by the number of teeth on each wheel. A small set of gears turns the hour hand once for every twelve turns of the minute hand. The weak link was the swinging bar, a poor oscillator: a typical clock of this kind drifted by about fifteen minutes a day.

A weight swinging on a string makes a far steadier oscillator than a swinging bar. Galileo Galilei's first biographer said his interest began in the cathedral of Pisa, watching a hanging lamp swing. By 1602 Galileo was writing that a pendulum's swings all take the same time, whether they are wide or narrow. This is isochronism, from the Greek for "equal time", and it holds closely as long as the swings stay small.

What does set the time of a swing is the pendulum's length. The weight of the bob makes no difference. A longer pendulum swings more slowly, but not in proportion: the time grows with the square root of the length, so a pendulum four times as long takes twice as long over each swing. One just under a metre long takes one second to swing across.

On its own a pendulum slowly dies away, as friction and air steal its energy, and nothing counts its swings. The Dutch scientist Christiaan Huygens joined one to an escapement. He finished his first pendulum clock at the end of 1656 and patented it the next year.

In his clock the jobs are split. The pendulum sets the pace, because the time of each swing is fixed by its length. The escapement lets the gears move on by one tooth per swing, and at the same time gives the pendulum a small push, just enough to replace the energy lost to friction. The result was a leap in accuracy. Clocks before the pendulum typically drifted by about fifteen minutes a day, and pendulum clocks brought that down to about fifteen seconds.

A pendulum has its limits. Anything that moves the clock disturbs it, and on a hot day its rod gets longer, so every swing takes a little more time. The next oscillator had no swinging parts at all.

Quartz is piezoelectric: bend a quartz crystal and a small voltage appears across it, and put a voltage across it and it bends. That two-way link lets a circuit keep a crystal vibrating. The circuit amplifies the crystal's own tiny signal and feeds it back, like pushes timed to a swing, and the crystal responds strongly at just one rate, its natural frequency, set by its size and shape. In a watch it is cut as a tiny tuning fork that vibrates 32,768 times a second. The first quartz clock was built at Bell Telephone Laboratories in 1927, and quartz wristwatches went on sale in 1969.

Why 32,768 vibrations a second? Because 32,768 is two multiplied by itself fifteen times, and that makes it easy to count down to seconds. A simple electronic stage halves a stream of pulses, passing on one for every two it receives. Chain fifteen stages together and 32,768 pulses a second come out as exactly one, which steps the second hand or changes the digits. This chain, a frequency divider, is the watch's counter.

The rate is also a compromise: high enough to sit above the highest pitch human ears can hear, so nobody hears the crystal hum, and to keep the crystal small, yet low enough to keep the battery drain low. The result is a steady oscillator on millions of wrists. A typical quartz watch drifts by about fifteen seconds a month, as much as the first pendulum clocks drifted in a day.

No two quartz crystals are cut exactly alike, so no two run at exactly the same rate. Atoms have no such flaw: every atom of caesium-133 is identical to every other. Each can switch between two energy states, and it makes that switch when it meets microwaves of one exact frequency, 9,192,631,770 cycles a second.

An atomic clock uses that as its reference. It bathes caesium atoms in microwaves and keeps adjusting the source until as many atoms as possible switch. The source is then running at the atoms' own frequency, and a counter counts its cycles. Louis Essen and Jack Parry built the first practical caesium clock at Britain's National Physical Laboratory in 1955. It was accurate to about a second in 300 years, roughly thirty times better than the best quartz clocks of the day and three hundred times better than the best pendulum clocks of the 1950s.

For most of history the second was simply a fraction of the day: one 86,400th of an average day. Precise clocks showed that this oscillator is unsteady. The Earth's spin speeds up and slows down slightly from year to year, and over the centuries the tides, dragging on it, are slowing it down, stretching the day by a couple of thousandths of a second each century. Caesium atoms are far steadier than the Earth.

So in 1967 the second was cut loose from the sky. It is now the time taken by 9,192,631,770 cycles of the radiation that makes caesium atoms switch, a number chosen to match the older second as closely as it could be measured. Clocks were kept in step with the slowing Earth by adding a leap second now and then, 27 since 1972, but in 2022 an international conference voted to stop adding them by 2035.

Every time a phone finds its place on a map, it is timing radio signals from space. Each GPS satellite carries atomic clocks and broadcasts the moment it sent each signal. The phone notes when the signal arrives, and the signal's travel time, multiplied by the speed of light, gives its distance from that satellite. Distances to three satellites fix its position, and a fourth lets it correct its own cheap clock.

Light travels about thirty centimetres in a billionth of a second, so tiny timing errors matter, and even Einstein's relativity counts. Moving fast in orbit, the satellite clocks run slow by about 7 millionths of a second a day; in the weaker gravity high above the Earth, they run fast by about 45. The net gain of about 38 millionths is corrected for. Left alone, it would throw positions off by about ten kilometres a day.