An hour that stretched with the season

The equation of time is the smallest of the terms between a clock and the Sun, and it is a correction of a particular kind: it converts the Sun’s uneven motion into the uniform motion of a clock. It presupposes that time is uniform, that an hour is a fixed length, and that a clock can keep it. Remove those assumptions and the equation of time does not become large or small. It stops existing.

For most of the history of timekeeping those assumptions did not hold, and not because clocks were poor. The hour itself was defined differently. From ancient Egypt to medieval Europe, the ordinary hour was a twelfth part of the daylight, and a twelfth part of the night was a night hour, so the length of an hour changed every day of the year.

A day with twelve parts whatever its length #

The arithmetic is simple and it is worth doing, because the answer is larger than intuition expects. The length of the daylight follows from the latitude and the Sun’s declination through the same geometry that makes the seasons: the Sun rises and sets where its altitude passes the horizon, and the fraction of its daily circle above the horizon grows with its declination. Divided by twelve, the day at Rome gives an hour of 76 minutes in June and 46 in December — a summer hour two thirds longer than a winter one.

The variation grows with latitude, because day length does. At Alexandria, where much of Greek astronomy was done, the hour varies by a factor of 1.4 across the year. At Stockholm it varies by a factor of three, and north of the Arctic Circle the system fails outright for part of the year, when there is no sunrise to count from or no sunset to count to. Temporal hours were a Mediterranean invention and they suited a Mediterranean latitude.

The night was divided the same way, and its hours ran opposite to the day’s. At Rome in midsummer, when a day hour lasted 76 minutes, a night hour lasted 44; in midwinter the day hour shrank to 46 and the night hour stretched to 74. The Roman night was also counted in four watches of three night hours each, the units in which sentries were relieved, so a December watch lasted about three and three quarter equal hours and a June watch a little over two. Anyone standing watch knew perfectly well that winter nights were long. What the system did not do was express that as a longer night of the same hours; it expressed it as the same number of longer hours, which is a different way of counting the same darkness.

Sunrise and sunset, the two ends of the day being divided, are themselves defined by the visible Sun, and the visible Sun rises before the geometric one because the air lifts its image. The figures use the almanac convention of a Sun whose centre is 0.83 degrees below the horizon, which lengthens every day by several minutes; a temporal hour inherits a twelfth of that.

A clock that the Sun always agreed with #

The consequence for timekeeping is the point of the whole construction. In temporal hours, the Sun is by definition a perfect clock. The first hour begins at sunrise, the sixth ends at noon, the twelfth ends at sunset, and all three are events the Sun itself defines. A sundial marked for temporal hours reads the time correctly on every day of the year, with no correction of any kind.

The ancient dials were built for exactly that. A hemispherical dial — a bowl with a pointer whose tip casts a shadow on the inside — carries eleven lines running from one rim to the other, and the shadow’s tip crosses the same line at the same fraction of the day in every season, because the bowl’s geometry divides each day’s arc into twelve equal parts whatever its length. Water clocks, which run at a steady rate, were the awkward instrument in this system, and they were given scales that changed with the month or flows adjusted by the season, so that they too would count twelve unequal hours.

There is no equation of time in this system, and the reason is not that its sixteen minutes were too small to notice. Noon is the end of the sixth hour, defined by the Sun, on every day. The idea that noon comes “early” or “late” requires something else to compare it with — a uniform clock, keeping an hour that does not stretch — and within the system of temporal hours there was nothing to compare it with. The equation of time is not a property of the Sun. It is a property of the Sun compared with a uniform clock, and until a uniform clock was the standard, the comparison was not being made.

The old hours on a mean-time clock #

Reading the temporal hours off a modern clock makes the size of what they built in visible.

The fan is symmetric about noon, and noon is the one boundary the season does not move — it moves only by the equation of time, the thin wave on the central line. Every other boundary is pushed away from noon in summer and pulled towards it in winter, by an amount that grows with its distance from noon. The end of the third hour, the hour a Roman would have called the third and a monastery marked with the office of terce, falls at 8:12 by a mean-time clock in midsummer and at 9:47 in midwinter.

At London the third hour ends as early as ten to eight and as late as five past ten. The equation of time’s thirty-minute annual range on the noon line is less than a quarter of that. A medieval English monastery keeping temporal hours carried, as a matter of definition, a departure from uniform time more than four times the entire equation of time.

The names of the hours outlived their lengths, and one of them moved. The ninth hour of the day, nona hora, was the hour of the monastic office of none, in mid-afternoon. Over the twelfth and thirteenth centuries the office, and the meal that followed a fast broken at the ninth hour, drifted earlier, and the English word that descends from it came to mean midday. Noon is named after an hour that fell three hours later, and the drift happened in the centuries when the hours themselves were being made equal.

Two corrections of different sizes #

The comparison can be made directly, as two departures from the same uniform clock.

The two curves have different shapes as well as different sizes. The temporal hour’s departure follows the day length, with one maximum and one minimum at the solstices and zeros at the equinoxes. The equation of time follows the orbit and the tilt, with two unequal maxima and two minima that fall at no solstice or equinox. A society that kept temporal hours lived with the first, as a definition; the second was hidden inside the first and, at a tenth of the size of its annual swing in the early hours, would have been invisible even had anyone looked.

The size of the departure depends on the hour and on the place.

The relation between the two Roman figures is exact and short. If sunrise falls $d$ hours before six o’clock in apparent solar time — negative in winter — then the end of the $k$-th temporal hour falls $d\,(6-k)/6$ hours before the equal hour it is named after. The first hour inherits five sixths of sunrise’s displacement, the third hour half of it, and the sixth none, which is why noon is the only boundary the seasons leave alone. At Rome sunrise swings by about an hour and a half either side of six o’clock, so the first hour’s end swings by five sixths of that, ±76 minutes, and the third hour’s by half, ±46. The equation of time is the same thin wave under every one of them, because it moves all the boundaries together.

At Alexandria the departure of the third hour is only twice the equation of time, and it was at Alexandria that astronomers did use equal hours — the equinoctial hours, a twenty-fourth of a whole day and night — for their calculations, while the city around them kept temporal ones. Ptolemy’s tables of the Sun’s motion include the correction now called the equation of time, because for a calculation of where the Sun should be at a given uniform time the correction is needed. It lived in the astronomers’ tables for fourteen centuries before it had any use outside them.

Equal hours, and a clock steady enough to disagree #

Two changes made the equation of time a public quantity, and they came centuries apart.

The first was the equal hour itself. Weight-driven mechanical clocks appeared in European towers in the late thirteenth and early fourteenth centuries, and a clock driven by a falling weight and regulated by an oscillating bar has no natural way to stretch its hours with the season. It strikes equal hours. Within a century towns that installed such clocks were counting equal hours, some from midnight and some, in Italy, from sunset — which gave equal hours whose count still moved with the season through the year. The polar-aligned gnomon, pointed at the celestial pole so that its shadow gives equal hours on a flat dial, spread through Europe in the fifteenth century to match.

Japan offers the controlled experiment. Its mechanical clocks, adapted from European ones in the seventeenth century, were modified to keep temporal hours: clockmakers added movable weights on the regulating bar, or two bars for day and night, and the weights were moved every few weeks to change the length of the hour with the season. Japan kept temporal hours on mechanical clocks until 1873. The equal hour was not forced by mechanism; it was a choice, and a mechanism could be built to refuse it.

The same institutions that kept the temporal hours kept the calendar, and they managed its unevenness in the same way: by rule rather than by measurement. The date of Easter came from a moon that had to be tabulated because no cycle of whole days matches the real one, and the year itself was a fraction chosen to approximate 365.2422. A monastery’s day of twelve stretching hours, its months of tabulated moons and its year of intercalated days were three separate compromises between counting and the sky, and only the first was given up when clocks improved.

The second change was a clock steady enough for the equation of time to show. The equation of time changes by at most half a minute a day on the Earth. A tower clock of the fourteenth century gained or lost a quarter of an hour a day, so it drifted from the Sun far faster than the Sun drifted from uniform time, and it was reset from a sundial as a matter of routine. The Sun was the better clock. Only when Christiaan Huygens built the pendulum clock in 1656, keeping time to something like ten seconds a day, did a clock hold a steadier rate than the apparent Sun for long enough to watch the two disagree by minutes over weeks. Huygens published a table of the equation of time in 1665 for setting pendulum clocks by the Sun, and John Flamsteed a more careful one in 1672. The day was shown to be four minutes short of a rotation long before; the unevenness of the solar day became a practical fact only when a clock could keep a better day.

An hour defined by the Sun, then by the Earth, then by an atom #

Once the hour was equal, it still had to be defined, and the definition walked away from the Sun in stages. The equal hour and its second were fractions of the mean solar day — the day of a fictitious Sun moving uniformly, the second of the two fictitious suns that the equation of time is built from. That made the mean Sun the reference and the real Sun the thing corrected, and it is the arrangement the equation of time describes.

It did not last, because the Earth’s rotation itself turned out to be uneven, slowing by tides and wandering by milliseconds. In 1960 the second was redefined as a fraction of the year instead, as measured by the Earth’s orbit, and in 1967 as a fixed number of oscillations of a caesium atom. The second that results is uniform to a degree no rotating planet can match, and the mean solar day has become a count of such seconds that drifts, by milliseconds a day, against it. The chain runs from the true Sun to a fictitious one to the planet’s orbit to an atom, and at each step the previous reference became the thing being corrected.

Not every equal hour was counted from midnight. Italian towns kept equal hours counted from sunset well into the eighteenth century, so that the hour a clock struck was uniform in length and yet moved through the year against the Sun: noon fell at the sixteenth hour in midsummer and the nineteenth in midwinter. The count carried the season that the length no longer did. And on a planet where the Sun itself does not move steadily in one direction, as it does not on Mercury near perihelion, neither kind of hour would help: a temporal hour there lasts about a week of Earth time, and at some longitudes the sixth hour would end three times.

What was actually measured #

The lengths of the temporal hours in the figures are computed from the Sun’s declination and the latitude, and they could have been checked at any time in antiquity with a water clock and a sunrise. The ancient record confirms their size: surviving tables of the length of the hour through the year, and hemispherical dials whose lines are spaced for a particular latitude, match the geometry. What the record does not give is precision — nobody timed a temporal hour to the minute, because the concept of a minute belonged to the equal hours of astronomy.

The equation of time is measured far better than the temporal hours ever were, and it is the same quantity in Ptolemy’s tables, in Huygens’s and in a modern almanac, known now to a fraction of a second.

What the figures leave out #

Night hours. The night was divided into twelve hours as well, lengthening in winter as the day hours shortened. The figures draw only the day.

Twilight. Some temporal systems counted the day from the first light of dawn rather than from sunrise, which adds an hour or more at mid-latitudes and changes the lengths of the early hours.

Local convention. How a day was divided varied from city to city and from monastery to monastery, and the fixed rule used here — twelve equal parts from the visible sunrise to the visible sunset — is the core of those conventions, not a record of any one of them.

Still open: whether an unequal hour is still the better clock for some purposes #

Temporal hours have not disappeared. Jewish religious law still reckons the times of prayer in proportional hours, each a twelfth of the day, and some daily schedules in agriculture and in regions far from their zone’s meridian are still set by sunrise rather than by the clock. Whether a clock that stretches with the light serves human sleep and work better than one that does not is argued with evidence from shift work, from schools and from the seasonal clock change, and the argument is not settled; what the figures settle is only that the unequal hour is the one a sundial keeps without correction, and the equal hour the one that made a sixteen-minute disagreement with the Sun a thing worth tabulating.