Marking Time (or at least calibrating it)

While I was reading some research material for the last post, I came across a statement that made me wonder: the statement was that the French group that went to Ecuador to measure the length of a degree at the equator, had pendulum clocks that had to be calibrated for the local version of one second.  Additionally they used their pendulums to measure the force of gravity.  The question that I came up with is, well, how did they do that?

First, calibrating a pendulum.  What would you calibrate it against?  Another pendulum that has already been calibrated?  But what would you have calibrated that one to, ad infinitum.  To understand how to calibrate a pendulum, a bit of time keeping history needs to be looked at.

So, to establish the beginnings of a “clock”, humans have relied on natural processes that happen outside of their control, that are regularly repeating events – oscillations, if you will – such as the rising and setting of the sun.  That represents the daylight portion of one day: the non-light night portion of the day is the only other portion, so that is a fairly simple oscillation to calibrate to.  How do you measure a day as a unit? You pick an arbitrary start – sunrise, sunset, midday, midnight – and each day unit is the duration between a rise and the next, or a setting and the next, etc.  A fairly simple cycle, and a likely starting place for time keeping.

What would be next?  There are some other cycles that were apparently untangled by several early cultures.  Herodotus in his Histories, Book 2, states:

As to human matters, they all agreed in saying that the Egyptians by their study of astronomy discovered the year and were the first to divide it into twelve parts…the Egyptians make the year consist of twelve months of thirty days each and every year intercalate five additional days, and so complete the regular circle of the seasons.1

However, he is not completely reliable and evidently didn’t learn from “they” about the Egyptian lunar calendar.  His informants, the “they” who “all agreed” he says were priests he questioned in Memphis, Thebes and Heliopolis.

As to how they did this, he does not address the question, but one would probably not be too far off-base by figuring that over a period of years, the pattern of the stars in the sky would have been recognized as being essentially unchanging.  Once that was figured out, counting the days between a repeat of the stellar positions would only have required some patience and diligence: one slash on a papyrus per day would do it.  By the time of Herodotus, though, this exercise had been performed by the Babylonians and the Chinese as well, for both cultures had calendars before 450 BC.

At this point we have the year roughly figured out, and the day as the unit.  How about dividing up the day somewhat finer than roughly half daylight and the other half dark?  Establishing a midpoint for the day could be done with a very simple tool: a gnomon, or a plain straight stick stuck into the ground vertically, would project a shadow that was always “longest” at the approximate middle of the day.  This way daylight could be divided into two portions of time, before “middle” and after “middle”.  The initial shadow of each day would also put boundaries on the time of daylight within a day, and would range from shorter times of sunlight in the winter to longer in the summer.  How would this time best be divided up into what would become “hours”?

Using a sundial would not necessarily suggest dividing the time of daylight or darkness into 12 portions, and I have found only one possible suggestion as to why there was such a division, rather than dividing it into, say, 4 portions or 6 portions, even 9 portions.  The suggestion comes from a Wikipedia article which I quote:

Astronomers in the Middle Kingdom (9th and 10th Dynasties) observed a set of 36 decan stars throughout the year. These star tables have been found on the lids of coffins of the period. The heliacal rising of the next decan star marked the start of a new civil week, which was then ten days. The period from sunset to sunrise was marked by 18 decan stars. Three of these were assigned to each of the two twilight periods, so the period of total darkness was marked by the remaining 12 decan stars, resulting in the 12 divisions of the night. The time between the appearance of each of these decan stars over the horizon during the night would have been about 40 modern minutes. During the New Kingdom, the system was simplified, using a set of 24 stars, 12 of which marked the passage of the night.2

Not exactly an intuitive way to break time into smaller units, but evidently pervasive enough to be considered as a standard.  By the way, “decan” as used above appears to be derived from astrology (!) indicating a time of about 10 days between first rising.  The rest of this reference contains some mystery, since the Middle Kingdom of Egypt, which I believe is being referred to, is the 11th through the 14th dynasties, and ran roughly from 2055 BC to 1650 BC.  If this is the correct timeline, then, days divided into 12 or 24 has been around a while.

We saw, in an earlier post, that Ptolemy gets credit for breaking the degrees on his maps into minutes and seconds.  It seems likely that when the effort was made to subdivide hours, minutes and seconds were adapted.  Not too sure when, however, since “minutes” and “seconds” hands on clocks don’t show up until about the 15th century, and their accuracy was questionable.

These are some apparent regularly repeating oscillations that could be starting points for calibration except for a few inconvenient facts about the main organ of calibration, the earth in relation to the sun, which raise some questions:

Question one: should the hours be equal all the time, or should daylight contain 12 hours and the nighttime 12 hours, whether it is summer or winter?  Based on the earth’s position in its course around the sun, the angle of its axis of rotation, approximately 23.5 degrees, has the southern hemisphere more directly aimed at the sun or the northern hemisphere more directly aimed at the sun.  This leads to the sun being in the sky much longer in the summer in the northern hemisphere.  Should that daylight time be divided into 12 hours, the same as the shorter daylight days of winter?

We are used to all hours being the same length and knowing that the sun rises at an earlier or later hour throughout the year, but our answer is not the only answer that has been used.  Dividing daylight hours and nighttime hours into 12 of each was used as the standard answer for thousands of years, and only began to change in Europe as mechanical clocks became better at telling equal hours.

Question two: how do you accommodate for the fact that there are times during the year when the sun appears to move faster and times when the sun moves slower, and that it does not appear in exactly the same place at midday?  Using the elapse of 24 hours, which is the mean solar day would be difficult for calibration.  Over the course of the year, the sun at midday would appear in the pattern of an analemma.  See the following website for an adequate explanation of this:

So, if you were trying to make sure you made a sighting of the sun at the exact highest point of the day on two consecutive days to establish the length of 24 hours, that would not be precise enough.  You might be able to do it by establishing the mean solar day, take the amount of time for each day and essentially average it.  But doing that too might be difficult – how can you tell if 24 hours today is not exactly the same as 24 hours two days ago?

Question three: If you decided to use the stellar background, you would run into a problem here as well.   The mean solar day is approximately 24 hours long.  The sidereal day (based on the rotation of the earth relative to the stars) is approximately 23 hours and 56 minutes long.  Why is that?  Over the course of a year, any reference star that you chose would reappear in exactly the same spot, relative to the earth, approximately 365 (365.2422 more precisely) days later.  In between, each day, it would arrive at the same spot a little bit earlier, by 3 minutes and 56 seconds.  That is very close to the difference of one degree per day, which makes sense with a 365 day year.  If it advances just a shade under a degree per day, the sum of all of those is 360 degrees in 365 days.  But that makes the calibration difficult as well.

There are some additional cycles that could be looked at, but either their period is too long (26,000 year precession) to be practical, or doesn’t line up quite right over the course of a day, such as the moon.  It would be possible to base a calendar on the moon’s reappearance in exactly the same spot, relative to the earth and the sun, which takes about 29.5 days, but it doesn’t evenly divide into 365.2422.  (365.2422 / 29.5 = 12.3811)  Even trying to set up a clock based on the tides, affected by the moon, would run into inconsistency problems.

With this information before us, we are now able to determine that there is no way to calibrate a second.  Right?

If the story by Vincenzo Viviani is correct, Galileo realized that the period of the swing of a pendulum was pretty close to equal from swing to swing.  The tale has him realizing it while noting the swinging of a chandelier in the cathedral in Pisa, probably in a service that was the usual boring service.  An active mind searching for things to occupy it might just have noticed the swinging, put a finger on the wrist to feel the pulse, and made the realization.  While I may be projecting my own activities during that sort of situation, the story goes on to describe the use of a plumb line with a bob to understand the details of the insight more.  When he found the formula that describes the period as being proportional to the square root of the length of the pendulum, it would not be too difficult to imagine trying to find a pendulum length that was close to one second long.  But to do so, you would have to have some idea of how long a second is.

Timekeeping devices, while initially based or calibrated on the sky, had been brought down to earth by Egyptians, Babylonians, Chinese and Greeks, all of whom had versions of water clocks, or clypsydrae (singular: clypsydra).  Other devices to employ the same or similar means (material being removed from a container by gravity or fire) were candle clocks, incense clocks and sand hour clocks.  Sand hour clocks, also known as hour glasses could be made fairly precise by either adding or removing a little bit of sand after determining the right amount of time for an hour: if twenty-four were made with equal volumes of sand, and one turned over as soon as the last one finished in succession for a day, presumably one could start as the sun reached its highest point and the last of the 24 would just be emptied as the sun reached its highest point on the next day.  (It could probably be done with one hourglass turned over 24 times?)  This could be repeated until, by adding or removing a few grains of sand, it made the hour glasses pretty close to accurate.  In fact, it turns out, Magellan, when his fleet circumnavigated the globe in 1522, had eighteen hourglasses on each ship.

This would at least give a fairly accurate version of one hour.  Now, to subdivide that into minutes and seconds wouldn’t be too complicated.  Set up a pendulum that is the length you think is a second, start it swinging at the same time as you turn over an hourglass, and count to 3600 (60 X 60).  Too hard?  Well, in the 17th century, a Jesuit, ‘…Father Giovanni Batista Ricciolli persuaded nine fellow Jesuits “to count nearly 87,000 oscillations in a single day.”‘3 3600 seconds in an hour times 24 hours is 86,400, so knowing Father Ricciolli’s reputation as a scientist, my guess is that the group would be broken into four groups of two, and each of the two would be responsible for their own count, which could be compared later.  They would rotate the shifts of counting.  That’s a guess, but the ability to do this, figure out that the pendulum was too long or too short, adjust it, try again might be an activity that would be used to calibrate the correct length, which it turns out is about a meter, or 39. 37 inches.

The first pendulum clock was designed and commissioned by Christiaan Huygens in 1656.  This clock and those made shortly thereafter increased the accuracy of mechanical clocks from about 15 minutes a day to about 15 seconds per day.  In 1671, Jean Richer took a pendulum clock that was accurate in Paris to Cayenne, French Guiana and discovered that it was 2.5 minutes per day slower.  His conclusion as to why was that gravity was not as strong in French Guiana.  A few years later, Newton speculated that due to the effect of centrifugal force on the rotation of the earth, there was a bulge at the equator and the poles were flattened.  The Cassinis of the French Royal Observatory disagreed, and a result was the trip to Peru to measure the shape of the earth.  So we are now almost to the point where we can answer the questions raised at the beginning of the post: how did the French workers go about calibrating their pendulum clocks?

Among the factors that affect the operation of a pendulum clock are gravity and temperature.  If the density of the material under the pendulum is high, as it would be under mountains, this will raise the gravity, thus speeding pendulums up a bit.  The opposite effect, that of Cayenne, also affects the rate of oscillation of pendulums.  Temperature can affect pendulums: Huygens used a solid rod for his pendulums, instead of a string or something similarly flexible, since the flex in flexible pendulums caused erratic swing periods.  But with a solid rod, heat can lengthen the rod, and cold cause it to contract, lengthening or shortening the period of its swing.  With these known factors potentially at play, a clock with a pendulum that had a one second period when it was taken to Quito, Peru, what would be the effect?  Depends on the density underneath, and the temperature.  By the time the French left for South America in 1735, at least two solutions to the temperature had been developed, and possibly a third.  However, from my research, I cannot tell if any of these had been employed in their pendulum clocks.

Presumably, they would have tried to control for temperature, and then assume that the differences would be a result of gravity.  So once with their calibrated-for-Paris clocks they had measured whether their clocks were slower or faster than the mean solar time in a few consecutive days, and used the result in two ways: one, to estimate the force of gravity beneath the clocks, and second, how much longer or shorter to set the pendulum to match 86,400 seconds to the time between two solar meridian crossings.  I’m certain that the amount of time their task took was lengthened by repeated calibrations.

There was an alternative that might have been used, but I cannot tell if they used it.  By the time they left, the transit of the moons of Jupiter had been precisely timed: when several of the moons went behind Jupiter and reappeared had been timed precisely enough that one astronomer, Ole Roemer in 1676, had proved that the speed of light was finite using the occultation of Io, one of Jupiter’s moons.  He timed the occultation when the earth in its orbit was nearer to Jupiter, then when earth was at a different part of its orbit, much further from Jupiter, and noted that the times of occultation took much longer when the earth was farther from Jupiter.  This he correctly assumed was because the distance was so much greater, and reasoned that because of that, light had a finite speed and was not infinitely fast.

Having precise timings for the transits or occultations of the Jovian moons meant that the French could have matched the seconds of their pendulum clocks to the amount of time for a transit or occultation, and adjust the clocks to match the standard timing.  Did they do that?  I don’t know for sure, but my guess is that they would have tried that in addition to meridian crossings by the sun.  They were well trained in the astronomical tasks of the day, so I’m sure that they would have tried any and possibly all methods available to them to establish the accuracy of their clocks, to insure the accuracy of their surveys.

The process of calibrating a pendulum clock, then, is not very straight forward.  At best it is an iterative process: check the timing against a known oscillation, if it is not correct, make a correction in the length of the pendulum that should either speed up the period or slow it down to match the known timing device, check it again.  If it is not correct, repeat until it is.

Choosing the reference oscillation is the key.  Pendulums were used to establish the standard until quartz oscillations were built into quartz clocks and then watches.  Now, the standard is based on the oscillations of cesium atoms in cesium clocks, and hydrogen oscillations in hydrogen maser clocks.  They are well beyond the accuracy of pendulum clocks.  The arrangement at the Naval Observatory is a Master Clock fed information by a set of cesium and hydrogen masers, which corrects itself based on the inputs.  The Master Clock “…keeps time to within one hundred picoseconds – one hundred trillionths of a second – over the course of each day, every day.  Had it been set when the dinosaurs went extinct, 60 million years ago, it would have gained or lost no more than about two seconds.”4

1 Herodotus, The Histories, Penguin Books, London, England.  Translated by Aubrey de Selincourt, 1954. Revised edition, 1972.  Revised edition with new introductory matter and notes by John Marincola 1996, further revision 2003. P. 96.  I had found so many “measurement” references to Herodotus that I bought a copy and am reading it.  Since Herodotus died around 425 and 420 BC, the book is surprisingly lively for 2400 years old.

4 Falk, Dan, In Search of Time, The Science of a Curious Dimension, Thomas Dunn Books, St. Martin’s Press, New York, N.Y. 2008. p.56.  Full of good information and really poor proofreading.

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