Home-grown astonomical time

Home grown time

Star-based time-keeping for amateurs

How do you set your clocks and watches? Time was once a very local matter, and every town had its own time according to its own specific longitude. Nowadays Europeans drink wine from Australia, North Americans eat apples from Argentina, and no one thinks it strange that people in Maine and Alaska set their watches to a clock running in Boulder, Colorado - if they are even aware that that Boulder is where the U.S. National Institute of Standards and Technology keeps its atomic time standard.

With the spread of global standard time in the nineteenth century, time like so many other things became a globalized, interchangeable commodity, and - as with so many other commodities, people lost their connection to it as part of their local identity.

These pages are meant to get you interested in growing your own local time, and, if you're as ambitious as I was, getting it down to the nearest second or better. Perhaps you would like to appoint yourself or your organization as your local time standard and give your town not only place on the map but on the clock as well.

I. Time by the Sun

The easiest way to get your own local time is with a sundial. This has been done since ancient times; in Renaissance Europe, many a town hall or cathedral carried a sundial that defined the 'standard' time of the locality. Here's one on the cathedral at Freiburg, Germany.

With the improvements in clock-making that took place in the eighteenth century, it undoubtedly became increasingly apparent that sun-dial time was somewhat erratic, jumping ahead by 12 minute, lagging by ?ß minutes, etc. This variation, known as the equation of time, has two causes.

By relating the sun's height to the time time of day, it is possible to read the correct mean solar time directly from the sun-dial. This appears to be the purpose of the lower dial, here, on the Hotel-Dieu, Église St.-Martin des Vignes, 25, Quai de Comtes de Champangne, in Troyes, France. Made by Bazin in 1778.

A sundial is hard to read to the exact minute, and in fact it would usually be wrong if it could be so read. Even though we regulate our daily lives by the sun, serious timekeeping can only be done by the stars.

Hence, many a town appointed someone, a jeweler, for instance, to set a clock regularly by the stars. Depending on how much time and money you can invest, your local time can be exact to a minute or so, or to a fraction of a second. Sextants. Riflescope. Theodolite.

Sextant method.

Using a sextant well takes considerable practice and a steady hand. If, like me, you need glasses to see the stars at night, it will be more difficult to get good sights through the sextant's telescope. If you live in or near a city, you may often not be able to see many stars at all. whereas with a rifle scope or theodolite, once everything is set up, you can get the time from stars that you can't see with the naked eye, sometimes even through cloud cover. The main advantage is that you get started with elementary astronomical time-keeping at a very low investment.

Artificial horizon

Paper sextant

Davis sextant

Metal sextants

Using the Davis sextant

An ideal artificial horizon would be a bowl of mercury about a foot in diameter, but mercury and its vapors are toxic, and it's not worth the health risks (not to mention problems of storing and disposing of the stuff). If you want to use the sextant method seriously, you might consider buying a small amount of a mercury substitute like NewMerc or Galinstan (both reg. trademarks). 100 ml can cost several hundred dollars, but you will spend as much for a good sextant, as well. In the sextant literature you will find mention of artificial horizons using water or oil to form the reflecting surface, but I found that these these liquids reflected too little light to let me sight on anything but the moon.

The alternative I settled for is a solid mirror, mounted on a plate with leveling screws. A leveling bubble is used to get it horizontal. Freiberger Instruments sells one, but it costs much more than the Davis sextant.

Measuring the altitude of a not-so-bright start with the artificial horizon is far more difficult than measuring the height of the moon from the sea's horizon. I suggest practicing first with the moon, then with a the brightest star or planet that is not more than 35° above the horizon. With any but the brightest stars you can easily mistake the reflected image of some other star for the one whose height you are trying to get. I found that I could get reliable results only by first setting the sextant to the expected angle at a given time, using the planetarium program. With practice I found I could aim the sextant with both eyes open. If the sky is not pitch black, though the left eye you may see a small disk that is slightly brighter than the sky. Bring this disk over the star you are trying to sight on, and at some point that star will suddenly appear within the disk, brighter than you see it with the right eye. The next task is to bring the reflected image into coincidence with the direct image. This you do by rocking the sextant a little to the left of vertical and then to the right. If you have set the angle correctly, at some point the reflected star will whiz past the directly seen star. Steady the image and then adjust the vernier knob to bring the two into coincidence. When you are sure the coincidence is a good as you can get, read the clock.

The higher the altitude of the star, the more difficult it will be to do this because the reflected star moves farther to the right or left as the sextant goes out of vertical. Another reason why low lying stars are to be preferred.

If you're looking for something specific, you might find it here:

II. Time by the Stars

An English Transit

The Zeiss Theo 010

III. Pendulum Time

Pepping up a Hermle regulator

One-Ipping Hipp

IV. Atomic Time

Recently (I'm writing in 2010)

A. Lines and Circles

Time present and time past are present in time future...

There is a curious paradox that arises in the mathematical theory of periodic functions, one that has reminded me of these lines that commence T. S. Eliot's Quartet No. 1.

An absolutely perfect time keeper, the best clock one could build, would have to be an object outside of time, like the eternal heavenly spheres of Ptolomaic astronomy. Take any earthly timekeeper, like the swinging pendulum in a grandfather clock, the balance wheel of an old-fashioned wrist-watch, or the tuning fork of a modern watch -- they are all approximations to what physicists call a driven, damped harmonic oscillator. Their motions are roughly described by the trigonometric sine and cosine functions; the instantaneous position of the oscillating member -- pendulum, wheel, or fork tine -- is given by these two trigonometric functions of the time it is marking out. The wavy line extending backwards and forwards in time simply records a motion around an endless circle. The back-and-forth motion of the oscillating member, traced out as a curve along this time line, is called a time domain description of the oscillator.

It is also possible to describe this motion in a more abstract way, by saying how often the line rises and falls in a given unit of time; this is called the frequency domain description. For example, the seconds pendulum of the grandfather clock, so-called because it ticks once a second, has a frequency of 0.5 Herz (Hz, or cycles per second) because in one-half second it is only half way through its full back-and-forth-and-back cycle; the typical wrist-watch ticks with a frequency of 5 Hz, and the tiny quartz tuning fork in your quartz watch hums along at 32,768 Hz.

Now, if these oscillators were perfect timekeepers, they would never vary in frequency, meaning that their frequency domain descriptions would each contain only a single frequency. But in fact the frequency of every really existing harmonic oscillator wobbles in various ways, some in response to environmental factors like temperature, and some of a purely random nature. For example, the frequency of your watch will change with changes in your body temperature and the position of your arm, and studies of high-precision pendulum clocks have shown that their frequencies change as the force of gravity changes with the tides and motions of the moon. The tuning fork in your quartz watch has a primary frequency of 32,768 Hz, but superimposed on that is probably a frequency of 0.0000116 Hz, or one cycle per day, from the daily variation in your body temperature or from taking the watch off in the evening and putting it on in the morning, as well as various other frequencies. Any change in the magnitude, or amplitude, of an oscillation (how far the pendulum swings, how far the balance wheel turns) also shows up as a separate frequency component in the spectrum or Fourier analysis of its motion, different from the primary frequency. The same applies to every start and stop of the oscillation; to describe a time-domain oscillation that starts or stops, frequencies additional to the primary frequency have to be introduced. The list of all frequencies needed to account for the wavy line drawn out in the time domain description is called the spectrum of the oscillation that the line depicts.

Hence, the mathematically pure, single-component frequency, in other words, the perfect timekeeper with just a single frequency in its spectrum, would have to exist in a world in which its running had no beginning and no end. The clockmaker's grail will not be found in this world, but for many centuries it was thought to exist above us, in the rotating heavenly spheres, whose timekeeping was the ultimate standard for all earthly timekeepers. The realization that the authoritative source of eternal time is not up there, but down here, is, I think, one of the more interesting stories of the last two hundred years.

Perhaps it is this implicit approach to eternity that draws folks in their last stretch of life in large numbers to hobbies like astronomy and clock building. In any case, after a my own early retirement from a high-tech sinking ship, I have found time to take up again a fascination of my school days. When I was in high school, getting seriously involved with positional astronomy would have been unthinkably expensive, but as technological revolutions have substituted electronics for the optical and mechanical tools of geodetic astronomy, these have become easily affordable. Discarded, high precision optical theodolites, for measuring the positions of stars, can be had almost for a song if you look hard enough, and the wonders of NC milling have made replicas of observatory clocks, which in their heyday would have cost you a house, available for the price of a used car; with some work of your own, an observatory-quality clock can be put together for much less. One other development has brought star-based time-keeping in closer reach of amateurs -- personal computers with appropriate software (http://www.usno.navy.mil/USNO/astronomical-applications/software-products), which eliminate most of the tedious calculation that used to be the lot of positional astronomers and their drudges.

Time by the stars

The notion that the stars and planets are attached to unchanging heavenly spheres was probably taken seriously by ever fewer savants from the seventeenth century onwards. By 1700?? careful astronomical observations had shown that some supposedly fixed stars actually changed their positions, if ever so slowly. By the end of the eighteenth century, the sometime physicist and philosopher Immanuel Kant could propose a cosmological theory in which the earth had condensed from a cloud of dust circling around the sun. Building on Newton's mathematical mechanics, eighteenth-century physicists undoubtedly realized that the rotation of the earth must be subject to small irregularities, as portions of its mass, like the ice caps, the oceans and the atmosphere are in constant motion, so that the distribution of mass in our planet is constantly shifting. Nevertheless, until the advent of atomic clocks, no clock convincingly showed itself to be so constant and reliable as the dial of stars passing over our heads every night. Since ???? the International Earth Rotation Service has been telling us, almost day to day, how fast or slow our heavenly spheres are turning.

Until the spread of telegraph lines in the middle of the nineteenth century, towns determined their own time by the sun or by star observations and often broadcast it from a tower clock, most famously from Big Ben in London, which was set from a telescope at the Royal Observatory a few miles downriver, in Greenwich, but even small towns had to provide a standard time so that their citizens could coordinate their activities. Fig. 2 shows a small transit instrument of the sort many a town would have kept on hand for setting its clock of reference on clear nights.

Alexis McCrossen - By the Clock - Reviews in American History 28:4 Reviews in American History 28.4 (2000) 553-559 By the Clock Alexis McCrossen

Ian R. Bartky. Selling the True Time: Nineteenth-Century Timekeeping in America. Stanford: Stanford University Press, 2000. 32 pp. Illustrations and notes.

Fig. 2. A Small Transit Instrument.

Fig. 3. The Transit Instrument of the Observatory of Neuchâtel. Reconstructed in the Musée Internationale de Horlogerie, La-cheaux-de-fonds, Switzerland.

The demand for accurate time was understandably most acute in areas where clocks and watches were manufactured, and it is no surprise that one of the most important time services was the observatory in Neuchâtel, in the watch-making region of French-speaking Switzerland. I have visited several times the reconstructed transit room in the Musée de Horologerie, La-cheax-de-fonds, with its 300mm?? transit telescope and a late version of the electric Hipp clock, running as it would have around 1900, when it may have been the most accurate clock in the world and helped to establish motions of the earth's poles that had been predicted by physicists but hitherto never observed.

The Hippoarchos star catalog (http://www.rssd.esa.int/index.php?project=HIPPARCOS) reduces the small but pesky uncertainties of pre-space age catalogs of up to half an arc second to the order of a thousandth of an arc second.

Using a Sextant

http://tycho.usno.navy.mil/sidereal.html

By far the easiest way to get the time from the stars is to use a sailor's sextant. At sea this is not possible unless your boat remains at anchor, or unless you cheat by getting your position via GPS. On land you also have to know your position before you can compute the time referenced to UTC. On land you generally can't sight the horizon, so you have to use a so-called artificial horizon, which is actually not a horizon at all but rather a mirror that reflects a star's image from a point below your sextant, so that you see the star once above the horizon and again, as a reflection, below the horizon. Instead of measuring the star's height above the horizon, you measure the angle between the real star and the virtual star. Dividing this angle by two gives you the angle above the horizon, which means you also divide the error by 2. With a plastic sextant, this should give an accuracy of 1 arc minute, which comes to 4 seconds of time, and a watch is going to have this much daily error in any case.

EBCO, Topoplastic, Davis Mark 15.

An English transit

The Zeiss Theo 010

Pepping up a Hermle regulator

Having learned how to find my meridian and observe star transits, I was ready to start keeping my own time, if I could find an affordable clock. Genuine observatory clocks are very high-priced collector's items, even when they're not in good condition, and my budget was limited to €2000. A number of European firms are now making what are more or less copies of famous regulators like those made by Riefler and Strasser & Rohde, most visibly the firm of Erwin Sattler in Munich [], which also offers some models as kits. But these start at around €4000, and the sky's the limit. They typically come with solid pendulums, using Invar to make them insensitive to changes in temperature, but for reasons I'll explain later, I wanted nothing to do with Invar.

The kind of clock you want is a simple, weight-driven wall clock, with a pendulum that counts seconds, possibly half-seconds, or anything in between. The driving power from a mainspring is so inconstant that all your further efforts at improvement to improve the clock's accuracy are doomed from the start. Chimes, gongs, and electrical contacts as provided in master clocks should be removed so that they do not influence the drive force that reaches the pendulum.

You may have better luck than I did, but after vainly scouting flea markets and antique shops for a number years, I ended up purchasing a small wall clock made by Hermle, a Black Forest company oriented more to the mass market for decorative clocks. They have, however, a small line of what they call Monatsläufer, German for 30-day clock, and I found a dealer nearby in the Black Forest who would sell me one for about €1700 if I would pick it up. That was a few years ago; this dealer now charges about €600 more. Note that the retail price spread for such clocks among dealers in Germany is more than 2 to 1; this may be similar elsewhere.

Many other kinds of weight-driven pendulum clocks will serve equally well. If you already have a good Vienna regulator or a weight-driven master clock of the sort used in large factories decades ago, most of what I say here will be relevant. You should free the mechanism all other duties, like activating chimes or electrical contacts, and make sure that it is competently cleaned and adjusted. The main point here is that good clock mechanism is relatively easy to come by -- but a good pendulum is not.

The Hermle clock is the strangest combination of fine workmanship and sheer incompetence I have ever encountered, but it gave me what I was looking for -- an excellent weight-driven movement with "maintaining power" (a spring-loaded wheel that keeps the gear train under tension while the clock is being wound), a fully adjustable, jeweled Graham escapement and, just for appearances, an observatory-style dial, with a large minute hand and smaller hands for seconds and hours (see Fig. 1), and no gongs or other frills. The case I didn't need, as I knew I would have to build a larger one, anyway, but I didn't think to ask if Hermle would supply the clock without a case. My dealer was able to order, for perhaps €10, an extra anchor and escape wheel, which I wanted for experimenting. The more expensive precision clocks from Sattler and others pride themselves on offering especially robust movements, with 3 and 4 mm thick plates, but I see no advantage in this. The job of the movement is to supply a milliwatt or so of energy to a moving pendulum rod, with a force one can hardly feel. In no case should the pendulum be supported via the movement, and with the driving weight supported close to the works' mountings, which it is in the Hermle movement, the movement does not need to transmit any large forces. At the few points where large forces are in play, which might distort or wear bearing holes in thin plates, as at the winding drum, the Hermle movement uses modern ball bearings, which distribute the load. There is another consideration: when a pallet of the Graham escapement releases its escape wheel, this wheel and the gear train driving it need to accelerate instantly to begin supplying energy to the pendulum (more about this later), which means that they should be as light as absolutely possible. Here, smaller and lighter is only better, as far as I can see.

This probably applies to the clock's size, as well. Observatory regulators were almost always built with seconds pendulums (having an oscillation, or period, of two seconds, but producing a 'tick' each second), perhaps because they were ultimately destined to supply time services with an impulse

Fig. 1. Hermle Monatsläufer, Model 70875-740761. The size is 90.3 x 25.5 x 12.5cm.

each second. The modern market for precision pendulum clocks follows this tradition, but modern experimental evidence, which I'll discuss below, speaks in favor of a faster pendulum, beating somewhere between 90 and 120 ticks per minute. The Hermle regulator beats with 78 ticks in the minute, a rather peculiar number, but by no means a horological disadvantage.

Where lightness is not better is in the pendulum. Most decorative pendulum clocks are sold with ridiculously light pendulum bobs, weighing 100 to 500 grams. I can only imagine that makers do this because they know they cannot get their customers to mount the clocks properly, and the light pendulum is less sensitive to the flimsiness of a wood panel or wallboard mounting than would be a proper heavy one. What I found most remarkable in the Hermle clock was the pendulum rod, made of nickel-plated brass, just about the worst material one could choose because of its high thermal coefficient of expansion. And indeed, I quickly noticed that the clock's rate went up and down with temperature changes in my living room.

But these and a number of other deficiencies are fairly easy to correct, and once that's done, you have a pretty good approximation to an observatory quality regulator.

My efforts were guided by three recent books that are well-known in horological circles: Phillip Woodward's My Own Right Time [xx], Robert Matthys' Accurate Clock Pendulums [xx], and Klaus Erbrich's Präzisionspendeluhren (in German and unfortunately out of print) [xx]. From Woodward I learned the importance of having as little friction as possible, and that a Graham escapement can be adjusted to compensate for the pendulum's circular error; from Matthys' I learned that very few manufacturers of precision clocks have ever done what is necessary to provide a good pendulum, and from Erbrich I learned that the importance of the escapement has been more than a bit overrated.

Clock makers, whose professional focus is on the intricacies of mechanism, like to claim that the going train and escapement are the most important parts of a clock, and they will urge you to direct your attention and money to these. However, Matthys has shown that the largest single source of inaccuracy in most pendulum clocks, even cheap ones, is likely to be the pendulum itself, not the mechanism. Fortunately, you don't need to have a machine shop to build a good pendulum, although you will probably need to have a foundry make the bob for you. Some fairly simple steps can lead to big improvements. Some of Matthys' suggestions are:

Use a low expansion material for the rod. Steel is better than brass, invar still better. The simplest alternative is carbon fibre rod, which is far less expensive than invar and readily available; Matthys' choice is quartz, which is also relatively inexpensive.

Attach the bob to the rod at a point low in the bob so that it will expand upwards when the rod expands downwards, introducing temperature compensation. With carbon fiber composite this is probably not necessary.

Use a low drag bob shape, lens-shaped or like that of Riefler (described below).

Enlarge the case so that no surface is closer than 60 mm (2.5 inches) to the bob.

It goes without saying (but must be said) that a sturdy, load-bearing wall is the only suitable mounting for a precision pendulum clock. Where I live, in southern Germany, newer dwellings are usually built in brick and concrete, and I have mounted my clock in a thermally stable hallway on a masonry wall. In wood frame houses, I imagine this requirement bans the clock to the cellar (which is where a real observatory clock belongs, anyway). For the Hermle regulator's pendulum, beating 78 seconds to the minute, the bob should weigh 3 kg (7 lbs), and it should hang from an absolutely rigid support fixed directly to the wall.

Hence, the first thing I tried out was a better pendulum. I bought a 1 m piece of carbon fibre tube, the kind used by kite-makers, at a local hobby store. Then I took the Hermle bob to a shop that makes stained glass windows and had it filled with lead solder, raising the weight to xx kg.

Just by replacing the pendulum, I expect you can get a big improvement in the accuracy of the Hermle clock. However, the pendulum support is also a significant problem because the clock is meant to be hung from a single hook or nail in the wall.

The pendulum support

In the English-speaking world, it has long been taken on faith that a pendulum must be supported from a spring, and that pivots or knife-edges are unreliable. But in fact the high-precision pendulums used by geologists to determine the force of gravity have nearly always been built with knife-edges, and the German firm Riefler used knife edges in its highest precision regulators over a time span of several decades [xx]. Matthys' investigations of the instabilities of materials used in pendulum construction show with astonishing clarity how much inaccuracy various metals must have introduced in even the best and most famous clocks. Unfortunately, his data do not factor out possible instabilities in his suspension springs, but given the large stress to which it is exposed, one might well suspect the spring of contributing instability as well. In any case, no spring is perfectly elastic, and measurements I have seen suggest that as much as 20 percent ?? of the pendulum's driving energy is lost in the spring. Finally, constructing a good pendulum spring is not a simple task for an amateur, and the springs you can by in clock shops are not worth the effort.

I thought it might be much easier to support the pendulum on a pivot made with the balls used in ball bearings. In any bicycle shop you can get steel balls that are technological marvels, so hard that you cannot nick them with a hack saw. Resting on a polished agate plate, it stands to reason that such balls would be the perfect pendulum support, introducing virtually no friction and no dimensional instability. The only drawback is that the ball adds a small amount of additional circular error.

The pendulum rod

Following Matthys, I opted for a quartz rod, the material with the greatest thermal stability and lowest coefficient of expansion (apart from expensive, zero-expansion glasses like Zerodur). For a pendulum beating seconds, Matthys recommends a quartz rod 1.63 cm (0.64 in) in diameter. This is a pretty thick rod, and the one regulator I have seen with a quartz rod is indeed so constructed (Fig. X). Such a rod contributes quite a bit of air resistance, so to obtain a better value of Q, I use two thinner capillary tubes with 7 mm outside diameter, lying in the plane of vibration and spreading slightly upwards, so as to be quite stiff in this plane.

Fig. X. Regulator by Sartori?? with Quartz Pendulum, Strasser-style Bob. On display at Karl Hofer & Sohn OHG, Vienna [info@uhrmachermeister-hofer.at]

Mr. Hofer, owner of the quartz pendulum shown here, told me that Vienna regulators with quartz pendulums were not as uncommon as the clock literature would have us believe. However, he did not have the impression that they had been significantly more accurate than other constructions. Indeed, a report on the state of the horological art from around 1918 [HK] remarks that the difficulty in producing such pendulums may well hamper their spread, despite their admirable physical properties. A nasty property of quartz is that it cannot be bored and threaded as easily as steel or wood. The support spring and bob in this clock are attached via set screws at each end, which project into ring-shaped grooves in the rod. I have followed Matthys' suggestion of boring transverse holes which carry cotter pins.

The pendulum bob

The most accurate pendulum clocks, those of Shortt and Fedchenko, placed their pendulums in vacuum vessels. After looking into it, I decided that vacuum technology was beyond what I was willing to attempt, and I appreciated why many important clock makers, like Hipp, Riefler and LeRoy, built clocks in sealed vessels with only slightly reduced (but constant) air pressure rather than vacuums.

Hence, any pendulum I was going to use would have to contend with the randomly changing barometric pressure of my middle European weather. This makes a bob with high density important, for two reasons. To keep the effects of air resistance low, we want as little surface area as possible; to keep the effects of varying buoyancy low, as little volume as possible. (Like a helium balloon, the bob is driven upwards by air pressure, so that its effective weight varies with the barometric pressure.) What to use? Depleted uranium is rather hard to come by; tungsten's melting point is so high that it must be formed by sintering, a process in which metallic dust is pressed together. As early as 19?? ?? Boys suggested using an iron-tungsten alloy with density of 16 kg/liter [xx], which is in fact available from ecvv.com (ECVV China) offers tungsten alloys in various forms. But would it be physically stable (and affordable)? Neither in my internet searches nor in the horological literature have I seen mention of a tungsten-alloy bob.

In a series of experiments extending over many years, Matthys tested the effects of annealing and temperature cycling on the stability of various materials suitable for constructing pendulums. For the rod, as mentioned above, he concluded that quartz was the best material. For a bob working in air, the density of quartz is too low. Matthys favorite is a bob of aluminum silicon bronze because of its unsurpassed thermal stability. I decided in favor of Linotype metal, although his data place it in third place. It is far easier to cast and anneal a Linotype bob than one made of bronze or of leaded brass (which was in second place). Further, Linotype metal has a density slightly over 10 kg/liter (1000 cc), as opposed to 8.7 kg/l for aluminum silicon bronze, which gives a Lintotype bob slightly less surface area and less buoyancy than the bronze bob and makes it a little less susceptible to changes in barometric pressure. The Shortt clocks had Linotype bobs, so I'm not terribly worried about this choice.

One-Ipping Hipp

Hipp's invention lived on in a slightly modified form well into the twentieth century, in master clocks built by the Swiss firm Favag.

The Hipp observatory clock

The Hipp clock demonstrated for the first time two principles of construction that were essential in later precision clocks: a sealed container for the pendulum and a constant-force, electrically-powered impulsing mechanism. A principle that did not enter the clockmaker's canon was that of amplitude control via feedback, and I have often wondered why. Even the famous Shortt clock, with its gravity-driven impulse, was subject to variations in pendulum swing. If Shortt couldn't eliminate amplitude variation without feedback, no one could. Shortt compensated to some extent by having the pendulum swing through a short arc of only 55 arc minutes. But the smaller the arc, the closer the oscillation comes to the inescapable vibrational noise of the the support.

This got me to thinking that more attention to this aspect of clock design could be worthwhile. Hipp's clock actually limits only the lower side of the pendulum's amplitude variation; the upper side (the maximum swing) is subject to variations in the impulse. Some modern builders of pendulum clocks have resorted to complex electronics, even computers, to measure and adjust the pendulum's swing (most notably Prof. Hall's Littlemore clock), but I have often wondered if the great masters of the past did not perhaps overlook things the technology of their time would have allowed, had they been even more persistent. Indeed, no one before Riefler seems to have thought to find experimentally the ideal aerodynamic shape for the bob, although even Harrison must have been in a position to do so. Nor does anyone seem to have thought of an obvious and simple scheme for applying feedback to control the pendulum's amplitude that would have been possible with the technology available to Hipp -- batteries, relays, and electromagnets.

As a thought experiment, consider how a small degree of amplitude feedback actually exists in the Graham escapement. When the pendulum's amplitude is small, its maximum velocity is also small. The fall begins close to the end of the swing; the anchor moves slowly. At the start of the fall the escape wheel can quickly accelerate to the speed necessary to deliver force to the pallet. If the pendulum is moving fast, the pallet will be moving away from the wheel tooth more quickly, and the force applied to it at the beginning of the impulse will be lower, reduced by the energy required to get the train moving. Above a certain pendulum velocity, the drive could disappear completely as the pallet races away from the sluggishly accelerating tooth of the escape wheel. By greatly increasing the mass of the driving train, this feedback effect could be strengthened. A complication is that the escapement error increases as the driving interval becomes limited to the end stretch of the pallet face. However, a decreasing escapement error with increasing velocity is precisely the effect that compensates for circular error in the best Graham escapements, and it is something we might not want to mess with.

Once we're allowed to use electrical impulsing, however, the second problem disappears. Imagine we had a Graham escapement with very shallowly cut pallets so that the fall is restricted to, say, the central twentieth of the average swing; imagine, further, that the impulse were delivered not by the pallets but by an electromagnet.

Cited literature

[HK] Kienle, Hans. "Untersuchungen über Pendeluhren. Mit besonderer Berücksichtigung der beiden luftdichten Riefler-Uhren R23 u. R33 der K. Sternwarte zu Münche"n. Neue Annalen der K. Sternwarte zu München. V:2 [undated, probably 1918]

[?H] Hirsch, ??. La pendule 'electrique de pr'ecision de M. Hipp. Bulletin Societ'e Sciences Naturelles de Neuch^atel, Vol. 14.