Vintage Watch Straps

Straps for vintage fixed wire lug trench or officer's wristwatches.

# Temperature effects in watches

Some time between 1658 and 1660 the English scientist Dr. Robert Hooke had the idea of using a balance spring to improve the timekeeping of watches. He thought that it would make a watch an even better timekeeper than a pendulum clock because the balance of a watch swings backwards and forwards in rotation around a fixed axis and therefore does not suffer from the problem of "circular error" caused by the unidirectional pull of gravity that prevents pendulums from being isochronous. Hooke realised that by adding to the watch balance a spring that obeyed his famous law, that the tension of a spring is in proportion to its extension, the conditions for an isochronous harmonic oscillator would be fulfilled.

Hooke showed a pocket watch with a balance spring to Lord Brouncker, Robert Boyle and Robert Murray, seeking their sponsorship in an application for a patent on the idea. A draft patent was drawn up in 1665, but then development of balance spring watches was put on hold and the patent was never granted. Hooke was very busy at the time with many scientific investigations, and from 1666 with supervising the rebuilding of London after the Great Fire. Christiaan Huygens successfully applied the balance spring to watches in 1675 and announced this as an invention of his, to Hooke's great annoyance. Whether Huygens conceived the idea independently or was told of Hooke's idea, perhaps by Henry Oldenburg, the secretary to the Royal Society, has been debated since. Oldenburg’s minutes record Hooke demonstrating a spring-regulated watch to the Royal Society in June 1670, and he is known to have corresponded with Huygens.

With the invention of the balance spring, watches became quite good timekeepers and even at the end of the seventeenth century a verge watch could be expected to keep time to within a few minutes a day, if it was kept at a constant temperature. However, its rate would alter by around 10 seconds per day for every degree centigrade change in temperature. This effect was probably too small to be noticed by Hooke, but by the early eighteenth century when John Harrison started to constructed marine timekeepers in an attempt to win the longitude prize the effect was well known. All of John Harrison's marine chronometers have devices that compensate for the effects of temperature, and they could not have achieved the accuracy that they did without them.

The story of improvement in the accuracy of balance controlled watches after Harrison had achieved the accuracy required by longitude act is very largely the story of reducing the effects of temperature on their rate.

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## Temperature effects

Compensation balance and spring: Click to enlarge

The timekeeping of a clock is usually determined by the pendulum swinging to and fro under the effect of gravity, with a little push, called an impulse, from the escapement every time the pendulum swings through its lowest point, accompanied by a gentle tick. A pendulum can't be used in a watch because it might be held at any angle but gravity only pulls downwards, so an alternative is needed. This is provided by the balance. The image here shows a watch balance, which comprises a central bar or spoke and a circular rim. The spoke has an axel passing through it at right angles about which it can rotate that is called the "balance staff".

The escapement mechanism pushes the balance in one direction and then the other. In early watches verge without balance springs, the rate at which the balance went backwards and forwards was determined solely by how hard it was pushed by the mainspring, which is why the fusee, or an alternative called a stackfreed, that equalises the torque from the mainspring was vital in such watches. Without a fusee or stackfreed the rate would change so much as the mainspring ran down that the watch would be useless as a timekeeper. Even with one, early watches without balance springs were poor timekeepers.

The addition of the balance spring transformed the timekeeping capabilities of watches by giving the balance a "natural frequency". The spring causes the balance to oscillate at this frequency, to which it returns after a disturbance. The less the balance and spring are disturbed the better the timekeeping, which is why detached escapements which only interfere with the balance over a short part of its arc, are better.

The image here shows a balance and balance spring. The balance spring is the blue spiral in the middle of the image. It is blue because it is made of high carbon steel that has been hardened and then heated until it turns blue to temper it to spring hardness. At its inner end it is attached to the balance staff by the brass collet. The spring enters a tangential hole in the collet and is fixed in place by a pin. The outer end of the spring is fixed to the movement, to the balance cock, by the brass stud.

The balance spring in the image is not a perfect spiral. This is not a fault, its outer turn is bent up above the plane of the spring and in towards the centre in a Breguet overcoil. This is to improve isochronism. The rim of the balance has a thin inner steel layer and a thicker outer layer of brass, and is cut through in two places near to the spoke. The two arms of the rim are bimetallic strips that bend in or out in response to temperature changes, to compensate for other temperature effects. Not all balance springs and balances are like this.

In a pendulum clock the major effect of an increase in temperature is that the pendulum becomes slightly longer; a similar effect occurs in watches, the diameter of the balance increases. Both of these effects are quite small, negligible unless an unusual level of precision is required.

Ferdinand Berthoud was the first to tabulate in 1773 the effects of temperature on one his marine watches. He recorded that a temperature change from 32°F to 92°F caused it to lose in 24 hours:

Berthoud's observation and (incorrect) apportionment
by expansion of the balance62 seconds
by loss of the spring's elastic force312 seconds
by elongation of the balance-spring19 seconds
Total loss per day393 seconds

The reasoning lying behind Berthoud's apportioning of the individual losses is not known, and is not correct. It was accepted until 1882 when Mr T. D. Wright, a teacher at the BHI, pointed out that the strength of a spring is proportional to its width and inversely proportional to its length, and as these two dimensions are affected in equal ratio by changes of temperature, there is no overall effect on the strength of the spring. The overall effect of 393 seconds in 24 hours over 60°F equates to 11.8 seconds per day per °C, which is in line with other observations.

In a watch a much more significant effect of increasing temperature is that the modulus of elasticity, also called Young's modulus, of the balance spring reduces. The effect of this is that the spring produces less force for a given angle of rotation. This effect is many times larger than that from the lengthening of a pendulum rod or increasing the diameter of the balance.

A watch that is carried in the pocket or worn on the wrist is kept at a fairly constant temperature by warmth from the body, which mitigates the problem to an extent, but it is usually taken off overnight and becomes cooler. However, precision time references are not normally worn and are therefore subject to all the temperature fluctuations of nature, which were more significant in a time before houses and workshops were heated.

The rate of a watch is determined by the rotational inertia of the balance and the strength of the balance spring. The period is given by

$$T = {\pi} \sqrt \frac {I}{S}$$

where I is the moment of inertia of the balance and S is the elastic moment of the balance spring. This equation gives the period of one excursion of the balance from its central position to its limit of rotation in one direction and back. This is half the time of a complete oscillation and is used by horologists because it is the time between ticks.

With an increase in temperature, thermal expansion causes the balance to increase in diameter, which increases its rotational inertia and causes the watch to run slower. Changes in elasticity of the material of the balance have no effect on timekeeping.

An increase in temperature causes the balance spring to expand in all directions, thickness, breadth and length. The increases in breadth and length have opposite effects on the strength of the spring and cancel each other out. The increase in thickness makes the spring slightly stronger, but this effect is vastly outweighed by the reduced modulus of elasticity of the metal, which makes the spring weaker as the temperature increases.

If a watch has a brass balance and carbon steel balance spring, it would lose over 10 seconds per day for a rise in temperature of just 1°C. It might be thought that a watch with a steel balance, which has nearly half the thermal expansion of a brass balance, would be better, but in fact a watch with a steel balance would lose "only" just under 10 seconds a day for the same 1°C temperature rise. The material the balance is made from has very little effect on temperature errors. The major source of error is the change in the elastic modulus of the spring.

I calculated the individual effects for both brass and steel balances using data for the thermal coefficients of expansion and modulus of elasticity give in A L Rawlings "Science of clocks and watches" and the spreadsheet that you can download from the section about Middle Temperature Error. The results are tabulated below.

Change in rate (seconds per day) for 1°C rise in temperature
Brass balance  Steel balance
Balance thermal expansion-1.64-0.95
Spring thermal expansion1.421.42
Spring decrease in Young's modulus-10.36-10.36
Totals (- indicates loss)-10.58-9.89

The thermal expansion of the spring and the brass balance somewhat compensate each other. The expansion of the thickness of the spring makes it stronger and, all other things being equal, would make the watch run faster by about 1.42 seconds per day. The expansion of the brass balance increases its radius of gyration, and hence its moment of inertia, which would make the watch run slower by about 1.64 seconds per day. These two effects, increasing spring thickness and increasing balance moment of inertia, thus oppose and partially cancel each other. In aggregate they contribute only 0.22 seconds per day to the overall loss of over 10 seconds. It is the change in Young's modulus of the balance spring that contributes the remaining amount.

With a steel balance the gain in rate due to the increase in strength of the spring is actually greater than the loss due to the increased moment of inertia of the steel balance, resulting in a gain in rate of 0.47 seconds per day, reducing the overall loss to less than 10 seconds per day.

Sir George Biddell Airy, the Astronomer Royal from 1835 to 1881, showed by experiment in 1859 that a chronometer with a plain brass balance lost 6.11 seconds in 24 hours for each degree Fahrenheit increase in temperature, equivalent to 11 seconds per degree centigrade, which is in very close agreement with the loss of 10.58 seconds calculated above.

The bottom line is that a watch that is not compensated for temperature variations can be expected to lose or gain around 10 seconds per day for every one degree change in temperature. If such a watch was adjusted to run correctly on the watchmaker's bench at, say, 20°C, which is the same temperature at which it might spend eight hours overnight on the bedside table, and then it was strapped to your wrist at 34°C for the remaining 16 hours of the day, you could expect it to lose over a minute and a half each day.

The fact that most watches do not show such alarming changes in rate is due either to aspects of their design (compensation balance) or materials (autocompensating balance springs) that compensate for or nullify the effects of temperature changes.

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### Temperature Compensation

John Harrison was the first person to successfully apply temperature compensation to a balance controlled timekeeper, in a pocket watch made for him in 1753 by John Jefferys to Harrison's specification. This was the first watch with temperature compensation, and also the first fusee watch with maintaining power, which kept the watch going as it was being wound. Before this invention, fusee watches without maintaining power stopped as they were being wound, losing accuracy. The Jefferys watch also has Harrison's version of the verge escapement, which includes verge pallets with cycloidal backs.

All of Harrison's marine timekeepers included temperature compensation. His first marine timekeeper now called H1, which was constructed between about 1728 and 1735, and the second H2, which was begun in 1737, had gridiron type bimetallic elements similar to the gridiron pendulum Harrison had invented for his land based clocks. For H3, which was begun in 1740, Harrison created a "brass and steel thermometer curb" which was a bimetallic element made from strips of brass and steel riveted together. Because of the differential thermal expansion of brass and steel - brass expands more than steel for a given rise in temperature - as the temperature rises the bimetallic curb will bend. The free end of the curb was fitted with two pins that embraced the balance spring near to the point at which it was attached to the plate, and the bending of the curb was arranged to shorten the effective length of the spring as the temperature rose, compensating for its loss of elasticity. This was the form of temperature compensation used in the Jefferys watch.

Harrison abandoned work on H3 before it was completed, and in 1755 began work on H4, essentially a large pocket watch with a verge escapement and plain steel uncut balance quite similar in overall design to the watch made for him by John Jefferys. H4 has temperature compensation by thermometer curb as used in H3 and the Jefferys watch, maintaining power, Harrison's version of the verge escapement with diamond pallets with cycloidal backs, and a train remontoire, which was its only essential difference from the Jefferys watch.

H4 was the timekeeper that successfully passed the tests stipulated by the 1714 Act of Queen Anne "An Act for Providing a Publick Reward for such Person or Persons as shall Discover the Longitude at Sea" and which resulted in Harrison eventually being awarded the prize for "finding the longitude". There is no question that H4 could not have achieved this feat without adequate temperature compensation, but Harrison was said to be unhappy with the compensation curb because he found that the balance, balance spring, and the compensation curb itself were not all affected at the same time by changes in temperature. The swinging balance and spring would have reacted to temperature changes more quickly that the stationary and more massive compensation curb, and Harrison considered that the compensation would be improved if it was in the balance itself.

The form of temperature compensation seen in nineteenth century pocket watches and early wristwatches uses a "compensation balance" with a cut bimetallic rim, which is discussed in more detail in the next section. The compensation balance was invented by Pierre Le Roy, son of Julien Le Roy, and improved to the form most widely seen by John Arnold and Thomas Earnshaw. The balance rim is made of steel with a layer of brass fused onto the outside, and it is cut so in two that the two arms of the rim can move inwards and outwards. When the temperature increases the extra expansion of the brass compared to the steel causes the parts of the rim to bend inward. This reduces the moment of inertia of the balance, compensating for the weakening of the spring. When the temperature falls the opposite effect occurs, the arms bend outwards to increase the moment of inertia and compensate for the increased strength of the spring.

Development of alloy steels meant that is was possible to make balance springs that did not change in elasticity with temperature, and springs and balances that did not change their dimensions with temperature, so temperature compensation was no longer required. This is discussed in the section below about on autocompensating balance springs.

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## Compensation balance

Marine chronometer compensation balance

The form of temperature compensation first seen eighteenth century marine chronometers, and later in pocket watches and early wristwatches uses a "compensation balance". This followed a principle suggested by John Harrison that rather than using a bimetallic curb to alter the effective length of the balance spring in response to changes it temperature it would be better if the compensation were in the balance itself.

This was achieved by making a balance with a split bimetallic rim that reacted to temperature changes. This balance was invented by Pierre Le Roy, son of Julien Le Roy, and improved to the form most widely seen by John Arnold and Thomas Earnshaw.

The general form of this balance is shown in the picture here. This is a sketch of the balance found in a marine chronometer, a large instrument mounted on gimbals in a cube shaped box. The balance rim is made of an inner layer of steel (coloured grey) with a layer of brass (coloured yellow) fused onto the outside. This turns the rim into a bimetallic strip and it is cut in two places near to the cross bar so that the two arms of the rim can move as shown by the dotted lines. This gives rise to the name and description of this balance as a "split bimetallic temperature compensation balance".

The operation of the balance is as follows. If the temperature increases, the brass on the outside of the rim expands more than the steel and this causes the arms of the rim to bend inward, carrying the weights mounted on the arms inwards towards the central axis of rotation of the balance. This reduces the moment of inertia of the balance, compensating for the weakening of the balance spring. If the temperature falls, the opposite effect occurs and the rims bend outwards, carrying the weights outward and increasing the moment of inertia of the balance.

Longines 13.34 compensation balance

The two large masses mounted part way along the arms were often called compensation weights, although it is their mass that contributes to the moment of inertia of the balance. The amount of compensation produced in response to a given change in temperature can be increased or decreased by sliding the masses along the arms. The further along the arms away from the cross bar the weights are positioned, the greater the effect of the compensation.

Experiments by Kullberg in 1887 had shown that the long arms of were significantly affected by centrifugal force. One of the balances he submitted to the Royal Observatory for testing was fitted with an "ordinary compensation balance" which had rims of thickness 0.038 inches (less than 1 millimetre) thick. The "length" of the acting laminae, the bimetallic strips, was given as 135° and the compensation masses were positioned 98° from the bar. A chronometer fitted with this balance was tested with the balance making long "arcs" of one turn and a fifth and short arcs three quarters of a turn, i.e. amplitudes of 432° and 270°. The mean daily rate in the short arcs was +2.6 seconds, in the long arcs it was +0.5 seconds. The centrifugal effect caused the compensation masses to bend the rims out further in the long arcs than the short, increasing the radius of gyration and slowing the rate by over 2 seconds a day. This shows how important the fusee was to the performance of a chronometer with a cut bimetallic balance, because by keeping the impulse constant it kept the amplitude constant.

In "The Marine Chronometer" Commander Gould implies that the compensation masses could be at 120°, which would make the centrifugal effect even more prominent. In the diagram here I have shown them at 98° as per Kullberg's data.

The two large screws at the end of the cross bar are mean time screws, used to adjust the rate when the best temperature compensation has been established, the two small screws next to them are for very fine adjustments to the rate.

In a smaller movement such as a pocket or wristwatch there is not enough room for the large compensation weights and meantime screws, so a number of small screws distributed along the length of the arms are used. Changes in the compensation are effected by moving some of the screws, and fine adjustments are achieved by fitting thin timing washers, or by reducing the size of some of the screw heads.

The balance shown in the photograph here is fitted to a Longines 13 ligne calibre 13.34 movement dated to 1913. It is a high grade version of the 13.34 calibre with the train jewels set in screw set chatons, cap jewels for the escape wheel, and jewelled to the centre, giving a total of 18 jewels.

The two different metals of the balance rim, steel on the inside and brass on the outside, are visible, and it is notable that the steel is much thinner than the brass; this was to get maximum movement from the bimetallic arms. One of the two splits in the rim that allow the bimetallic arms to move is visible at the top of the photograph near to the steel stud carrier. The other split is concealed by the centre wheel. The timing screws are visible, distributed along the arms of the balance. These screws are made of gold, which was used because of its greater density compared to steel; a screw made of gold weighs more than twice as much than the same screw made of steel, and therefore is more effective in adjusting the compensation.

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### Middle Temperature Error

HSN article about Middle Temperature Error. A313S152

In the late eighteenth century a phenomenon was observed in marine chronometers with temperature compensation. It was found that if the device was brought to time at a certain temperature it would lose at higher and lower temperatures. This effect was not observed in watches without temperature compensation because it was dwarfed by other effects, principally the change in rate with temperature due to variation in the elasticity of the balance spring. The effect became observable in marine chronometers with compensation for this major, primary source, of temperature caused error.

To minimise the total error over the range of temperatures a marine chronometer was expected to encounter, the timing was adjusted so that it was fast at a "middle" temperature and correct at two temperatures either side of this. This gaining rate at the middle temperature was called the "middle temperature error".

Because the effect only became noticeable once the primary source of error, the change in the elasticity of the balance spring with temperature, had been compensated, it was also called "secondary error".

The spreadsheet that accompanies the article allows you to interactively explore the effects of MTE. I strongly recommend that you download and try it. You don't need to do any spreadsheet programming, it is already set up. You just alter the values of the thermal coefficients of expansion and elasticity, and charts built into the spreadsheet immediately show you the effect on timekeeping. It's really simple so give it a go, and if you have any problems just drop me an email. If you don't have the Microsoft Office Excel spreadsheet application, then Libre Office contains an excellent alternative and is available absolutely free from Download Libre Office.

The spreadsheet can also be used to investigate the effects of the individual components. For example, to see the effect of thermal expansion of the balance alone, then set all the coefficients apart from the "Thermal expansion/ºC – balance:" to zero. As a check, a brass balance should show a loss of 1.64 seconds per day for a temperature increase of one degree C, a steel balance 0.95 seconds per day per degree C.

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#### Explanations of Middle Temperature Error

An explanation for middle temperature error was developed by the Reverend George Fisher and published in The Nautical Magazine of 1842 under the name of E. J. Dent, of the chronometer makers Frodsham & Dent. Fisher's name is not mentioned in Dent's article, probably because Fisher had incurred the displeasure of the Astronomer Royal by publishing a paper that suggested that the going of chronometers could be affected by magnetism in iron ships. The Astronomer Royal refuted this in polite but strong terms and Fisher probably decided to keep a low profile as a result.

The explanation published by Dent was based on the observation that the equation for the period of a balance controlled timepiece can be written as:

$$T = {\pi} \sqrt \frac { m k^2}{S}$$

where mk2 is the moment of inertia of the balance, m being the mass and k the radius of gyration, and S represents the force of the balance spring. Observations by Dent had indicated that the elasticity of the balance spring varied linearly with temperature.

Fisher surmised that it was the ratio within the square root of k2 to a linear S that was the cause of middle temperature error. Clearly a curve produced by plotting the k2 term against temperature on a chart could only intersect with a straight line representing a linear S term at either one or, at most, two points. This is the explanation related by Commander Gould in "The Marine Chronometer".

Fisher's explanation considers the relationship of inertia to spring force within the encompassing square root sign of the equation for period. An alternative way of visualising the effect is by recognising that the period is proportional to the square root of the square of the radius of gyration k and rewriting the equation as:

$$T \propto \frac { k} {\sqrt S}$$

If both k and S vary linearly with temperature, as was supposed, then a plot of k against temperature on a chart would be a straight line which could only intersect with a curve representing the square root of the linear S term at either one or, at most, two points. This is the explanation related by A. L. Rawlings in "The Science of Clocks and Watches".

The two different versions use the same geometric or mechanical phenomenon to explain middle temperature error, the only difference is that the Fisher / Dent version considers the ratio k2 to S inside the square root sign, whereas Rawlings looks at the whole equation for period including the square root and considers the ratio of k to √ S .

Middle temperature error or secondary error: seconds per day.
1: Blue line; square root effect alone.
2: Orange line; square root plus non-linear elasticity of balance spring.

This "square root" explanation is rational, logical and correct, and it held sway as the only explanation for middle temperature error for many years. However, although the reasoning is correct, the magnitude of the middle temperature error produced by this effect is smaller than observed in practice. This is not a criticism of Fisher and Dent and others, it is just that until there was an easy way of calculating the effect and plotting it, there was no reason to seek an alternative explanation.

The shortcoming in the square root explanation was noticed by Peter Baxandall during the updating of Rawlings' classic work by the BHI, and subsequently investigated by Philip Woodward in a paper published in the Horological Journal of April 2011. Woodward showed that the middle temperature error produced by the square root effect alone was around one tenth of a second per day, only a small fraction of observed values.

There is no doubt that the square root effect does exist, but a cause for the remaining majority of the error must be sought.

In the early twentieth century Dr Guillaume explained that middle temperature error, which he called "secondary error" or "Dent's error", is due to the fact that the elasticity of the balance spring does not vary linearly with temperature but has a curvature. It is not clear that Guillaume ascribed any of the middle temperature error to the square root effect, he doesn't mention it in his writings. From his work on changes in the dimensions and elastic moduli of nickel steels with changes in temperature he would have known that any such effect would be small, it appears that he probably thought it would be negligible. Dr Guillaume's explanation of the effect can be found at The Guillaume "integral" balance.

The chart here shows the two effects on the timekeeping of a machine brought to time at 5°C and 35°C, with a middle temperature of 20°C. The blue line shows the middle temperature error that results from the square root explanation alone. It is very small, of the order of one tenth of a second per day. The orange line shows the error that results when the curvature in the temperature response of the elasticity of the balance spring is also taken into account. This results in an error of more than two seconds per day, which is in accordance with observed values.

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## Auto-compensation

Development of alloy steels meant that is was possible to make balance springs that did not change in elasticity with temperature, and springs and balances that did not change their dimensions with temperature, so temperature compensation was no longer required.

In the 1890s Dr. Guillaume of the Bureau of Weights and Measures in Paris was searching for a material that did not change size with temperature to make standard lengths, and in 1896 discovered that a nickel steel alloy which he called Invar had a very low temperature coefficient of expansion. When Dr. Guillaume announced this discovery a Swiss watch timer Paul Perret of La Chaux-de-Fonds immediately asked him for a sample. Perret had been making watches with palladium balance springs with good results in the 1896/97 trials in Berne and was always on the lookout for new materials.

Perret made the sample of Invar into a balance spring and was very surprised when a watch fitted with this spring increased in rate with increasing temperature. With this, Perret had discovered that the thermoelastic coefficient of the sample was positive, whereas normal steel balance springs had a negative thermoelastic coefficient. This meant that the Invar spring got stronger as it got hotter, compensating for the increase in dimensions of itself and the balance. Paul Perret and Dr Guillaume agreed to collaborate on the study of thermoelastic effects in nickel steels and discovered two nickel steel alloys with 28% and 43% nickel content whose thermoelastic coefficient at normal temperatures is zero. For compositions between 28% and 43% nickel the thermoelastic coefficient is positive, outside this range it is negative, as in normal steels.

Paul Perret patented the use of a nickel-steel balance spring with a positive temperature coefficient of elasticity in in Switzerland on 6 May 1897 as CH 14270, in Great Britain on 5 February 1898 as GB 25,142 and in the USA on March 12, 1901 as US 669,763. Perret set up a company to market these balance springs, as shown by the advert from 1901 reproduced here. The springs were manufactured by the Swiss company La Société des fabriques de spiraux réunies.

Paul Perret 1901 advert for compensation balance springs. Click image to enlarge.

This patenting of nickel steel balance springs by Paul Perret did not lead to any problems between him and Dr. Guillaume. Throughout the rest of his career until his untimely death in 1903 at only 49 years old, Paul Perret and Dr Guillaume worked together studying the properties of nickel steels and autocompensating balance springs (springs which compensate for the thermal expansion of themselves and the balance by having an elastic modulus that increases with temperature) and balances.

The Paul Perret nickel steel balance springs had one drawback. Although the thermoelastic coefficient was zero at normal temperature it varied at higher or lower temperatures. The elastic modulus of the alloy followed a curve that reached a maximum at room temperature. The decrease in the modulus either side of this peak resulted in a secondary error that Dr. Guillaume said was of 20 to 25 seconds in twenty-four hours over a temperature range of 30 degrees. The advert for Paul Perret balance springs shows this effect.

The advert says that watches fitted with "ordinary non magnetic" balance springs varied their rate by 15 to 18 seconds per degree centigrade, which is rather a lot. A well made steel balance spring resulted in a variation of 9 to 11 seconds per degree centigrade. Two classes of Paul Perret compensation balance springs are then listed, a half compensation at 3 to 5 seconds per degree centigrade, and full compensation at 0 to 1 second. Although not stated in the advert, all these figures are over 24 hours.

Although the Paul Perret nickel steel balance springs did not provide perfect compensation they were considerably better than plain steel balance springs. They enabled cheaper watches to be compensated for temperature for the first time. The performance of such watches was better than Dr. Guillaumes comments about 20 to 25 seconds in 24 hours would suggest, because watches worn or carried on the person do not experience 30 degree changes in temperature. The temperature compensation of watches with nickel steel balance springs was not as good as those with bimetallic compensation balances, which more expensive watches continued to be fitted with.

Marine chronometers are expected to perform very accurately over a wide range of temperatures. Nickel steel balance springs could not ecause of this, Dr Guillaume invented an alternative means of compensation for chronometers where very high accuracy was required, the Guillaume or "integral" balance.

In 1912 Dr Guillame, working together with Pierre Chenevard of the Imphy steelworks, discovered that the addition of 12% chromium to the nickel steel used to make balance springs reduced the sensitivity of the thermoelastic coefficient to the precise composition of the alloy, and also eliminated most of the secondary error.

A problem with nickel steel balance springs was that they were relatively soft and could be distorted if they were not handled very carefully. In 1933 Dr. Reinhard Straumann, technical director of Thommen S.A. working in conjunction with Heraeus Vacuumschmelze G.m.b.H. invented and patented an auto-compensating balance spring material that he called "Nivarox". This was a nickel-iron alloy with beryllium in place of carbon in steel, and with molybdenum, tungsten and chromium. Nivarox could be made non-magnetic and the thermal coefficient of its modulus of elasticity controlled by heat treatment.

Today Vacuumschmelze is a leading global manufacturer of advanced magnetic materials and related products, still making an alloy called Nivarox CT® which is used for the balance springs of mechanical watches. Straumann used the knowledge that he gained of corrosion resistant products to found a medical implant company, which is also still thriving.

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## The Guillaume "integral" balance

After Dr Guillaume published the discovery of invar in 1896, Paul Perret found that it had a positive thermoelastic coefficient. This led to the creation of nickel steel balance springs which improved the temperature compensation of ordinary watches considerably. But although the thermoelastic coefficient was zero at normal temperatures, at higher or lower temperatures it varied, which resulted in a secondary error of 20 to 25 seconds in twenty-four hours over a temperature range of 30 degrees.

Figure 1: Ordinary bimetallic compensation

Marine chronometers with bimetallic compensation balances already achieved a better performance than this, but their accuracy was affected by middle temperature error. Many auxiliary compensation devices had been invented to counter middle temperature error, but they were delicate and difficult to adjust. Dr Guillaume realised that the properties of nickel steel could be exploited to overcome middle temperature error, or secondary error as he called it, and invented the Guillaume or "integral" balance.

To understand how Dr Guillaume created the integral balance it is useful to understand his view of how middle temperature error arose. The two charts here are based on figures presented by Dr Guillaume in his Nobel lecture. I find those a little misleading because effect of the ordinary bimetallic compensation is magnified so that the curves showing spring tension are on the same scale in both diagrams. I have redrawn the diagrams using coefficients quoted by Dr Guillaume but without any artificial magnification. The scales on the y axis are the same in both figures so they are directly comparable.

Figure 1 shows the expansion of the brass and steel parts of the bimetallic rim plotted against temperature, the blue and red curves. The yellow curve, which is virtually a straight line, is the difference between these and represents the radial movement of the compensation masses. Dr Guillaume explained that this was linear because the coefficients that described the curvatures of the brass and steel lines were almost exactly the same.

The green line on Figure 1 represents the decrease in elasticity of the balance spring with increasing temperature. Dr Guillaume explained that rather than being straight this had a curvature as shown. This should not be a surprise, the strength of a material is determined by inter-atomic or inter-molecular forces. As increasing temperature expands the material by causing the molecules to move further apart the strength of those bonds must decrease by an inverse square law type effect. In fact, Dent's data from his experiments with a chronometer fitted with a glass balance show this curvature.

The rate of the chronometer is determined by how well the compensation matches the change in strength of the balance spring. It is apparent that the straight yellow line representing the movement of the compensation masses does not exactly mirror the green curve of the balance spring, resulting in the curved rate line showing the middle temperature error at 15 degrees.

Figure 2: Guillaume integral balance

Dr Guillaume realised that by exploiting the unusual properties of nickel steel he could alter change the rate of movement of the compensation masses to mirror the change in strength of the balance spring. Brass and ordinary steel, like most metals, have a small positive curvature in their rate of temperature expansion as shown in Figure 1.

Some nickel steel alloys have the unusual property of a negative curvature in their temperature expansion at normal room temperatures and Dr Guillaume used a nickel steel alloy that expanded as shown by the red curve on Figure 2. This alloy also had a much lower linear coefficient of expansion than plain steel, resulting in the nearly horizontal downward curving line. Dr Guillaume called this material "Anibal", short for "acier-nickel pour balanciers" (nickel steel for balances).

The steel inner layer of a bimetallic compensation balance was replaced by the nickel steel Anibal alloy. The pale blue curve shows the difference between the expansion of the brass layer, indicated by the dark blue line, and the Anibal layer, shown by the red line. Compared to the yellow line of Figure 1 the pale blue line shows a much greater response to temperature change because of the difference in the expansibilities of steel and the nickel-steel Anibal.

Because the difference between the expansion of brass and nickel-steel is much greater than between brass and steel, therefore a bimetallic strip made from brass and nickel-steel deflects more for a given change of temperature than a brass / steel one does.

To show this effect the pale blue line in Figure 2 is scaled down by a factor of 3.65 to produce the yellow line. This line is similar to the yellow line of Figure 1, and as in Figure 1 the the difference between the yellow line and the green line of the spring tension shows the compensation in action. However, in Figure 2 the yellow line has an upwards curve that matches the downwards curve of the green spring line, eliminating the secondary error and producing a flat rate over the temperature range.

Guillaume integral balance

The Guillaume Integral balance was designed to be used with carbon steel balance springs, the same as ordinary brass / steel compensation balances. This meant that in both types of balance the compensation masses had to move essentially the same distances, although the masses in the Guillaume balance moved with the non-linearity needed to eliminate secondary error. Because the deflection of the arms of the Guillaume balance was greater for the same temperature change, a shorter length of rim produced the necessary movement of the masses. Essentially the masses could be placed closer to the spoke of the Guillaume balance.

To produce the necessary movement of the compensation masses, an ordinary brass / steel compensation balance had to have long thin arms. Kullberg had shown in 1887 that these long arms were affected by centrifugal force. One of the balances he submitted to the Royal Observatory for testing was an "ordinary compensation balance" which had rims of thickness 0.038 inches (less than 1 millimetre) thick. The "length" of the acting laminae, the bimetallic strips, was given as 135° and the compensation masses were positioned 98° from the bar. With such long thin bimetallic arms it is hardly surprising that the ordinary bimetallic balance could be significantly affected by centrifugal force

The Guillaume compensation balance produced the required movement of the masses with much shorter arms, which were also less susceptible to centrifugal force. Because the arms were shorter they could be cut at 90° to the bar and four smaller compensation masses used. The distance these had to be moved to provide the compensation was essentially the same as in the ordinary compensation balance, but instead of being position 100° to 120° along the laminae they could be at about 45°.

The drawing of a Guillaume compensation balance shown here can be compared to the drawing of an ordinary marine chronometer compensation balance in the section about the compensation balance.

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