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An equation clock is a mechanical
clock A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and the ...
which includes a mechanism that simulates the
equation of time In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
, so that the user can read or calculate
solar time Solar time is a calculation of the passage of time based on the position of the Sun in the sky. The fundamental unit of solar time is the day, based on the synodic rotation period. Two types of solar time are apparent solar time (sundial ti ...
, as would be shown by a
sundial A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a flat ...
. The first accurate clocks, controlled by
pendulum A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the ...
s, were patented by
Christiaan Huyghens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists o ...
in 1657. For the next few decades, people were still accustomed to using sundials, and wanted to be able to use clocks to find solar time. Equation clocks were invented to fill this need. Early equation clocks have a pointer that moves to show the equation of time on a dial or scale. The clock itself runs at constant speed. The user calculates solar time by adding the equation of time to the clock reading. Later equation clocks, made in the 18th Century, perform the compensation automatically, so the clock directly shows solar time. Some of them also show
mean time ''Mean Time'' is the second studio album by English rock band The Barracudas, released in 1983 by record label Closer. The incorporation of Chris Wilson from The Flamin' Groovies gave this new formation a paramount weight in the eighties' garag ...
, which is often called "clock time".


Simulation mechanisms

All equation clocks include a mechanism that simulates the equation of time, so a lever moves, or a shaft rotates, in a way that represents the variations of the equation of time as the year progresses. There are two frequently-used types of mechanism:


Cam and lever mechanism

In this type of mechanism, a shaft is driven by the clock so it rotates once a year, at constant speed. The shaft carries a
cam Calmodulin (CaM) (an abbreviation for calcium-modulated protein) is a multifunctional intermediate calcium-binding messenger protein expressed in all eukaryotic cells. It is an intracellular target of the secondary messenger Ca2+, and the bind ...
, which is approximately "kidney shaped" such that its radius is essentially a graph of the annual variation of the equation of time. A follower and lever rest against the cam, so that as the cam rotates the lever moves in a way that represents the changing equation of time. This lever drives other components in the clock.


Double shaft mechanism

To a close approximation, the equation of time can be represented as the sum of two sine waves, one with a period of one year and the other with a period of six months, with the relative phase varying very slowly (marginally noticeable over the course of a century). See the explanation in ''
Equation of time In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
'' for more detail. The double shaft mechanism has two shafts rotating at constant speeds: one turns once a year, and the other twice a year. Cranks or pins attached to the two shafts move the two ends of a
combining lever A whippletree, or whiffletree, is a mechanism to distribute force evenly through linkages. It is also referred to as an equalizer, leader bar, or double tree. It consists of a bar pivoted at or near the centre, with force applied from one direct ...
(sometimes referred to as a whippletree) sinusoidally; if the dimensions are chosen correctly, the midpoint of the rod moves in a way that simulates the equation of time.


Types of equation clock

During the period when equation clocks were made and used, all clocks were made by hand. No two are exactly alike. Many equation clocks also have other features, such as displays of the phase of the moon or the times of sunrise and sunset. Leaving aside such additions, there are four different ways in which the clocks function. The following paragraphs are intended, not as detailed descriptions of individual clocks, but as illustrations of the general principles of these four different types of equation clock. The basic workings of particular clocks resemble these, but details vary. Pictures and descriptions of various equation clocks, which still exist in museums, can be accessed through the ''
External links An internal link is a type of hyperlink on a web page to another page or resource, such as an image or document, on the same website or domain. Hyperlinks are considered either "external" or "internal" depending on their target or destination ...
'' listed below.


Clocks without solar time displays

Many equation clocks, especially early ones, have a normal clock mechanism, showing mean time, and also a display that shows the equation of time. An equation of time simulation mechanism drives the pointer on this display. The user has to add the equation of time to clock time to calculate solar time.


Clocks that directly display solar time

Most later equation clocks, made in the 18th Century, directly display solar time. Many of them also display mean time and the equation of time, but the user does not have to perform addition. Three types exist:


Clocks with movable minute markings

Clocks have been constructed in which the minute markings are on a circular plate that can be turned around the same axis as the hands. The axis passes through a hole in the centre of the plate, and the hands are in front of the plate. The minutes part of the time shown by the clock is given by the position of the minute hand relative to the markings on the plate. The hand is driven clockwise at constant speed by the clock mechanism, and the plate is turned by the mechanism that simulates the equation of time, rotating anticlockwise as the equation of time increases, and clockwise when it decreases. If the gear ratios are correct, the clock shows solar time. Mean time can also be shown by a separate, stationary set of minute markings on the dial, outside the edge of the plate. The hour display is not adjusted for the equation of time, so the hour reading is slightly approximate. This has no practical effect, since it is always easy to see which hour is correct. These clocks are mechanically simpler than the other types described below, but they have disadvantages: Solar time is difficult to read without looking closely at the minute markings, and the clock cannot be made to strike the hour in solar time.


Clocks with variable pendula

These clocks include a device at the top of the pendulum that slightly changes its effective length, so the speed of the clock varies. This device is driven by a simulation mechanism which moves to simulate the rate of change of the equation of time, rather than its actual value. For example, during the months of December and January, when the equation of time is decreasing so a sundial runs slower than usual, the mechanism makes the pendulum effectively longer than usual, so the clock runs slower and keeps pace with sundial time. At other times of the year, the pendulum is shortened, so the clock runs faster, again keeping pace with sundial time. This type of mechanism shows only solar (sundial) time. Clocks using it cannot easily be made to show mean time unless a separate clock mechanism, with its own pendulum, is included. There are some equation clocks in which this is done, but it requires the clock case to be very sturdy, to avoid
coupling A coupling is a device used to connect two shafts together at their ends for the purpose of transmitting power. The primary purpose of couplings is to join two pieces of rotating equipment while permitting some degree of misalignment or end mov ...
between the pendula. Another disadvantage of variable pendulum clocks is that the equation of time cannot be easily displayed.


Clocks that do mechanical addition

In some later equation clocks, a pendulum swings at a constant
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, controlling a normal clock mechanism. Often, this mechanism drives a display showing mean (clock) time. However, there are additional components: an equation of time simulation mechanism as described above, and a device that automatically adds the equation of time to clock time, and drives a display that shows solar time. The addition is done by an analogue method, using a
differential gear A differential is a gear train with three drive shafts that has the property that the rotational speed of one shaft is the average of the speeds of the others, or a fixed multiple of that average. Functional description The following descr ...
. This type of equation clock mechanism is the most versatile. Both solar and mean times can be easily and clearly displayed, as can the equation of time. Striking the hours in both kinds of time is also easy to arrange. After its invention in 1720, this mechanism became the standard one, and was used for much of the 18th Century, until the demand for equation clocks ceased.


Slow changes in the equation of time

Slow changes in the motions of the Earth cause gradual changes in the annual variation of the equation of time. The graph at the top of this article shows the annual variation as it is at present, around the year 2000. Many centuries in the past or future, the shape of the graph would be very different. Most equation clocks were constructed some three centuries ago, since when the change in the annual variation of the equation of time has been small, but appreciable. The clocks embody the annual variation as it was when they were made. They do not compensate for the slow changes, which were then unknown, so they are slightly less accurate now than they were when new. The greatest error from this cause is currently about one minute. Centuries in the future, if these clocks survive, the errors will be larger.


Similar modern devices

Equation clocks, as such, are no longer widely used. However, components functionally the same as those in equation clocks are still used in, for example,
solar tracker A solar tracker is a device that orients a payload toward the Sun. Payloads are usually solar panels, parabolic troughs, fresnel reflectors, lenses or the mirrors of a heliostat. For flat-panel photovoltaic systems, trackers are used to mi ...
s, which move so as to follow the movements of the Sun in the sky. Many of them do not sense the position of the Sun. Instead, they have a mechanism which rotates about a polar axis at a constant speed of 15 degrees per hour, keeping pace with the average speed of the Earth's rotation relative to the Sun. Sometimes, a digital representation of this rotation is generated, rather than physical rotation of a component. The equation of time is then added to this constant rotation, producing a rotation of the tracker that keeps pace with the apparent motion of the Sun. Generally, these machines use modern technology, involving electronics and computers, instead of the mechanical devices that were used in historic equation clocks, but the function is the same.


See also

*
Equation of time In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
*
Sundial A sundial is a horological device that tells the time of day (referred to as civil time in modern usage) when direct sunlight shines by the apparent position of the Sun in the sky. In the narrowest sense of the word, it consists of a flat ...
*
Clock A clock or a timepiece is a device used to measure and indicate time. The clock is one of the oldest human inventions, meeting the need to measure intervals of time shorter than the natural units such as the day, the lunar month and the ...
*
Differential (mechanical device) A differential is a gear train with three drive shafts that has the property that the rotational speed of one shaft is the average of the speeds of the others, or a fixed multiple of that average. Functional description The following descr ...


References and footnotes

{{reflist


External links

Note: In some of these historical materials, clock time is called "equal time", and sundial time is called "apparent time" or "true solar time".
Variable pendulum clock
in the
British Museum The British Museum is a public museum dedicated to human history, art and culture located in the Bloomsbury area of London. Its permanent collection of eight million works is among the largest and most comprehensive in existence. It docum ...
.
British Museum equation clocks, with descriptions.

Pocket watch that works as an equation clock.
The description suggests that it does mechanical addition.

* ttps://archive.org/stream/philtrans05753149/05753149#page/n0/mode/2up Letter from Joseph Williamsonwritten c.1715, claiming the invention of clocks showing solar time, working by movable minute marks or variable pendula.
The Equation of Time
gives illustrated examples of the development of equation clocks Clocks