Readers interested in this subject should also refer to the page on this site, titled "Mathematically breaking down the Equation"


Showing how the Equation of Time can be generated by one gear rotating around another.

Showing how the Equation of Time can be generated in a practical manner by 2 Tusi Couples.

Kurt Descovich - 2016 - Schwartenau Heliochronometer : Ref NASS Compendium v23(1) Mar 2016 (see also page on Moving Hour Lines Sundials) The Left end of the blue slider is fixed (except when adjusting for Daylight Savings time, The Right end is connected to the rotatable equatorial ring
Schwartenau Heliochronometer. the knob at the bottom adjusts for the year in the leap year cycle. The little pointer in the middle of the picture allows the tiny adjustment of the movement of perihelion towards the vernal equinox.
Click on images to enlarge & view captions

Mark Frank - 2014 - Astro Skeleton Clock. Short speeded-up video showing how the differential drive outboard wheel dips and rises throughout the year controlled by a equation kidney cam. This action is one year compressed into only a few seconds.

The Rittenhouse Clock at Drexel University - 1773 

David Rittenhouse : 1732 – 1796 - Politician, Surveyor, Inventor, Mathematician Astronomer & Clockmaker 

Probably the first significant early gear driven Equation clock was made in Pennsylvania by the polymath David Rittenhouse. His astronomical musical clock is a masterpiece. The three images below are from "The Most Important Clock in America" by Ronald R. Hoppes : The American Philosophical Society : 2009
The Rittenhouse Clock in Drexel University
Face of the Clock
The Equation Face
Rittenhouse was aware that the generation of the Equation of Time using two gears would, at best,  have an accuracy of around 60 seconds of the true value, using an epicyclic mechanism - with a single fixed gear with planet gear of the same size rotating about it. See animation below
Click to Play
In the above animation, the distance between the stationary magenta gear is proportional to the Elliptical component of the EoT, while the length of  the cyan arm is proportional to the Obliquity component. The relative positions of the green gear and cyan arm on 1st January reflects the different phases of these two signals : one originating at Perihelion (ca 3 January) the other at the Equinox (ca, 20 March).
In the graph of the EoT generated by this mechanism, the fine blue line represents the true value of the  Equation
Rittenhouse's improvement was to add the orange gear - rotating at twice the speed of the yellow planet-carrying wheel. This carried the black pin operating on the blue arm. This, in turn rotated the previously fixed magenta central gear. In the animation, the effect of this change has been greatly exaggerated to show the effect.
This final animation show the tiny increment that was actually added by Rittenhouse's correction.

Jean-Baptiste Schwilgué 1776-1856

The Strasbourg Cathedral Clock was built on 1353. It was restored in 1574. By 1788, it stopped completely : worn out and no longer astronomically correct. It 1842, the clock mechanics were completely rebuilt by Jean-Baptiste Schwilgué, who has admired the 'dead' clock as a boy
Strasbourg Clock - The Equation mechanism is at the bottom right corner.Illustration from Schwilgué's own documentation.
The Clock today
The Clock's Solar & Lunar Dial. Also showing the time of sunrise and sunset.
The Solar and Lunar Computer
The Equation Mechanism works through both gears and profiles (profiles being the vertical equivalent to a cam), The advantage of using a profile/cam - (provided it can be machined accurately) is that it can remove the error inherent in the Obliquity component - which is only approximates a sine function rather badly..
The astronomically technical advance in this mechanism is the introduction of both the anomalistic year (365.259636 days between successive mean perihelions) and the tropical year (265.242189 days between successive mean vernal equinoxs). Since these are different, perihelion is moving towards the vernal equinox, so over the long term, the Equation will vary. 

Jens Olsen 1872-1945

Almost certainly, the most significant Equation modelling has been accomplished by the famous Jens Olsen clock that can be seen in the town hall in Copenhagen. The clock - comprising more than 15,000 parts, was completed after the death of Jens Olsen in 1955.

The Front of the Jens Olsen Clock. The top left face shows the Equation of Time, local time and Solar Time

Detail of Equation Clock Face. TOP - (A) Equation of Time & (B) EoT longitude corrected (these hands are fixed 9 mins 47 second apart ; the longitude difference between CET's 15° East and Copenhagen.  LEFT Local Mean Time.  RIGHT True Solar Time. The clock was not running when this photo was taken (see the vast difference between local and true solar time).

The Rear of the Jens Olsen Clock. The rectangular section (bottom centre) generated the 5 sine components of the lunar motion and, at the right , the 2 main & 1 small sine components of the Equation of Time. The connection between the signal generation and the Dial (now top right) is via a steel ribbon.

Note that the black & white illustrations, following, were taken from Otto Mortensen's book "Jens Olsen's Clock" - 1957 - Copenhagen Technological Institute. No longer in print. Otto Mortensen was Olsen's  friend and collaborator. He was the chief clockmaker who completed the clock, following Olsen's death.
Moon & Equation Works
The 2 Tusi Couples
Details of Equation Mechanism,

Behind the Tusi Couples, is the gear work to generate the anomalistic and tropical year

Jan Olsen's own diagram showing the generation of Tropical and Anomalistic year rotations

Detail of the 2 Tusi Couples and the 3rd order correction

The two Tusi couples, having been summed by a pulley system are connected via a steel ribbon to the Equation face, seen below.
Detail of Rear of Equation Face : Whole works - top left is the quarter-round sector with steel ribbon from Equation mechanism and to left centre, the connection rod to main pendulum mean time mechanism.
Further Detail

Prof. Dr.-Ing. Hans Scheurenbrand

Prof Scheureenbrand worked as Director of Research and Development at the Festo Group - a leader in automation technology. This remarkable modern clock is in the Festo headquarters in Esslingen. It comprise a calendar clock, an astrolabe in the form on an astronomical clock, an artistically designed glockenspiel featuring 76 bells, 40 tuned bars and a claviatur. The Equation of Time mechanism is a double sine mechanism based on circular cams added with differentials.
The illustrations are from Festo Group publications 
Festo Harmonices Mundi Astronomical Face
Festo Harmonices Mundi, showing Solar Hand and Not Indicator
Festo Harmonices Mundi Auxiliary Dials showing summer & winter Civil Time (left) and Solar Time & the Equation of Time (right) with gearing for the Ellipticity effect driven by the anomalistic year gearing (bottom left) and the gearing for the Obliquity effect driven by the tropical year gearing (bottom right)
Festo Harmonices Mundi EoT Gearing for anomalistic and tropical years and the summation with differentials to produce the Equation
Click on images to enlarge & view captions
The sun once more corrects the Clock
The Clock of the Long Now is an extraordinary project to build a clock which - with appropriate maintenance - will run for 10,000 years. A prototype (some 3 metres tall) has been completed and has toured the science museums of the world. A second prototype is being built inside a mountain in Texas. The final clock will be built in Mount Washington in Nevada, The clock runs with a torsion pendulum, but needs to be corrected every now and then, since it is a mechanical device. This will be done whenever the sun is shining at noon. A lens focuses the sunlight onto a bar which heats and bends. This resets the clock to mean noon, via an Equation of Time cam, which must reflect the 10,000 year design life of the clock.
Prototype 1 - Solar Noon Correction Lens and the 10,000 years EoT Cam
Prototype 1 - 10,000 year EoT Cam Replica
Model of Prototype 2 - showing the scale of Project.
Model of Prototype 2 - showing the scale of Project.
Prototype 2 - showing the scale of part of the mechanism
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