Readers interested in this subject should also refer to the page on this site, titled "Mathematically breaking down the Equation"
GENERATING THE EQUATION WITH GEARS
The Rittenhouse Clock at Drexel University - 1773
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
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
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.
THE SCHWILGUÉ STRASBOURG CLOCK 1838
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
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 only rather badly approximates a sine function.
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.
THE JENS OLSEN WORLD CLOCK 1955
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.
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.
Behind the Tusi Couples, is the gear work to generate the anomalistic and tropical year