Skip Navigation Links
 Products Expand  Products 
 Reference Expand  Reference 


A cometarium is a mechanical illustration of Johannes Kepler's 2nd Law, which states that an object in orbit sweeps out equal areas in equal time intervals. These were used in lecture demonstrations beginning in the 18th Century, not long after Halley and Newton proved Kepler's assertion about orbits, thanks in part to the timely return of what we now call "Halley's Comet".

Cometarium Picture

Understanding Orbits

The cometarium was invented at a time when man's understanding of orbital mechanics was evolving rapidly and the middle class began to ask for lecture demonstrations of the new findings.

The story of today's understanding of orbits has more chapters than can be described here, but one must first credit the work of the Danish astronomer Tycho Brahe, who worked to test the Copernican model of the solar system by measuring the parallax of Mars. If Mars were to come closer to the Earth than the Sun, the Ptolemaic system would be disproved. Brahe's instruments were the best available of the time but not sufficiently accurate to resolve this parallax. Beginning in 1600 Johannes Kepler worked with Brahe analyzing some of the new Mars observations. By nature, Kepler defended heliocentrism, and he was skeptical of Brahe's "geo-heliocentric" system now known as the Tychonic system. When Brahe died in 1601, Kepler was appointed his successor, and now had full access to 20 years of Mars data.

Kepler calculated and recalculated various approximations of Mars' orbit, eventually creating a model that generally agreed with Brahe's observations to within two arcminutes (the average measurement error). He was not satisfied with the complex and still slightly inaccurate result; at certain points the model differed from the data by up to eight arcminutes.

Based on measurements of the aphelion and perihelion of the Earth and Mars, Kepler created a formula in which a planet's rate of motion is inversely proportional to its distance from the Sun. Verifying this relationship throughout the orbital cycle, however, required very extensive calculation; to simplify this task, by late 1602 Kepler reformulated the proportion in terms of geometry: planets sweep out equal areas in equal times — Kepler's second law of planetary motion. This is the law the cometarium illustrates.

Much of Kepler's work through 1604 involved a complete calculation of the orbit of Mars. Calculations of three different eccentricities at 1 degree intervals in steps of 1 degree to 5 digit accuracy was most tedious. That these calculations disagreed with Brahe's observations are in effect a tribute to the accuracy of Brahe's observational data. After about 40 failed attempts, Kepler hit upon the idea of an ellipse, which he had previously assumed to be too simple a solution for earlier astronomers to have overlooked.

Finding that an elliptical orbit fit the Mars data, he immediately concluded that all planets move in ellipses, with the Sun at one focus — Kepler's first law of planetary motion. Because he employed no calculating assistants, however, he did not extend the mathematical analysis beyond Mars.

In the late 17th century, a number of physical astronomy theories drawing from Kepler's work — notably those of Giovanni Alfonso Borelli and Robert Hooke — began to incorporate attractive forces and the Cartesian concept of inertia. This culminated in Isaac Newton's "Principia Mathematica" (1687), in which Newton derived Kepler's laws of planetary motion from a force-based theory of universal gravitation.

Lecture Demonstrations

At the end of the 17th century science had been a topic for the elites of society and courts. This began to change quickly in the first decade of the 18th century with the combination of coffee houses, a growing middle class, and the availability of lecture demonstrations. By the middle of the 18th century the achievements of astronomers were becoming widely accepted and a culture of lectures using demonstration apparatus evolved greatly. Between 1745 and 1770 the London paper "Daily Advertiser" featured advertisements from twelve lecturers, including Stephen Demainbray, James Ferguson, and Benjamin Martin.

The demand for these lectures created a small market for makers of scientific instruments. Simpler models were driven by a hand crank; elaborate ones were driven by clockwork.


The two best known mechanical models used in lecture demonstrations were the orrery (a mechanical model of planets going around the Sun) and the cometarium – a model of a comet's orbit. The orrery and the cometarium have to make compromises. An orrery cannot show correct relative size or spacing of planets in their orbits, nor do they show elliptical orbits. A cometarium is constrained by mechanical limits to show only modest eccentricities to fit on a tabletop.

The origins of the cometarium trace to a device first made by J T Desaguliers and demonstrated to the Royal Society in 1732 to facilitate a discussion of the orbit of the planet Mercury. Mercury's orbit is the most eccentric (0.21) of the planets in our solar system, so the changes in velocity of Mercury between aphelion and perihelion were sufficiently large to attract interest. Later, in the 1740s as Halley's comet was predicted to return soon, interest in comets surged and the name "cometarium" was coined to name the device by Benjamin Martin. Martin sold a cometarium along with his book "The Theory of Comets" which was published in 1757.

Early Cometaria used elliptical wheels and a figure 8 shaped cord as the essential device to create the differential motion of the comet and the calendar indicator.

Cometarium Picture

Today the Cometarium by Armstrong Metalcrafts uses elliptical gears to achieve the correct motions.

Cometarium Gears

The gears in this model have an eccentricity of about .65, which is close to matching some models found in museums.

Early cometaria had varying methods for moving the comet ball around the orbit. Example ideas included a rod pushing the comet ball in the track, or a pin and slot mechanism. This cometarium is the first to have the ball mounted on the shaft driven by the gears.

For Sale

Armstrong Metalcrafts has built a limited number of these for sale. The price is $1250, and this includes a small booklet featuring an in-depth description of the history and how this mechanism works. To inquire about a purchase, please use our contact form or send an email to "sales" at

Copyright © Armstrong Metalcrafts 2011-2024    Privacy Notice