The Cometarium

Definition: A cometarium is a mechanical device designed to illustrate the motion of a comet about the Sun. The key feature of such instruments resides in their ability to show an apparent increase in orbital velocity as the comet marker moves from aphelion to perihelion. In this manner, cometaria offer a demonstration of Kepler's second law of planetary motion.

Alternate names: Equal area machine, Mercurium (see the discussion below).


  1. The orbit of each planet is an ellipse with the Sun at one focus.

  2. The line drawn from the Sun to a planet sweeps out equal areas in equal time.

  3. The cube of the orbital semi-major axis is proportional to the square of the planet's sidereal period.

History and background

The device that ultimately became known as the cometarium was first constructed by J. T. Desaguliers in the early 1730s. Showing his invention to the assembled Fellows of the Royal Society of London on March 8th, 1732, Desaguliers device described the orbit of the planet Mercury. The term cometarium was apparently first employed against Desaguliers-like machines by Benjamin Martin in the early 1740s. In the mid 1750s, James Ferguson used the expression equal-area machine for the device.

The key features of a cometarium are shown in the figure to the right. The diagram is from Astronomy Explained Upon Sir Isaac Newton's Principles, by James Ferguson (Published in 1757). 'Figure III' shows the front of the cometarium and its two dials. The upper circular dial shows the time and is divided in to 30 degree (zodiacal) divisions or signs. The 30 degree spacing is echoed in the larger, lower circle in the diagram. Interior to the large circle is the elliptical track within which the comet marker ball moves. The Sun is located at position S, the center of the large circle (and focus of the elliptical track). The radial lines from S mark out equal areas as swept out from the Sun as the comet-ball moves around the elliptical track. By turning the crank N the pointer in the upper circle traces out equal divisions in equal time, and the comet-ball likewise traces out equal areas in equal time about the Sun at S.

It is the tracing out of equal areas in equal time feature of the cometarium that provides a demonstration of Kepler's second law of planetary motion. Since the motion of the comet about S is non-uniform (faster at perihelion and slower at aphelion), the uniform circular motion of crank N (as displayed in the upper circular dial) must be converted into non-uniform drive motion about S (through the pointer SYW). 'Figure V' illustrates how the non-uniform drive motion is achieved through the use of two elliptical cams (or meshed gears - see below) connected via a figure-of-eight cord.

Without going into details here, it transpires that the elliptical cam system does not provide an exact reproduction of Kepler's second law. The system only provides the correct perihelion to aphelion velocity ratio. The details of the motion produced by elliptical pulleys is discussed in J. R. Millburn's article Demonstrating the Motion of Comets [Space Education, 1(2), 1981, 55-58] and in my article The mechanics and origin of cometaria [Journal of Astronomical History and Heritage, 5(2), 2002, 155-163]. I have also discussed the correct pulley profile for generating a true demonstration Kepler's 2nd law in Cometaria and the demonstration of Kepler's 1st and 2nd Law (Bulletin of the Scientific Instrument Society, No. 82, 29 - 33, 2005a). A further discussion on the design options for cometaria are presented in On Ptolemy's Equant, Kepler's Second Law, and the Non-existent "Empty Focus" Cometarium published in the Journal of the Royal Astronomical Society of Canada (99 (4), 120 - 123, 2005b). A discussion of how several cometaria were used as public display demonstration pieces is given in my BSIS article (94, 18 - 24, 2007) On two lost American cometaria.

Extant cometaria

The list below indicates the location and provides brief notes on the various known cometaria. Any additions to this list will be gratefully received (see email link at end of page).

Stationary cometaria

The term stationary cometaria is introduced here to describe those three and two-dimensional instruments that illustrate cometary orbits, but do not attempt to describe Kepler's second law. By default they may illustrate Kepler's first law if the entire cometary orbit is displayed. At the present time I am aware of one extant and one historical, but now (apparently) lost, stationary cometaria.


Additional lost cometaria


1. Description: Length 50-cm, width 39-cm, height 11.5-cm. "The instrument ... forms the top of an oak box and has a glass cover. There are two dials: the upper one gives hours and minutes; the lower one shows the motion of the comet. When the handle is turned a brass pointer moves round each of the dials. The upper dial shows equal arcs being covered in equal times; the lower pointer consists of a brass rod on which slides the ball representing the comet. The tip of the pointer moves over the circle representing the great circle of the heavens, while a groove carries the comet in an elliptical path, giving equal areas in equal times but not equal arcs. The Sun is represented by a large brass ball at the centre. Strangely, the upper dial is divided into 24 but the lower one is divided into 22". Details from Public and Private Science by A. Q. Morton and J. A. Wess (OUP, Oxford, 1993. pp. 160-161). See also by the same authours The Historical Context of the Models of Stephen Demainbray in the King George III Collection [Journal of the History of Collections, 7(2), 1995. pp. 171-178].

2. Description: "Overall dimensions: 9-cm high, 22-cm wide, 15-cm long. A rough metal box 9.8-cm wide, 15.2-cm long, 1.3-cm deep, with cover hinged at one end, fastened by hasp at opposite end; small dial with one hand; oval inner ring; larger outer circle, all of brass, on outside of cover, operated by mechanism inside of box. Iron tripod to support box. Maker's signature [J. Rix] marked on oval brass plate in center of box top. Apparatus in damaged condition when acquired. Restored in the museum in early 1960's. Box is bluish green, with an antiqued finish. Iron legs pitted from rust (rust now removed). Weight of instrument is 435 grams (15.3 oz.)". Information from the curator of Physical Sciences Collection, Smithsonian Institution. An image of the cometarium can be found on the Instruments For Natural Philosophy webpage by Professor T. B. Greenslade, Jr (follow the Astronomy link).

3. Description: "Mahogany box with lock and key contains gear and cam drive. Three-part brass plate - a circle inside an oval inside a rectangle - fastened lid. Mechanism, operated by hand-crank, turns two brass pointers. Longer one pushes brass ball along elliptical orbit. The other measures time on scale marked off in yearly units from 1 to 75 1/2. Engraved 'A Cometarium B. Martin London.' Dimensions: length 10-in; width 6 1/2-in; height 2-in". Information from The Apparatus of Science at Harvard 1765 - 1800 by David P. Wheatland. (Harvard University Press, Harvard, 1968. pp. 60 - 62). An image of this cometarium can be seen in Beech (2005a).

4. "Vertical demonstration cometarium. Made by a local Edinburgh instrument maker [John Miller] this model was used to teach students at the University of Edinbugh when astronomy formed part of the ordinary arts curriculum. It was listed in the inventory of 1789". Information from A Heavenly Library, an exhibition held at the National Museum of Scotland, 1994. See also Science Preserved: A directory of Scientific instruments in collections in the United Kingdom and Eire by M. Holbrook (HMSO, London 1992. p. 64).

5. "This instrument is made in the more usual horizontal form. It was probably constructed by the instrument maker Robert Fidler (fl. 1810 - 1822) in association with the Rev. William Pearson (1767 - 1847). It was used in the teaching of students at the University of St. Andrews. The chief mechanical interest of this device is the skillful cutting of the elliptical gear mechanism". Information from A Heavenly Library, an exhibition held at the National Museum of Scotland, 1994. [See also references in 4 and reference 6]

6. Construction: In wood and brass and ivory measuring 30 x 23 x 15-cm. "Constructed by William Jones ... it uses an eccentrically-mounted circular gear quite different from the elliptical-pulley-cord device and was more expensive [the catalogue price was five guineas]. While its gear is relatively easy to make, it requires a complicated driving mechanism with sliding parts and means for keeping the gear constantly in mesh with its driving pinion". Information from Fire in the Sky: Comets and Meteors, the Decisive Centuries, in British Art and Science by R.J.M. Olson and J. M. Pasachoff (CUP, Cambridge, 1998. pp. 44 - 48). See also L. Taub's cometarium article in Instruments of Science: An historical encyclopedia [R. Bud and D. J. Warner (eds). Galand Publishing Inc. New York, 1998). The gearing mechanism for the cometarium is shown in Abraham Rees's The Cyclopedia (plate II, Volume 9, 1819). In reference 4, the cometarium by Fidler is described as being "superior in construction to the one by W & S. Jones".

7. Construction: Thick cardboard faceplate, fixed to a wooden cross, but missing all moveable parts. The dimensions are 37 x 23 cm. The maker and date of construction are unknown. A picture can be found at the following location: history of scientific instruments section [use the 'Search Engine' with input A-387].

8. Arjen Roelofs (1754 - 1828) is said to have made a cometarium after sighting the great comet of 1807 (comet C/1807 R1), but it has apparently not survived to the modern era [H. Van Gent, University of Utrecht - private communication].

9. This cometarium was designed by John Taylor in 1828, and manufactured by R. Adie in 1835 at his Liverpool, England workshop. See Beech (2006).

10. Nicolaas Struyck (1687 - 1769) described a cometarium in his Vervolg van de beschryving der staartsterren (Amsterdam, 1753). The cometarium displays the partial orbits of 14 comets. All of the comets displayed were prominent visual objects in the northern sky, most were long period comets, but two of the orbits displayed correspond to short period comets 6P/d'Arrest (1678 R1) and 55P/Tempel-Tuttle (1699 U1). Surprisingly, perhaps, Halley's comet (1P/Halley) due to return in 1758 is not one of the orbits displayed. [H. Van Gent, University of Utrecht - private communication] . A picture of this cometarium is shown in Beech (2006).

This page last modified September 14th, 2012

Page maintained by Martin Beech, Campion College, The University of Regina.