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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Commentariolus | 3/3 | https://en.wikipedia.org/wiki/Commentariolus | reference | science, encyclopedia | 2026-05-05T08:32:52.246082+00:00 | kb-cron |
Like the Moon's motion, that of the outer planets, represented in the diagram to the right, is produced by a combination of a deferent and two epicycles. The centre of the first, and larger of the two epicycles, represented by the point e1 in the diagram, revolves uniformly from west to east around the circumference of a deferent whose centre is the centre of the Earth's orbit, represented by the point S in the diagram, with a period relative to the fixed stars as given in the section The order of the spheres above. The centre of the second epicycle, represented by the point e2 in the diagram, revolves uniformly from east to west around the circumference of the first, with the same period relative to the radial line joining S to e1. As a consequence, the direction of the radial line joining e1 to e2 remains fixed relative to the fixed stars, parallel to the planet's line of apses EW, and the point e2 describes an eccentric circle whose radius is equal to that of the deferent, and whose centre, represented by the point O in the diagram, is offset from that of the deferent by the radius of the first epicycle. In his later work, De revolutionibus orbium coelestium, Copernicus uses this eccentric circle directly, rather than representing it as a combination of a deferent and an epicycle. The planet itself, represented by the point P in the diagram, revolves uniformly from west to east around the circumference of the second epicycle, whose radius is exactly one third of that of the first, at twice the rate of revolution of e1 about S. This device enabled Copernicus to dispense with the equant, a much-criticised feature of Claudius Ptolemy's theories for the motions of the outer planets. In a heliocentric version of Ptolemy's models, his equant would lie at the point Q in the diagram, offset along the line of apses EW from the point S by a distance one and a third times the radius of Copernicus's first epicycle. The centre of the planet's deferent, with the same radius as Copernicus's, would lie at the point C, mid-way between S and Q. The planet itself would lie at the point of intersection of this deferent with the line QP. While this point only coincides exactly with P whenever they are both at an apsis, the difference between their positions is always negligible in comparison with the inaccuracies inherent to both theories. For the ratios of the radii of the outer planets' deferents to radius of the Earth, the Commentariolus gives 113⁄25 for Mars, 513⁄60 for Jupiter, and 97⁄30 for Saturn. For the ratios of the radii of their deferents to the radii of the larger of their epicycles, it gives 6138⁄167 for Mars, 12553⁄606 for Jupiter, and 11859⁄1181 for Saturn.
=== Venus === In the last two sections Copernicus talks about Venus and Mercury. The first has a system of circles and takes 9 months to complete a revolution.
=== Mercury === Mercury's orbit is harder than any of the other planets' to study because it is visible for only a few days a year. Mercury, just like Venus, has two epicycles, one greater than another. It takes almost three months to complete a revolution.
== Notes ==
== References ==
=== Bibliography === Bardi, A. (2024). Copernicus and Axiomatics. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-031-40846-5_110 Copernicus, Nicolaus (1992), Czartoryski, Pawel (ed.), The manuscripts of Nicholas Copernicus' minor works; facsimiles, Krakow: Polish Academy of Sciences, ISBN 83-01-10562-3 Dreyer, John Louis Emil (1890). Tycho Brahe; a picture of scientific life and work in the sixteenth century. Edinburgh: Adam and Charles Black. Gingerich, Owen (2004). The Book Nobody Read. London: William Heinemann. ISBN 0-434-01315-3. Goddu, André (2010). Copernicus and the Aristotelian tradition. Leiden, Netherlands: Brill. ISBN 978-90-04-18107-6. Koyré, Alexandre (1973). The Astronomical Revolution: Copernicus – Kepler – Borelli. Ithaca, NY: Cornell University Press. ISBN 0-8014-0504-1. Rosen, Edward (2004) [1939]. Three Copernican Treatises: The Commentariolus of Copernicus; The Letter against Werner; The Narratio Prima of Rheticus (Second Edition, Revised ed.). New York, NY: Dover Publications, Inc. Swerdlow, Noel M. (December 1973), "The derivation and first draft of Copernicus's planetary theoryA translation of the Commentariolus with commentary.", Proceedings of the American Philosophical Society, 117 (6): 423–512 Thoren, Victor E. (1990). The Lord of Uraniborg. Cambridge: Cambridge University Press. ISBN 0-521-35158-8.