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The Wikipedia article for Cassini ovals claims in the introduction that "Cassini believed that the Sun traveled around the Earth on one of these ovals, with the Earth at one focus of the oval." This claim doesn't have an associated citation, but it appears that Wikipedia may have gotten it from this website, which doesn't cite any sources. Thus, my question:

Does there exists a reliable source (i.e., one with citations tracing back to a primary source) which backs up that claim that Cassini attempted to prove that the Sun traveled around the Earth in a Cassini oval?

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The history of Cassini's curve is summarized by Curtis Wilson, along with citations to seventeenth-century sources, in his chapter 'Predictive Astronomy in the Century after Kepler", in "General History of Astronomy" vol. 2A, eds R Taton & C Wilson (Cambridge, 1989). Wilson's account includes this about the eldest Cassini and his curve:

"... around 1690 he [Cassini] invented the cassinoid, with the aim of obtaining a possible orbit for the planets in which the superior focus would serve as equant point. The hypothesis was first mentioned in 1691 by Jacques Ozanam in his Dictionnaire mathématique, and was desribed by Cassini in 1693 in his treatise "Sur l'origine et le progrès de l'astronomie". In the cassinoid the product of the distances from any point to the two foci is a constant." Curtis Wilson's chapter goes on to recount the reaction of astronomical contemporaries to the 'cassinoid' as a pattern for planetary orbits.

Cassini aimed to arrive at a model for the orbits from which the angles could be calculated directly for a given point in time, without guesswork or approximation. The background to this was one of some dissatisfaction, during the later 17th century, with Kepler's hypothesis of ellipses and constant areal-velocities. Such dissatisfaction may be hard to appreciate now, when much of the talk of Kepler for almost three centuries has been about "Kepler's laws". An early source that sums up contemporary complaints about Kepler's hypotheses is a letter by Nicolas Mercator that appeared in Philosophical Transactions (of the Royal Society) in 1670, vol.5, pp.1168-1175. Mercator wrote (at p.1174,in Latin, translated here):

"No-one has been found up to now who would deny that Kepler's areas can satisfy the appearances; but since neither he himself nor anyone after him could determine them by direct calculation, some have criticized Kepler for having parted ways with geometry while indulging too much in [speculation about] physical causes."

(It appears that mathematical conservatives of that time required proofs that went directly to a final certain result by successive valid steps of demonstration, each leading to a true intermediate statement. Those who thought like this would not accept that truth could emerge from a succession of approximations, thus, a succession of errors or at best of uncertainties. Kepler himself had admitted that to find the orbital angles for a given time required either guesswork or approximation, e.g. a series of approximations: (Kepler, 'Epitome Astronomiae Copernicanae', book 5 (1621), pt 2, at p.695, "Hic via directa nulla est..." ('Here there is no direct way....'), see also N M Swerdlow, 'Kepler's Iterative Solution to Kepler's Equation', Journal for the History of Astronomy v.31 (2000) pp.339-341.) More about the background can be found in another paper by Curtis Wilson: "From Kepler's laws, so-called, to universal gravitation: Empirical factors", Archive for History of Exact Sciences 1970, Volume 6, Number 2, pp.89-170.)

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In his article Traité de la comete, qui a paru en Decembre 1743 (see here) G. Cassini writes that the orbits of comets and planets around the sun are ovals. At the end of this paragraphe he writes that the mathematicians call these ovals ellipses.

The claim that Giovanni Cassini used his ovals for describing the orbits of planets was made by his son Jean-Dominique on p. 149 of his Elements d'Astronomie,

"Since the exact observations of the apparent size of the diameter of the sun, my father has found another curve different from the ellipse that may serve as representing rather precisely the true movement of the sun" (Depuis l'observation exacte de la grandeur apparente des diamètres du soleil, mon père a trouvé une autre courbe différente de l'ellipse, qui sert à représenter fort exactement les mouvements vrais du Soleil).

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