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On Kepler’s system of conics in Astronomiae pars optica

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Abstract

This is an attempt to explain Kepler’s invention of the first “non-cone-based” system of conics, and to put it into a historical perspective.

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Notes

  1. Erazmus Ciolek Witelo (c.1230, after 1275), was author of Perspectiva, printed for the first time only in 1535, a treatise on optics largely based on the Latin version of Kitab-al-manazir (The book of optics) by the Persian Ibn al-Haytham (Alhazen). It is to Witelo’s Perspectiva to which Kepler was referring in the title of his work. Likely Kepler used Risner’s Opticae thesaurus (1572) containing the works of Witelo and Ibn al-Haytham.

  2. See for instance the detailed (Buchdhal 1972), and the more recent (Knobloch 2000).

  3. A similar figure also appeared in Supplementum ad Archimedem, de sectonibus coni, a section of Kepler’s Nova stereometria (Kepler 1615), where he stated again that the centre and the second focus of the parabola are at infinity.

  4. See also the notes on this topic that Taylor inserted in his (1881) and (1883).

  5. See footnote n. 16 in Sect. 3.

  6. Also (Risner 1572, pp. 398–402).

  7. Kepler in Astronomia nova (1609, p. 189) attested to have read this work.

  8. This term, which means “hearth”, “fire-place” and “burning-point”, certainly arise from investigations on burning-mirrors. In the later Astronomia nova, Kepler called each of such points puncto eccentrico.

  9. See also (Davis 1975, p. 679).

  10. Here Kepler wrote of length “AC duplicata” (p. 96, line 1), i.e. 2AC, which is inaccurate. This was already pointed out by Taylor (1900, p. 202).

  11. The Harmonices mundi was planned since 1599, but not completed and published until 1619. At p. 39 of this work, Kepler explicitly referred to the wrong construction of the heptagon offered by Dürer in (1525, fig. 9), and so that he had read his treatise. We thank Aldo Brigaglia for having brought this to our attention.

  12. Kepler studied Euclid’s Elements and the works of Archimedes, Apollonius and Pappus translated into Latin by Federico Commandino (Commandino 1558, 1566, 1588). Kepler quoted Commandino’s works in Astronomia nova, p. 286, see also (Cronwell 1997, p. 140). In particular he knew the definition of conics through the focus-directrix property, as in Pappus’ Collectiones, book 7, prop. 138. The constant e is the eccentricity of the conic.

  13. Francesco Maurolico in Photismi de lumine et umbra, completed in 1521, proved that any conic is the perspective image of a circle (1611, theorems XII, XIII). In this work Maurolico also studied problems concerning the camera obscura. It seems that Blaise Pascal, in the lost Traité des coniques, affirmed that all conics are the perspective image of a circle, see (Taton 1962, p. 237). We also observe that Isaac Newton, in section 29 “Genesis curvarum per umbra” of his short treatise Enumeratio linearum tertii ordinis, see (1704, p.157), wrote that all conics can be seen as shadows of a circle. It is interesting to notice that Taylor (1900, p. 217), wrote that such genesis may have been suggested to Newton by some of Kepler’s problemata observatoria (Kepler 1604, pp. 201, 203).

  14. This is the system obtained when one allows \(\lambda \) to assume the value \(\infty \), or, which is the same, \(\varPhi \) is put in the projective form \(\mu (z -2) + \lambda (2x + z - 0)= 0\).

  15. In fact for \(\lambda \rightarrow -1\), the two arms (or branches) of the hyperbola become closer and closer, until they flatten onto the line \(x = 1\).

  16. We note in passing that Hofmann, see p. 336 of the reprint of his work, wrote the following equation \(y^{2}=2px-(1-e^{2})x^{2}\), where p is a fixed number and e the eccentricity. This equation does not represent a system of semi-confocal conics. For instance, for \(p = 1\) the coordinates of the foci are \((1/(1+e),0)\) and \((1/(1- e),0)\), and clearly both depend on the eccentricity. Let us remark that when \( e \rightarrow +\infty \), the limiting position of the foci is the point (0, 0); so in this system both foci coincide for the degenerate hyperbola and fall on it.

  17. See for instance (Ayoub 2003).

  18. Desargues in the Brouillon project (1639) adopted the term “foyer”, used for the first time by Kepler. Taylor (1900, p. 205) wrote: “Desargues must have learned directly or indirectly from the work in which Kepler propounded his new theory of these points, first called by him Foci (foyers)”.

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  1. Dipartimento di Matematica e Informatica, Università di Ferrara, Via Machiavelli, 35, 44100, Ferrara, Italy

    Andrea Del Centina

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Correspondence to Andrea Del Centina.

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Communicated by : Noel Swerdlow.

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Del Centina, A. On Kepler’s system of conics in Astronomiae pars optica . Arch. Hist. Exact Sci. 70, 567–589 (2016). https://doi.org/10.1007/s00407-016-0175-2

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