Celestial Charts and Spherical Triangles: The Unifying Power of Symmetry

We have checked the relevant compendious dictionaries of Greek, Latin, English, French, and German, as well as a great many texts not cited in any of them. See, e.g., Hon G. Goldstein B. R. , “From proportion to balance: The background to symmetry in science”, Studies in history and philosophy of science , xxxvi (2005), 1–21. In this essay we examined various usages of the term, symmetry, as an aesthetic category (mainly in architecture) up to the end of the eighteenth century.

See Hon G. Goldstein B. R. , “Symmetry in Copernicus and Galileo”, Journal for the history of astronomy, xxxv (2004), 273–92.

Legendre A.-M. , Éléments de géométrie (Paris, [1794]/1817), 155, 163 (italics in the original); cited in Hon G. Goldstein B. R. , “Legendre's revolution (1794): The definition of symmetry in solid geometry”, Archive for history of exact sciences, lix (2005), 2005–55, pp. 126, 129.

We take issue with Walter Kambartel who identifies the definition of Claude Perrault (1613–88) with the modern concept of symmetry (“des modernen Symmetriebegriffs”, as Kambartel puts it). See Kambartel W. , Symmetrie und Schönheit: über mögliche Voraussetzungen des neueren Kunstbewusstseins in der Architekturtheorie Claude Perraults (Munich, 1972), 43. For our view on this matter, see Hon and Goldstein (ref. 1), 2–3, 9, 19. For an early recognition of anachronism in the attribution of the modern concept of symmetry in pre-modern texts, see Roche J. J. , “A critical study of symmetry in physics from Galileo to Newton”, in Symmetries in physics (1600–1980): Proceedings of the 1st International Meeting on the History of Scientific Ideas, Sant Feliu de Guíxols, Catalonia, Spain, September 20–26, 1983, ed. by Doncel M. García (Barcelona, 1987), 1–28.

Ptolemy , Almagest, II.8 and XI.11; Toomer G. J. , Ptolemy's Almagest (New York and Berlin, 1984), 100–3, 549–53.

The kind of transformation under consideration is now called “reflection” and is described mathematically in group theory: See, e.g., Weyl H. , Symmetry (Princeton, 1952), 4–5.

The celestial sphere is merely a sphere with a unit radius onto which the stars are projected radially. The unit may be chosen fairly freely without affecting any argument. For the purpose at hand the Earth is taken to be very small compared to this unit radius, and is considered to be a point at the centre of this sphere (cf. Ptolemy, Almagest, I.6; Toomer, op. cit. (ref. 5), 43). The notion of ‘orb’ (i.e., a spherical shell) should not be confused with the modern term ‘orbit’ (in the sense of a path in space) coined by Kepler: See Goldstein B. R. Hon G. , “Kepler's move from orbs to orbits: Documenting a revolutionary scientific concept”, Perspectives on science, xiii (2005), 74–111.

See, e.g., Comes M. , “Al-Ṣūfī como fuente del libro de la ‘Ochava Espera’ de Alfonso X”, in “Ochava espera” y “astrofísica” , ed. by Comes M. (Barcelona, 1990), 11–113, p. 22.

This passage was translated from Arabic by Goldstein B. R. : See Abu 'l-Ḥusayn ^cAbd al-Rahman al-Ṣūfī, Kitāb ṣuwar al-kawākib, edited from the oldest extant Mss . (Hyderabad, 1954), 28; cf. Schjellerup H. C. F. , Description des étoiles fixes par l'astronome persan Abd al-Rahman al-Sufi (St Petersburg, 1874; reprinted Frankfurt/M, 1986), 45–46. See also ^cAbd al-Raḥmân al-Ṣūfī, The book of constellations (Kitāb ṣuwar al-kawākib), reproduced from Oxford, Bodleian Library, MS Marsh 144 (copied in 1009–10 A.D.), ed. by Sezgin F. (Frankfurt/M, 1986); An Islamic book of constellations, ed. by Wellesz E. (Bodleian Picture Book, no. 13; Oxford, 1965).

See Sezgin , op. cit. (ref. 9), 345, and Wellesz , op. cit. (ref. 9), 342, Fig. 22b.

Sezgin , op. cit. (ref. 9), 122–3.

We thank Paul Kunitzsch for drawing our attention to this fact; subsequently, we found a similar comment in a text of Euler (see Section 4, below). Projecting the entire sphere onto a plane is a completely different issue. Note further that medieval celestial globes were not transparent. Al-Ṣūfī gives appropriate instructions for a maker of such globes. For a survey of medieval celestial globes, see Savage-Smith E. , Islamicate celestial globes: Their history, construction, and use (Washington, DC, 1985).

Sezgin , op. cit. (ref. 9), 368–71.

Kunitzsch P. , “The astronomer Abu 'l-Ḥusayn al-Ṣūfī and his book on the constellations”, Zeitschrift für Geschichte der Arabisch-Islamischen Wissenschaften, iii (1986), 56–81, pp. 68–69, n. 40. Reprinted as Essay XI in idem, The Arabs and the stars: Texts and traditions on the fixed stars, and their influence in medieval Europe (Northampton, 1989). Moreover, the Arabic text had almost certainly been consulted by Peter Apian (d. 1552), professor of mathematics at the University of Ingolstadt: Kunitzsch P. , “Peter Apian and ‘Azophi’: Arabic constellations in Renaissance astronomy”, Journal for the history of astronomy, xviii (1987), 1987–24.

Flamsteed J. , Historiae coelestis britannicae (3 vols, London, 1725); idem, Atlas coelestis (London, 1729). See also The Preface to John Flamsteed's Historia coelestis britannica, or British catalogue of the heavens (1725), ed. and introd. by Chapman A. , based on a translation by Johnson A. D. (Maritime monographs and reports, no. 52; London, 1982), 8–15. Curiously, Flamsteed wrote his preface in English, completing it shortly before he died but, in the posthumous publication of 1725, there is only a Latin translation of it. According to Chapman's introduction, Flamsteed's English version is only partially preserved in manuscripts, and some passages of the original version were incorporated into the English translation by Chapman and Johnson. But see ref. 16, below.

Flamsteed , Atlas (ref. 15), 3–6 (spelling, italics, and capitalization as in the original); idem, Historiae (ref. 15), iii, 156–60; cf. Chapman and Johnson, op. cit. (ref. 15), 157–60. The preface to Flamsteed's Atlas (9 pages in all) is in English and this passage corresponds very closely to the Latin published in 1725, suggesting that this English version was in fact Flamsteed's original. The Atlas of 1729 was prepared by Flamsteed's widow, Margaret Flamsteed, together with James Hodgson, one of Flamsteed's assistants.

Bayer J. , Uranometria: Omnium asterismorum continens schemata, nova methodo delineata, aeris laminis expressa (Augsburg, 1603).

Flamsteed , Historiae (ref. 15), iii, 33; Chapman Johnson , op. cit. (ref. 15), 54.

Göttingische Zeitungen von gelehrten Sachen auf das Jahr MDCCL [1750], 76.

“Auszug eines Briefes von Hrn. Euler über die Vorstellung der Sternenbilder auf der Himmelskugel”, in Göttingische Zeitungen von gelehrten Sachen auf das Jahr MDCCL [1750], 475–7; reprinted in Leonhardi Euleri Opera omnia, ser. 2, vol. xxx, Commentationes astronomicae: Ad praecessionem et nutationem pertinentes, ed. by Courvoisier L. (Zurich, 1964), 101–2.

Euler L. , “Principes de la trigonométrie sphérique”, Histoire de l'Académie Royale des Sciences et des Belles-lettres de Berlin, année 1753 (Berlin, 1755), 223–57.

Cf. Hon G. , “Kant vs. Legendre on symmetry: Mirror images in philosophy and mathematics”, Centaurus, xlvii (2005), 283–97.

Kant I. , “Von dem ersten Grunde des Unterschiedes der Gegenden im Raume”, in Immanuel Kants Werke , ed. by Cassirer E. , ii: Vorkritische Schriften, ed. by Buchenau A. (Berlin, [1768]/1912), 393–400, p. 398; Kant I. , “Concerning the ultimate ground of the differentiation of directions in space”, in I. Kant, Theoretical philosophy 1755–1770, transl. by Walford D. Meerbote R. (New York, 1992), 363–72, p. 370. Cf. I. Kant, Prolegomena zu einer jeden künftigen Metaphysik, die als Wissenschaft wird auftreten können, ed. by Vorländer K. , 6th edn (Leipzig, [1783]/1920), 39–40 (§13); Kant I. , Prolegomena to any future metaphysics that will be able to come forward as science, ed. and transl. by Hatfield G. (New York, [1997]/2004), 37.

Kant , “Von dem ersten Grunde” (ref. 23), 398 (italics in the original); Kant , “Concerning the ultimate ground” (ref. 23), 370. Cf. Kant , Prolegomena zu einer jeden künftigen Metaphysik (ref. 23), 40–41, and Kant , Prolegomena to any future metaphysics (ref. 23), 38.

We have searched many books on spherical geometry and trigonometry in Latin, French, English, and German prior to 1768 without finding this theorem; see also ref. 21.

Heath T. L. , Euclid's Elements (3 vols, New York, [1926]/1956), i, 224–31.

Cf. Kant , Prolegomena zu einer jeden künftigen Metaphysik (ref. 23), 39 (§13), and Kant , Prolegomena to any future metaphysics (ref. 23), 38 (§13).

Kant's remark in his Prolegomena may shed light on the way he thought about spherical triangles in 1768: “e.g., two spherical triangles from each of the hemispheres, which have an arc of the equator for a common base, can be fully equal with respect to their sides as well as their angles, so that nothing will be found in either, when it is fully described by itself, that is not also in the description of the other, and still one cannot be put in the place of the other (that is, in the opposite hemisphere).” See Kant , Prolegomena zu einer jeden künftigen Metaphysik (ref. 23), 39–40 (§13), and Kant, Prolegomena to any future metaphysics (ref. 23), 37 (§13). We may consider any of the circles in Fig. 2 to correspond to Kant's equator. Kant does not explicitly say that the third vertex of one spherical triangle lies to the north of the equator whereas the corresponding vertex of the other spherical triangle lies to the south of the equator.

Hon , op. cit. (ref. 22), 292–5. One problem with Kant's analysis is that he considers objects in various domains from a viewer's perspectives. But his goal, namely, to identify an intrinsic property which will support Newton's concept of absolute space, should not, in principle, depend on a viewer.

Legendre , op. cit. (ref. 3), 203 (italics in the original).

Legendre , op. cit. (ref. 3), 214–15 (italics in the original). For Legendre's two senses of equality, “absolute” and “of superposition”, see Hon and Goldstein, op. cit. (ref. 3), 126–7.

Legendre , op. cit. (ref. 3), 305. Cited in Hon and Goldstein, op. cit. (ref. 3), 133.

Kant , “Von dem ersten Grunde” (ref. 23), 398; Kant , “Concerning the ultimate ground” (ref. 23), 370.