Abstract
Isaac Barrow did not spare laudatory adjectives to the « excellent Method of Indivisibles, the most fruitful mother of new inventions in Geometry ». According to him, there is a small, select group of productive methods, probably unknown to the Ancients, to which the method of indivisibles belongs. They are used
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Notes
- 1.
I. Barrow, Mathematical Works, p. 166–167 (Mathematical Lectures pagination). The Mathematical Lectures were translated as The Usefulness of Mathematical Learning Explained and Demonstrated, J. Kirkby tr. (London, 1734). Translated quotations are A. Malet's with the benefit of Kirkby's.
- 2.
Ibid, pp. 212–213.
- 3.
Ibid., p. 135.
- 4.
Ibid., p. 139.
- 5.
Ibid., pp. 141–142.
- 6.
Torricelli, Opere, I-2, 320. See the chapter on Torricelli in this book.
- 7.
Ibid., p. 142.
- 8.
Ibid., p. 260.
- 9.
In fact Barrow did not specify whether he was arguing against a finite or an infinite-kind of mathematical atomism. In one of the few occasions in which he did so, he deemed the hypothesis of the finite number of indivisibles to be still “more repugnant to the laws of Geometry”. In this occasion he pointed out that Galileo held the view that magnitude is composed of an infinite number of atoms. See Barrow (M.W.), p. 147; on Galileo’s indivisibles, see in this book, chapter Indivisibles in the work of Galileo.
- 10.
Ibid., p. 144.
- 11.
Ibid., p. 146.
- 12.
Ibid.
- 13.
Ibid., p. 147.
- 14.
Newton to Bentley, 17 January 1692/3, in Newton, Isaac Newton’s Papers…, pp. 293–296.
- 15.
Ibid., p. 146.
- 16.
Ibid., p. 148.
- 17.
Ibid. Compare with Newton’s words on the soundness of the analysis performed by means of infinite series in 1669 De Analysis: “…we, mere men possessed only of a finite intelligence, can neither designate all their terms, nor so grasp them as to ascertain exactly the quantities we desire from them”, in Newton, Mathematical Papers…, II, p. 240.
- 18.
Pascal, Œuvres, VIII, 352–3. The fragment belongs to one of the letters to Carcavi. See in this book, chapter Pascal’s indivisibles, p. 216.
- 19.
Barrow, (M.W.), p. 284. The appendix includes, along with the proof discussed here, arguments to support the legitimacy of substituting an indefinitely small segment of the tangent for the arc; ibid, pp. 284–285.
- 20.
Ibid., pp. 285–6.
- 21.
Malet (1996, pp.57–65, 68–72, 75–100).
- 22.
I. Barrow, (M.L.)
- 23.
Barrow, (M.L.), p. 106.
- 24.
Barrow, Ibid., pp. 108–110. That hypotheses in natural philosophy need not be a priori true was of course a common feature of Cartesian natural philosophy. “About the Excellency and Grounds of the Mechanical Hypothesis”, in 1772, Boyle, IV, p. 77, contains a forceful defense of the advancing of hypotheses that cannot be proved a priori to be true. What is new in Barrow’s discussion is that such hypotheses can be used as mathematical axioms.
- 25.
Barrow, Ibid., p. 116.
- 26.
Barrow, Ibid., p. 109.
- 27.
Barrow, Ibid., pp. 103–107.
- 28.
Barrow, Ibid., p. 112. In Barrow’s words, a definition “is really a complete proposition predicating, concerning the proposed subject, some property of itself which is useful for deducing other properties.” See also pp. 120–1.
- 29.
Malet (1997, pp. 265–287).
- 30.
Barrow, Ibid., pp. 134–135, 139–141, 175; Augustine’s quotation on p. 131.
- 31.
Ibid., p. 20.
- 32.
Ibid., pp. 9–21, and passim.
- 33.
Ibid., p. 309.
- 34.
Ibid., pp. 306–307.
- 35.
Ibid., pp. 139–141; quotation on p. 141.
- 36.
“magnitudinem ωσ ετυχε contemplantibus, se sensui nostro, et cogitationi subjicit Terminatio. Nullam certe rem sensu attingimus, nisi ceu terminatam; nullius corporis interiora viscera penetramus, sed externam tantum cutem oculo perlustramus, manu contrectamus”, ibid., p. 142.
- 37.
Ibid., p. 139.
- 38.
Ibid., pp. 142–163.
- 39.
Ibid., pp. 142–147.
- 40.
Ibid., pp. 143–144, 146–147.
- 41.
Ibid., p. 144.
- 42.
Ibid., p. 21.
- 43.
Malet (1996), pp. 18–9, 23–31.
References
1658, Pascal, Blaise, Lettre de A. Dettonville à Monsieur de Carcavy, en luy envoyant une Méthode generale pour trouver les Centres de gravité de toutes sortes de grandeurs : Un Traitté des Trilignes et de leurs onglets ; Un Traitté des Sinus du quart de Cercle… ; Un Traitté des Arcs de Cercle ; Un Traitté des Solides circulaires ; Et enfin un un Traitté general de la Roulette, qu’il avoit proposez publiquement au mois de Iuin 1658. A PARIS, M.DC.LVIII, Paris, Guillaume Desprez, 1658.
1772, Boyle, Robert, The Works of the Honorable Robert Boyle, ed. Thomas Birch, (6 volumes), London., Works, London, 1772.
Malet, Antoni (1996), From Indivisibles to Infinitesimals. Studies on Seventeenth-Century Mathematization of Infinitely Small Quantities, Barcelona, Servei de Publication de la Universitat Autònoma de Barcelona, 1996.
Malet, Antoni, (1997), “Isaac Barrow on the Mathematization of Nature: Theological Voluntarism and the Rise of Geometrical Optics”, Journal of the History of Ideas, 58, 1997.
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Malet, A. (2015). Isaac Barrow’s Indivisibles. In: Jullien, V. (eds) Seventeenth-Century Indivisibles Revisited. Science Networks. Historical Studies, vol 49. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-00131-9_12
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