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Huygens, Van Rekeningh in Spelen van Geluck (Amsterdam, 1660), refers to these problems as ‘voorstellen’, not ‘vraeg-stucken’, but apart from this they are stated in precisely the same words. The first and third originated from Femat, the fifth from Pascal and the second and fourth from Huygens (Œuvres XIV, pp. 88–91). Spinoza had a copy of the Latin edition of Huygens’ treatise (1657) in his library (no. 53). Huygens’controversy with Jan Hudde in 1665 (Œuvres V, nos. 1374–1450) made it apparent that both Huygens’ problems were ambiguous. In the case of no. 2, for example, Huygens assumed that each time a black chip is drawn it is put back, whereas Hudde assumed that it is not. Jakob Bernoulli (1654–1705), Ars conjectandt (Basel, 1713, pp. 57–65) pointed out that one can also assume that each of the three players begins with his own hat of twelve chips and draws without replacement.
Fermat thought up this problem and communicated it to P. de Carcavy, who sent it by letter to Huygens on June 22nd, 1656. Huygens wrote back with the correct solution, showing his working in detail, on July 6th, 1656 (Œuvres I, pp. 433–446). Apart from Spinoza’s, solutions were subsequently provided by P.R. de montmort (1678–1719), Essay d’analyse sur les jeux de hazard (2nd ed., Paris, 1713), pp. 216-217; reekening der Kannsen in het Spelen (Amsterdam, 1716), see his Œuvres (Amsterdam, 1912), pp. 32–34. A modern treatment is to be found in Jacques Dutka, ‘Spinoza and the Theory of Probability’, Scripta Mathematica, vol. XIX, no. 1 (March, 1953), pp. 24–33, cf. R. McKeon, ‘Spinoza on the Rainbow and on Probability’, in H.A. Wolfson. Jubilee Volume (Jerusalem, 1965), vol. II, pp. 555–556.
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© 1985 Martinus Nijhoff Publishers, Dordrecht
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Petry, M.J. (1985). Calculation of Chances. In: Petry, M.J. (eds) Spinoza’s Algebraic Calculation of the Rainbow & Calculation of Chances. Archives Internationales D’Histoire des Idées / International Archives of the History of Ideas, vol 108. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5097-9_2
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