APPENDIX

THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS ON THE RAINBOW AND PROBABILITY

A. SPINOZA'S CONTEMPORARIES

The most concrete and clear-cut information concerning the development of Spinoza's ideas on theoretical optics is to be found in his correspondence. It is evident from the letters that it was probably during the mid-1660's, the four years following the publication of his book on the principles of Cartesianism, that he was most actively involved in exploring the theoretical foundations of his professional concern with the grinding of lenses and the construction of optical instruments. During these years he was co-operating with Jan Hudde (1628-1704) in the preparation of lenses for telescopes. Hudde also composed a small treatise on dioptrics, which if it had been preserved would have provided us with invaluable insight into the precise manner in which Spinoza was then thinking about light and optics. It is apparent from the surviving letters not only that he thought very highly of Hudde's expertise in dioptrics, but that as might have been expected, they were both approaching the problems it presented from a basically Cartesian standpoint. In the summer of 1666, for example, when Spinoza wrote asking for advice on the possible advantages of plano-convex lenses, he also took the trouble to re-state Hudde's geometrical presentation of spheri• cal aberration in algebraic terms.! It was also during these years that JarigJelles (1619120-1683), a close friend of Spinoza's who had paid for the publication of the book on the principles of Cartesianism, wrote asking him to explain an apparent anomaly in Descartes' Dioptrics. After some delay, Spinoza replied in two letters, the second of which con- 1. See Spinoza, BrieJwlsselmg (ed. F. Akkerman, H.G. Hubbeling, A.G. Wester• brink, Amsterdam, 1977), letters 34 (7 January 1666), 35 (10 April 1666), 36 Gune 1666). It used to be thought that these letters were written to Huygens.

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tains a not too successful attempt at explaining the working of the human eye.2 During the 1650's, several members of this Amsterdam circle, together with their associates, had had close connections with the University of Leiden through the professor of mathematics there, Frans van Schoo ten (1615-1660). Hudde, together with Christiaan Huygens (1629-1695) and Jan de Witt (1625-1672), had studied mathematics privately at Leiden under van Schoo• ten, who in 1649 had published an annotated Latin edition of Descartes' Geometry. This work had proved to be such a suc• cess, that he was encouraged to prepare a second edition, in which he included a number of contributions by his pupils: articles by Hudde on the reduction of equations and the rule of extreme values, by Huygens on the intersections of a parabola with a circle and an improved method of constructing tangents to the conchoid, and by de Witt on conic sections.3 The pro• fessor of philosophy at Leiden at this time was Adriaan Heere• boord (1614-1661), also an enthusiastic Cartesian, and with an interest in introducing the new methods of enquiry into the natural sciences. Initially, he met with uncompromising opposi• tion to this in the University, but in 1653 the curators appointed his pupil Joannes de Raey (1622-1702)to the chair of physics.4 Claes van der Meer (1594-1654), a Leiden financier and insur• ance broker, was then treasurer of the University, and his son Jan van der Meer (1639-1686), who was corresponding with Spinoza on probability in 1666, and in 1678 was thought by Spinoza's friend the Amsterdam physician G.H. Schuller to be in business in Amsterdam, was evidently one of the links be• tween Leiden and the Amsterdam circle during the 1660's.5

2. BriefwlSSeling, letters 39 (3 March 1667), 40 (25 March 1667). 3. Geometria, a Renato Des Cartes (2 vols., Amsterdam, 1659/61). Spinoza evi• dently had both the first and the second edition of this work in his library. 4. H. De Dijn, 'Adriaan Heereboord en het Nederlands Cartesianisme', Algemeen Nederlands TiJdschrtft voor Wtjsbegeerte, vol. 75, no. 1 Qanuary 1983), pp. 56-69; E.G. Ruestow, Phystcs at seventeenth and etghteenth century Letden (The Hague, 1973), pp. 34-72. 5. Brtefwisseling, letter 38 (1 October 1666); J. Freudenthal, Dte Lebensgeschlchte Spinoza's (Leipzig, 1899), p. 207. There is no evidence that van der Meer was ever in business in Amsterdam, see K.O. Meinsma, Spmoza en ziJn krmg Cs-Gravenhage, 1896), p. 259, note 2.

92 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

In respect of the background to Spinoza's texts on the rain• bow and probability, it is certainly worth noting that there is evidence that the dioptrics of the rainbow and the theory of probability were topics of general discussion in this circle during the 1650's. Huygens, for example, was investigating refraction with reference to Descartes' explanation of the rainbow as early as December 1652, and five years later published the Latin version of his work on the theory of probability, a copy of which was in Spinoza's library. Since Rene-Fran~ois de Sluse (1622-1685) is mentioned in the Treatise on the Rainbow, it is also worth noting that Huygens was corresponding with him in 1657. 6 As well as providing us with insight into the ways in which Spinoza's involvement in practical and theoretical optics led to his exchanging ideas with his fellow countrymen, the corre• spondence also seems to indicate that prior to the publication of the Tractatus theologico-politicus in 1670, this involvement con• stituted the main basis of his international reputation. It is certainly of significance that when Henry Oldenburg (c. 1620- 1677), secretary of the Royal Society, wanted to re-open his correspondence with him, he did so by informing him of the recent publication of Boyle's Experiments and Considerations concerning Colours and Hooke's Micrographia. What is more, when Spinoza returned the compliment, he did so by sending news of Huygens' activities and information concerning discov• eries in astronomy being made by means of Campani's tele• scopes. 7 When Leibniz was informed that it was Spinoza who had written the Tractatus theologico-politicus, he introduced himself to him not by commenting upon the work, but by indicating that he was aware of the importance of Hudde's dioptrical writings, and asking for Spinoza's opinion on his own recently-published Note on advanced optics. Spinoza replied

6. Huygens, CEuvres XIII, 1, pp. 146-153; letters 396-404 (II, pp. 39-53). Huygens' Tractatus de ratiOClnllS In aleae ludo was published in F. van Schooten's Exerclta• tzonum mathematicarum (Leiden, 1657), which was in Spinoza's library: Freudenthal, op. cit., p. 161, no. (53). 7. Briefwlsseling, letters 25 (28 April 1665), 26 (May 1665); d. the enquiry and response concerning Huygens' DIOptrics in letters 29 (c. 20 September 1665) and 30 (c. 1 October 1665)

93 APPENDIX

expressing doubts concerning what Leibniz called his 'pan• dochal' lenses, and informing him that he had sent the second copy of the work on to Hudde, who had replied that he had had no time to look at it, but that he hoped he would be able to do so in a week or two.8 lt is not surprising, therefore, that the mention of a lost Treatise on the Rainbow in the preface to the collected works of Spinoza, published immediately after his death in 1677, should have aroused widespread curiosity abroad. When Jelles wrote to Spinoza enquiring about the apparent anomaly in Descartes' Dioptrics, only he and a small circle of friends in Amsterdam knew that something resembling the geometrical method then being used in dealing with light and colours was also being applied, in the re-drafting of the Ethics, to questions concerning nothing less than God and mind, the emotions and human servitude, the power of the intellect and human freedom. 9 Ten years later the full text of the Ethics was about to be presented to the world in both Latin and Dutch. Anyone who knew anything of Spinoza's interest in theoretical optics was bound to search the collected works for new light on the precise connection between the geometrical order of presentation employed in his philosophical masterpiece and the analytical geometry then being applied so successfully in the various branches of the exact sciences. Unfortunately, it was by no means apparent from the material included in the Opera posthuma what this connection might be. lt was J arig J elles who wrote the preface to the Dutch edition of the collected works, and he was evidently aware that this absence of any clear indication of what the connection might be between Spinoza's metaphysics and his mathematics was almost certain to give rise to disappointment. This probably accounts, 8. Briefwzsselmg, letters 45 (5 October 1671), 46 (9 November 1671); G.W. Leibniz, Sdmtliche Schnften und Bnefe, Zweiter Reihe, Erster Band (Darmstadt, 1926), pp. 155,184,185. Leibniz read the Tractatus soon after it was published and by the April of 1671 knew that it was by Spinoza: L. Stein, Leibmz und Spinoza (Berlin, 1890), pp. 32-34. His Notztia optzc£ promot£ was published at Frankfurt-on-Main in 1671. 9. In the draft as it existed in June 1665, the work was divided into three main sections, not five: Ethica (tr. and ed. N. and G. van Suchtelen, Amsterdam, 1979), pp. 7-8. Cf. F. Mignini's edition of the Korte Verhandeling, in Spinoza: Korte Geschriften (Amsterdam, 1982), pp. 221-436.

94 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

in part at least, for his statement that the Treatise on the Rain• bow was the only manuscript of any significance that the editors had been unable to obtain a copy of:

This is all the material, of any value, that we have been able to gather from the papers he left behind, and from certain copies, preserved by his friends and acquaintances. It is not improbable that some of our author's material, now in the keeping of someone or other, has not been included: the reader may rest assured, however, that this material contains nothing not to be found in various parts of the present publication; the only exception to this being, perhaps, a small Treatise on the Rainbow, which he is known to have com• posed, and which, if he did not burn it, as he is believed to have done, is still in the keeping of someone or other, although it is not known who.lO J elles' preface was translated for the Latin edition by Lodewijk Meyer (1629-1681), another close and lifelong friend of Spi• noza's, who had helped him with the preparation of the book on the principles of Cartesianism. Since there is a very good chance that Meyer was well-informed concerning the history of Spi• noza's interest in optics, it is important to note that in translating Jelles' account of the Treatise on the Rainbow, he altered it in two important respects: he added that the work had been writ• ten 'a number of years' before Spinoza's death, and changed the statement that he was 'believed' to have burnt it into the slightly more positive statement that he had 'probably' done SO.11 Naturally enough, Jelles and Meyer wanted to create the impression that their edition was as complete as could reason• ably have been expected. There are, however, further grounds for thinking that their statements concerning the possible fate of the Treatise on the Rainbow are not entirely reliable. They must have been aware that they were not justified in maintaining that if Spinoza had burnt the Treatise it no longer existed. They certainly knew that various copies and versions of the Ethics had

10. F. Akkerman and H.G. Hubbeling, 'The Preface to Spinoza's Posthumous Works 1677 and its Author JarigJelles (c. 1619/20-1683)', in Lias, vol. VI (1979). pp. 103-173; see especially p. 112. 11. See Akkerman and Hubbeling, loc. cit., p. 113: 'nisi forte sit Tractatulus de Iride, quem ante aliquot annos, ut quibusdam notum, compo suit, quique, nisi eum igni tradidit, ut probabile est, alicubi delitescit.'

95 APPENDIX

circulated among Spinoza's friends and acquaintances prior to their publication of the work.12 If one confines oneself to the information they supply in their preface, it is therefore difficult to imagine what reasons they might have given for supposing that this had not also been the case with the Treatise on the Rainbow. What is more, it is quite evident from their edition of Spinoza's correspondence that they had had great difficulty in gathering much material dating from the Voorburg period. Only eight of the letters in the correspondence as it now stands were written between December 1665 and January 1671, and all but one of them were written to members of the Amsterdam circle to which J elles and Meyer belonged. The exception is a letter concerned with the theory of probability, written by Spinoza to Jan van der Meer on October 1st 1666. There is evidence that after 1666 Spinoza's ideas on theoretical optics were less sought after by his friends and acquaintances in the Netherlands, and it is by no means unlikely that some had begun to suspect his competence in the field at an even earlier date. Despite I!lescartes' success in publicizing the sine law and explaining the rainbow, the book on the principles of Cartesia• nism, published in 1663, excellent though it is in many respects, contains nothing of any value or interest on dioptrics, and this may well have been a disappointment to many of those who shared Spinoza's dual interest in philosophy and optics. Others almost certainly failed to appreciate the philosophical motiva• tion behind his preoccupation with light and colours. Hudde, for example, was not the sort of person to enter very enthusi• astically into an exchange of letters devoted to exploring the philosophical significance of reducing practical optics to a series of algebraic equations. As he once observed to Frans van Schoo• ten, he set no store by, 'fruitless enquiries, not worth a dough• nut.'13 Even J arig Jelles was quite evidently dissatisfied with the way in which Spinoza had explained the apparent anomaly in Descartes' Dioptrics. 14 It is, however, the letters which Chris• tiaan Huygens wrote to his brother Constantijn between 9

12. Ethlca, ed. cit., p. 8. 13. Huygens, CEuvres completes, II, p. 101; a letter written on 1 December 1657. 14. Briefwisselmg, letter 40, paragraph 4.

96 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

September 1667 and 11 May 1668, which provide us with the clearest evidence that by then, those engaged on actual research into dioptrics had begun to take a somewhat patronising attitude to Spinoza's theorizing on the subject. They make it perfectly clear that although Huygens valued 'our Israelite's' practical skill in producing first-rate lenses, he thought it very unlikely that he was capable of adding anything of value to the under• standing of optical phenomena. IS Although Spinoza's fellow countrymen seem to have lost interest in his ideas on theoretical optics by about 1667, this was by no means the case abroad. Leibniz concerned himself exten• sively with the Dutch Cartesian school during the course of 1668, and it was then or soon afterwards that he first discovered that Spinoza was one of its leading lights. What fascinated him about the book on the principles of Cartesianism was the at• tempt to apply the rigorous methodology of geometry to philo• sophical speculation.16 After the exchange of letters with Spi• noza in 1671, he spent several years concerning himself principally with the natural sciences. In the course of these enquiries he noticed the importance of Edme Mariotte's (1620- 1684) criticism of Descartes' explanation of the colours of the rainbow, based as this was upon the discovery that the red and yellow rays have a smaller index of refraction than the blue and violet ones.17 During the late autumn of 1676 he spent more than two months in the Netherlands, informing himself concerning Spinoza's circle of friends in Amsterdam and eventually manag• ing to arrange a meeting with him in The Hague, during which he was shown the manuscript of the Ethics and gained some insight into Spinoza's conception of the identity of God and nature. He received a copy of the Opera posthuma through Schuller early in 1678, and immediately set about studying and

15. Huygens, CEuvres completes, VI, pp. 148,151,155,158,164,168,205,213,215. 16. Dze philosophlschen Schrlften (ed. C.]. Gerhardt, 7 vols., Berlin, 1875/90), vol. I, pp. 16, 58, 71; vol. IV, p. 163. 17. Letter to Oldenburg, 8 March 1673: Samtlzche Schrzften und Brzefe, Dritte Reihe, Erster Band (Berlin, 1976), p. 43; Edme Mariotte, CEuvres (2 vols., La Haye, 1740), vol. I, pp. 244-268.

97 APPENDIX

annotating it closely.18 The preface may well have aroused his curiosity concerning the way in which Spinoza had expounded the rainbow. It is therefore perfectly possible that Frederik Muller (1817-1881) and D. Bierens de Haan (1822-1895) were justified in thinking that he made attempts to obtain a copy of the Treatise, although I have been unable to discover any con• crete evidence of this.19 Had he succeeded, the history of his intellectual development after 1679 might well have been some• what different: it is almost certainly the case that he would not have reacted against Cartesianism and Spinoza's pantheism in quite the same way as he did. The complexity of motives which eventually gave rise to his mature monodology led him to re-read Plato and Aristotle in order to confirm the notion that it was essential to criticize Spinoza for regarding God as being in the universe rather than vice versa, for formulating no idea of a final cause, having no adequate conception of the necessity of thinking teleologically. Drawing upon his insights into the natural sciences, he felt justified in postulating force as being inseparably inherent in substance, and the individuation of substance as being in direct contradiction of Spinoza's pantheism. Since every individual soul is as a self-contained world, independent of everything other than God, and since substance is both individual and immaterial, matter has to be regarded as devoid of corporeal interaction. He discussed this extraordinary conclusion with Antoine Arnauld (1612-1694) in a series of letters. Arnauld manoeuvred him into concentrating upon its implications for the natural sciences, and Leibniz found that he was obliged to hesitate before maintaining that corporeality is nothing more than a phenomenon, like the rainbow. He wrote to Arnauld as follows on July 14th 1686:

18. Sdmtliche SchriJten und BneJe, Sechster Reihe, Dritte Band, pp. 578-580 (Berlin, 1980); K. Muller and G. Kronert, Leben und Werk von G. W. Letbmz. Eine Chronik (Frankfurt/M., 1969), pp. 40-46; L. Stein, op. CU., pp. 284--296; G. Fried• mann, Leibmz et Spmoza (Paris, 1946), p. 190. 19. A.C. Kruseman, Fredenk Muller . .. In Memonam (Haarlem, 1881), pp. 43-44; D. Bierens de Haan, in Verslagen en Mededelmgen der KomnkliJke Akademte van Wetenschappen, Afd. Natuurkunde, Derde Reeks, Vierde Dee!, Eerste Stuk (Amster• dam, 1887), p. 66.

98 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

It may cause no little surprise that I should deny what seems to be so evident, namely, that one corporeal substance acts on the other. It should be remembered, however, that others have also maintained this, and that it is a flight of the imagination rather than a clear conception. If body is a substance, and neither a simple phe• nomenon like the rainbow nor a being united by chance or aggrega• tion like a heap of stones, it cannot consist of extension, and has to be conceived of as something called substantial form, which in some way responds to the soul. 20

It may well have been on account of such speculations that Leibniz had attempted to find out what Spinoza had had to say on the rainbow, and what we now know of the Treatise indicates that he was indeed justified in trying to acquire a copy of it. The next record of there having been any interest in Spinoza's work on the rainbow was not printed until twenty eight years after his death, and has an entirely different background.21 In October 1693, the German Lutheran pastor Johannes Kohler (1647-1707) moved into the house on the Stille Kade in The Hague where Spinoza had lived as a lodger from 1670 until 1671. His study was the room in which Spinoza had lived and worked, and in the course of carrying out his parochial duties and making other social contacts in The Hague, he met a number of people who had known the philosopher personally, including a certain master-decorator Hendrik van der Spyck and his wife, in whose house in the Paviljoensgracht he had lodged for the last six years of his life. Kohler set about gathering all the information he could about Spinoza, from both books and hearsay, and in 1705 published his Short but veracious account of the life of Benedict

20. L. Stein, op cit., Ch. VI. For this letter, see: DIe phzlosophzschen Schriften (ed. Gerhardt), vol. II, p. 58; d. pp. 65, 71. Much later (1706-1716), when Leibniz was corresponding with the Jesuit Bartholomeus des Bosses, he maintained unequivocally that 'Massa nihil aliud est quam phaenomenon, ut Iris'. Stein, op. Cit., p. 175. 21. The author of La vIe et l'esprzt de Mr BenOIt de Spmosa, probably Jean Maximilien Lucas, a physician practising in The Hague, knew Spinoza personally and must have written the work between 1678 and 1688. He makes mention of Spinoza's expertise in optics, but not of the Treatlse on the Rambow: the entry in the bibliogra• phy 'Traite de l'Iris ou de I' Arc-en-ciel, qu'il a jete au feu' has recently been shown to be an addition made in 1719, when the work was first printed: M.e. Jacob, The RadIcal Enlzghtenment (London, 1981), pp. 277-278. Cf. Freudenthal, op. CIt., p. 14.

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de Spinoza, based upon authentic sources and the oral testimony of persons still living. 22 His original motivation in undertaking this work seems to have been a mixture of curiosity and antipa• thy arising out of his subject's reputation as an atheist, but as a result of his researches he quite evidently developed a certain admiration for Spinoza's personal qualities, and although his account of his early life is sketchy and unreliable, he gathered many valuable snippets of information concerning his last years in The Hague. In chapter twelve of his book he gives the following account of the Treatise on the Rainbow: The editor of Spinoza's published works includes among the un• published writings his treatise De Iride, on the Rainbow. I am acquainted here with people of standing who have seen and read this treatise, but who dissuaded him from publishing it. Those in whose house he lived inform me that this so vexed him that six months before he died he burnt the work.23

Kohler's basic information was quite obviously derived from Jelles' and Meyer's preface. It looks, moreover, as though the further details concerning the fate of the Treatise which he may have thought he was gathering from his informants, were also derived from this source. From what we know of Spinoza's character and intellectual preoccupations, it seems unlikely that he was so out of touch with current developments in dioptrics while living in The Hague, that he should still have been solicit• ing the opinion of friends and acquaintances concerning the Treatise. One can imagine him burning papers and those living in the same house being aware of his doing so. It is rather more difficult to imagine him being disturbed by criticism of the algebraic manner in which he had expounded the rainbow, and discussing his state of mind with his landlord. In 1680 Christian Kortholt (1633-1694), professor of theology at the University of Kiel, published a work in which he classified Spinoza, together with Hobbes and Lord Herbert of Cherbury, as one of the three great impostors of the age. Eighteen or

22. Freudenthal, op. cit., pp. 35-104; Meinsma, op. CIt. pp. XII-XV. 23. Freudenthal, op cit., p. 83. The crucial question raised by this passage is the nature of Kohler's 'acquaintance' with the 'people of standing'. Is he simply inform• ing us that he knew of them through van der Spyck?

100 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS nineteen years later his son Sebastian Kortholt (1675-1760) vis• ited The Hague in order to gather further information con• cerning Spinoza. What he collected was subsequently incorpor• ated into the preface he wrote for the second edition of his father's book, published in 1700, which evidently helped him to gain the appoinment of professor of poetry at his father's uni• versity during the following year.24 Kortholt also consulted Hendrik van der Spyck, who appears to have been stimulated by this interest from afar into giving further rein to his imagination concerning the fate of the Treatise on the Rainbow, the follow• ing account of which appeared in Kortholt's preface:

The treatise de /ride, which according to the editor of the Opera posthuma may even now be lying hidden somewhere or other, was not among those of Spinoza's works found among the manuscripts he left behind. I consider it to be certain, however, that during the year that was to be his last, the author consigned it not to the light of day but to the flames. On that very day, nearly all the streets of The Hague were ablaze with the light of a festive illumination. He said scoffingly that he was imitating the fireworks, recreating the cele• brations and festive lights in his own home, and added: 'Long have I laboured and fervently at expounding this and thinking it out, but now it is certain that no one will ever read what I have written.' His other works would deck with darkness the most excellent light of truth, products as they are of a wayward imagination, horrible apparitions of the Gate of Horn, deserving to be banished to hell, from whence they came. 0 that he had cast them too into the avenging fire, that none might have read them and been swept away into flames eternal.

The firework display in The Hague, which by the turn of the century Van der Spyck was associating in some way or another with the burning of the Treatise on the Rainbow, had taken place on December 14th, 1677, ten months after Spinoza's death. It is

24. De Tribus Impostoribus Magms (Kiel, 1680, 17002); F. Volbehr and R. Weyl, Professoren und Dozenten der Christian-Albrechts-Universitat zu Kiel (Kiel, 1934), pp. 1, 158; Freudenthal, op. cit., pp. 26-28. Freudenthal (pp. 246-247) tends to blame Sebastian Kortholt for the inaccuracy of the information included in the preface, but his main ground for doing so would appear to be nothing more than Kortholt's hostility to Spinoza.

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interesting to note, however, that this connecting of the two events may not have been due simply to his desire to impress his German visitor. The fireworks and illuminations he remem• bered were part of the celebrations for the wedding of William III and Mary Stuart. They accompanied their public entrance to The Hague, and Van der Spyck was certainly aware of Spinoza's republicanism and sympathy for the de Witts.25 Although it was not until 1703 that the lawyer Gottlieb Stolle (1673-1744) and his friend le. Hallmann visited the Nether• lands and recorded further reactions to questions relating to the Treatise, their sources of information concerning Spinoza's writings were much more reliable than those of Kohler and Kortholt, and they quite evidently approached them with more openness of mind. What is more, we have not only a note jotted down after a conversation with the son of Spinoza's publisher, we also have an expanded version of this note, a memorandum, which has the merit of never having been re-worked in a pub• lishable form. 26 The Amsterdam publisher and bookseller Jan Rieuwertsz the elder (1616/17-1687), who had marketed not only Spinoza's works but also the writings of many of his friends, left his business to his son of the same name in 1685.27 In 1703 Stolle and Hallmann visited Jan Rieuwertsz the younger (165112-1723), and it is evident from the notes they took and the account of their visit which they subsequently wrote out, that they ques• tioned him in great detail concerning the precise history of the manuscripts and editions of Spinoza's works. Their note con• cerning the fate of the Treatise on the Rainbow is as follows: Spinoza did not burn the treatise de !ride; it was not found among his manuscripts after his death, however, and it must therefore be

25. Freudenthal, op. at., p. 65; H. and B. van der Zee, Wtllem en Mary (Den Haag, 1975), p. 120. There was an even more impressive firework display in The Hague in 1690, in celebration of the coronation in London: H.E. van Gelder, 's-Gravenhage in zeven eeuwen (Amsterdam, 1937), p. 179. 26. Freudenthal, op. ett., pp. 227, 301; Meinsma, op. at., Appendices V and VI. 27. M.M. Kleerkooper and W.P. van Stockum, De Boekhandel te Amsterdam voomameliJk in de 17e eeuw (2 vols., 's-Gravenhage, 1914/16), vol. I, pp. 622-631; H. van Eeghen, De Amsterdamse Boekhandel1680-1725 (6 vols., Amsterdam, 1960179, vol. IV, pp. 63-66.

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somewhere or other in the keeping of a friend. 28 It is worth noting that as Stolle and Hallmann understood and recorded him, Rieuwertsz referred to the treatise by its Latin title, and was not of the opinion that Spinoza had made a copy of it. His argument was that it must be somewhere or other in the keeping of a friend, since he had not burnt it and it had not been found among his manuscripts. The note seems to indicate, there• fore, that the only real ground he could have had for this opinion was inside information concerning the reason for J elles' and Meyer's having made a point of mentioning the Treatise in their preface. This was in fact the case. Fortunately, in the full account of the interview, Stolle and Hallmann record what he knew of their reasons for writing as they did: With regard to the treatise de Iride, he assured us that Spinoza had never burnt it, but that nevertheless it had not been found among the manuscripts, and that it must therefore be somewhere or other in the keeping of a good friend. Care had been taken not to omit mentioning the treatise de !ride in the preface, however, in order to prevent someone else coming along and publishing the work under his own name. 29 It seems reasonable to suppose that this does in fact account for Jelles' and Meyer's quite evidently untenable assertion in the preface to the collected works that the Treatise could only have survived if Spinoza had not burnt it. Had they simply admitted that they had not found the original among Spinoza's papers, they would have had no grounds at all for claiming that the text in the keeping of 'someone or other' was a copy of Spinoza's, especially if the 'good friend' had also translated a Latin original into Dutch. Rieuwertsz knew of the problem and the way in which Jelles and Meyer had tackled it, but he did not realize what this implied in respect of the possibility of there having been more than one copy of the work. It was, of course, per• fectly possible that Spinoza should have given a copy of his treatise 1:0 a friend and subsequently lost or destroyed the origi• nal. In fact this was almost certainly what happened. It is inter• esting to note, moreover, that Jelles' and Meyer's anxiety and 28. Meinsma, op. nt., Appendix VI. 29. Freudenthal, op. nt., p. 227.

103 APPENDIX

scrupulousness concerning the point must have been a result of their having been pretty certain that no one else in the N ether• land at that time was likely to have expounded the rainbow in the same way as Spinoza. Had they been less certain of the uniqueness of his approach, they would have been less anxious concerning possible plagiarism. It is certainly of some significance that apart from the J elles• Meyer preface of 1677, there would appear to be no explicit record of any Dutchman having taken any initiative in con• cerning himself with Spinoza's ideas on dioptrics after March 1667. Leibniz, Kohler, Kortholt, Stolle and Hallmann were all Germans. Among the general run of those of Spinoza's fellow countrymen who might have been expected to show some inter• est in his ideas on light and colours, there appears to have been a complete indifference, a total lack of any knowledge or interest. When on July 18th 1703 Stolle and Hallmann visited Burchardus de VoIder (1643-1709), professor of mathematics and physics at Leiden, they found that although he had certain well-defined ideas on the shortcomings of Spinoza's general metaphysics, he was completely unaware that he had written anything on the rainbow: He had heard nothing of Spinoza's treatise de Iride, and when we told him that it is mentioned in the preface to the Opera posthuma, he was not prepared to believe it.30

One can only assume, therefore, that even if de VoIder and his peers had been presented with the treatise after Spinoza's death, they would have had no ready means of relating it to their conception of his general manner of thinking, no terms of reference by which they might have been able to identify it as being Spinozistic.

B. THE 1687 EDITION

Ten years after Spinoza's death and some twenty years after his immediate acquaintances appear to have lost interest in his ideas on dioptrics, an anonymous booklet containing not only an

30. Freudenthal, op. czt., p. 229.

104 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

Algebraic Calculation of the Rainbow but also a Calculation of Chances was published at The Hague.3! Since Spinoza is known to have concerned himself with both these topics, and since this is the first Dutch work on probability published after 1660 and the only monograph on the rainbow published in the N ether• lands prior to 1713,32 there would appear to be some reason for anyone interested in the fate of Spinoza's writings to enquire into the background, content and origin of the booklet. It was published by Levyn van Dyck (d. 1695), who since 1682 had been official printer to the town council in The Hague.33 Although there is no evidence that van Dyck knew anything of the origin of the texts he was publishing, it is of course possible that he was aware that they were by Spinoza and insisted on their being published anonymously in order to avoid running into difficulties with the authorities. Eighteen years previously he had published a work on physics and Cartesianism by the eccentric soldier of fortune Baron von Nulandt, an acquaintance of Huygens and Tschirnhaus and probably also of Spinoza.34 In 1687, however, he published an anti-Spinozistic work by the pious and extremely peculiar physician ].F. Helvetius (1629- 1709), and in 1693/4 books by Johannes Kohler, to whose con-

31. Stelkonstige Reeckening van den Regenboog: Reeckening van Kanssen (In 's Gravenhage, Ter Druckerye van Levyn van Dyck, M.De. LXXXVII). Like Le Journal des S~avans (53 vols., Amsterdam, 1666-1713), the title-page is ornamented with the emblem of an armillary sphere. 32. The Dutch version of Huygen's Tractatus de ratlOcinils m aleae ludo (1657) was published at Amsterdam in 1660. The DlSputatlO de Inde by the clergyman Isaac Samuel Chatelain, was written under the supervision of Jacques Bernard (1658-1718), professor of philosophy and mathematics at Leiden, and defended there for a doctor• ate on June 17th 1713. 33. Curiously enough, Spinoza's publisher Jan Rieuwertsz fulfilled the same func• tion in Amsterdam; see the sources mentioned in note 27 of this Appendix. Van Dyck first published in 1662: E.F. Kossmann, De Boekhandel te 's-Gravenhage tot het emd van de 18de Eeuw (,s-Gravenhage, 1937), pp. 113-117; ].A. Gruys and e. de Wolf, Typographl & Blbliopolae NeerlandlCl usque ad annum MDCC Thesaurus (Nieuw• koop, 1980), p. 31. 34. Elementa Physica, sive Nova Philosophiae PrincIpia, ubi Cartesianorum Prm• ClplOrum Jalsitas ostenditur tpsiusque errores ac Paralogtsmi ad oculum demonstrantur ac reJutantur (1669): see Huygens, CEuvres VI, p. 304, a letter on dioptrics; K. Lasswitz (1848-1910), Geschichte der Atomtstik vom Mtttelalter bis Newton (2 vols., Hamburg - Leipzig, 1889/90), vol. II, pp. 500-502.

105 APPENDIX

gregation he belonged,35 so although he certainly had long• standing contacts with those who were concerned with Spinoz• ism, there is no reason to believe that he would ever have gone out of his way to propagate its doctrines. The booklet, which consists of no more than thirty two pages in quarto, set in various large-size types and padded out with five reproductions of one of the diagrams on five separate pages, is little more than a pamphlet. The pagination is not continuous, and there are sev• eral mistakes in the texts and diagrams which could easily have been corrected by a proof-reader knowing nothing at all about the subject-matter. It certainly looks as though there was no one to cast an effective eye over the final version, and that it could well have been published semi-privately, van Dyck having un• dertaken to print for someone who wanted the text primarily for personal reasons. There is no evidence that van Dyck ever put the booklet into commercial circulation, and the fact that there is no record dating from earlier than 1693 of anyone's having been aware of its existence would certainly seem to indicate that he made no attempt to publicize it.36 Only four copies are known to have survived.37 The content of the booklet, in so far as it is concerned with the rainbow and probability, fits in perfectly with what we can deduce from other sources concerning the way in which Spinoza must have been approaching such topics between the publica• tion of the Renati Des Cartes Principiorum Philosophite in 1663 and the correspondence with Jarig Jelles in March 1667. In expounding the principles of Cartesianism, he had emphasized

35. Adams oud Graft, Opgevult met jonge Coccei Cartesiaenschen en Descartts Spinosistischen beenders (1687); Wonderen van Gods Heyligdom (1693). 36. He retired as official printer to the town council in 1690, and thereafter, apart from the works by Kohler, published little more than a few pamphlets: d. Kossmann, op. cit., pp. 115, 117. 37. J. Kingma and A.K. Offenberg, 'Bibliography of Spinoza's works up to 1800' (Studza Rosenthaliana, vol. XI, no. 1, p. 31). These copies are in the Royal Library at The Hague, the University Libraries of Leiden and Amsterdam and the Bibliotheque Nationale, Paris. Written on the title-page of the Amsterdam copy (U.B.A. 288 D 13) is, 'Hunc Librum possidet Js Schilling'. The hand is that of Johannes Schilling (1746-1820), town-surveyor of Amsterdam from 1772 until his death, who co-operated with J .F. van Beeck Calkoen (1772-1811), professor of mathematics, physics and philosophy at

106 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS that thought is known more directly than what is corporeal, that all that is known clearly and distinctly is true, and that the matter and motion giving substance to extension are to be most clearly and distinctly comprehended in geometrical terms.38 He had not shown how Cartesian geometry enables matter and motion to be resolved into algebraic equations, however, and although he had referred in passing to the Dioptrics,39 he had given no account of the highly successful way in which Descartes had employed the sine law in providing a geometrical explanation of the rainbow.40 The second edition of van Schooten's translation of Descartes' Geometry, together with the essays by Hudde, Huygens and de Witt illustrating the power of the new analyti• cal methods, was published between 1659 and 1661. Since we know that Spinoza had a copy of this work in his library, it seems reasonable to assume that he was encouraged by it, either directly or through his friends and acquaintances, to do what he had not done in his book, and show how Descartes' explanation of the rainbow might be reduced to a series of algebraic equa• tions. Since Descartes' explanation was clearly superior to that

Leiden (1799) and Utrecht (1805/9), in preparing a report on Amsterdam harbour: G. van Suchtelen, private correspondence, May 1984. Since Schilling was by no means a common surname in the Netherlands during the seventeenth and eighteenth cen• turies, it is not unlikely that this Johannes was related to the clothworker of the same name who originated from Waldeck in the Palatinate, settled in Leiden on June 2 1651 (Raadhuis, inteekenregister, br. D. Fo 89), and had two sons, Peter and Paulus, christened in the Lutheran church there in August 1655. Some evidence of a further Leiden connection is provided by the Paris copy (B.N. V 6645 (4)), which is bound together with three other works, possibly deriving from the library of W.]. 's-Gravesande (1688-1742), professor of mathematics, physics and philosophy there from 1717 until his death: Thomas Fantet de Lagny (1660-1734), Methodes nouvelles et abregees pour l'extractlOn et I'approximatzon des racines (Paris, 1692); the copy of the Methodus Incrementorum Dtrecta et Inversa (London, 1715) Brook Taylor (1685-1731) sent to 's-Gravesande, and a copy of the Commercium Epistolicum (London, 1712), inscribed as being the gift of the Royal Society of London. Written on the title-page of this Paris copy, in a seventeenth century hand which has not, as yet, been identified, are the words: 'Voor de Heer de Ruyter' (For Mr. de Ruyter). A Cornelius Ruyter from Enkhuizen, aged twenty two, was ma• triculated at Leiden to read Philosophy on May 10 1688. 38. Op. cit., Pt. I, Props. 4, 14; Pt. II, Def.l: Korte Geschriften (1982), pp. 47, 68,78. 39. Op. ext., Pt. II, Props. 19,24, 31. 40. La Dioptrique, Discours Second; Les Mheores, Discours Huitiesme.

107 APPENDIX

provided by Aristotle in his M eteorologica, 41 and since the battle between the Aristotelians and Cartesians at Leiden was then in full swing, academic politics may well have played their part in encouraging him to work out his ideas on the subject. The more purely philosophical significance of the exercise was that the physical phenomenon should be thoroughly intellectualized, reduced by means of geometrical representation and algebraic logic to something that was clearly and distinctly understood, immediately and unequivocally comprehended. This reduction of physical phenomena to geometry and al• gebra had its exact counterpart in the attempts then being made to deduce a theory of probability from first principles, develop it as a purely mathematical procedure which cannot be falsified by empirical data, and so transform the rational conception of a random event. Aristotle had formulated the frequency theory of probable events: A probability is a thing that usually happens; not, however, as some definitions would suggest, anything whatever that usually happens, but only if it belongs to the class of the 'contingent' or 'variable'. It bears the same relation to that in respect of which it is probable as the particular bears to the universa1. 42

Prior to the 1650's no substantial progress was made beyond this conception, with the result that the theory of probability simply consisted of classifying events as certain, probable and un• knowable. It was evidently Antoine de Mere (1607-1684) who first drew Pascal's attention to the probability calculus implicit in games of chanceY During the course of 1654 Pascal ex• changed letters with Fermat on the subject and informed the Parisian Academy of his intention to compose a treatise on 'the

41. III, 2-5. By taking into consideration the relative positions of the sun, the raincloud and the eye of the observer, Aristotle had been able to give a broadly satisfactory geometrical account of both the size and form of the rainbow. Since he regarded it simply as a reflection from the whole mass of the raincloud, however, and had no conception of the refraction and reflection taking place within the individual raindrops, the reason he gives for the whole area of the primary and secondary bows not being uniformly illuminated (375a 29-375b 9) is not very satisfactory. 42. Rhetorica 1357a 43. O. Ore, 'Pascal and the invention of probability theory' (American Mathematz• cal Monthly, vol. 67, no. 5 (1960), pp. 409-419, p. 409.

108 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

geometry of chance', which would, he claimed, 'link the rigour of scientific demonstrations to the uncertainty of chance, and so reconcile these apparently incompatible matters'.44 It was, how• ever, Huygens' Calculation in games of chance, published in 1657, which first managed to draw a clear distinction between the inductive or classificatory aspect and the mathematics of probability, and so open up this new discipline for the general public.45 Pascal did not hesitate to apply it to nothing less than commit• ment to God:

Let us then examine this point, and say, 'God is, or He is not'. But to which side shall we incline? Reason can decide nothing here. There is an infinite chaos which separated us. A game is being played at the extremity of this infinite distance where heads or tails will turn up. What will you wager? ... Let us weigh the gain and the loss in wagering that God is. Let us estimate these two chances. If you gain, you gain all; if you lose, you lose nothing. Wager, then, without hesitation that He is.46

Antoine Arnauld and Pierre Nicole (1623-1695) agreed with the logic of the famous wager and also suggested that calculation concerning certitude might justifiably be introduced into rea• soning on ethical and legal matters.47 In the Netherlands, it was not long before the a priori calcu• lation of probability was combined with the empirical statistical approach employed by John Graunt (1620-1674) in analyzing mortality rates in London,48 in developing a new approach to life insurance. Life insurance and wagers made on voyages had been

44. P. Fermat, CEuvres (ed. C. Henry and P. Tannery, 4 vols., Paris., 1891-1912), vo!' II, pp. 288-307; B. Pascal, CEuvres completes (Paris, 1963), pp. 101-103 45. CEuvres completes, XIV, pp. 49-91. The publication of the Opera Omnia of the Italian mathematician Girolamo Cardan (1501-1576) in Amsterdam in 1663 showed that Huygens and his acquaintances were not the first to develop the new discipline, although they were certainly the first to publish an account of it. O. Ore, Cardano: The Gambling Scholar (Princeton, 1953). 46. Pensees, no. 233. 47. L'Art de penser (Paris, 1662), Pt. 4, Chs. 15, 16. 48. Natural and Political Observations Mentioned in a following Index, and made upon the Bllls of Mortality (London, 1662); d. G. Keynes, A Bibliography of Sir Wilham Petty (Oxford, 1971), pp. 75-97.

109 APPENDIX

expressly prohibited by the Amsterdam Ordinances of 1598 and the Rotterdam Ordinances of 1604 and 1635, evidently on ac• count of their having been regarded as akin to gambling and therefore immoral.49 Nevertheless, such insurance was by no means uncommon in the Netherlands in the first half of the seventeenth century, and under de Witt both the central govern• ment and the town councils made use of it in order to raise capital. Jan Hudde, for example, who did very little work on pure mathematics after 1663, on account of his devoting most of his time and energy to the service of Amsterdam as a member of the city council, juror and chancellor, became extensively in• volved after that date in the theory and practice of selling life annuities.50 The most important theorist of this practical appli• cation of the theory of probability was, however, Jan de Witt himself, whose Value of life annuities in proportion to redeema• ble annuities was published at The Hague in 1671.5l It is not surprising, therefore, that we should find Jan van der Meer, who was corresponding with Spinoza on the theory of probability in 1666, also corresponding with de Witt on life insurance a few years later. 52 As we have already seen, his father Claes van der Meer, treasurer of the University of Leiden, had also been involved in life insurance, and it is quite likely that his son became involved in the family business in some way or another soon after 1654. Spinoza moved from Amsterdam to Rijnsburg near Leiden in 1661 and from there to Voorburg near The Hague

49. F. Hendriks, 'Contributions to the history of insurance, etc. with restoration of De Witt's Treatise on life annuities', Assurance Magazine, vol. 2 (1852), pp. 121-150, 222-258; vol. 3 (1853), pp. 93-120, 229; D. Houtzager, Hollands LIJ!- en Losrente• lemngen v66r 1672 (Thesis, Rotterdam, 1950); E.5. Pearson and M.G. Kendall, StudIes In the History of probability and statistICS (London, 1970), pp. 19-34. 50. Huygens, (Euvres V, pp. 305-311, 348-351; VII, pp. 95-98; Brieven van Johann de WItt (ed. R. Fruin, G.W. Kernkamp, N.Japikse, 4vols., Amsterdam, 1906/13), vol. IV; K. Haas, 'Die Mathematischen Arbeiten von Johann Hudde (1628-1704) Biirger• meister von Amsterdam', Centaurus, vol. 4, no. 3, pp. 235-284. 51. See the English translation in Hendriks, op. Cit., pp. 232-249. The title of the Dutch original is Waerdye van lzjf-renten naer proportze van los-renten (In's Graven• Hage. By Jacobus Scheltus, 1671). 52. Bneven aanJohann de WItt (ed. R. Fruin and N.Japikse, 2 vols., Amsterdam, 1919122), vol. II, p. 530 (24 March 1670); Brieven van Johann de Witt, ed. cit., vol. IV, pp. 230-232 (April-May 1671).

110 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS in April 1663, so it may well have been between these dates that he first made the acquaintance of Van der Meer. Many of those who followed Pascal in developing the new mathematics of probability saw it as complementing Cartesian algebra. The general assumption was that just as analytical ge• ometry enables us to reason with absolute certainty concerning non-ideal points, lines and surfaces, so the calculation of chances enables us to explicate a priori the mathematical expectation of a random event. Spinoza seems to have shared common ground with Pascal in more ways than one, and his correspondence with Henry Oldenburg between August 1661 and August 1663 shows how he brought his basic assumptions concerning the relation• ship between mathematics and theology to bear upon the prob• lems presented by the natural sciences. Oldenburg opened the correspondence by asking Spinoza for further explanation of certain points they had discussed when he had visited him in Rijnsburg, notably Spinoza's conception of God as infinite extension and infinite thought, and the way in which this conception had determined his critical attitude to• ward Cartesianism and Baconianism. Spinoza was evidently not averse to being drawn on the point, and a month later wrote back telling Oldenburg that neither Descartes nor Bacon had had an adequate conception of, 'the prime cause of all things', that they had both failed to grasp the nature of the human mind, and that they knew nothing of the true cause of human error. Oldenburg, intrigued by this sweeping criticism, asked how God's existence might be deduced from a conception, how one might progress from the perfections perceived in 'people, ani• mals, plants, minerals' to asserting the existence of absolute perfection. He informed Spinoza, moreover, that 'in our philo• sophical society we are applying ourselves zealously, in so far as we are able, to experimentation and observation'. Spinoza wrote back to say that although he had been too busy to give much thought to experimentation, he could assure Oldenburg that a careful consideration of thought and extension would neces• sarily provide him with a self-evident definition of God. Assum• ing that anyone so thoroughly acquainted with the nature of the Deity ought to be able to say a thing or two worth remembering about the nature of saltpetre, Oldenburg sent a Latin translation

111 APPENDIX

of Boyle's investigations into the nature of this substance, the main practical use of which was in the manufacture of gun• powder. Boyle had described his experiments carefully and concluded from them, quite correctly, that chemical substances might be composed of intrinsically distinct elements having entirely different properties. As a result of such work, he is rightly regarded as one of the pioneers of modern analytical chemistry. It is certainly significant, therefore, that in his sub• sequent correspondence with Oldenburg, Spinoza should have argued so persistently that Boyle's experiments had not proved his point. Failing to see that Boyle had provided a destructive analysis of traditional terminology, he maintained that since the 'spirit' of saltpetre was 'driven out' of the 'pores' of the original crystals, the residue was simply the 'base part' or 'excrement' of the saltpetre, which was redintegrated into its original state once the 'spirit' had 'repenetrated' its 'pores'.53 After Spinoza had failed to respond constructively to Boyle's criticism of his conceptions, Oldenburg realized that there was little point in pursuing the matter, although he did inform Spinoza once again of the aims of the Royal Society in co• ordinating knowledge and encouraging experimentation, and promise to send him further publications by 'the illustrious Boyle'. Spinoza replied by informing Oldenburg that he was about to publish a work on the principles of Cartesianism. Oldenburg responded diplomatically, expressing the conviction that the use Spinoza was making of his, 'acute mathematical mind in establishing the basic principles of things' was in fact complementary to the experimentation and observation being carried out by Boyle and his colleagues. Here the matter rested until Oldenburg re-opened the correspondence two years later. Spinoza can hardly have been unaware that his encounter with the Baconian principles being institutionalized by the Royal Society had not been too successful, and Oldenburg, despite the politeness of tone which he managed to maintain in correspond• ing with Spinoza, gave a clearer indication of his real opinion of

53. Briefwisseling, letter 6 (April, 1662); R. Boyle, Certain physzologzcal essays (London, 1661; Latin tr. Amsterdam, 1667); A.R. and M.B. Hall, 'Philosophy and Natural Philosophy: Boyle and Spinoza', in : Melanges Alexandre Koyre (Paris, 1964), pp. 241-256.

112 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

him when he wrote to Sir Robert Murray (d. 1673) that he was in contact with, 'an odd Philosopher, that lives in Holland, but no Hollander'.54 Although it certainly seems reasonable to assume that the original incentive for composing the Treatise on the Rainbow came from those of Spinoza's friends and acquaintances who were disappointed on account of his not having dealt with the subject in his book on Cartesianism, the actual decision to undertake the task could well have been the outcome of this exchange of letters with Oldenburg. It is interesting to note that he had in his library a copy of the Optica Promota published by James Gregory (1638-1675) in 1663. This in itself is not par• ticularly significant, since the book is in many respects a fairly standard work on dioptrics and catoptrics. What is significant is that it contains no mention of the rainbow, and that when he wrote it, Gregory was unaware that the sine law had already been formulated by Descartes. He illustrates refraction and reflection not by means of a circle as Descartes had done, but by means of an ellipse, which makes his calculations unnecessarily complicated. The main reason for the clumsiness and outdated• ness of the work was the backwardness of the instruction he had received at the University of Aberdeen. Aware of the lack of scientific opportunities north of the border, he had come down to London in 1662 in order to get his book published, and it was only after he had seen it through the press that he became acquainted with the work of Descartes. Sir Robert Murray was so impressed by his ability, however, that he attempted to bring him into contact with Huygens.55 Spinoza must have been struck by the fact that when Gregory had attempted to work out his geometrical explanation of re• fraction, he had known nothing of Descartes' Dioptrics. He may also have known that Sir Robert Murray, one-time president of

54. Letter, October 7th 1665: The Correspondence of Henry Oldenburg (ed. A.R. and M.B. Hall, 11 vols., University of Wisconsin Press, 1965/75), vol. II, p. 549. 55. Opttca Promota, seu Abdlta radlOrum reflexorum et refractorum Mysterta, Geometricae Enucleata (Londini, 1663), pp. 134; Freudenthal, op. at., p. 161, no. 62. Soon after the publication of the book Gregory wrote out a still unpublished adden• dum to it (David Gregory, B29, Edinburgh), revising his discussion of reflection and refraction in accordance with the sine law.

113 APPENDIX

Oldenburg's much-lauded Royal Society, had been so im• pressed by his work that he had attempted to introduce him to Huygens. Still disturbed by the unsatisfactory outcome of his correspondence with Oldenburg, he may well have concluded that the English Baconians were in dire need of some enlighten• ment as to the superior insight being provided by continental mathematical physics, and that a Treatise on the Rainbow would be an excellent means of bringing this home to them. 56 If this was in fact the motivation behind the composition of the treatise, it does not seem likely that Spinoza began work on it before the late summer of 1663. There is no evidence that during this period Spinoza was prepared to re-think his basic philosophical position as a result of the exchange of ideas with Oldenburg. Consideration of infinite thought and infinite extension can provide us with a rational conception of the nature of the Deity. The fundamental

56. The first part of Gregory's book is concerned with optics, the second part (pp. 109-134) with astronomy. The authorities he mentions are Alhazen, Kepler, Galilei, Kircher and Seth Ward. It is possible that the elliptical diagrams he uses for illustrating refraction had some connection in his mind with Kepler's first law of planetary motion. It is not known how Spinoza came by the book. It is interesting to note, however, that the only copy of it now in the Netherlands is in the University Library at Leiden, and that this copy used to belong to Isaac Vossius (1618-1689), who knew Spinoza well (BrzeJwIsselmg, letter 40, 25 March 1667), and was at that time deeply involved in criticizing Cartesian conceptions concerning light and colours: see his controversy with Jan de Bruyn (1620-1675), professor at Utrecht and Pierre Petit (1598-1677) of Paris, arising out of the publication of his De luCls natura et proprzetate (Amsterdam, 1662): De Bruyn, EpIstola ad Isaacurn Vosslurn (Amsterdam, 1663); Vossius, Responsurn ad ob;ecta (Hagae-Comitis, 1663), AppendIX ad Scrzpturn de Natura et Proprzetate LucIs (Hagae-Comitis, 1666). The rainbow is mentioned in all these works, and Voss ius also illustrates refraction by means of ellipses (1663 p. 34). Cf. Huygens, CEuvres III, p. 364; IV, pp. 158, 159, 163. Gregory's reputation seems never to have been very high among Huygens' friends and acquaintances. Huygens questioned the originality and validity of Gregory's Vera Clrcuiz et hyperbola? quadratura (Padua, 1667), although the book was reviewed enthusiastically by John Collins (1625-1683) in the Philosophical Transactions of the Royal Society (vol. 3, no. 33, pp. 640-644, 16 March 1668). Gregory replied to Huygens in his ExerCltatlOnes geornetrzca? (London, 1668); see Huygens, (Euvres VI, pp. 313-323. In 1676 the lawyer Salomon Dierkens (1641-1703) informed Huygens of a work on the rainbow he had received from Leiden which employed an ellipse rather than a circle in order to illustrate the nature of refraction, CEuvres VIII, p. 13 (5 September 1676).

114 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS principles of thought are mathematical, and in working out a rational conception of the Deity as revealed in the content of extension, it is therefore essential that one should proceed 'mathematically'. It was evidently this basic conviction which led him to transform the looser arrangement of the Short Treatise on God, man and his well-being into the apparently rigorous structure of the Ethics proved in geometrical order,57 and to re-state Hudde's geometrical presentation of spherical aberration in algebraic terms.58 It is perfectly understandable, therefore, that the central theme of the Treatise on the Rainbow should be the transformation of geometrical physics into al• gebraic equations, and that even the Calculation of Chances should involve imposing a Cartesian analytical approach upon Huygens' mathematical procedures.59 When Spinoza was working out this a priori mathematical method in dioptrics and probability theory, the state of empiri• cal optics was still predominantly Cartesian. It was generally assumed that Descartes had distinguished satisfactorily between the primary truths of physics, on the basis of which one could demonstrate with absolute certainty, and the hypothetico-de• ductive method by means of which explanations could be given of particular phenomena. The propagation of light was regarded as taking place directly, by reflection or by refraction, and as being an instantaneous succession of shocks. The individuality of the various colours was attributed to the rotational velocity of the corresponding globules. There was no widespread objection to assuming that the laws of reflection could be deduced from the postulates of the impenetrability of matter and the con• servation of the absolute quantity of motion. Refraction was taken to be the result of the incident motion being increased or decreased in a given order, the velocity of light to be a property of the medium it is traversing.6o The plausibility of this Cartesian framework diminished rapidly after 1665. Experts were no longer certain of their basic

57. According to F. Mignini (Korte Geschrzften (1982), p. 240), the decision to re-work the Short Treatise was probably made early in 1662. 58. Letter 36 aune 1666); see Appendix note 1. 59. Korte Geschrzften, ed. cit., p. 524. 60. A.I. Sabra, Theories of Light from Descartes to Newton (Cambridge, 1981).

115 APPENDIX

assumptions concerning light and colours, fresh empirical ob• servations made it necessary to revise the whole theory of optics. The age-old belief that light could only be propagated in three ways was shattered by the announcement of the discovery of diffraction in the very first proposition of a book on light, colour and the rainbow, written by the Jesuit Francesco Maria Grimaldi (1618-1663), and published in 1665. Grimaldi had noticed that the propagation of light was not uniformly recti• linear, that there was, for example, a certain periodic distribu• tion in the coloured fringes bordering the shadow cast by a disc placed in a cone of sunlight from a small aperture, and that this seemed to indicate that light was a wave-phenomenon, propaga• ted as ripples are in water, rather than atomistic. 61 In the same year, Robert Hooke (1635-1703) published an account of his investigations into the colours which appear in thin sections of mica, the coloured fringes and rings which can be produced by pressing together plates of glass. He had come to the correct conclusion that these colours were produced by the interaction of light reflected from the front and back surfaces of the plates, and so come very close to formulating the principle of inter• ference. 62 In 1672 an account of Newton's theory of prismatic colours was published in the Philosophical Transactions of the Royal Society. 63 In 1662 Pierre de Fermat (1601-1665), who had always been critical of both the mathematics and the physics of Descartes' Dioptrics, postulated that the finite speed of light varies as the rarity of the medium through which it passes and that 'nature operates by the simplest and most expeditious ways and means', and on this basis worked out a more satisfactory mathematical derivation of Descartes' sine law. Two years later, in a letter to an unknown acquaintance, he summarized his objections to Descartes' derivation of the law as follows: it was based on a mere analogy, it presupposed that light passes more easily through a denser medium and it assumed that the hori-

61. Physlco-mathesls de lumme, coloribus et irzde (Bonomae, 1665). 62. Mlcrographia, or some physiological desmptwns of minute bodies made by magnifying glasses, with observations and mquiries thereupon (London, 1665). 63. No. 80, pp. 3075-3087, 19 February 167112.

116 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS zontal motion is unchanged by refraction.64 In 1669 Rasmus Bartholin (1625-1698), professor of mathematics and medicine at Copenhagen, published a work in which he described for the first time the double refraction of light in Iceland spar. Bartholin noticed that when a stream of light enters this crystal, one pencil of it assumes a direction given by the ordinary law of refraction, whereas another pencil is bent in accordance with another law, which he was unable to determine.65 Huygens had taken the problem up by 1673 and by 1677 had managed to explain some of the phenomena connected with double refraction by applying his wave theory of light. 66 We have direct evidence that Spinoza was aware of the importance of these developments, since in a letter written not long before his death he asks Walther von Tschirnhaus (1651-1708) if he can inform him as to the signifi• cance of the latest discoveries being made in the field of refrac• tion.67 Although Spinoza did not revise his basic metaphysical posi• tion as the result of these developments, he does appear to have given up any attempt to treat optics as a wholly integral part of his philosophical system. Once the advances made in the empiri• cal sciences had called in question the validity of the Cartesian principles on which he had based his conception of optics, he seems to have made no further attempt to submit the one field of natural science of which he had first-hand expert knowledge to the general methodology of his philosophical system. In his Ethics,68 he confines himself to dealing with the empirical sci• ences only in the most general terms: to criticizing those who,

64. CEuvres, ed. cit., I, pp. 170-172, AnalYSIS ad refractlOnes (d. III, pp. 149-151); II, pp. 457-463, the letter to La Chambre of 1 January 1662; II, pp. 485-489, the letter to the unknown person, 1664. 65. Expenmenta crystalh IslandlCl dlsdlaclastlcl, qUlbus mira & msolatz refractlO detegltur (Hafniae, 1669). Bartholin's assistant Ole Romer (1644-1710) was the first to calculate the speed of light from observations of Jupiter's satellites: see his Demon• stratIOn touchant Ie mouvement de la lumlere (journal des Savants, December 1676). 66. CEuvres XIII, Fasc. II, pp. 739, 743; VIII, pp. 36, 37, letter to Colbert, 14 October 1677. 67. Bnefwlsseling, letter 83 (15 July 1676). 68. Ethics, Bk. II, Props. X-XIII; d. D.R. Lachterman, 'The Physics of Spinoza's Ethics', The Southwestern Journal of Phzlosophy, vol. VIII, no. 3 (December 1977), pp.71-111.

117 APPENDIX

'believe the objects of the senses and not knowledge of God to be prior to all things', and to providing no more than the broadest of systematic frameworks for the subject-matter of the natural sciences. It is almost certain, therefore, that the Treatise on the Rainbow was completed and circulated prior to the exchange of letters with Jarig Jelles in March 1667. This circumstantial evidence is essential to an understanding of the texts on the rainbow and probability, since it enables us to date with some accuracy what was published by Levyn van Dyck in 1687, and to show that it fits in well with what we can gather from other sources concerning Spinoza's general re• sponse to natural science. It also makes it essential, however, that a sharp distinction should be drawn between the main body of these texts and what was added to them when the booklet was prepared for the press. The difficulties involved in identifying, dating and analyzing this editorial material are, if anything, even more challenging than those presented by the texts themselves, and it is, therefore, absolutely essential that the two fields of enquiry should not be confused. Although Jan Rieuwertsz the younger evidently referred to the Treatise on the Rainbow by its Latin title, the Dutch in which the published version of it is printed does not read like a translation. With his closest friends, such as Meyer, Spinoza habitually corresponded in Latin, and we know that not only the book on the principles of Cartesianism but also the Short Treatise were originally written in this language. Nevertheless, since the letter on probability which he sent to Jan van der Meer on October 1st 1666, evidently after having discussed the matter with him, was written in Dutch, it seems quite likely that the communications constituting the basis of the published texts of the Calculation of Chances and the Algebraic Calculation of the Rainbow were also written in this language. If this was in fact the case, it is not difficult to determine the scope of the editorial work involved in preparing them for the press. In the main, it must have consisted of drawing up the title• page, together with its motto concerning the book's 'serving to unite physics more closely with mathematics', selecting and translating the quotation from Cicero, indicating the historical background and didactic purpose of the publication in the pre-

118 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS face, introducing the marginal references to Euclid, Descartes and Barrow, and inserting the lengthy quotation from the Dutch edition of Huygens' work on probability at the beginning of the Calculation of Chances. It may also have included adding the opening and closing paragraphs of the rainbow text and the references to the works of Philip Lansbergen (1561-1632) and Frans van Schooten the elder (1581-1645), and it must have involved inserting the reference to the death of Rene-Fran<;ois de Sluse (1622-1685).69 Since Sluse died on March 19th 1685, this piece of editorial work enables us to date the preparation of the published text fairly accurately. Bearing in mind this distinction between the main body of the text and the editorial work, it is possible to form a fairly clear idea of the sort of person the editor must have been. Since he published anonymously, there must have been reasons for his not having wanted it to be known that the texts were by Spinoza and that he had edited them. He did go to the trouble of preparing them for the press, however, and there is no reason why we should not accept the explanation he gives when he says that he wants to awaken the young to the need for 'uniting physics more closely with mathematics'. If he did insert the references to Euclid, Descartes, Barrow, Lansbergen, van Schooten, de Sluse and Huygens, he must have had a fairly lively interest in what was going on in the world of mathematical physics, although one does not get the impression that he was in any respect an expert. There is every likelihood that he was in contact with Spinoza between 1663 and 1667, and since he is certainly justified in calling attention to the corresponding work of Hudde, Huygens and de Witt in his preface, he must have been aware of the historical background to the texts he was editing. On the other hand, if he had been a very close friend, or if he had kept up his contact with Spinoza after 1670 when he moved to The Hague, he would have been known to JelIes, Meyer or Van der Spyck. It is not very likely that Hudde edited the texts, since he would probably have published them in Amsterdam, he took little interest in such matters after 1663, and

69. See the notes to the text in the Korte GeschrzJten, op. cit., pp. 527-533 and Baruch de Spinoza, Algebralsche Berechnung des Regenbogens. Berechnung von Wahrscheinhchkezten (tr. and ed. H.-C. Lucas and M.J. Petry, Hamburg, 1982).

119 APPENDIX

Jelles and Meyer, when they were preparing the collected works, seem to have had access to the material he had in his possession. Huygens is not likely to have been interested in publishing such outdated and elementary material; had he been, he would certainly have edited it more carefully and effectively, and in any case, we know that he did not possess a copy of the booklet.?o In 1687 De Witt had been dead for fifteen years. The most likely possibility is, therefore, Jan van der Meer. In fact all the available evidence points to the fact that it was he who prepared the texts for the press between the spring of 1685 and his death on October 3rd 1686. After the fall of De Witt in 1672, he certainly had reasons for not wanting it to be known that he had had dealings with Spinoza or been in very close contact with the former Grand Pensionary of Holland. As a result of a prolonged depression in the cloth industry, there was tension in Leiden at that time between the workers, who generally sup• ported the House of Orange, and the ruling oligarchy of the town, which by and large had republican sympathies.?! Van der Meer had held the post of government tax-collector under De Witt, and although he continued to do so under William III he could easily have run into difficulties had he been suspected of sympathising with Spinoza's republican ideals. In 1675, for ex• ample, the curators of the University, with the personal ap• proval of the Prince of Orange, drew up a list of twenty one propositions, many of them Cartesian, which were thereby prohibited from being taught, disputed or dealt with in the University.?2 When Jelles and Meyer were gathering material for the collected works, Van der Meer was, therefore, in no position to come forward and advertize the fact that he had previously had connections with the Amsterdam Cartesians and Spinoza.

70. Catalogus Vartorum et mstgntum m omnt Facultate et Lmgua Llbrorum, PrtRcipue Mathematlcorum, Polztlcorum et MlScellaneorum Amplisslmt ac NobtllSSlmt Vm Christiant Hugenil Zuylzchemll (Hagae-Comitum, 1695), 70pp. The books were auctioned on October 25, 1695. 71. N.W. Posthumus, Bronnen tot de Geschzedents van de Letdsche TextzelntJver• held, Pt. V, 1651-1702 (,s-Gravenhage, 1918); De Geschtedemsvan de Leldsche Laken• mdustrle (2 vols., Den Haag, 1939), vo!' II, p. 1142. 72. P.e. Molhuysen, Bronnen tot de Geschiedems der Leidsche Umversttett (7 vols., 's-Gravenhage, 1913/24), vo!' Ill, pp. 317-321,259.':-

120 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

As a member of one of the foremost patrician families of Leiden he was in any case part of the establishment of the town, and from 1669 onwards had played an active and prominent part in civil administration, - as captain in the home guard (1669/74), governor of the town orphanage (1675/6) and churchwarden (1679/86).73 The situation in Leiden changed somewhat after 1680, how• ever, as the political tensions associated with Cartesianism re• laxed, and a new attitude toward the relationship between phi• losophy, mathematics and the natural sciences began to develop at the University. This new attitude was largely the result of the eclectic teaching of Wolpherdus Senguerdius (1646-1724)/4 and the attempts made by Burchardus de VoIder (1643-1709) to combine traditional Cartesian methodology with a new aware• ness of the significance of the Baconian-Newtonian approach being practised and propagated by the Royal Society.75 In 1682 De VoIder, who had been appointed professor of philosophy in 1670, was also appointed professor of mathematics. He assumed his new post with an address on the union of mathematics and natural philosophy in which he dwelt upon the usefulness, the practical applicability of the new mathematical physics, the way in which the 'illustrious Huygens' had 'joined to the other sciences in which he excels a recondite knowledge of the mathe• matical arts' and by his 'most elegant' doctrine of the pendulum, for example, made possible a great improvement in the con• struction of clocks, the achievement of greater accuracy in the measuring of time. 76 One of De VoIder's pupils from The 73. Naamwyser, waar In vertoond werden de Naamen van de Ed. Achtb. H.H. Regenten der Stad Leyden. Tsedert den fare 1641 to 1687 (Leyden, 1688). 74. See his Phtlosophla naturalls (Lugduni Batavorum, 1680, 1685 2). 75. In 1674, two years after the publication of Newton's theory of prismatic colours, De VoIder visited England to investigate the new experimental philosophy, and on his return successfully petitioned the curators to be allowed to give a course on experimental physics at the University: Molhuysen, op. Cit., vol. III, pp. 298, 301. He also undertook a close reading of Newton's Phzlosophlae Naturalzs PrInCIpia Mathe• matlca (London, 1687) soon after it was published: Jean Ie Clerc (1657-1736), 'Eloge de feu Mr. de VoIder', in: Blblzotheque cholsle (Amsterdam, 1703/13), vol. XVIII, pp. 347,379,380; M.e. Pitassi, 'Jean Ie Clerc bon tache ron de la philosophic', Lias, vol. X (1983), no. 1, pp. 105-122. 76. Oratzo de conJugendls phzlosophlClS et mathematlcls dlSClplInlS (Lugduni Bata• vorum, 1682).

121 APPENDIX

Hague, a certain Hermannus Schuyl, in a doctoral disputation defended in 1688, undertook to reduce a variant of the Cartesian laws of impact to algebraic formula:. 77 I t seems very likely, therefore, that Van der Meer was encour• aged by the way in which mathematics and physics were being taught at Leiden during the 1680's, to think afresh about the Spinoza texts he had in his possession. It is not improbable that he was first encouraged to prepare them for the press as a result of the spate of rather second-rate articles on the rainbow pub• lished in Germany in 1685. 78 The final decision to do so may well have been determined by the didactic purpose he mentions in his preface, however, for there are good reasons for thinking that he actually prepared the texts for his son Claes, christened at Leiden on March 31st 1675. Within the town, it was usual for children from better-class families to receive instruction from registered schoolmasters in their own homes, probably in small groups. Van der Meer may well have planned to have the book• let printed so that it could be used in such a group.?9 It has been suggested that in accordance with a Ciceronian tradition, he

77. Dtsputatio phtlosophica mauguralls de VI corporum elastica (Lugduni Bata• vorum, 1688); d. Molhuysen, op. at., vol IV, p. 20P. 78. See the Miscellanea Curwsa slve Ephemerzdum Medico-Physlcarum Germa• mcarum Academia? Natura? Curwsorum. Decuria: II. Annus Tertius, Anni MDCLXXXIV (Norimberga:, 1685). Christian Mentzel (1622-1701), physician to the Grand Elector in Berlin, was the author of four of the articles: De Iride Solarz alba (pp. 20-22); De Irzde Aureo-lutea Solari, et alba Lunarl msequente (pp.23-24); Iride coelestl rubra Solarz, item de Aurora matutma et vespertma, Halore, nube replendente pauca (pp. 24-27); Colorum !ridis coelestls comparatione cum Colorzbus hypostatlcls slve Pigmentzs horum ordme, et nature (pp. 27-33). Dr. Georgius Francus, in his De !ridl Lunari (pp. 41-43), quotes Isaac Voss ius as an authority, and like Newton distinguishes seven colours in the lunar rainbow. Georg Caspar Kirchmaier (1635- 1700), professor of rhetoric at Wittenberg, in giving account of a lunar rainbow he had seen on January 24th 1684, refers to both Aristotle and Descartes: De Admlranda Lunari !ride (pp. 44-47). 79.]. van Groenendijck, Ordonnantie, Van het Schoolmeesters Gtlde, Bmnen de Stad Leiden (Leiden, 1689), c. VI. The ordinance dates from 20] anuary 1683. It seems probable that Van der Meer did not arrange to have the texts published in Leiden because he wanted to avoid speculation concerning their origin. There is ample evidence in the notarial records of the time held by the Municipal Archive in The Hague, approximately a third of which have now been indexed, that he had numerous business contacts in the town: there are ten documents concerning property and loans etc. dating from the period 6 December 1675 to 15 ] anuary 1685.

122 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS may have dedicated it to his son, but both were dead before it was published.8o Finally, it should be remembered that we have concrete evi• dence in the letter of October 1st 1666 of Van der Meer's having been in contact with Spinoza during the period when he is most likely to have been concerned with the preparation of the Treatise on the Rainbow, that he was then corresponding with him on the calculation of chances, and that by 1678 he had so passed from the minds of those who had known Spinoza best, that Schuller could do no better than inform Leibniz that he was in business in Amsterdam. It is perfectly understandable, there• fore, that Jelles and Meyer, a year earlier, should have known nothing of his whereabouts or of the manuscripts he had in his possession. 81 If the 1687 edition was prepared for semi-private publication, for the use of Claes van der Meer's schoolfellows, it can hardly be regarded as surprising that there should be no record of any notice having been taken of it at the time. Van Dyck almost certainly made no attempt to advertise or distribute it. What is surprising is that it should have been published at all. Who saw it through the press after Jan van der Meer and his son were dead? We have already noticed that the fact that there are several mistakes in the text and the diagrams which could easily have been corrected by a proof-reader knowing nothing at all about the subject matter, would seem to indicate that there was no one to cast an effective eye over the final version. 82 Who could have taken the trouble to deliver what had been prepared to the printer's and keep the copies he produced? Attempting to answer this question could have been regarded as a matter of pure speculation were it not for the curious fact that the booklet was reviewed by Pieter Rabus (1660-1702), notary and master at the Erasmus Grammar School in Rotterdam, in his periodical The European Library, - not immediately, but some

80. N. van der Bjorn, private correspondence, March 1984. Cf. Van der Bjorn's article 'Een vader draagt zijn boek op aan een zoon, of: Erasmus imiteert Cicero' (Hermeneus, vol. 44 (1972/3), pp. 205-210). CJaes van der Meer died three weeks after his father in October 1686. 81. Cf. Appendix notes 5, 12. 82. See the Lucas-Petry German edition, op. cit., pp. 15, 22, 33, 35, 45, 46, 47, 65.

123 APPENDIX

six years after it had been printed.83 This delay is not only curious in itself, but doubly curious in that all the other works reviewed by Rabus in this and other numbers of his periodical were very recent books, marketed by commercial publishers, and obtainable in the Rotterdam bookshop run by Pieter vander Slaart (c.l662-c.l711), who financed and published The Euro• pean Library in order to advertise his wares. 84 No other books published by Van Dyck were reviewed by Rabus. It is almost certain, therefore, that he acquired this one not from the publisher but through a private contact, and from what we know of the journalistic procedures of the time, we are able to determine fairly accurately how this contact must have been made. His periodical was intended to be the Dutch coun• terpart to Pierre Bayle'S highly successful Nouvelles de fa Repu• bfique des Lettres, and we know that even Bayle, during the early stages of his venture, had some difficulty in getting enough books to review. 85 In the foreword to the first number of his European Library, therefore, published in July/August 1692, Rabus requested that all, 'lovers of the arts and sciences, of theology, mathematics, philosophy, law, medicine, physics, languages, history, indeed, of all commendable intellectual ac• tivity, should send to the publisher of this periodical anything of any importance that they may have in their possession'. 86 We know from the preface to the fourth number, published in January/February 1693, that this request had brought the re• quired results. It is not often repeated in subsequent numbers of the periodical, presumably on account of Rabus' having had enough books to review. Although Rabus was a notary and a classicist, and although

83. De Boekzaal van Europe, no. 4 (Rotterdam, January/February, 1693), pp. 153- 157. 84. A.P.F. Wouters and P.H.A.M. Abels, Pzeter vander Slaart. Boekdrukker en boekverkooper te Rotterdam (1691-1702) (Nijmegen, 1981). Lists of the books he had for sale are to be found in several of his publications: De HlstortSche Gedenkschriften van den Ridder W. Temple (Rotterdam, 1692), 20 titles; Petro Chauvin, De Naturalz Religione (Roterodami, 1693), 94 titles; De Boekzaal, no. 6 (May/June 1693), 274 titles. None of these lists includes the Treatzse on the Rambow. 85. C. Serrurier, Pzerre Bayle en Hollande (Lausanne, 1912), pp. 77-78. The periodi• cal was published at Amsterdam from 1684 until 1710. 86. Voorwoord (p. 5).

124 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS the great bulk of his work reflects his literary and historical bias, he made a brave attempt to cover mathematics and physics in his reviewing. In the first number of the periodical, for example, he gave an extensive write-up to Etienne Chauvin's (1640-1725) recently published compendium of Cartesian science, paying particular attention to its treatment of the rainbowY Although he was aware that Cartesianism was being forced to give way to the sort of experimentalism being practised in England, he had a respect for it and usually reviewed Cartesian works appre• ciatively. He was highly critical of Spinozism, however, and made a point of not reviewing works he suspected of being influenced by it, in order to avoid giving it gratuitous publicity. This attitude seems to have been the result partly of a religious and intellectual conviction and partly of the social inconve• nience his wife's family had suffered in Rotterdam, on account of her uncle's having been associated with Spinoza.88 The very fact that Rabus reviewed the Treatise on the Rainbow is there• fore evidence that he was unaware of its ultimate origin, and may well indicate that whoever passed it on to him, presumably as a result of his request for such books and the way in which he had reviewed Chauvin's compendium, knew that it would be unad• visable to enlighten him on the point. Like nearly all Rabus' journalistic work, the review is quite obviously a rushed job. 89 The greater part of it simply consists of word for word excerpts from the texts, which are, incidentally, not indicated as such. Since he was under the impression that one hand had written the whole thing, he quotes the preface, the main text and the extract from Huygens as if they were all the work of this one person. He evidently regarded the writer as a Cartesian, and yet when he refers to the account of the way in

87. LeXIcon ratzonale (Pieter vander Slaart, Rotterdam, 1692); De Boekzaal, no. 1, pp. 148/9. 88. Rabus' wife was Elisabeth Ostens, the niece of Jacob Ostens (1625-1678), surgeon of Rotterdam, who corresponded with Spinoza: BrzeJwisselzng, letters 42, 43 Qanuary/February 1671); Jan de Vet, 'Spinoza's afwezige aanwezigheid', in: Hans Bots, Pleter Rabus en de Boekzaal van Europe 1692-1702 (Amsterdam, 1974), pp. 287-294. 89. He occasionally admIts that he has rushed his work, see: De Boekzaal, no. 19 Qanuary/February 1695), p. 172.

125 APPENDIX

which Descartes explained the difference in brightness between the primary and the secondary bow, he misquotes the original and gets the whole business hopelessly wrong. There are pas• sages in which he might appear to be providing us with extra information concerning the origin of the text, but in which it is just as likely that he was simply colouring things up by drawing upon his general knowledge. For example, when he informs us that the work was: 'published by a certain gentleman, for the use of his son, who was at that time gaining practice in mathematics in Leiden, and primarily, moreover, in the basics of the art of singing and playing the keyboard', it is not unlikely that he had in mind a work he had reviewed a few months earlier.90 This emphasis upon the supposedly musical interests of the gentle• man in question occurs again at the end of the review. Rabus rounds it off by informing the readers of The European Library that he has been told: 'on good authority, that the mathematical gentleman who has composed the abovementioned Calculation of the Rainbow, has no objection to hearing what those who are earnestly concerned with the matter think of it, in order that he may know whether his further observations on points of suspen• sion, midpoint inclinations, carillons and so on, might appeal to inquiring experts. These all being highly ingenious matters, requiring a scientific grounding in the works of Archimedes, Euclid's De Sectione Canonis, and Ptolemy's De Harmonicis.' If this was in fact the case, one wonders why this gentleman should have published his texts anonymously, why he should have left it so long before getting them reviewed, and what particular advantage he hoped to gain from making his contacts through Rabus. There is, of course, no evidence that anyone responded to what Rabus had written, and he makes no further mention of the matter in the pages of his periodical. Nevertheless, despite this journalistic waffle, understandable, perhaps, from someone without much knowledge of natural science who was attempting to build-up the scientific reputation of his journal, the review does contain some worthwhile hints as to who might have given the booklet to the publisher. It looks as

90. Salomon van Til (1643-1713), Dzgt- Sang- en Speelkonst (Dordrecht, 1692): De Boekzaal, no. 2 (September/October 1692), pp. 269-278.

126 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

though Pieter vander Slaart probably received his information orally and passed it on in the same manner. The arrangement was, doubtless, that if he should receive orders for the publica• tion as a result of the review, they could be met by his sending them on rather than by his actually keeping the booklet in stock.91 In any case, Rabus seems to have been accurately in• formed on a number of basic facts which he could hardly have derived from the text alone. He knew, for example, that the work was associated with Leiden, that it had been prepared by a father for his son, and that the son had not been studying mathematics as a matriculated student, but that he had been 'gaining practice' in the subject. Judging from his choice of words, he would also seem to have known that although the father had 'composed' and 'published' the work, and, therefore, 'written' it out, he was not in fact the 'author' of it. It may well have been the case that Jan van der Meer was also interested in mechanical devices and musical theory, and that he had left papers relating to these subjects. And if this was so, it is of course very likely that mention was made of the fact when approaching Vander Slaart, in order to encourage him to accept the booklet for review. But there is, at present, no available evidence which might enable us to confirm or deny that this was the case.92 The most likely source of such inside information concerning the origin of the 1687 edition is Jan van der Meer's wife Anna Verboom (c. 1647-1697). She, too, came of a patrician family, and since a number of her relations had studied philosophy or law at Leiden, it is not difficult to imagine her taking over the task of seeing the booklet through the press after her husband and son had died, and keeping the copies once it had been printed. Some of her family connections may even account for Jan van der Meer's having been so involved in matters philo• sophical. Her father's cousin, for example, had married Franco van Burgersdijck (1590-1635), who had held the chair of phi• losophy at Leiden just prior to the advent of Cartesianism, and

91. See Appendix note 84; J.].V.M. de Vet, Pleter Rabus (1660-1702). Een weg• berezder van de Noordnederlandse Verlzchtmg (Amsterdam, 1980), p. 128. 92. The quite extensive legal documentation relating to the death of Jan van der Meer in October 1686 throws no light upon the subject: Gemeentelijk Archiefdienst Leiden: Register van Seclusien 1668-1811: Inv. nr. 142, F 454, fol. 306-.

127 APPENDIX

two of her uncles had studied philosophy at the University while he was teaching there. 93 Two of her cousins had studied philosophy and law at Leiden during the second half of the 1650's, when the tensions between the Aristotelians and the Cartesians were at their height.94 When her husband and only surviving son died in 1686, she was left with an eight year old daughter. In July 1690 she married again, Adriaen van Kruijs• kercke (1654-1705), magistrate in Leiden. 95 Of more immediate importance for the light it throws upon the connection with Rabus is the fact that she came of a Rotterdam, not a Leiden family, and that we know that she and her husband were back in the town on September 19th 1692, when they witnessed the christening of one of her cousin's children.96 The first number of The European Library, with its request for works worth review• ing in the field of mathematics and physics and its enthusiastic account of Chauvin's treatment of the Cartesian theory of the rainbow, had just been published. It is not difficult, therefore, to imagine them contacting Pieter vander Slaart, and sending on a copy of the booklet once they were back in Leiden.

C. BACKGROUND AND EDITORIAL WORK SINCE 1860

If it is extraordinary that there should be no record of anyone's having taken any notice of the 1687 edition until six years after it was published, essential that we should assume it to have been printed semi-privately, it is almost incredible that prior to 1973

93. Ursela Verboom Jacobsdochter married Burgersdijck and their son was ap• pointed pensionary of Leiden in 1670. Willem Verboom was matriculated at Leiden on 11 December 1621, and Francois Verboom (1607-1676) on 20 October 1626: Album StudlOsorum Academiae Lugduno Batavae 1575-1875 (Hagae Comitum, 1875). 94. Leonardus Verboom (1637-1681), matric. 22.3.1655; Willem Verboom, matnc. 21.10.1658. 95. J.H. Croockewit 'Fragmem-Genealogie Verboom', Maandblad van het Genealoglsch-heraldlek genootschap 'De Nederlandsche Leeuw', Jrg. XVII (1899), nos. 7, 9, 10, cols. 97-101, 138-154. 96. Willem Heindrickse Verboom, the son of Will em Verboom, who had matricul• ated at Leiden in 1658; a lawyer, widely involved in the administration of Rotterdam: E.A. Engelbrecht, Brannen voor de Geschiedems van Rotterdam. V: De Vroedschap van Rotterdam 1572-1795 (Rotterdam, 1973), pp. 45-47.

128 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS no one should have paid any attention to Rabus' review of it.97 Despite the high probability of their being connected in some way or another with Spinoza's Treatise on the Rainbow, no one looked into the matter, and both booklet and review sank into complete oblivion. It was simply assumed that since the treatise had been burnt or lost, there was no longer any evidence of the way in which Spinoza had applied himself to this branch of the exact sciences. This is not to say, however, that there was no interest abroad. The change which took place in the general assessment of Spinoza's philosophical significance in Germany at the end of the eighteenth century encouraged H.E.G. Paulus (1761-1851), professor of evangelical theology at the University of Jena, to undertake the re-editing of his works. Hegel helped in the preparation of this edition, and in the lectures on the history of philosophy which he delivered at Berlin some twenty years later, he observed that: Spinoza gained a livelihood for himself by grinding optical glasses. It was no arbitrary choice that led him to occupy himself with light, for it represents in the material sphere the absolute identity which forms the foundation of the Oriental view of things.98

He may have been reading more into the matter than is war• ranted by present-day evidence, and he was of course obliged to base the observation not upon any knowledge of the 1687 edi• tion or of Rabus, but upon Spinoza's correspondence and the various accounts of his life. It was, however, neither a philosopher nor an historian of philosophy, nor even an antiquarian in any proper sense of the word, who took the first major step forward in opening up this aspect of Spinoza's work to scholarly enquiry. The Treatise on the Rainbow was eventually discovered by a bibliophile, - a collector, cataloguer and marketer of books, a man who was, as

97. J. de Vet, in: Hans Bots, op. cit., pp. 288-289. 98. Hegel's Lectures on the History of Philosophy (tr. E.S. Haldane and F.H. Simson, 3 vols., London-New York, 1963), vol. III, p. 253. Evidence that Hegel helped with the preparation of Paulus' Benedicti de Spinoza Opera quae supersunt omnia (2 vols.,Jena, 1802/3) is forthcoming in the editorial apparatus to volume five of Hegel's Gesammelte Werke (Hamburg, 1968-): Schriften und Entwurfe 1799-1808 (ed. K.R. Meist).

129 APPENDIX

he openly admitted, much more interested in the rarity, the outward form and appearance of what has been printed than in its academic significance or intellectual content. Frederik Muller (1817-1881) was of German descent, and earnt his living in Amsterdam by acquiring and selling books and manuscripts. Since he was more of a businessman than a scholar, he was quite prepared to exchange anything he had as soon as he had come by it. He evidently came across the treatise in the course of prepar• ing his monumental survey of Dutch pamphlet literature,99 and subsequently wrote as follows of his discovery:

It seems to me that not enough attention is paid to the difference between the outer appearance of a book and its intellectual content. I am familiar with the appearance of a large number of theological, legal, medical, mathematical works, the content of which has sel• dom attracted my attention and never been the object of my study. For the history of science it may, perhaps, be important to know what Spinoza thought about the rainbow, while what the booklet looks like, why it is such a rarity, when it was discovered and reprinted, will be matters of little importance; - it is, however, precisely these things which constitute the essence of its interest to the bibliographer. Spinoza's work, listed in a bibliographical hand• book, will appear as follows: [Spinoza (B. de)]. Algebraic Calculation of the rainbow, serving to unite physics more closely with mathematics. The Hague, printed by Levyn van Dijck, 1687, 20 pp. 4°. (For the precise collation of the book see Bierens de Haan, Materials p. 61). This extremely rare booklet, published anonymously, was dis• covered in 1860 by Frederik Muller, who, - knowing that Leib• nitz, a few years after Spinoza's death, had suspected the exist• ence of the work and made fruitless attempts here in the Nether• lands to acquire a copy of it, - passed it on for expert advice to Dr. J. van Vloten, who recognised it as being in fact the work which Spinoza, on the authority of his biographers, was thought to have burnt. In actual fact, they simply indicate that certain persons were aware that Spinoza had written a treatise on the rainbow some years previously, 'which may still be hidden away some• where, if he did not burn it, as he probably did'.

99. Btblwtheek van Nederlandsche Pamfletten (3 vols., Amsterdam, 1858/61), which lists 9668 items for the period 1500-March 1702.

130 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

This copy of the original edition is now in the Royal Library in The Hague. Prof. Bierens de Haan of Leiden is also in possession of a copy, bound with the works of Claes Janz. Voogt, to whom he ascribes it in his Materials. lOo

From 1855 until 1876 Johannes van Vloten (1818-1883) was professor of Dutch language and literature at Deventer. After having studied theology at Leiden he had developed violently anti-clerical views, for which he sought intellectual support in the writings ofD.F. Strauss (1808-1874), L.A. Feuerbach (1804- 1872) and Spinoza. As he made perfectly clear in the book he published on Spinoza in 1862, he saw him as a forerunner of nineteenth century materialism.lol In the same year he also published an edition of Spinoza's works, in which he included the text of the newly discovered Treatise on the Rainbow, together with a Latin translation. In the preface to this work he informed the reader that Muller had discovered the booklet 'some years previously', and that he had been persuaded of its genuineness on account of its evidently Cartesian background, the mention made of Hudde, Huygens and De Witt in the foreword, the accounts given of its fate in the preface to the Opera posthuma and the writings of Kohler and Kortholt, and Spinoza's letter on probability of October 1st 1666.102 In a couple of footnotes to his Latin translation, Van Vloten also noted that the misquotation of Horace in the preface was evidently inten• ded to be a reference to the ten years that had elapsed between the death of Spinoza and the publication of the booklet, and that the contrasting of the theological and physical explanations of the bow in the second paragraph of the work was typical of Spinoza.lo3

100. A.C. Kruseman, op. cit., pp. 42-44. 101. Baruch d'Espmoza, ZIJn Leven en schrlften, m verband met ziJnen en onzen tlJd (Amsterdam, 1862). For his assessment of the Treatzse on the Rambow, see p. 217. 102. Ad Benedictt de Spinoza Opera quae supersunt omma suppLementum (Amstelodami, Apud Fredericum Muller, 1862), pp. iii, 252-285. 103. Op. cit., pp. 256, 260:

131 APPENDIX

J.P.N. Land (1834-1897), professor of philosophy at Leiden, in the course of co-operation with Van Vloten in preparing the collected edition of Spinoza's works published in 1882/3 104 dis• covered a second copy of the Treatise on the Rainbow in The Royal Library at the Hague. It was bound together in parch• ment with the Calculation of Chances. Since this second work had not been attached to the copy discovered by Muller, and since it had a separate pagination and was printed in a larger type, there was some uncertainly as to whether the two had been printed as one work or whether they had simply been bound together at some later date. In 1883 Land discussed the matter with D. Bierens de Haan (1822-1895), who was then in posses• sion of the copy discovered by Muller, and who five years previously had published a work in which he had ascribed the Treatise on the Rainbow to the Amsterdam mathematician and teacher Claas Jansz. Vooght. IOS Closer investigation now con• vinced Bierens de Haan that the two treatises had in fact been published as one work, and a year later, in a new edition of the contents of Van Dyck's booklet, dedicated to the University of Edinburgh on the occasion of its tercentenary celebrations, he gave the following account of what had decided him:

Was this Calculation of Chances due to the same author? At first sight, it seemed this question ought to be answered in the negative: moreover it seemed most probable that here we had to do with a fortuitous joining of two separate tracts, of the same dimensions only, perhaps of the same time, here still-what not always happens - of the same, mathematical, tendency. Now such a conglomeration often occurs, as is well known to Bibliophiles, and very often it has been my first care, when getting hold of such a one, to disperse again every separate tract to where its place should be, according to its contents and the distribution of my own library.

104. Benedicti de Spinoza Opera quotquot reperta sunt (ed.J. van Vloten andJ.P.N. Land, 2 vols., The Hague, 1882/3). The two treatises were published in vol. II, pp. 507-524. 105. 'Bouwstoffen voor de Geschiedenis der Wis- en Natuurkundige Wetenschap• pen in de Nederlanden', no. 1 (1878); cf. no. 24 (1884), p. 66 (1887), in: Verslagen en Mededeelmgen der Koninklijke Akademie van Wetenschappen. Afdeeling Natuur• kunde.

132 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

But on closer inspection of this last tract I remembered having seen it somewhere; and in truth after having searched for it in vain, I found another copy of it in my own collection of tracts on Proba• bility, amidst the anonymous ones in 4°. This one had the indubita• ble marks of being detached from such a collection of different tracts: and the same was the case with my copy of the first men• tioned tract. Now these two, nearly unique, tracts going together in the Hague-collection, and being separated in my possession, probably from one and the same conglomeration, - it was clear, that I had nothing better to do than to reunite them, after their prolonged separation to their former intimacy.106

Despite Bierens de Haan's exhaustive bibliographical knowl• edge of Dutch works on mathematics and physics published during the seventeenth century, and despite his being com• pletely convinced that the two treatises published in the 1687 edition were by Spinoza,107 he made little progress in exploring their scientific background or finding out how they might have been preserved and brought to the printer. In 1882 he suggested that Spinoza's letter on probability may have been written to a certain Johannes van Meerveen (b. 1639), a student from Utrecht, matriculated at Leiden to read mathematics on Febru• ary 28th 1667, and that it might therefore have been he who had the works printed. This surmisal was, however, shown to be unwarranted when the Schuller-Leibniz correspondence was published in 1890. 108 This failure to look more closely into the archives and unpublished sources which might have provided

106. Benedlctus de Spinoza, 'Stelkonstige Reeckenzng van den Regenboog' and 'Reeckening van Kanssen'. Two nearly unknown treatises (Leiden, 1884). The English here is Bierens de Haan's own translation of a Dutch text published in his Bouwstoffen (2 vols., Leiden, 1878/84), vol. II, xxiv pp. (174-178). 107. Bouwstoffen, op. cit., vol. II (1884), p. 176: 'And truly, further research showed that there were no objections to accepting their authenticity, that there was no danger of being proved wrong on that account: on the contrary, that there was absolutely nothing to be discovered which might call in question Spinoza's authorship.' 108. Album Studiosorum Academlae Lugduno Batavae 1575-1875 (Hagae Com• irum, 1875), col. 535; De Nederlandse Spectator, 1882, p. 54; L. Stein, op. Cit., p. 293. Cf. the further edition of the two works by Bierens de Haan: 'Twee Zcldzame Werken van Benedictus Spinoza' in: Nieuw Archlef voor Wiskunde, XI (1884), pp. 49-82.

133 APPENDIX

new information, in fact the general stagnation of this branch of Dutch Spinoza research, led no less an authority than Freu• denthal to argue that since there was no mention of a Calculation of Chances in seventeenth century reactions to Spinoza's writ• ings, there were no grounds for ascribing this work to him.,09 The situation was much the same twenty years later, when Carl Gebhardt prepared his critical edition of the works of Spinoza under the auspices of the Heidelberg Academy of Sciences. Gebhardt was, however, fully aware that a distinction had to be drawn between the editorial work that had gone into the prep• aration of the booklet and the basic Spinoza texts it included. He suggested that the opening and closing paragraphs of the Treatise on the Rainbow probably owed more to the editor than to Spinoza. In attempting to refute Freudenthal's opinion con• cerning the authenticity of the Calculation of Chances, he was reduced to observing that in the copy of the booklet now in the Royal Library in The Hague, 'the same worm has eaten its way through the pages of both works',"o The general background to the Calculation of Chances in contemporary work on probability by Pascal, Fermat, Huygens and Hudde was indicated in the notes to the French translation of the treatise published in the collected works of Huygens in 1920, and much the same was done in connection with the English translation published in 1953.1" In 1963 G. ten Does• schate produced a facsimile of the 1687 text of the Treatise on the Rainbow held by the University Library at Leiden, together with a general historical introduction written in extraordinary English.ll2 At about the same time, the American scholar Richard Mc Keon published a substantial article on Spinoza's

109.]. Freudenthal, Spinoza; sem Leben und seine Lehre (2 vols., Stuttgart, 1904), vol. I, pp. 296-298, 34. 110. Spinoza Opera 1m Auftrag der HeIdelberger Akademle der WIssenschaften herausgegeben von Carl Gebhardt (4 vols., Heidelberg, 1925), vol. IV, pp. 345-362, 431-434. 111. Huygens, CEuvres XIV, pp. 29-31 Calcul des chances; J. Dutka, 'Spinoza and the Theory of Probability', Smpta Mathemattca, vol. XIX, no. 1 (March 1953), pp. 24-33. 112. The Facslmtle of Sptnoza 's Stelkonsttge reeckentng van den Regenboog (ed. G. ten Doesschate, Nieuwkoop (Holland), 1963), volume five of the Dutch Classics on HIstory of Science series.

134 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS treatment of both the rainbow and probability, showing in some detail how the treatises relate to what we know of Spinoza's other interests and preoccupations, and how they fit into the developments then taking place in mathematics and physics. l13 I had much the same sort of objective in the lecture on Spinoza's treatment of the rainbow which I delivered on March 25th 1977 at the Erasmus University, Rotterdam, as part of the tercenten• ary celebrations of the philosopher's death. As a result of pre• paring this lecture, however, I became aware of how much more research was required, both into the Dutch sources and into contemporary developments in probability theory and diop• trics, before the preservation of the basic texts could be satisfac• torily accounted for, and their scientific and philosophical sig• nificance brought into perspective. I was therefore most grateful for the opportunity to give some account of the results of my subsequent researches in the two popular editions of the 1687 booklet published in celebration of the three hundred and fiftieth anniversary of Spinoza's birth in November 1982,l14 The bibliographical work on the Dutch pamphlet literature of the seventeenth century, carried out by Muller and Bierens de Haan a century ago, was so thorough and so comprehensive, that there is little likelihood of anything new turning up and throwing fresh light upon the Treatise on the Rainbow and the Calculation of Chances. Given the present state of knowledge concerning the origin and significance of the 1687 edition, it looks, therefore, as though research ought now to be concen• trated upon the details of Jan van der Meer's career as a financier and insurance agent, developments in probability theory and dioptrics between 1654 and 1667, and the way in which the relationship between mathematics and physics was being treated in Leiden between 1680 and 1687. There is plenty of scope here for historians of Dutch philosophy. It is certainly interesting,

113. Richard Mc Keon, 'Spinoza on the Rainbow and on Probability', in: Harry Austryn Wo/fson.}ubtlee Volume Gerusalem, 1965), vo!' II, pp. 533-559. 114. M.J. Petry, 'De regenboog', in Spinoza. Kernmomenten in zlJn denken (ed. J. Sperna Weiland, Baarn, 1977), pp. 31-43; Stelkonstlge Reeckening van den Regenboog en Reeckemng van Kanssen, in: Spinoza, Korte Geschnften, op. cit., pp. 495-533; Algebraische Berechnung des Regenbogens. Berechnung von Wahrschemlichketten (tr. and ed. H.-C. Lucas and M.J. Petry, Hamburg, 1982).

135 APPENDIX

therefore, that the most erudite and challenging piece of work published in this field since the appearance of the popular edi• tions in 1982 is an analysis of the Rabus review of 1693. The author of this analysis, Joannes de Vet, has recently published a comprehensive work on Rabus in which he gives a convincing account of him as a herald of the enlightenment, as the Dutch counterpart to Bayle, as a valiant opponent of cant and superstition and a generous admirer of all that was en• lightened and progressive in the European culture of his time. When de Vet was beginning on the researches which eventually bore fruit in this book, he contributed an informative chapter on eSpinoza's absent presence' to Pieter Rabus and The European Library, calling attention, for the very first time, to the review of Van Dyck's booklet, and so confirming that the two treatises had in fact been published together.115 When he wrote this chapter, de Vet had no doubt that Spinoza was the author of the booklet. He was amused by the fact that Rabus should have unwittingly reviewed a work by Spinoza, since it was Rabus' policy to avoid giving any gratuitous publicity to Spinoza's ideas by making mention of him in the periodical. In his book, however, he notes that 'precisely the fact that the Treatise on the Rainbow etc. was published in both 1687 and 1693 without any of the friends of Spinoza, who died when he was relatively young, indicating who had written it, argues against the philoso• pher's being the author of the work'. He adds, moreover, that: 'Rabus' Library can provide us with a more likely candidate for the writing of the Treatise', and that, 'the author of the present study intends, in due course, to produce an article on this' y6 De Vet's main point in the article is that no one came forward to claim Spinoza as the author after the printing of the booklet in 1687 and the publishing of the review in 1693, that Rabus had inside information indicating that the author was still alive, and that the author was probably the lawyer Salomon Dierkens

115. J.J.V.M. de Vet, 'Was Spinoza de Auteur van Stelkonstige Reeckening van den Regenboog en Reeckening van Kanssen?', Tijdschrift voor Filosofie, 45e Jrg., no. 4 (Dec. 1983), pp. 602-639. 'Spinoza's afwezige aanwezigheid', in: Hans Bots, Pieter Rabus, op. cit., pp. 287-294. 116. Pieter Rabus, op. cit., p. 387.

136 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

(1641-1703), an acquaintance of Christiaan Huygens', who had a son Nicolaas Dierkens (1670-1745), who was matriculated at Leiden in November 1686 and subsequently showed ability as a mathematician. Although this is certainly an intriguing thesis, it clashes rather awkwardly with what we know of Huygens' knowledge of these matters. By 1660 he was aware, as many other experts were by 1670, that Descartes' explanation of the rainbow was almost certainly not original. One can only as• sume, therefore, that if the Calculation ofthe Rainbow had been written in consulation with him at any time after that date, he would have advised the author to revise the references to Des• cartes. What is more, although he certainly sent copies of his Traite de la Lumiere and Discours de la Pesanteur to Dierkens in 1690, he seems not to have been in possession of the treatise on the rainbow when he died five years later .117 It should not be overlooked, moreover, that although Rabus in his review makes mention of someone's having 'composed' and 'published' the work, he does not actually say that this person was the 'author' of it.

D. THE SIGNIFICANCE

The internal evidence provided by the 1687 texts has enabled us to reconstruct the history of Spinoza's ideas on probability and dioptrics. It now remains for us to examine these ideas and gauge what relevance they might have to an understanding of the general relationship between mathematics and physics. Spinoza, like so many of his contemporaries, was fascinated by the dis• covery of an a priori theory of probability evidently directly relevant not only to the throwing of dice and the playing of cards, but also to life-insurance, ethics, the law and even theol• ogy. He was encouraged by Descartes' preoccupation with the

117. In 1653 Huygens still thought that Descartes' explanation was original. By 1660 he was aware that it had almost certainly been borrowed from a work published by Marco Antonio de Dominis (1564-1624) as early as 1611: (£uvres I, p. 238; VIII, pp. 13-18; IX, p. 380; XXII, p. 541; Appendix note 70. Cf. R.E. Ockenden, 'Marco Antonio de Dominis', 1m, vol. 26 (1936), pp. 40-49; A. Ziggelaar, 'Die Erklarung des Regenbogens', Centaurus, vol. 23 (1979/80), pp. 21-50.

137 APPENDIX

duality of thought and extension, the use he had made of the ontological argument for God's existence in overcoming this duality, the success with which he had applied geometry in explaining the rainbow, to regard a revised version of his meth• odology as capable of yielding truth, not only in theology and ethics but also in physics and mathematics. It was the intellectual confidence engendered by this state of affairs which gave rise to the sweeping criticism of Baconianism in the early correspondence with Oldenburg, the re-casting of the main themes dealt with in the Short Treatise in the geometri• cal mould of the Ethics. Spinoza was only too ready to inform the Secretary of the Royal Society that neither Bacon nor Des• cartes had an adequate conception of, 'the prime cause of all things', that they had both failed to grasp the nature of the human mind, that they knew nothing of the true cause of human error. What is more, the final version of the Ethics makes it perfectly clear that he never seriously considered retracting any of the central metaphysical postulates on which this confidence was based. Oldenburg, wholly involved as he was in the Baco• nian programme of co-ordinating the findings of experimenta• tion and observation in the interest of improving man's mastery over nature, asked Spinoza how one could possibly deduce God's existence from a conception. Spinoza, totally absorbed as he was in the methodology basic to a rational conception of God, mind, the emotions, servitude and freedom, could only react by maintaining that whatever mastery man might gain over nature, it must remain subordinate to the, 'love of God with which God loves himself'. The treatises on probability and the rainbow were composed when Spinoza was still confident that the geometrical approach central to a rational theology was essentially identical to that basic to a rational physics. Philosophy was capable, not only of comprehending the nature of God, but also of enunciating the regularities of natural occurrences in the pure language of mathematics, expounding an apparently random event in the terminology of an apodictic logic, re-thinking light itself in a series of algebraic equations. In his mature conception, in the 'geometrical' expositions of the Ethics, the main thesis is still the same. All causality except that of God himself is finite. All that

138 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS is, is therefore in God, and in the last resort all contingency is an illusion: 'In the nature of things nothing contingent is granted, but all things are determined by the necessity of divine nature for existing and working in a certain way'. In dealing with nature, however, a philosopher can do no more than provide a general systematic framework for the consideration of bodies in motion or at rest.ll8 When we compare the treatises with the Ethics, therefore, it is evident that although the developments which took place in empirical optics after 1665 did not encour• age Spinoza to qualify his main metaphysical position, they did force him to draw a distinction between the certainties of a 'geometrical order' and those of a mathematical physics, and a very definite distinction between mathematics and the con• tingencies of natural science. It is important to note, therefore, that although the mathe• matical techniques employed in the treatises in analyzing events into numbers or refractive angles into equations are certainly based on the meta-mathematics of his general philosophical method, the 'geometrical order' employed in expounding the Ethics, they are by no means identical with it. Towards the end of his life, in July 1676, Spinoza was asking Tschirnhaus to inform him as to the significance of the latest discoveries relating to refraction. By that time he was no longer capable of writing as Leibniz had to Oldenburg some seven months earlier:

This algebra (of which we deservedly make so much) is only part of that general system. It is an outstanding part, in that we cannot err even if we wish to, and in that truth is as it were delineated for us as though with the aid of a sketching-machine. But I am truly willing to recognise that whatever algebra furnishes to us of this sort is the fruit of a superior science which I am accustomed to call either Combinatory or Characteristic ... After this way of philosophising has been accepted the time will come, and come soon, when we shall have no less certainty about God and the mind than about figures and numbers, and when the invention of machines will be no more difficult than the construction of geometrical problems.ll9

118. EthICS, Bk. I, Props. 28, 29; Bk. II, Props. 11-13. 119. BrieJwlsselmg, letter 83 (15 July 1676); Leibniz, Sdmtllche SchrzJten und BrzeJe (ed. J.E. Hofmann, Berlin, 1976), III. Reihe, 1. Band, pp. 327-334, 18128 December 1675.

139 APPENDIX

Spinoza had based his conception of such a unified science upon Van Schooten's interpretation of Descartes' analytical geome• try. When Leibniz wrote to Oldenburg he had already perceived the potentialities of the integral and differential calculus. There was, therefore, a marked difference between the mathematical techniques which encouraged the two philosophers to entertain such a vision. For Spinoza, however, certainty about God and the mind was no less in 1676 than it had been fifteen years earlier, whereas for Leibniz it had yet to be attained and had simply been foreshadowed by the certainty of the methodology being introduced into science and technology by the new algebra. It is safe to assume, therefore, that when Spinoza was putting the finishing touches to the Ethics shortly before his death, he would have maintained that despite his having cast the work in the external form of Euclid's geometry, its contents were not to be subjected to analysis by the techniques and methods of ordinary mathematics. Although Spinoza continued to assert the truth of the central propositions of the Ethics, he was, therefore, aware that he had failed to demonstrate their validity in the one branch of the exact sciences of which he had first-hand knowledge, and he must have realized that to some extent at least, this had vindicated the criticism of his metaphysics implicit in Oldenburg's letters. He must have seen the significance of the fact that it was develop• ments in the natural sciences which were throwing the potential contingency of his general metaphysical position into relief, and he must have found it particularly disconcerting that it should have been the same developments which were bringing out the basic soundness of the Baconian approach. He had criticized Bacon for knowing nothing of the true cause of human error, yet it was principally through the fruits of Baconianism that the limitations of his own philosophy were becoming apparent. It might be regarded as somewhat curious, that a philosophy which made so much of mathematics and of the identity of God and nature should have drifted so rapidly into such a predica• ment. Looked at from Oldenburg's standpoint, however, it was perfectly understandable. In the opening aphorisms of the Novum Organum, Bacon had reminded his readers that:

140 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

The subtilty of Nature is far beyond that of sense or of the under• standing: so that the specious meditations, speculations, and theo• ries of mankind are but a kind of insanity, only there is no one to stand by and observe it.

When assessed from a Baconian standpoint, therefore, Spinoza had to be regarded as having been deluded by mathematics into exaggerating the capabilities of human reason, thinking that tautological constructions had some sort of automatic ontologi• cal significance: For if any man shall think by view and inquiry into these sensible and material things to attain that light, whereby he may reveal unto himself the Nature or Will of God, then indeed is he spoiled by vain philosophy: for the contemplation of God's creatures and works produceth (having regard to the works and creatures themselves) knowledge, but having regard to God, no perfect knowledge, but wonder, which is broken knowledge. 120

Although Spinoza is undoubtedly part of the mainstream of seventeenth century thought, the difficulties he creates for him• self by modelling his basic philosophical position exclusively on pure mathematics and so excluding himself from any easy di• alogue with the empirical sciences, are by no means typical of the period. Most of his peers displayed remarkable versatility in bringing mathematics and physics together in more or less con• structive combinations. Bacon himself was no mathematician, and there is therefore every reason for contrasting his philo• sophical programme with that of the Cartesians. Nevertheless, the Baconian plan for the development of mathematics drawn up by John Pell (1611-1685) was approved of by Descartes, and was well-known to Spinoza's fellow countrymen on account of Pell's having taught in the Netherlands for a number of years. 121

120. Novum Organum (1620), Bk. 1, Aphorism x; The Advancement of Learnmg (1605), Bk. 1, Section i. 121. At Amsterdam from 1643 until 1646 and then at Breda until 1652: see his An Idea of Mathematicks (London, 1638); P.]. Wallis 'An Early Mathematical Manifesto - John Pell's Idea of Mathematics', Durham Research Review, No. 18 (1967), pp. 139- 148; D. Langedijk, 'De Illustre Schole ende Collegium Auriacum te Breda', Taxan• dria, 42" Jrg., Reeks 5 (1935), pp. 128-136.

141 APPENDIX

Hobbes had his conflicts with the professionals, but the way in which he subordinates mathematics to a systematic classifica• tion of the various forms of motion in De Corpore (1655), the fruitful use he makes of it in progressing from the physical to the mathematical conception of the propagation of light in his Op• tiques (1646), and his awareness of the necessary contingency of all science show clearly that his grasp of the real potential of the discipline was much sounder than Spinoza's.122 Even the seven• teenth century Behmenists, in both England and the N ether• lands, managed to combine their curious vision of the seven levels of the macrocosm with constructive work in the mathe• matical and empirical sciences, in a rather more fruitful manner than the followers of Spinoza.123 Extraordinary though it may seem if we only consider the significance of Descartes' Geometry, there are good reasons for thinking that Spinoza's failure to apply his methodology effec• tively in the Treatise on the Rainbow was the direct outcome of the one sided way in which he interpreted Cartesianism. Des• cartes had dealt with the passions and with God, and had de• veloped a powerful branch of pure mathematics by showing how points, lines and surfaces might be expressed in algebraic terms. He had also been deeply involved in the empirical sci• ences, however, and when entering into the details and contro• versies then occupying professional astronomers, meteorolo• gists, physicists, anatomists etc., had relied heavily upon ex• perience, observation and experimentation. In practice, if not always in theory, he had used mathematics not in order to extend his knowledge, but in order to express the insights he had gained through his empirical work. What is more, he had taken a

122. 'Few sciences can be demonstrated, not physics, because natural actions generally escape the senses, not ethics, because of the inconsistency of the human will; not politics because of the ignora~ce of ethics.' Opera Philosophtca, IV, p. 5. Cf. M.J. Petry, 'Hobbes en de natuurwetenschappen', Wzjsgerig Perspecttef, 20e J rg., no. 1, pp. 1-7; J. Medina 'Les mathematiques chez Spinoza et Hobbes', Revue Phdosophique, no. 2, 1985, pp. 177-188. 123. M.J. Petry, 'Behmenism and Spinozism in the religious culture of the Nether• lands, 1660-1730', in Spznoza in der Fruhzeit seiner religiosen Wzrkung (cd. K. Grunder and W. Schmidt-Biggemann, Heidelberg, 1984), pp. 111-147.

142 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS very largely uncritical attitude to the hypotheses, heuristic con• cepts and analogies he had employed in the initial co-ordination of his empirical data. He had paid little attention to the necessity of attempting to distinguish between basic facts and theoretical frameworks. By postulating the motion of particles as the cen• tral principle of natural events he had had some marked suc• cesses in explaining natural phenomena in mathematical terms, but the heuristic concepts basic to his expositions were often confused and arbitrary. His successful geometrical explanation of the rainbow is a good example of this. On the one hand he insisted that the transmission of the light was instantaneous. When he attempted to explain its propagation and refraction, however, he asked the reader to think of the finite motions involved in wine dripping from the bottom of a vat and a tennis ball changing speed as it passes from one medium to another. Elsewhere he insisted that a distinction should be drawn be• tween motion and tendency to motion. In attempting to explain the motion of light, however, he asked the reader to imagine a blind man who feels the sensation in his hand the moment he taps an object with his cane. Such an instantaneous transmission can only take place if the cane is rigid, however, and a rigid cane cannot transmit a tendency to motion without also transmitting the motion itself. When faced with the necessity of attempting to explain the colours of the bow, he conceived of light as the action or movement of an extremely subtle matter, the parts of which are tiny globules capable of rolling through the pores of terrestrial bodies and rotating with varying velocities. As Huygens subsequently observed, there was 'nothing less likely' than this explanation.124 It is certainly significant that in the Treatise on the Rainbow (p. 37), Spinoza should refer to Descartes' explanation of colour as 'a fine discovery'. He seems to have been completely unaware of the weaknesses of the empirical work on which Descartes had imposed his geometrical exposition. Had he had more experi• ence in dealing with the natural sciences, he would have been on the look-out for unwarranted hypotheses, or, like Hobbes, for

124. La Dzoptrzque, Discours 1 and 2; d. Le Monde, Ch. 12., Huygens CEuvres X, p. 405.

143 APPENDIX

the levels of physical complexity implicit in the regularities being mathematicized. All that really interested him, however, was the resolution of Descartes' lines and triangles into algebraic equations. It is hardly surprising, therefore, that even in his own lifetime he should have become aware of the shortcomings of the Treatise on the Rainbow, and that his general philosophy should have shared the fate of Cartesianism in being forced to give way to the new experimental approach championed by the Royal Society. As the seventeenth century drew to its close, a growing awareness of the ramshackle foundations on which Descartes had erected so much of his physics became one of the main factors in the general decline of interest in his manner of phi• losophizing. After the publication of Newton's Principia in 1687, this soon became a popular prejudice in the Netherlands, and even Rabus was well aware that Cartesian physics was no longer to be taken very seriously. Since Spinoza was never confronted with any evidence which might have caused him to revise or abandon the main theses and methodology of his Ethics, it is not surprising that those who were impressed by this work should have continued to attempt to work out its ethical and theological implications long after developments in the natural sciences had outdated the Cartesian presuppositions out of which it had arisen. The court-preacher and logician Petrus van Balen (1643-1690), for example, in a Spinozistic work composed during the 1680's, praises the clarity and certainty of the geometrical method, and provides an exten• sive demonstration of its use in Biblical exegesis. 125 The mystic Jakob Bril (1639-1700), in his Foundations of a Good Life, mathematically demonstrated on natural grounds, leaving no room for doubt, attempted to show how the principles of mathe• matics could be used to overcome the passions and establish ethics as a self-evident science. He provides the following ac• count of his method:

This work has a mathematical certainty, that is to say, it puts forward nothing which is not self-evident, which could possibly be

125. De Verbetering der Gedagten, ontrent waarhetd en valsheid: OfWaare Logica (2 vols. Rotterdam, 1684/91), vol I, pp. 264-286; vol. II, pp. 185-232.

144 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

doubted by natural reason. If anything is deduced from what is by its nature clear, this is done in such a certain and definite manner, that no one who makes use of his natural reason can deny, cast doubt upon, or fail to understand, any part of it.126

The popular preacher Frederik van Leenhof (1647-1712), who was as well acquainted with the works of Descartes as he was with those of Spinoza, and who evidently composed his Heaven on Earth at the request of the Church, had no doubt that mathematics could enable us to discover certainties in both nature and social affairs: The intellect conceives of things as they are within the eternal order, or in their immediate causes, and as they are deduced from certain knowledge and basic truths. This is the same as in mathematics, which is extremely clear, and in which one would never think of certain truths were they not clearly demonstrable. 127

As one might have expected, by the end of the seventeenth century, the conception of mathematics among those aware of the implications of Baconianism in the natural sciences was quite different. Despite the way in which Newton's Opticks was usually interpreted during the eighteenth century, he was in fact extremely cautious about making any pronouncements con• cerning the mathematics or the intrinsic nature of light.128 In a letter written to Oldenburg on December 7th 1675, for example, he observed that since experimentation alone was capable of extending our knowledge of light and colours, the assumption of any hypothesis in order to explain the properties of light was

126. De Cronden van een Coed Leven, Wis-konstzg aangewesen uyt NatuurlzJke Cronden, Daar Niemand aan twzjffelen kan, in: De Werken van den Hoog-Ver• lichten Jacob Bril (Amsterdam, 1705), pp. 359-369. 127. Den Hemel op Aarden opgeheldert van de Nevelen van Mzsverstand, en Vooroordeelen (Zwolle, 1704), pp. 48-53. 128. John Robison (1739-1805), professor of Natural Philosophy at the University of Edinburgh, was a notable exception. In the lectures on optics he delivered in the mid 1770's, he claimed that Newton had not pronounced on the nature of light, 'because he did not choose to make any hypothesis whatever the foundation of reasoning'. Edinburgh University Library, Dc.7.24 and Dc.7.31.

145 APPENDIX

unnecessary.129 As is evident from his Opticks, all he was pre• pared to do in respect of explaining optical phenomena hypo• thetically was to apply his second law of motion, according to which, 'change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed' .130 He attributed reflection, for example, to, 'some power of the Body which is evenly diffused all over its Surface, and by which it acts upon the Ray without immediate Contact', and refraction to, 'one and the same power, variously exercised in various Circumstances'. He also suggested that the inflexion or diffraction of light discovered by Grimaldi might be explained on this principle: 'Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is not this action strongest at the least distance ?'131 He was evidently encouraged to postulate these short-range forces acting on light-rays as a result of his extensive work in alchemy, his 'inquiring very far downwards into the ultimate component parts of matter' as Stukeley put it.132 A basic motiva• tion behind his alchemical and optical work was the search for an exact analogy between the gravitational force operating be• tween gross bodies and these short-range 'forces'. He never discovered such an analogy, however, and even when expound• ing the world at large in the Principia, he was careful not to confuse his mathematical reasonings with his pronouncements concerning what is physical. In his speculations concerning the structure of matter there is, therefore, a marked absence of mathematical formulation, and throughout all his published works on physics, a constant awareness of the fundamental difference between mathematics as an abstract discipline and the natural philosophy being expressed in mathematical terms. After dealing with centripetal forces in the Principia, for exam• ple, he adds in a scholium that: In the same general sense I use the word impulse, not defining in this

129. The Correspondence of Isaac Newton (ed. H.W. Turnbull, etc., Cambridge, 1959-) vol. I, pp. 363/4. 130. Prmcipla, Bk. 1, Axiom 2. 131. Optlcks, Bk. 2, Pt. iii, Props. 8 and 9; Bk. 3, Pt. i, Obs. 11, query 1. 132. William Stukeley (1687-1765), MemOIrs of Sir Isaac Newton's LIfe (1752; London, 1936), p. 56.

146 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

treatise the species or physical qualities of forces, but investigating the quantities and mathematical proportions of them ... In mathe• matics we are to investigate the quantities of forces with their proportions upon any conditions supposed; then, when we enter upon physics, we compare those proportions with the phenomena of Nature, that we may know what conditions of those forces answer to the several kinds of attractive bodies. And this prep• aration being made, we argue more safely concerning the physical species, causes, and proportions of the forces. 133 It may well have been his awareness of the way in which the manipulations made possible by analytical geometry had caused this essential distinction to be blurred and overlooked which lay behind his interest in restoring the lost writings of Euclid, his enthusiasm for Halley's work on Apollonius, his appreciative comments on a Spanish treatise concerning Analysis geometria sent to him by the Royal Society in April 1699. He found the latter work valuable since it: laid a foundation for restoring the Analysis of the Ancients, which is more simple, more ingenious and more fit for a Geometer than the Algebra of the Moderns.134

In the Netherlands, it was Bernard Nieuwentijt (1654-1718) who first grasped the central philosophical implications of these de• velopments in mathematics, physics, ethics and rational theol• ogy, and made use of them in working out a full-scale and radical critique of Spinozism. Nieuwentijt had been trained as a physician in a Cartesian environment, and as founder and chairman of a scientific body modelled on the Royal Society, had occasion to follow closely the developments taking place in most branches of mathematics and in an astonishingly broad spectrum of the exact sciences. The inspiration behind this interest was religious. Like so many of his contemporaries, he was disturbed by the turn Spinoza had given to Descartes' use of the ontological argument for the existence of God. Jacobus Leydekker (1656-1729), for example, one of the most philosophically incisive of the orthodox the- 133. Prmczpia, Bk. 1, Section xi scholium. 134. The Mathematical Papers of Isaac Newton (ed. D.T. Whiteside, 8 vols., Cambridge, 1967/80), vol. 7, p. 198.

147 APPENDIX

ologians of the Dutch Reformed Church, in a work published in 1692, pointed out that Spinoza's central philosophical thesis was fundamentally confused in that it failed to allow for any definite distinction between Creator and created, God and man, infinite and finite:

He acknowledged no God other than that composed of the material universe, no divine decrees other than natural laws. Each person is, according to him, a part of the Godhead, and he himself a God. He knows of no other origin for things than nature, no other Divine Spirit than human reason, no other goods than those of this life, no other natural law than that imposed by someone who has got the upper hand, and who is therefore able to institute the religious services he desires in order best to be able to rule the public.135

Some years later, Nieuwentijt's friend Carolus Tuinman (1659- 1728) made the same point in a hard-hitting and polemical refutation of the Spinozists:

We deny that the comprehension of all substantial being necessarily includes a complete independence. We take this to be the case only in respect of the Infinite and Supreme Being, and maintain that the same Being has freely brought forth from itself inferior and extra• neous beings which are less perfect ... In all that is extraneous to the Supreme Being, one has to grasp much nothingness, that is to say, all which there falls short of infinite perfection. There is therefore most certainly an infinite divide between finite and infinite, the one being quite incapable of passing over into the other: but between nothing and something there is only a difference on account of something's having degrees of being. No finite force can make nothing into even the most minimal something, since it cannot createY6

Unaware of the existence of the Treatise on the Rainbow, of Spinoza's having concerned himself with any aspect of the exact sciences, Nieuwentijt's initial concern was the theological and philosophical background to the development of the integral and differential calculus, Leibniz's claim that, 'after this way of

135. Dr. Bekkers Phdosophise Duyvel (Dordrecht, 1692), p. 117. 136. De hezllooze Gruwelleere der VriJgeesten (Middelburg, 1714), pp. 109-110.

148 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS philosophising had been accepted, the time would come, and come soon, when there would be no less certainty about God and the mind than about figures and numbers'. The way in which he approaches the problems presented by this back• ground is a remarkable anticipation of the critique of the cal• culus put forward by Berkeley, forty years later, in The Analyst. In a work published in 1694 he criticizes Barrow's tangent procedure in that it involves a (dy) and e (dx) being regarded as zero, the initial lemma of Newton's Principia on account of the supposition that the continual convergence of quantities invol• ves their becoming 'ultimately equal', the contradictions im• plicit in Leibniz's employment of higher-order infinitesimals.137 During the following year he published a substantial work in which he elaborated upon this criticism and made a particular point of rejecting higher-order infinitesimals on account of its being implicit in their acceptance that man is capable of grasping the infinite. According to Nieuwentijt, to postulate that this is so, is to overlook the irrefutable fact that: Our Maker has willed that we should be created in such a way that although our intellect can grasp a quantity greater or smaller than any perceived quantity, we are only capable of perceiving finite and determinate objects, the human understanding being incapable of rising to a true and adequate grasp of the infinite itself. 13s

What is more, to apply the mathematical infinite to the physical world is to lend credence to the atheistic or Spinozistic doctrine that matter is eternal. It is, therefore, essential to accept as the basic axiom of applied mathematics, that anything multiplied by an infinite quantity, if it does not become a magnitude, is a mere nothing. By doing so, one acquiesces in the double truth implicit in the theological necessity of postulating the infinite power of God and the transitoriness of all matter, namely, that any quan• tity can be created from nothing, and that infinite power is

137. Constderationes circa Analyseos ad quantitates infinite parvas applicatae prm• cipia, et calculi dtfferentialis usum in resolvendts problemattbus Geometrlm (Amster• dam, 1694). 138. Analysts infinitorum, seu curvilmeorum Proprietas ex polygonorum natura deductt£ (Amsterdam, 1695), p. 4.

149 APPENDIX

capable of transforming any such quantity back into nothing,139 In his Foundations of Certitude (1720), a work written during the last years of his life, Nieuwentijt made use of these insights in developing an elaborate and comprehensive critique of the sup• posedly scientific and mathematical foundations of Spinozism. Unfortunately, although the work proved to be very popular in the Netherlands, it has never been translated into any foreign language. In the opening chapters he provides numerous instances, drawn mainly from the experimentally based disciplines of his day, but also from the general history of philosophy and the sciences, of the ways in which mathematical procedures may be used in order to clarify, simplify and give greater cogency to scientific exposition. He draws a very sharp distinction between this applied aspect and the principles of pure mathematics, however, arguing that the ideas with which pure mathematics operates need only to be possible, not real, and that it is only by applying and testing them, showing them to be pragmatically effective in experimental work, that one can demonstrate their objective validity. After exploring the scientific and mathematical implications of this crucial distinction, he notices that the followers of Spin• oza have made much of the fact that he professed to demonstrate his philosophy mathematically, starting, like Euclid, with defi• nitions and axioms, and progressing to what appear to be log• ically deduced propositions and corollaries. He then asks whether this procedure can be regarded as involving the princi• ples of pure or of applied mathematics. The answer is, of course, that since it is 'imaginary and in no respect material', it has to be regarded as an example of the former. As Nieuwentijt sees it, however, the difficulty is that this system of pure mathematics has been applied, Spinoza has claimed an ontological signifi• cance for his expositions. It has already been shown that mathe• matics can only have a genuinely ontological significance in conjunction with expenmentation. Consequently, since

139. Op. ezt., p. 4; cf. B.P. Vermeulen, 'Nieuwentijts contraverse met Leibniz', Algemeen Nederlands TZJdsehrzJt voor WiJsbegeerte, 75eJrg., no. 1 Oanuary 1983), pp. 88-94.

150 THE AUTHENTICITY AND SIGNIFICANCE OF THE TEXTS

Nieuwentijt has no difficulty in demonstrating that the Spinoza of the Ethics was not concerned with expounding any form of experimental philosophy, he has good reason to conclude that he was not warranted in claiming any ontological significance for his philosophical system. Nieuwentijt's positive doctrine concerning the relationship between theology and natural science is clearly Baconian: the knowledge to be gained from the natural world is not perfect but broken, and can only yield us a refracted image of the Author of nature:

For the greatest philosopher, as for the best mathematician, the world lies open in all its grandeur and beauty, and by making basic and experimental discoveries through his own observations and those of others on whom he may rely, he can trace, find and materially demonstrate the perfections of the glorious Creator and Ruler of the Universe.14o

It can hardly be regarded as surprising, that as the popularity of the experimental philosophy spread throughout the Nether• lands during the opening decades of the eighteenth century, and as the prestige and effectiveness of N ewtonianism set their mark upon European culture as a whole, this powerful and incisive criticism of Spinozism should have been regarded by the great majority of Nieuwentijt's countrymen as conclusive.141

140. Gronden van Zekerhezd, of de Regte Betoogwyse der Wtskundigen So tn het Denkbeeldtge, als tn het Zakelyke: Ter Wederleggtng van Sptnosaas Denkbeeldtg Samenstel; En Ter aanletding van eene Sekere Sakelyke Wysbegeerte (Amsterdam, 1720,17282,17393, 17544), p. 455. 141. J. Bots, Tussen Descartes en Darwtn. Geloof en Natuurwetenschap tn de achttiende eeuw tn Nederland CAssen, 1972); M.J. Petry, NieuwentlJt's CrzttClSm of Sptnoza, Mededelingen XL vanwege Het Spinozahuis (Leiden, 1979).

151 Index

Abels, P.H.A.M., 124 130, 131, 132, 133, 135 Adam, c., 31 Birch, T., 61 Akkerman, F., 9, 91, 95 blindfold, 75 Alexander's band, 39 Blom, N. van der, 123 algebra, 17, 20, 31, 53, 91, 111, 122, Boehme, J., 142 140, 144: algebraic proof, 41, 55, Bosses, B. des, 99 59, 63: algebraic skill, 67 Bots, H., 125, 129 Alhazen, 114 Bots, J., 151 Amsterdam, 29, 92; ordinances Boyer, C.B., 14 (1598), 110; University Library, Boyle, R., 9, 61, 93, 112 89, 106 Bril, J., 144, 145 annuities, 29 Bruyn, J. de, 114 anti-clericalism, 131 Burgersdijck, F. van, 127, 128 Apollonius, 147 Archimedes, 126 calculation of probability, 11, 16, Aristotle, 16, 98, 108, 122; M eterol- 85: calculus, 16, 140 ogy, 14, 108; Rhetoric, 16, 108 Campani, G., 93 Arnauld, A., 98, 109 Carcavy, P. de, 79 atheism, 7 card, and probability, 75 atomism, 105, 116 Cardan, G., 17, 109 Cartesian algebra, 111; conception Bacon, F., 138, 140, 141: Baconia• of light, 9, 114; geometry, 11, 115; nism, 111, 114, 121, 138, 140, 141, physics, 144; science, 125: Carte• 145 sianism, 91, 105, 108, 120, 121, Balen, P. van, 144 127, 128, 142, 144 Barrow, I., 61, 119, 149 category, 16 Bartholin, R., 117 causality, 53, 98 Bayle, P., 124, 136 chances, calculation of, 73 Beeck Calkoen, J.F. van, 106 Chatelain, IS, 105 Behmenism, 142 Chauvin, E., 125, 128 Bekker, B., 148 Chauvin, P., 124 Berkeley, G., 149 chemistry, analytical, 112 Bernard, J., 105 Cicero, 27, 118, 122, 123 Bernoulli, J., 18, 73, 79 circle, 114; tangent to, 45 Biblical exegesis, 144 Clerc, J. Ie, 121 Bierens de Haan, D., 10,21, 98, Cocceju5, J., 106

153 INDEX

Colbert, J.B., 117 Eeghen, H. van, 102 Collins,]., 114 ellipse, 114 colour, 37, 39, 97 Engelbrecht, E.A., 128 conchoid, 92 equation, 29, 55, 92 conic sections, 92 Erasmus, D., 123 contingency, 20 Euclid, 31, 47, 49, 55, 5~ 59, 61, 63, Coolidge, J.L., 29 119,126,140,150 counter, 75, 77 extension, 111, 138 Covenant of God, 35 eye, 39 Creation, 31, 35 Crombie, A.C, 14 Fermat, P. de, 17, 18, 19,79,87, Croockewit, J.H., 128 108, 109, 116, 134 Feuerbach, L.A., 131 Darwin, C, 151 fireworks, 101, 102 De BoekzaaL van Europe, 10, 124, fluidity, 33 125, 136 Francus, G., 122 Descartes, R., 14, 19, 37, 53, 113, Freudenthal, ]., 8, 10, 92, 93, 99, 119,122,126,137,138,141,144, 100, 101, 102, 103, 104, 113, 134 145, 151: and Mersenne, 12; Snel, Friedmann, G., 98 47; telescope, 45: diagrams, 43, Froidment, L., 14 49: Discours de La Methode, 31, Fruin, R., 13, 110 35,81: empiricism, 142, 143: ex• planation of rainbow, 15, 37, 39: Galilei, G., 12, 31, 114 Geometrie, 10, 15, 17, 29, 49, 71, gambling, 110 92, 140, 142: La Dioptrique, 45, Gebhardt, C, 8, 21, 134 91,94,96,107,113,116,143: Les Gelder, H.E. van, 102 Meteores, 9,14,37,49,51,59, Genesis, 35 107: methodology, 19: ontologi• geometry, 27, 108; analytical, 147; cal argument, 147: theory of Cartesian, 17: geometrical light, 9, 115, 143 method, 94, 97, 144; order, 27, dice and probability, 77, 81, 83, 137 59 Dierkens, N., 137 Gerhardt, C]., 97, 99 Dierkens, S., 114, 136 glass, 33, 35 dioptrics, 93 Glazemaker, J.H., 12, 31, 35 Dijn, H. De, 92 God, 31, 35 Doesschate, G. ten, 21, 134 Graunt, ]., 12, 109 Dominis, M.A. de, 137 Gravesande, W.J. 's-, 107 Dutch language, 35 Greeks, 27 Dutka, ]., 22, 79 Gregory, D., 10, 113 Dyck, L. van, 7, 13, 25, 105, 106, Gregory,]., 10, 113, 114 118, 123, 124, 130, 132 Grimaldi, F.M., 9, 116, 146 Groenendijck, J. van, 122

154 INDEX

Grosseteste, R., 14 insurance, 12, 27, 29, 109, 137 Grunder, K., 142 interference, principle of, 116 Gruys, J.A., 105 isosceles triangle, 57

Haas, K., 29, 110 Jacob, M.C, 99 Hacking, 1., 12, 18 Japikse, N., 13, 110 Haldane, E.S., 129 Jelles, J., 8, 10, 91, 94, 95, 96, 100, Hall, A.R., 112, 113 103, 104, 106, 118, 119, 120, 123 Hall, M.B., 112, 113 Jewish literature, 87 Halley, E., 147 Hallmann, J.C, 102, 103, 104 Kendall, M.G., 110 Heereboord, A., 92 Kepler, ]., 114 Hegel, G.W.F., 129 Kernkamp, G.W., 13, 110 Helvetius, J.F., 7, 105 Keynes, G., 109 Hendriks, F., 110 Kingma, ]., 8, 106 Henry, C, 109 Kircher, A., 35, 114 Herbert, E., Baron of Cherbury, Kirchmaier, G.C, 122 100 Kleerkooper, M.M., 102 Hobbes, T., 142, 144 Kohler, ]., 8, 99, 100, 102, 104, 105, Hofmann, J.E., 139 106, 131 Holland,29 Kortholt, C, 100 Hooke, R., 9, 16, 93, 116 Kortholt, S., 8, 100, 101, 102, 104, Horace, 12, 29, 31, 131 131 Houtzager, D., 110 Kossmann, E.F., 7, 105, 106 Hubbeling, H.G., 9, 91, 95 Koyre, A., 112 Hudde, ]., 10, 11, 15, 18, 27, 29, 91, Kronen, G., 98 92,93,94,96,110,115,119,131, Kruijskerke, A. van, 128 134; and Huygens, 73; Spinoza, Kruseman, A.C., 98 63: Dioptrics, 63: rule, 67 human error, 111 La Chambre, M.C de, 117 Huygens, Chr., 9, 10, 11, 18, 19, 22, Lachterman, D.R., 117 27, 29, 63, 67, 71, 91, 93, 96, 105, Lagny, T.F. de, 107 110, 113, 114, 115, 120, 121, 125, Land, J.P.N., 8, 21, 132 131,134,137,143: and Descartes, Langedijk, D., 141 37: library, 120: on probability, Lansbergen, P., 49, 119 10, 11, 17, 73, 79, 87, 105, 109, Laplace, P.S., 17 119: theory of light, 117; rainbow, Lasswitz, K., 105 61 law, 35, 39 Huygens, Const., 96 Leenhof, F. van, 145 Leibniz, G.W. von, 16, 93, 94, 97, Iceland spar, 117 99, 104, 123, 130, 133, 139, 140, insanity, 141 148

155 INDEX

Leiden, 13, 127: administration, 121: 11, 18, 92, 110, 111, 118;]. de Witt, cloth industry, 120; schools, 122, 29,110: as editor of Spinoza, 120, 126: university, 11, 12, 92, 104, 121, 122, 123, 127: careeG 12, 13, 105, 108, 110, 121, 127, 128: uni• 135: death, 123 versity library, 89, 106, 114, 134 Meerven,j. van, 133 Leurechon, J., 14 Meinsma, K.O., 92, 100, 102, 103 Leydekker, j., 147 Meist, K.R., 129 Liege, 61 Mentzel, c., 122 life insurance, 29, 109, 110 Mere, A. de, 17, 108 light and shadow, 39; diffraction Mersenne, M., 12 of, 116; new ideas on, 116; propa• metaphysics, and mathematics, 94 gation of, 115; rays, 33; reflection Meyer, L., 95, 96, 100, 103, 104, 118, of, 35, 37, 39, 43, 47, 57; refrac• 119, 120, 123 tion of, 14, 35, 37, 39, 43, 45, 47, mica, and light, 9 53, 61, 113, 117; double refraction Mignini, 94, 115 of, 117; speed of, 117; wave the• mirror, 35 ory of, 9, 116 Molhuysen, P.c., 120, 121, 122 logarithms, 71 Montmort, P.R. de 79 logic, 144 Moreau, P.-F., 22 London, 109 Miiller, K., 98 Lucas, H.-C., 22, 119, 123, 135 Muller, F., 7, 98, 130, 131, 132, 135 Lucas, ].M., 99 Murray, R., 10, 113 lunar rainbow, 122 Newton, 1., 9, 16, 105, 115, 116: Maimonides, M., 87 Opticks, 145, 146: Principia, 12, Marci, M., 35 121, 144: second law of motion, Marriote, E., 97 146: seven colours, 122: Newto• Mary Stuart, 102 nianism, 121, 151 materialism, 131 Nicole, P., 109 mathematics, 7, 19, 25, 35, 145; and Nieuwenburg, j. van, 13 empiricism, 143; metaphysics, 94: Nieuwentijt, B., 147, 148, 149, 150, Baconian and Cartesian, 141: 151 mathematical infinite, 149 Nispen, M. van, 49 maxima and minima, 29, 67 Noah,35 McKeon, R., 22, 79, 134, 135 Nulandt, F.W. von, 105 measuring, 27 Medina, j., 142 Ockenden, R.E., 137 Meer, C. van der (1594-1654), 12, Offenberg, A.K., 8, 106 92, 110 Oldenburg, H., 9, 10, 61, 93, 97, Meer, C. van der (1675-1686), 122, 111, 112, 113, 114, 138, 139, 140, 123 145 Meer,]. van der, and Spinoza, 10, one and many, 87

156 INDEX ontological argument, 138, 147 ary, 37, 39, 47, 51,67, 69 Ore, 0., 17, 108, 109 raindrop, 36, 37 Ostens, E., 125 ray, of light, 35, 39, 43 Ostens,]., 125 reflection, of light, 35, 37, 39, 43, 47, 57 pantheism, 98 refraction, of light, 14, 35, 37, 39, parallels, 63 43,45,47,53,61,113,114,117: Paris, Academy, 108: Bibliotheque double refraction 117 Nationale, 89, 106, 107 Rieuwensz, J. (1616/17-1687), 102, Pascal, B., 17, 18, 87, 108, 109, 111, 105 134 Rieuwertsz,]. (165112-1723), 102, Paulus, H.E.G., 129 103, 118 Pearson, E.S., 110 Rijckhuysen, G. van, 13 Pell,]., 141 Robison, ]., 145 Petit, P., 114 R0mer, 0., 117 Petry, M.]., 21, 22, 119, 123, 135, Romans, 27 142, 151 Rotterdam, Erasmus Grammar Petty, W., 109 School, 123: Erasmus University, physics, 25, 35 135: Ordinances (1604,1635),110: Pitassi, M.e., 121 Spinozism in, 125 Plato, 98 Royal Library, The Hague, 89 Posthumus, N.W., 120 Royal Society of London, 9, 10, 93, prismatic colour, 116 107, 112, 114, 116, 121, 138, 144, probability, 75, 77,105,108,109, 147 111: and dice, 83: proof of, 81, 85, Ruestow, E.G., 92 87: theory of, 10, 16, 17, 93 rule, 59: Hudde's, 67: of three, 45 proof, algebraic, 41, 55, 59: geo• Ruyter, e., 107 metrical, 47, 59: of probability, 81, 85, 87 Sabra, A.I., 9, 115 Ptolemy, 126 saltpetre, 111 Pythagoras, 49 Scheltus, ]., 110 Schilling,]., 106, 107 Rabinovitch, N.L., 87 Schilling, P., 107 Rabus, P., 10, 123, 124, 125, 126, Schmidt-Biggemann, W., 142 127, 128, 129, 136, 144 Schooten, F. van (1581-1645),49, Raey, ]. de, 92 119 rainbow, 33: and theology, 35: Schooten, F. van (1615-1660), 11, 12, antediluvian, 35: articles on, 105, 15, 29, 49, 92, 93, 96 122: colours, 37: explanation of, Schuller, G.H., 13, 92, 123, 133 113: God's, 35: Huygens' analy• Schuyl, H., 122 sis, 61: lunar, 122: phenomenon Senguerdius, W., 121 of, 98, 99: primary and second- senses, 33

157 INDEX

Serrurier, C., 124 Struyck, N., 79 shadow, 39, 67 Studia Rosenthaliana, 8, 106 sight, 33 Stukeley, W., 146 Simson, F.H., 129 Suchtelen, G. van, 94, 107 sine, 47, 69, 71, 116 Suchtelen, N. van, 94 Slaart, P. vander, 124, 125, 127, 128 sun, 33, 35, 37, 43 Sluze, R.-F. de 61, 93, 119 surface, 33 Snel, W., 14, 47 Sperna Weiland,]., 22, 135 tables, sine, 71 Spinoza, B. de, and Boyle, 112; tangent, to circle, 45 Cartesianism, 95, 96; Huygens, Tannery, P., 31, 109 71; Jan van der Meer, 122; Taylor, B., 107 Oldenburg, 10; Sluze, 61; Calcu• teleology, 98 lation of Chances, 16, 18, 19, 79, telescope, 45 87, 132, 137; conception of, Temple, W., 124 chemistry, 112; colour, 37; di• The Hague, 99, 101, 102, 105: Royal optrics, 104; God, 111, 114, 138, Library, 89, 106, 132, 134 140,141,148; light, 117; mathe• Theodoric of Freiberg, 14 matics, 141; optics, 91; science, theology, 35 140: Correspondence, 9, 10, 11, three, rule of, 45 20, 63, 91, 92, 93, 96, 112, 114, 125 Til, S. van, 126 131,133,138,139: Ethics, 7,19, Treatlse on the Rainbow, burning 20, 27, 94, 96, 97, 107, 115, 117, of, 101: editing of, 118, 119, 120 138,139, 140, 144, 151: language, triangle, isosceles, 57 118: library, 10, 31, 49, 73: meta• Tschirnhaus, E.W. von, 20,105, physics, 117: ontological argu• 117, 139 ment, 147: personality, 100: Tuinman, c., 148 Posthumous Works, 8, 13, 94, 97, Turnbull, H.W., 146 101,103,104,131: Renati Des Cartes Principiorum Phi• Verboom, A., 127, 128 losophiae, 91, 106, 113: Short Verboom, F., 128 Treatise, 94, 107, 115, 138: Trac• Verboom, L., 128 tatus theologico-politicus, 35, 93: Verboom, U., 128 Voorburg period, 96 Verboom, W., 128 Spinozism, 125, 147, 150 Verboom, W.H., 128 Spyck, H. van der, 8, 99, 100, 101, Vermeulen, B.P., 150 102, 119 Vet, J.].V.M. de, 22,125,127,129, Stein, L., 94, 98, 99, 133 136 stochastic reasoning, 87 Vloten,]. van, 7, 8,21, 22, 130, 131, Stockum, W.P. van, 102 132 Stolle, G., 102, 103, 104 Volbehr, F., 101 Strauss, D.F., 131 VoIder, B. de, 104, 121

158 INDEX

Voogt, c.]., 131, 132 William III, 102, 120 Voorburg, 96 Witt, C. de, 102 Vossius, 1., 114, 122 Witt, J. de, 12, 13, 15, 27, 29, 92, 102,110,119,120,131 Wallis, P.]., 141 Wolf, C. de, 105 Ward, S., 114 Wolfson, H.A., 22,79 water, 33, 39, 41: refractive index, Wouters, A.P.F., 124 43, 45 waterdrop, 35, 39,41,47 Zee, B. van der, 102 Westerbrink, A.G., 91 Zee, H. van der, 102 Weyl, R., 101 Ziggelaar, A., 137 Whiteside, D.T., 147

159