Table Of ContentALGEBRAIC CALCULATION OF THE RAINBOW
&
CALCULATION OF CHANCES
ARCHIVES INTERNATIONALES D'HISTOIRE DES IDEES
INTERNAn ONAL ARCHIVES OF THE HISTORY OF IDEAS
108
SPINOZA'S
ALGEBRAIC CALCULA nON OF THE RAINBOW
&
CALCULA nON OF CHANCES
Directors: P. Dibon (Paris) and R. Popkin (Washington Univ. St. Louis)
Editorial Board: J.F. Battail (Uppsala); F. Duchesneau (Montreal);
T. Gregory (Rome); J.D. North (Groningen); M.J. Petry (Rotterdam);
Ch.B. Schmitt (Warburg Inst. London).
Advisory Editorial Board:.J. Aubin (Paris); J. Collins (St. Louis Univ.); P.
Costabel (Paris); A. Crombie (Oxford); H. de la Fontaine Verwey (Amster
dam); H. Gadamer (Heidelberg); H. Gouhier (Paris); K. Hanada (Hokkaido
University); W. Kirsop (Melbourne); P.O. Kristeller (Columbia Univ.);
Elisabeth Labrousse (Paris); A. Lossky (Los Angeles); J. Malarczyk
(Lublin); E. de Olaso (C.LF. Buenos Aires); J. Orcibal (Paris); Wolfgang
Rod (Munchen); J. Roger (Paris); G. Rousseau (Los Angeles); H. Rowen
(Rutgers Univ., N.J.); J.P. Schobinger (Zurich); G. Sebba (Emory Univ.,
Atlanta); R. Shackleton (Oxford); J. Tans (Groningen).
SPINOZA'S
Algebraic Calculation of the Rainbow
&
Calculation of Chances
Edited and translated
with an introduction, explanatory notes
and an appendix
by
M.J. PETRY
Professor of the History of Phzlosophy,
Erasmus Unzverstty, Rotterdam
1985
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Lihrary of Congress Cataloging In Publication Data
Sptnc~a, Senec,ctu, de, 1632_1611.
Splnoza's Aigebealc calculat,on of the ra,nbow.
(ln~erna~ional archives of ~he his~ory of Ideas 108)
TranslAtIon of: S,elkonstlge Reeckening van de
Regcnbcog and S~mcnknoplng der Natuurkunde met d~
Ihskonsun.
Illbliogeaphy: p.
Inelud", index.
1. Ralnbow __ Eady ,"orks 1.0 1800. 2. Probabillou-
Early work. to 1800. ). Sc,cnce __ Ph,losophy __ F.arly
,",ork. to 1800. I- p.,tey, ~Ichael John. 11. Splnoza,
Senedlctus de, 1632-1677. Samenknop.ng der n~tuurkundc
met de wiskon'ten. Engl,.h. 198~.
!11. Title: Splnoza'. Algebraic calculation of the ra!n
bow. IV. Title: Algebra!c calculation of the rainbow.
v. T.tle: C.lcuhtlon of chancu. VI. Series: Archives
intetna~ionales d'hl$tolre des idees 108.
QC976.R2~6\3 1985 551.~'67 85-8166
1SDN-\3: <J78-9.WIM753-7
1XX:1J7W1.ro).5007-9
Copyright
© 1985 by Martinus Nijhoff Publishers, Dordrecht
Sotlo::MY n-pirtofdlChnUM.YIstWti.rJ 1985
All rights reserved. No part of this public:ltion may bc reproduced, stored in a
retrieval system, or transmitted in any form or by any means, mechanical, photocopy
ing, recording. or utherwise, without the prior written permIssion of the publishers,
Martinus Nijhoff Publishers, P.O. Box 163, 3300 AD Dordrecht, The Netherlands.
4
CONTENTS
Introduction .................................... . 7
A. The text ................................... . 7
B. The background ............................. . 13
C. The significance ............................. . 19
Bibliographical note .............................. . 21
Algebraic Calculation of the Rainbow ............... . 25
To the Reader ................................. . 29
Calculation of the Rainbow ..................... . 33
Calculation of Chances ........................... . 73
First proposition ............................... . 81
Second proposition ............................ . 87
The Text ....................................... . 89
Appendix: The authenticity and significance of the texts on
the rainbow and probability ..................... . 91
A. Spinoza's contemporaries ..................... . 91
B. The 1687 edition ............................ . 104
C. Background and editorial work since 1860 ...... . 128
D. The significance ............................. . 137
Index .......................................... . 153
5
INTRODUCTION
A. THE TEXT
The main importance of these two treatises lies in the insight
they provide into Spinoza's conception of the relation between
mathematics and certain disciplines not touched upon elsewhere
in his major writings. The mathematics they involve are not the
same as those of the Ethics however, and the precise connection
between the geometrical order of this work and these excursions
into optics and probability is by no means obvious. Add to this
difficulty the knotty problems presented by their editorial his
tory, dating and scientific background, and it is not perhaps
surprising that in spite of the fact that they provide such an
excellent illustration of Spinoza's reaction to certain important
developments in the history of physics and mathematics, they
should not, so far, have attracted much attention.
They were first published in 1687 by Levyn van Dyck (d.
1695), official printer to the town council in The Hague. Printing
anything by Spinoza was not without its risks, and it is probably
significant that during the same year van Dyck should also have
published a lengthy and elaborate refutation of Spinozism by
the pious and eccentric physician ].F. Helvetius.1 Spinoza's
name was omitted from the title-page, possibly because the
editor or publisher thought that his reputation as an atheist
might prejudice the sale of the booklet, and it was not until 1860
that the Amsterdam bookseller Frederik Muller (1817-1881)
identified him as its author. The treatise on the rainbow was
subsequently included, together with a Latin translation, in van
Vloten's edition of Spinoza's minor writings (1862), and twenty
years later both treatises made their first appearance in the
collected works. The printing in 1687 must have been very small,
1. ].F. Helvetius (1629-1709), Adams oud Graft (,s-Gravenhage, 1687), see
especially pp. 1-59. On Levyn van Dyck, see E.F. Kossmann, De Boekhandel te
's-Gravenhage tot het emd van de l8de Eeuw (,s-Gravenhage, 1937), pp. 113-117.
7
M.J. PETRY
for only four copies of the original appear to have survived.2
The first edition cannot have been prepared for the press by
any of Spinoza's closest associates, nor can it have attracted
much attention, for although those who wrote about him soon
after his death knew that he was the author of a treatise on the
rainbow, they were generally of the opinion that he had burnt it
and that it was therefore lost. Jarig Jelles, for example, in his
preface to the Posthumous Works, published in the year of
Spinoza's death, concluded his discussion of the completeness
of this edition of his friend's literary remains by assuring his
readers that it was: 'not improbable that some of our author's
material, now in the keeping of someone or other, has not been
included'. He continued: 'The reader may rest assured, how
ever, 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 composed, 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.' The Van der Spyck family,
with whom Spinoza had lodged during the last years of his life,
subsequently confirmed this. They informed Sebastian Kort
holt, for example, that although Spinoza had expended much
labour upon the treatise, he had burnt it during the last year of
his life. Johann Kohler, after having made a comprehensive
enquiry some sixteen years after Spinoza's death, published the
following account of its fate: '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.'3
One can well imagine those aware of the problems then facing
anyone attempting to provide a theoretical framework for ex-
2. J. van Vloten, Ad Benedtctt de Spmoza Opera quae supersunt omma supplemen
tum (Amsterdam, 1862), pp. 252-285; Opera quotquot reperta sunt (ed. J. van Vloten
and J.P.N. Land, 2 vols., The Hague, 1882/3), vol. II, pp. 507-524; J. Kingma and
A.K. Offen berg, 'Bibliography of Spinoza's works up to 1800', Studia Rosenthaliana,
Vol. XI, No.1 Uanuary 1977), p. 3l.
3. All these sources are to be found in J. Freudenthal, Dte Lebensgeschtchte
Spmoza's (Leipzig, 1899), and C. Gebhardt, Baruch de Spmoza. Sdmtliche Werke (3
vols., Leipzig, 1914122), Vol. 3, pt. 2.
8
INTRODUCTION
perimental optics dissuading Spinoza from publishing the
treatise, for in spite of the fact that for more than a decade prior
to his death the Cartesian theory of light and colours was no
longer acceptable to informed investigators, the conception of
colour basic to his exposition is still wholly Cartesian. Descartes
had assumed, for example, that there are only three ways in
which light may be propagated, - directly, refractively and by
reflection, whereas by the middle of the 1660's Grimaldi's dis
covery of diffraction had been made known, and probably
encouraged Huygens to develop the wave theory of light he first
announced publicly in 1679. In 1665 Robert Hooke published an
attempt to explain the colours produced in thin plates of mica by
conceiving of light as a vibrative phenomenon, and soon after
wards Newton first performed the famous series of prismatic
experiments which led him to regard white light as consisting of
dissimilar components which can be separated out by refrac
tion.4 Descartes (Les Meteores viii, 1637) had attempted to ex
plain colour by conceiving 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. Although Huygens (CEuvres X,
p. 405) subsequently observed that there was nothing less likely
than this explanation, Spinoza, evidently wholly unaware of the
developments taking place in the theoretical foundations of
experimental optics, blithely refers to it as, 'a fine discovery'
(p. 37).
It is unlikely that Spinoza would have adopted such an un
critical attitude to Cartesian optics had he composed his treatise
much later than 1667. It is apparent from his correspondence
moreover, that between the early summer of 1665 and the spring
of 1667 theoretical dioptrics occupied a great deal of his atten
tion. In April 1665 for example, Henry Oldenburg, Secretary of
the Royal Society, informed him of the publication of Robert
Boyle'S Experiments and Considerations concerning Colours
(1663), a work which he also discussed with Huygens.5 It was
4. A.1. Sabra, Theones of Light from Descartes to Newton (London, 1967).
5. Spmoza. Briefwlsselmg (ed. F. Akkerman, H.G. Hubbeling, A.G. Westerbrink,
Amsterdam, 1977), nos. 25, 26, 36, 39, 40.
9
M.J. PETRY
probably at about this time that he acquired his copy of James
Gregory's (1638-1675) Optica Promota (London, 1663), and it
may well have been the outdatedness of this work which first
encouraged him to try his hand at expounding Descartes' expla
nation of the rainbow by means of analytical geometry. When
he wrote his book, Gregory evidently knew nothing of Des
cartes, for he presents the law of refraction as an independent
discovery and makes no attempt to explain the rainbow. Robert
Murray, the first president of the Royal Society, was so im
pressed by his abilities however, that he attempted to bring him
into contact with Huygens, and this may well have led Spinoza
to assume that Oldenburg and his colleagues needed enlighten
ing with regard to the merits of Cartesian optics.6 During the
following year Spinoza corresponded with Jan Hudde on lenses,
and in March 1667 with J arig J elles on Descartes' dioptrics and
telescopes. In spite of this exchange of ideas however, it is
evident from the way in which Huygens refers to him in his
correspondence (CEuvres VI, nos. 151-215, Sept. 1667-May
1668), that no very illuminating reaction to post-Cartesian op
tics was to be expected from him, and Huygens was in fact well
advised to confine himself to consulting him on the technicali
ties of lens production.
It is certainly significant that although all Spinoza's early
biographers refer to his treatise on the rainbow, not one of them
mentions the Calculation of Chances. It is apparent however,
from a letter he wrote to Jan van der Meer (no. 38) that during
the middle years of the 1660's he was also engaged in discussing
the theory of probability with his friends and acquaintances.!
Huygens had first opened this up as a field of mathematical
enquiry in 1657, when he had published a work which was to
remain the standard textbook on the subject for the rest of the
6. Spinoza's library list no. 62; Freudenthal, op. CU., pp. 161,279. Soon after his
arrival in London in 1663 Gregory wrote out a still unpublished addendum to his
OptiCS (David Gregory, B. 29, Edinburgh) revising his discussion of reflection and
refraction in accordance with the sine law.
7. Briefwisselmg, no. 38; d. Freudenthal, op. Clt., p. 207. Prior to the evidence
brought forward by D. Bierens de Haan in his edition of the two treatises (1884), there
was some doubt as to whether they had originally been published as a single booklet.
It is worth noting, therefore, that they were reviewed as such by Pieter Rabus in De
Boekzaal van Europe, no. 14 (Rotterdam, Ja nuary/February, 1693), pp. 153-157.
10
INTRODUCTION
century. A Dutch translation of it had appeared in 1660, and had
given rise to intense and widespread discussion. In 1665 Huy
gens was corresponding with Jan Hudde on the subject and
already reconsidering several of his basic conceptions ((Euvres
XIV, pp. 96-150; V, nos. 1374-1450). At the end of his book he
had stated five problems in terms of games of chance, and it is
quite evidently these posers that constitute the background to
the letter which Spinoza wrote to van der Meer. What is more,
the Dutch version in which they were published in 1660 is
reproduced, word for word, in the Calculation of Chances. It
seems reasonable to assume, therefore, that this work was not
originally a separate treatise, but that it was put together by
reprinting Huygens' problems and appending a solution to the
first of them which had been communicated in one of Spinoza's
letters.8
When we remember that these two treatises were first pub
lished in 1687, the most remarkable thing about the editorial
work they involve is the faithfulness and accuracy with which it
reflects the interests and preoccupations of the preceding gener
ation. Although the editor indicates that he is publishing them
mainly for the benefit of the young and the layman (p. 29), he
seems to have been only vaguely aware of the extent to which
natural philosophy had been transformed during the preceding
thirty years. On the title-page for example, he recommends the
work to the public as 'serving to unite physics more closely with
mathematics' (p. 25). In 1657, when it was still something of a
novelty that Frans van Schooten (1615-1660), professor of
mathematics at Leiden, should be publicizing the merits of
Cartesian geometry, the desirability of this objective could
hardly have been questioned. In 1687 however, what was needed
from those capable of thinking in general terms about the natu
ral sciences was not further emphasis upon the proven value of
such a unification, but an effective analysis of the foundations
and results of experimental work, and of the principles involved
in translating these results into the language of mathematics.
Twenty years had passed since Spinoza had written the treatises
8. Huygens' RatioclnllS In aleae ludo was first published in F. van Schooten's
ExercitlOnum Mathematlcorum (Amsterdam, 1657), and subsequently as Van Reke
nlngh In Spelen van Geluck (Amsterdam, 1660). Cf. CEuvres XIV (1920).
11
