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Conservation and Contrariety: The Logical Foundations of Cartesian Physics

  • Peter Damerow
  • Gideon Freudenthal
  • Peter McLaughlin
  • Jürgen Renn
Part of the Sources and Studies in the History of Mathematics and Physical Sciences book series (SHMP)

Abstract

The general theory of matter presented by Descartes in the second book of the Principia Philosophiae is the first well founded systematic physical theory of modern science; for it explicitly introduces the logical presuppositions necessary for a system of causal explanations of physical phenomena using equations. While it is true that Descartes himself takes very little advantage of the possibilities created by the introduction of these prerequisites (there is, for instance, very little mathematics, no formal equations, and few proportions in the Principia itself), he nonetheless determines basic requirements of such a system of explanations and provides conceptual instruments adequate for the formation of such a physical theory.

Keywords

Conceptual System Tennis Ball Oblique Impact Tennis Racket Oblique Collision 
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References

  1. 1.
    We shall speak of “principles of conservation” as the logical or philosophical prerequisites of a physical theory, and of “conservation laws” as the specific fulfillment of the requirements in a particular physical theory.Google Scholar
  2. 2.
    An extensive discussion of the relation between conservation principles and the principle of causality in mechanics (with special reference to Descartes’ and Leibniz’s contributions) can be found in Wundt 1910, pp. 84–114. The principle of equivalence of cause and effect is the last of the six axioms or hypotheses on which mechanics is founded (according to Wundt). For a discussion of conservation principles with many historical examples, but from a point of view very different from ours, see Meyerson 1962.Google Scholar
  3. 3.
    The conceptual link between change and invariance is discussed by Kant under the heading “The Permanence [of Substance]” in the Critique of Pure Reason (A182–189). He later explicitly refers to this discussion in his proof of the law of conservation of mass in the Metaphysical Foundations of Natural Science (1786). The law states: “First Law of Mechanics. In all changes of corporeal nature, the total quantity of matter remains the same, neither increased nor diminished” (Gesammelte Schriften, vol. 8, p. 541; Kant 2002, p. 249).Google Scholar
  4. 4.
    “Equipollence” is a synonym for equivalence used in logic at least up to the time of Carnap. Leibniz gives it the particular technical meaning that we have adopted in a letter to l’Hopital (Jan. 15, 1696). Leibniz, GM II, 305–307, translated in document 5.2.1. See also GP IV, 370–72; and in general “Initia rerum mathematicarum metaphysica,” GM VII, 17–29. For a discussion of the development of Leibniz’s views on equipollence and a collection of sources, see Fichant’s commentary in Leibniz 1994, pp. 277–302.Google Scholar
  5. 5.
    Traditionally, velocity was the most important intensive magnitude; and as late as Kant mass was also sometimes taken as an intensive magnitude.Google Scholar
  6. 6.
    “The law of energy directs us to coordinate every member of a manifold with one and only one member of any other manifold, in so far as to any quantum of motion there corresponds one quantum of heat, to any quantum of electricity, one quantum of chemical attraction, etc. In the concept of work, all these determinations of magnitude are related to a common denominator. If such a connection is once established, then every numerical difference that we find within one series can be completely expressed and reproduced in the appropriate values of any other series. The unit of comparison, which we take as a basis, can arbitrarily vary without the result being affected. If two elements of any field are equal when the same amount of work corresponds to them in any series of physical qualities, then this equality must be maintained, even when we go over to any other series for the purpose of their numerical comparison.” (Cassirer 1923, p. 191) Cassirer calls the one-to-one correspondence of values in different series “equivalence” (p. 197).Google Scholar
  7. 7.
    We have intimated that the formation of concepts for objects that are invariant in our manipulation of them is similar in structure to the formation of concepts of scientific entities and conservation laws. We cannot, however, further discuss the extent and significance of the similarity between these two processes of concept formation.Google Scholar
  8. 8.
    Robert Mayer, for instance, argued that the conservation of energy follows from this philosophical principle. See Freudenthal 1983.Google Scholar
  9. 9.
    As E. P. Wigner maintained in his Nobel Prize Lecture of 1963, the most important function of invariance principles is “to be touchstones for the validity of possible laws of nature. A law of nature can be accepted as valid only if the correlations which it postulates are consistent with the accepted invariance principles” (Wigner 1967, p. 46).Google Scholar
  10. 10.
    «We can put the question: what would be changed in physics if a perpetuum mobile were to be discovered today? Our conviction of the universal subjection of nature to law would not be shaken... the validity of the law of the conservation of energy would be restricted to certain limits, and perhaps we could hope to recognize it ultimately as a special case of a still more general law“ (von Weizsäcker 1952, p. 64–5).Google Scholar
  11. 11.
    This does not mean, however, that any particular scientist had to realize that science demands a scientific world view nor that he suscribed to such a view. But even scientists who did not conceive of the universe as a closed system determined by conservation laws had nonetheless to presuppose (in practice) a constant quantity of the relevant magnitudes. In such a case, only inconsistency between the philosophy espoused and the science actually practiced allowed science to be pursued on the basis of unscientific presuppositions. Newton provides a good example of this state of affairs; see Freudenthal 1986, pp. 44–76. The politically and ideologically motivated attempt to construe the universe as a closed system, the states of which can be related only to physical entities, in conjunction with the attempt similarly to construe society was, according to Lefèvre (1978, pp. 45–79), an important factor in the construction of modern science.Google Scholar
  12. 12.
    Max Planck began his book, Das Prinzip der Erhaltung der Energie (1913, p. 1) by isolating two conservation laws as somehow more fundamental than other laws: “There are two propositions which serve as the foundation of the current edifice of exact natural sciences: the principle of the conservation of matter and the principle of the conservation of energy. They maintain undeniable precedence over all other laws of physics however comprehensive; for even the great Newtonian axioms, the law of inertia, the proportionality of force and acceleration, and the equality of action and reaction apply only to a special part of physics: mechanics — for which, moreover, under certain presuppositions to be discussed later they can all be derived from the principle of conservation of energy.” Planck seems to mean that the fundamental character of the conservation of energy and matter has to do with their being scalar magnitudes and thus not limited to mechanics.Google Scholar
  13. 13.
    Numbers in parentheses refer to part and paragraph of the Principia philosophiae (AT VIII); a translation of bk. II, §§36–53 is given in document 5.2.2.Google Scholar
  14. 14.
    AT VII, 40 (Meditations, III).Google Scholar
  15. 15.
    principia, II, §39. It should be noted, however, that Descartes’ concept of “speed” (celeritas, velocitas) does not refer to an instantaneous quantity but rather denotes the space traversed in a finite time. Determination, on the other hand, is introduced explicitly as an instantaneous magnitude: “...that each part of matter, considered in itself, always tends to continue moving, not in any oblique lines but only in straight lines... For [God] always conserves it precisely as it is at the very moment when he conserves it, without taking any account of the motion which was occurring a little while earlier. It is true that no motion takes place in an instant; but it is manifest that everything that moves is determined in the individual instants which can be specified as it moves, to continue its motion in a given direction along a straight line, and never along a curved line” (emphasis added). On the development and the systematic consequences of Descartes’ concept of speed, see Chap. 1, above.Google Scholar
  16. 16.
    In the Dioptrics and in a series of letters for Fermat and Hobbes; section 2.5 of this chapter discusses the concept of determination in great detail.Google Scholar
  17. 18.
    See Aristotles Physics V, 1, 225a34ff and the commentaries of Aquinas (1963, §670) and Scotus (Opera, 1891, vol. 2, p. 326).Google Scholar
  18. 19.
    Our discussion of Aristotle relies heavily on H. Maier, Die Syllogistik des Aristoteles and J. Lukasiewicz, “Aristotle on the Law of Contradiction.” For a contemporary grounding of the principle of noncontradiction on the more general principle of inconsistency, see Strawson 1952.Google Scholar
  19. 20.
    See also Categories, Chap.10, 12b16f: “Nor is what underlies affirmation or negation itself an affirmation or negation. For an affirmation is an affirmative statement and a negation a negative statement whereas none of the things underlying an affirmation or negation is a statement... For in the way an affirmation is opposed to a negation, for example, ‘he is sitting’ — ‘he is not sitting’, so are opposed the actual things underlying each, his sitting — his not sitting.”Google Scholar
  20. 21.
    We believe that Lukasiewicz has the same argument in mind when he writes that “it is in general impossible to suppose that we might meet with a contradiction in perception; for negation, which is part of any contradiction, is not perceptible. Really existing contradictions could only be inferred” (Lukasiewicz 1979, §19b). The most prominent opponent of this point of view is Geach (1972, p. 79), who remarks: “What positive predication, we might well ask, justifies us in saying that pure water has no taste? Again, when I say there is no beer in an empty bottle, this is not because I know that the bottle is full of air, which is incompatible with its containing beer.” While it is clear that one need not know what else is in the bottle in order to state that there is no beer in it, positive knowledge is nonetheless necessary to justify this statement. One cannot simply see or feel that there is or isn’t beer in the bottle. Mistaken judgments which involve no optical illusions illustrate this clearly; and many of the standard tricks of magicians depend precisely on such mistaken inferences (e.g., that the hat is empty or that there is nothing up his sleeve). Although we may readily admit that a gestalt can be built up over time so that we begin to perceive immediately what we originally had to infer (e.g., the absence of something), this psychological process does not effect the inferential justification of the statement.Google Scholar
  21. 22.
    This enables Aristotle then to turn around and draw inferences about the state of the world from the consistency of language: “If then it is impossible to affirm and deny truly at the same time, it is also impossible that contraries should belong to a subject at the same time...” (1011b13f).Google Scholar
  22. 23.
    See the following observations of Sigwart: “Which representations are incompatible cannot be derived from a general rule, but rather is given with the factual nature of the content of the representations and their relations to one another. We can conceive our sense of sight as so constituted that we could see the same surface shine in different colors, as it in fact emits light with different refractibility, just as we hear different overtones in a single tone and distinguish different tones in a chord; it is purely factual that the colors are incompatible as predicates of the same visible surface, but different tones as predicates of the same source of sound are not, no more than sensations of pressure and temperature, which can be ascribed to the same subject in very different combinations (cold and hard, cold and soft, etc.)” (Sigwart 1904, vol. I, p. 179).Google Scholar
  23. 24.
    The relation between the real incompatibility of properties and the logical inconsistency of propositions attributing them to the same subject caused considerable difficulties in the early development of Logical Empiricism. The problem arose, why not all “elementary” or “atomic” propositions are compatible with one another. For instance, Ludwig Wittgenstein in his Tractatus logico-philosophicus (6.3751) wrote: “For example, the simultaneous presence of two colours at the same place in the visual field is impossible, in fact logically impossible, since it is ruled out by the logical structure of colour.” In his “Some Remarks on Logical Form” (1929), Wittgenstein then introduced a distinction between contrariety and contradiction in his own terminology: “I here deliberately say ‘exclude’ and not ‘contradict,’ for there is a difference between these two notions, and atomic propositions, although they cannot contradict, may exclude each other” (p. 35). The process of exclusion is then described: “The propositions ‘Brown now sits in this chair’ and ‘Jones now sits in this chair’ each, in a sense, try to set their subject term on the chair. But the logical product of these propositions will put them both there at once, and this leads to a collision, a mutual exclusion of these terms” (p. 36; emphasis added). Wittgenstein’s “collision” and “exclusion” correspond to the “struggle” and “expulsion” adduced by most scholastic logicians up to the 17th century to explain the incompatibility of contraries (see the quotation from Toletus in section 2.3.4 below). The difference between Wittgenstein’s position and that of a scholastic logician like Toletus lies of course in Toletus’ insight that the relevant struggle takes place between physical entities, not between propositions, and that their incompatibility is an empirical fact and therefore stated on the basis of empirical knowledge. For an analysis of the role of this problem in Wittgenstein’s eventual rejection of the Tractatus, see Allaire 1966.Google Scholar
  24. 25.
    Such predicates have often been called contradictory predicates. This can lead to confusion since, strictly speaking, terms cannot be contradictories; only statements or propositions can be contradictories. Sigwart (1904, pp. 23–25) deals with some of the problems associated with the use of such predicates; and Wundt (1906, vol. II, pp. 62f. and 80f.) gives examples of their use in empirical sciences.Google Scholar
  25. 26.
    See also Strawson 1952, p. 8.Google Scholar
  26. 27.
    The informativity of this logic is of course not confined to cases where two species exhaust the genus. If there are a limited number of determinate species, a limited number of negations are necessary to establish the sole remaining species.Google Scholar
  27. 28.
    Contrary qualities were conceived in general as incompatible. It seems that Buridan was the first to allow that some contrary qualities, for instance, hotness and coldness, may coexist in the same substance and that their sum is constant. See Maier 1952, pp. 304f. Oresme first represented this thesis by means of the method of configuration of qualities and motions:Google Scholar
  28. 29.
    In a true mixture two or more materials generate a new one with determinate qualities out of the contrary qualities of the components (327b10–238b25). For a discussions of these topics in scholastic natural philosophy see Maier 1952, pp. 1–140.Google Scholar
  29. 30.
    See Heninger 1977, esp. pp. 103–108, for a number of such tetrads. A tetrad almost identical to the one given here from the Cosmographia of Oronce Fine can be found in Leibniz, Dissertatio de arte combinatoria, (1666) GP IV, 34. The Square of Opposition is taken from De Soto Summulae (1554), p. 52.Google Scholar
  30. 31.
    See Cronin (1966, pp. 32–3) for a discussion of the texts used at the Jesuit school La Flêche.Google Scholar
  31. 32.
    Toletus, Logicam (1589), p. 105r (emphasis added); almost verbatim the same discussion can also be found in the Summa of Eustace of St. Paul (Part I, pp. 38–39), which Descartes once recommended as a competent scholastic presentation (AT III, 251). For an analysis of relevant late scholastic doctrines see Des Chene 1996, esp. pp. 55–64.Google Scholar
  32. 33.
    Toletus 1589, p. 176.Google Scholar
  33. 34.
    Hobbes, letter to Cavendish, Jan. 29/Feb. 8, 1641 (HC I, 82–83;Works 7, pp. 458459). For Galileo’s use of contraries, see Chapter 3 below, sections 3.2.1, 3.2.3, 3.3.1, and 3.7.3.Google Scholar
  34. 35.
    Marcus Marci, De proportione motus (1639), no pagination, sigs. A3v, A4v, BlvB2r.; see also Gabbey 1980, p. 245.Google Scholar
  35. 36.
    Marci 1639, Blv-B2r. On the quantification of contraries by means of statics, see our discussion below in 2.4.3.Google Scholar
  36. 37.
    Descartes criticized scholastic philosophers who “attribute to the least of these motions a being much more solid and real than they do to rest, which they say is nothing but the privation of motion. For my part I conceive that rest is just as much a quality, which must be attributed to matter while it remains on one place, as is motion, which is attributed to it while it is changing place” (Le Monde, AT XI, 40).Google Scholar
  37. 38.
    Although determination is defined as a mode of motion, it is also sometimes used as if it were a mode of the bodies themselves. For instance, in the section of the Principia in which Descartes first introduces the concept it seems that a body can have a determination in an instant (“in that instant at which it is at point A”), although it can only have motion during some finite length of time: “It is true that no motion takes place in an instant; but it is manifest that everything that moves is determined [determinatus esse] in the individual instants which can be specified as it moves, to continue its motion in a given direction along a straight line, and never along a curved line” (II, §39; AT VIII, 64; emphasis added).Google Scholar
  38. 39.
    principia, II, §§53 and 56; AT VIII, 70 and 71.Google Scholar
  39. 40.
    The one area of scientific knowledge, to which Descartes could and did appeal, namely statics, had to be interpreted and its adaptation itself had to be justified.Google Scholar
  40. 41.
    Leibniz clearly saw that Descartes was deriving physical theorems by applying the logic of contraries and presented one of them (concerning firmness) in syllogisms; in these syllogisms the common relation of the predicates is “maxime adversatur.” Leibniz’s criticism of Descartes is not directed against this procedure in general (in fact Leibniz applies it too), but against two specific points. On the one hand, Leibniz doubts that the characterization of certain concepts as most opposed is correct (thus he remarks that contrary motion is more opposed to a specific motion than is rest) and on the other hand he doubts the validity of Descartes’ implicit axiom, that the cause of what is most opposed to something is also most opposed to the same thing. (see. Leibniz, “Critical Thoughts on the General Part of the Principles of Descartes,” on Articles 54, 55, GP IV, 385–388, PPL, 403–407.Google Scholar
  41. 42.
    See Aristotle, 225b10–11 and 3b24–27.Google Scholar
  42. 43.
    There have been numerous studies of Descartes’ physics, which we shall not be able to deal with here. For a discussion of various previous interpretations of Descartes’ physics consult Gabbey 1980, which has set qualitatively new standards of analysis for the study of Cartesian science. Gabbey’s interpretation emphasizes especially the opposition of modes as a key to understanding Descartes’ system. Our debt to his work will be evident both in this and the next section.Google Scholar
  43. 44.
    Princ. II §55; AT VIII, 71.Google Scholar
  44. 45.
    Although the context of §43 makes it is unequivocally clear that we are dealing with the interaction of two and only two bodies and although the Latin text itself makes it clear that the surface in question is only the surface between the two colliding bodies, most commentators have nonetheless interpreted this passage as making some vague reference to the entire surface area of the body, including its back and sides. This has even led them to mistranslate the passage to fit the interpretation. The Latin reads: “Visque illa debet aestimari tum a magnitudine corporis in quo est, et superficiei secundum quam istud corpus ab alio disjungitur; turn a celeritate motus, ac natura et contraritate modi, quo diversa corpora sibi mutuo occurunt.” The key phrase is ab alio; Descartes speaks of the surface that separates the colliding body from the other body. All three published English translations have Descartes talk about the surface that separates a body from all the surrounding bodies, not just from the one it hits. See Descartes 1964, 1983, 1985f, as well as almost every commentator on the subject. The French translation of the phrase (“separé d’un autre”) appears to be ambiguous and has been cited in support of the usual interpretation, e.g., by Costabel 1967. There are other passages in the Principles, Le Monde, and an often cited letter to de Beaune on “natural inertia” (AT II, 543–544), where Descartes also deals with the physical significance of the surface of a body and may be interpreted to mean more than just the front end. But how much of the surface is significant depends on how much is involved in interactions: the dynamically relevant cross section.Google Scholar
  45. 46.
    Descartes, letter to Clerselier, Feb. 17, 1645; AT IV, 185–186 (transl. adapted from Gabbey 1980, p. 236). Spinoza argued that this principle directly follows from the first law of nature (inertia); see Renati Des Cartes Principiorum Philosophiae Pars I & II, bk. II, §25.Google Scholar
  46. 47.
    Gabbey 1980, p. 263f.Google Scholar
  47. 48.
    However, when calculating the amount of change in the various possible results, it should be remembered that the quantity of determination changes whenever the quantity of motion changes. Thus, although each mode changes with equal difficulty, a change in determination involving only change of direction is in fact easier than a change in motion which also necessitates a change in the quantity of determination and thus counts double (i.e., reversing four units of determination without changing motion is equal to transferring two units of speed because two units of determination are attached to them).Google Scholar
  48. 49.
    In fact there are four possibilities: two kinds of opposition, each of which can be absolute or admit of degrees. As Descartes says: “strictly speaking, only a two-fold contrariety is found here. One is between motion and rest, or also between swiftness and slowness of motion (that is, to the extent that this slowness partakes of the nature of rest); the other is between the determination of a body to move in a particular direction and the encounter in this direction with a body which is at rest or moving in a different manner; and this contrariety is greater or lesser in accordance with the direction in which the body that encounters the other is moving” (Princ. II, §44; AT VIII, 67). The fourth possibility — oblique oppositions will be dealt with in the next section.Google Scholar
  49. 50.
    See Westfall Force, pp. 121–22.Google Scholar
  50. 51.
    As we shall discuss below, two of the rules Decartes presents actually deal with the same case so that one rule is actually missing.Google Scholar
  51. 52.
    Even this, the simplest case and the only “empirically correct” one, is not strictly implied by Descartes’ premises. Since the two bodies are equal in force, this case does not really fall under the Third Law of Nature, which holds for the relations between stronger and weaker bodies. See also Spinoza (1925, vol. 1, pp. 211–212) Renati Des Cartes Principiorum Philosophiae, bk. II, prop. 24, who appeals implicitly to symmetry considerations when the two bodies are equal.Google Scholar
  52. 53.
    For details see McLaughlin 2001Google Scholar
  53. 54.
    Descartes deals with this last possibility only in the French edition, but it can be derived in analogy to Rule 6.Google Scholar
  54. 55.
    See Gabbey 1980, p. 269. This rule is further complicated by the fact that a body at rest with respect to its contiguous bodies must actually be considered to be part of them, since “our reason certainly cannot discover any bond which could join the particles of solid bodies more firmly together than does their own rest... for no other mode can be more opposed to the movement which would separate these particles than is their own rest” (II, §55; AT VIII, 71). It is thus hard to see how we can even estimate the size of a resting body without contradicting ourselves.Google Scholar
  55. 56.
    Hobbes’ objection is cited by Descartes in a letter to Mersenne of Jan. 21, 1641, in which he answers the objection; AT III, 289. See document 5.2.11; also 5.2.9.Google Scholar
  56. 57.
    See especially the figures in the letter to Clerselier of Feb. 17, 1645 (Descartes 1657, p. 651). The versions of the figure published in AT IV, 185 and in the first edition of our book are slightly inaccurate: Body B should be smaller and come from the right. See also Princ. Phil. II, §46, and AT III, 79; see documents 5.2.2, 5.2.3, and 5.1.12. William Neile (1637–1670), who like Descartes defined as one body all matter sharing the same motion, stated among the presuppositions of his discussion of impact that the colliding bodies are cubes and that “the whole square surface of the one meets in the same instant of time with the whole square surface of the other” (Neile, “Hypothesis of Motion,” (May, 1669), in: The Correspondence of Henry Oldenburg, vol. 5, pp. 519–524).Google Scholar
  57. 58.
    On instantaneous direction see the Second Law of Nature, Principia, II, §39; AT VIII, 63–64. The mind or will cannot increase or decrease the amount of motion in the world; it can only determine the motion of the body. Descartes uses the term “determine” on a number of occasions to describe the action of the mind on the body (see especially AT VII, 229 and XI, 225–226). This is an adaptation of a traditional way of speaking about freedom of the will: in decision the will determines itself to action. See Suarez, Opera vol. 10, pp. 459ff; Descartes (Passions de l’âme, §170; AT XI, 459) also uses the term in a similar sense; see McLaughlin 1993.Google Scholar
  58. 59.
    Gabbey (1980, p. 248) provides many examples of the logical use of “determination” and points out that it was derived from the Greek 2rpoa&toptay6g meaning division, distinction, definition.Google Scholar
  59. 60.
    Hobbes objects that “towards a certain side” does not unequivocally determine a motion (letter to Mersenne, March 30, 1641; AT III, 344–5). See document 5.2.12.Google Scholar
  60. 61.
    See Gabbey 1980, p. 258; Schuster 1977, p. 288Google Scholar
  61. 62.
    See Crowe 1967, chap. 1.Google Scholar
  62. 63.
    There were, however, serious technical difficulties involved in applying the compounding of motions to infinitesimals and accelerations, e.g., in calculating tangents to curves and ellipses. See Costabel 1960.Google Scholar
  63. 64.
    See Dijksterhuis 1970, pp. 48–63.Google Scholar
  64. 65.
    Other contemporary attempts to deal with the problems that arise from the notion of compounding and resolving the forces of bodies in motion show them to be inherent in the shared knowledge of the scientific community of the time, not just peculiar to Descartes. For two later examples, John Wallis and Honoré Fabri, see Freudenthal 2000.Google Scholar
  65. 66.
    Fermat, letter to Mersenne, Nov. 1637; AT I, 464–74. See document 5.2.6.Google Scholar
  66. 67.
    Hobbes later applied the same reasoning to determinations, which he took in the traditional sense of the points from which and to which the motion is directed: “Thirdly, it is to be objected that one motion cannot determinations of two motion of two bodies, one of which goes from AB to DC, the other of which from AD to BC.” (AT III, 344–5; see document 5.2.12.)Google Scholar
  67. 68.
    The significance of the question, whether the entities represented by the sides of the parallelogram may be considered as parts of that represented by the diagonal, can be illustrated by the consequences drawn by Bertrand Russell. In his A Critical Exposition of the Philosophy of Leibniz (p. 98) he wrote: “It has not been generally perceived that a sum of motions, or forces, or vectors generally, is a sum in a quite peculiar sense — its constituents are not parts of it. This is a peculiarity of all addition of vectors, or even of quantities having sign. Thus no one of the constituent causes ever really produces its effect, the only effect is one compounded, in this special sense, of the effects which would have resulted if the causes had acted independently.” In The Principles of Mathematics (p. 477) he formulated a “paradox of independent causal series”: “The whole has no effect except what results from the effects of the parts, but the effects of the parts are non-existent.”Google Scholar
  68. 69.
    These assumptions are: (1) that light is transmitted instantaneously; (2) that it is an instantaneous action or inclination to move that can be taken to follow the same laws as an actual motion in time; (3) that the amount of impetus necessary to traverse a particular medium instantaneously with a particular intensity is analogous to the speed with which a body with a particular force traverses the medium, so that the speed of a ball is comparable to the ease of passage of a light ray.Google Scholar
  69. 70.
    Harriot, Snel, and Mydorge were all working on the problem. See Schuster 1977, pp. 308–321.Google Scholar
  70. 71.
    See Gabbey 1980; Sabra 1967, chaps. 3 and 4; and Schuster 1977, chap. 4. The standard work on Descartes’ Dioptrics has long been the excellent analysis in Sabra 1967. While we follow in basic outline much of Sabra’s presentation, there are important differences, especially in the analysis of Fermat’s objections. A recent monograph on Descartes’ Dioptrics (Smith 1987), while somewhat unclear about the concept of determination, also presents some interesting material on the backgound in perspectivalist optics.Google Scholar
  71. 72.
    See Ferrier’s letter to Descartes, Oct. 26, 1629 (AT I, 38–52) and Descartes’ letter to Ferrier, Nov. 13, 1629 (AT I, 53–59); also Descartes to Golius, Feb. 2, 1632 (AT I, 259); and Descartes to Huygens Dec., 1635 (AT I, 335–6).Google Scholar
  72. 74.
    AT VI, 94–5; emphasis added. This is the point that caused most of the misunderstandings which Descartes tried unsuccessfully to clarify in his letters. See AT II, 1820; AT III, 163, 250–51; see documents 5.2.8, 5.2.17, and 5.2.18.Google Scholar
  73. 75.
    Descartes, letter to Mersenne, Oct. 5, 1637; AT I, 452. See document 5.2.6. The italics indicating where Descartes is quoting himself have been removed.Google Scholar
  74. 76.
    Descartes and his contemporaries sometimes used the term “angle of incidence” for the angle made by a ray with the surface and “angle of refraction” for the deviation of a refracted ray from its original line. See documents 5.2.4 and 5.4.3.Google Scholar
  75. 77.
    In the original formulation of the Third Law of Nature in Principia philosophiae,cited in section 2.4.2 above, Descartes even says merely that the colliding body loses (amittit) its determination and only later adds that it acquires a new one.Google Scholar
  76. 78.
    In the Dioptrics Descartes says that light passes “more easily” in one medium than in another. As early as the manuscripts now known as the Cogitationes Privatae (1619–1621) Descartes had maintained that light passes more easily through a dense medium than through a rare medium (AT X, 242–3). Later, Descartes avoids the terms dense and rare when speaking of the different media. Since in his physics there is no vacuum, all matter should be equally dense. The media differ in that one is “more solid” (durior) or more fluid (fluidior) than the other. In a later letter for Hobbes (Jan. 21, 1641; AT III, 291) Descartes explains that less impetus is required in water than in air. See document 5.2.11.Google Scholar
  77. 79.
    Letter to Mydorge, March 1, 1638 (AT II, 19–20). See document 5.2.8.Google Scholar
  78. 80.
    Historical evidence strongly suggests that Descartes, like Harriot, Snel, and Mydorge originally worked with an altered radius, formulating the law of refraction in terms of a constant ratio of the lengths of the radii (cosecant form). See Schuster 1977, pp. 268–368, for a highly plausible reconstruction of the original path of discovery.Google Scholar
  79. 81.
    Descartes does not stipulate that the racket “hits the incident ball perpendicularly, thus increasing its perpendicular velocity,” as Sabra (1967, p. 124) assumes; he says nothing about the slant of the racket, asserting merely that it should be thought to increase the scalar speed and not to affect the parallel component of the determination.Google Scholar
  80. 82.
    Descartes does not use the terms dense and rare for the different media since there is no empty space; but in this context he also avoids his own terms, more solid and more fluid. He just speaks of air and water.Google Scholar
  81. 83.
    For a detailed explanation and the relevant equations see Joyce and Joyce (1966), who also give a list of modern physics textbooks that argue (wrongly) along the lines of Descartes.Google Scholar
  82. 84.
    The exchange with Fermat consisted of three letters: Fermat to Mersenne April or May 1637 (AT I, 354–363); Descartes to Mersenne Oct. 5, 1637 (AT I, 450–54); Fermat to Mersenne Nov. 1637 (AT I, 463–74). Descartes later discussed Fermat’s comments in a letter to Mydorge March 1, 1638 (AT II, 15–23). Fermat continued to consider motion and determination as independent magnitudes, specifically, he thought that motion can change without the determination’s changing. He admitted (at least for the sake of argument) this misunderstanding 20 years later in his correspondence with Clerselier and Rohault. But even in a letter after this admission, he still interprets Descartes’ determination as direction. See Fermat 1891, vol. 2, pp. 397–8, and 486; see also Sabra 1967, p. 129, fn. 77.Google Scholar
  83. 85.
    Fermat, letter to Mersenne for Descartes, Nov. 1637; AT I, 466 and 468. See document 5.2.7.Google Scholar
  84. 86.
    “Whereby it should be obvious that AF makes an acute angle with AB; otherwise, if it were obtuse, the ball would not advance along AF, as is easy to understand” (AT I, 359). Fermat’s insistence that the angle BAF be acute makes it clear that he is talking about projections and not about the parallelogram rule. A line can only be projected on another line that makes an acute angle with it. This restriction does not apply to the side and the diagonal of a parallelogram; here, the angle made by the diagonal with either side may be obtuse as long as their sum is less than 180°. On this point we differ significantly with Sabra’s interpretation. Sabra attempts to interpret Fermat (Fig. 2.13) as applying the parallelogram rule; this compels him to treat line HB as the line of opposition to the surface. Not only is there no textual basis for this, but it represents a position that would have been unique in the 17th century. See document 5.2.5.Google Scholar
  85. 87.
    Fermat, letter to Mersenne, April or May 1637; AT I, 359. See document 5.2.5.Google Scholar
  86. 88.
    Fermat, letter to Mersenne, Nov. 1637; AT I, 470. See document 5.2.7.Google Scholar
  87. 89.
    Fermat, letter to Mersenne, Nov. 1637; AT I, 473–474.Google Scholar
  88. 90.
    Descartes, letter to Mydorge, March 1, 1638; AT II, 17–18. Letter to Mersenne, July 29, 1640; AT III, 113. For an analysis of this problem see McLaughlin 2000.Google Scholar
  89. 91.
    The exchange with Hobbes consisted of eight letters, starting with the two extracts from the lost letter of Hobbes made by Mersenne and sent to Descartes by way of Huygens: Mersenne to Huygens (received Jan. 20 and Feb. 18, 1641); Descartes to Mersenne, Jan. 21, 1641 (AT III, 287–392); Hobbes to Mersenne, Feb. 7, 1641 (AT III, 300–313); Descartes to Mersenne, Feb. 18, 1641 (AT III, 313–318) — this letter deals with Hobbes’s own optical work (see Shapiro 1973); Descartes to Mersenne, March 4, 1641 (AT III, 318–333); Hobbes to Mersenne, March 30, 1641 (AT III, 341348); Descartes to Mersenne, April 21, 1941 (AT III, 353–357). This exchange overlapped with Hobbes’s “Objections” to Descartes’ Meditations. Google Scholar
  90. 92.
    The Optica is part of Mersenne’s compilation Universae Geometriae... synopsis (Mersenne 1644b) OL V, 215–248. The actual title given by Mersenne to Hobbes’s work is “Opticae, liber septimus,” but since it was published by Molesworth in Hobbes’s Opera Latina under the title Tractatus opticus, it is now known under that name. We have been very much helped in sorting out a number of the technical details concerning Hobbes’s manuscripts by the sound advice of Frank Horstmann.Google Scholar
  91. 93.
    A preliminary transcription of this manuscript was published by Ferdinand Alessio in 1963 under the title Tractatus opticus, but since another work of Hobbes published by Mersenne in 1644 had long been known under that title, this manuscript has usually been referred to as Tractatus opticus II. Scholars were long uncertain as to its actual date, though most of them, following Brandt 1928, dated it later than the exchange with Descartes. There is, however, a great deal of evidence both internal and external indicating that the Tractatus opticus II dates from 1640: The manuscript is in the hand of a scribe not in the regular employment of the Cavendish family, who also copied a number of other manuscripts in 1640; as pointed out by G. C. Robertson in 1886, the figures and many corrections to the manuscript are in Hobbes’s own hand; and some of the corrections on the basis of their content can only have been made by Hobbes, who left England in Nov. 1640 for ten years. For details about this manuscript, see Malcolm in HC I, liv—lv, and Tuck 1988; for a reconstruction of the content of the lost letter, see Schuhmann 1998. Some important passages from this manuscript can be found in document 5.2.9. Our translation there is based on a new transcription made by Karl Schuhmann to be published in the near future, which he has generously made available to us prior to publication.Google Scholar
  92. 94.
    The Ballistica is part of Mersenne’s compilation Cogitata physico-mathematica (Mersenne 1644a), some parts of which are unquestionably derived from Hobbes. The editors of Mersenne’s Correspondance (MC 10, 577) attribute the content of this argument to Hobbes, but this remains conjectural. See document 5.2.10.Google Scholar
  93. 96.
    Hobbes, letter to Mersenne, Feb. 7, 1641; AT III, 304–5: “In as much as the motion from A to B [i.e., B to A; see Fig. 2.15] is composed of the motions from F to A and from F to B [i.e. B to F], the compounded motion AB does not contribute more speed to the motion from B towards C than the components FA, FB can contribute; but the motion FB contributes nothing to the motion from B towards C: this motion is determined downwards and does not at all tend from B towards C. Therefore only the motion FA gives motion from B to C...” (emphasis added). Hobbes makes a number of minor technical mistakes in this letter (which Descartes harps on and corrects); they do not however affect the substance of his argument (to which Descartes also replies). See documents 5.2.12 and 5.2.13.Google Scholar
  94. 97.
    Descartes, letter to Mersenne, March 4, 1641; AT III, 324.Google Scholar
  95. 98.
    Descartes, letter to Mersenne, March 4, 1641, AT III, 324–5.Google Scholar
  96. 99.
    This point is made more clearly in an earlier letter to Mersenne (July 29, 1640; AT III, 113), from which the details of the example are taken. See document 5.2.16. and for a more detailed analysis of this argument see McLaughlin 2000.Google Scholar
  97. 100.
    Letter to Mersenne, April 21, 1641, AT III, 354–6. See document 5.2.15.Google Scholar
  98. 101.
    See Kneale and Kneale 1969, p. 258ff.Google Scholar
  99. 102.
    Letter to Mersenne, April 21, 1641, AT III, 354–6. J. M. Keynes (1906, p. 469) still calls the components of a “complex term,” e.g., “A and B and C...” determinants of the term.Google Scholar
  100. 103.
    See AT II, 370; VIII, 187; and XI, 100.Google Scholar
  101. 104.
    “Since the continuous motion of these [balls] brings it about that this action is never, in any period of time, received simultaneously by two, and that it is transmitted sucessively, first by the one and then by the other” (AT VIII, 187; Principia, III, §135). Google Scholar
  102. 105.
    April 26, 1643, AT III, 648–655; see document 5.2.19.Google Scholar
  103. 106.
    AT III, 651–2; document 5.2.19.Google Scholar
  104. 107.
    This condition implies that both bodies can be conceived as points (i.e., that Fig. 2.17 [Descartes’] and Fig. 2.18 [ours] are equivalent); it is a conclusion that is difficult to reconcile with Descartes’ definition of material bodies.Google Scholar
  105. 108.
    Spinoza 1925, vol. 1, pp. 213–216; Renati Des Cartes..., II, Prop. 27.Google Scholar
  106. 109.
    Clerselier, letter to Fermat, May 13, 1662; Oeuvres de Fermat, vol. 2, pp. 478–9. See the Epilogue (section 4.2) and document 5.4.2.Google Scholar
  107. 110.
    Descartes cites Bourdin’s remarks in a letter to Mersenne (July 29, 1640; AT III, 105–119). For Descartes comments on Bourdin, see documents 5.2.16–5.2.18.Google Scholar
  108. 111.
    See AT VI, 95–97 and 591–92. On the terminology of indirect causality in the 17th century, see Specht 1967, pp. 29–56.Google Scholar
  109. 112.
    Letter to Mersenne, Dec. 3, 1640 (AT III, 251; second emphasis added). See document 5.2.18. Unaccountably, Gabbey (1980, p. 259) cites this passage as “the nearest Descartes came to a clear definition of the notion.”Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Peter Damerow
    • 1
  • Gideon Freudenthal
    • 2
  • Peter McLaughlin
    • 3
  • Jürgen Renn
    • 1
  1. 1.Max Planck Institute for the History of ScienceBerlinGermany
  2. 2.The Cohn Institute for the History and Philosophy of Science and IdeasTel Aviv UniversityTel AvivIsrael
  3. 3.Philosophisches SeminarUniversität HeidelbergHeidelbergGermany

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