the impact rules of descartes' physics

The Impact Rules of Descartes' PhysicsAuthor(s): Desmond M. ClarkeSource: Isis, Vol. 68, No. 1 (Mar., 1977), pp. 55-66Published by: The University of Chicago Press on behalf of The History of Science SocietyStable URL: http://www.jstor.org/stable/230373 .

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The Impact Rules of Descartes'

Physics

By Desmond M. Clarke*

IN THE PRINCIPIA PHILOSOPHIAE OF 1644 Descartes develops seven rules for predicting what happens when one body collides with another

and changes its speed or direction as a result of the impact. Since Descartes implies that these rules are derived from the three "laws of nature" of the Principia, and because the rules themselves appear to be evidently counter- experiential, two problems arise in attempting to understand the significance of these rules for Descartes' science: how does Descartes deduce the rules from the laws of nature, and to what extent are the rules compelling evidence that Descartes' scientific method is nonempirical?

To facilitate discussion of the first question, translations of the relevant sections of the laws of nature are provided (they are referred to as L1-L3).1

LI: "Everything, insofar as it is simple and undivided, always remains in the same condition as long as it can, and it is never changed except by external causes."2

L2: "No piece of matter, considered on its own, tends to continue its movement in a curved line, but only in a straight line, although many pieces of matter are often forced to deflect because of contact with others so that . . . in every movement there is a kind of circle created from all the (pieces of) matter simultaneously moved."3

L3: "When a moving body encounters another, if it has less force to continue in a straight line than the other body has to resist it, then it is deflected somewhere else, and while it retains its motion, it loses only the determination of its motion; if however it has a greater force, then it moves the other body with itself, and it loses as much of its own motion as it transfers to the other body."4

Rather than translate the full text of Descartes' rules, the following summary is provided, and the abbreviated rules are named R,-R7. The original text is considered in more detail in the discussion which follows whenever it is relevant to deciphering Descartes' argument.

Consider two bodies, B and C, which are moving with initial speeds of Vb

and Vc before impact. After colliding with one another their speeds will be

Received September 1974: revised/accepted December 1974. *Department of Philosophy, University College, Cork, Ireland. 'The references to Descartes' works are, unless otherwise indicated, to the Oeuvres, ed.

C. Adam and P. Tannery (new ed., Paris: Vrin, 1964). The translations throughout are my own. 2Ibid., Vol. VIII, Pt. 1, p. 62. 3Ibid., p. 63. 4Ibid., p. 65.

Isis, 1977, 68 (No. 241) 55



56 DESMOND M. CLARKE

V' and VW. The symbol Q, with appropriate subscripts for B and C, is used to refer- to what Descartes calls the size or quantity of matter of a body. Using these symbols, the impact rules are as follows:

R1: If Qb = Qcl Vb = Vc, and if B and C are moving in opposite directions, they are reflected on impact with no change in speed.

R2: If Qb 'Q,, and if the other conditions are identical to the previous case, then both bodies travel in B's initial direction, after impact, with no change in speed.

R3: If Qb = QC, Vb> VC, and if B and C are moving in opposite directions, then both bodies travel in B's original direction after colliding, and Vb = VC

R4: If Qb <Q, and if V, = 0, then B will always be reflected on impact with C, no matter what its initial speed, and Vb = Vb.

R5: If Qb> QC and if the other conditions are the same as in R4, then both B and C move in B's initial direction after impact, and Vb= V'

R6: If Qb= =Q and the same conditions prevail as in R4, then B and C are both reflected on impact, and V' = 1 /4 Vb and Vl= 3/4 Vb.

R7: If B and C move in the same direction, and if B follows C in such a way that Vb> Vc, then three possibilities arise: (a) if Qc < Qb or if Qc> Qb and QC/Qb< Vb/VcQ5 then B and C continue to move in the same initial direction after impact, and V' = '

(b) if Qc> Qb' and Q,/Qb> Vb/Vc, then B is reflected from C on impact and retains all of its original motion; (c) if Qc> Qb' and Qc/Qb = Vb / VcI then B transfers part of its motion to C on impact and is reflected from C with the other part of its initial

6 motion.

A number of commentators have suggested that the rules cannot be derived from the laws without introducing some auxiliary hypotheses or principles.7 Descartes himself, however, contends that "all the particular causes of the changes that occur in bodies are contained in the third law." 8 Hence, to determine "from these [laws] how individual bodies increase or,decrease their motion or change direction as a result of colliding with other bodies, it is only necessary to calculate the amount of force in each one to move or to resist motion, and to decide that the more powerful one will always produce its effect."9 This implies that once Descartes' concept of force is clarified, and a method of measuring both the force of resisting motion and causing motion is provided, it is relatively easy to determine the results of collisions under different conditions. The plausibility of this suggestion is examined in the discussion which follows.

Cartesian science is constructed on the foundation of every-day sensory

5This possibility is considered only in the French version of the Principia (see Oeuvres, Vol. IX, Pt. 2, p. 92).

6R7(c) occurs only in the French edition (ibid., p. 93). 7See, e.g., D. Dubarle, "Remarques sur les regles du choc chez Descartes," Cartesio (Supplement

to Rivista di Filosofia Neo-scolastica) (Milan: Vita e Pensiero, 1937), pp. 325-334, and Richard J. Blackwell, "Descartes' Laws of Motion," Isis, 1966, 57:220-234.

80euvres, Vol. VIII, Pt. 1, p. 65. 9Ibid., p. 67.



IMPACT RULES OF DESCARTES' PHYSICS 57

experience. Among the consequences of this methodological procedure one finds that the fundamental explanatory concepts of Descartes' dynamics are derived in a rather uncritical manner from our ordinary experience of physical bodies in motion. Descartes is extreme in his zeal to avoid the postulated entities which were disputed in the schools of philosophy, and he plans to construct his science within the scope of a rather limited conceptual framework.'0 Not only does he shun the multiplicity of scholastic forms, he also avoids theoretical concepts which are not easily understood and empirically known by the average observer of nature.

In the absence of well-defined theoretical concepts, it is inevitable that the apparent clarity of Cartesian science masks deep conceptual confusions which only come to light on close inspection. Thus, Descartes does not distinguish clearly between matter and mass, nor between the mass and weight of a body. He notoriously defines matter in terms of extension, which seems to preclude any distinction between more or less dense bodies. Despite this, one finds frequent references to the relative density of different bodies or, more often, to their relative solidities. " There is no evidence in Descartes' writings that he noticed any conceptual confusion in this, and consequently one finds no attempt on his part to explain the consistency of the concepts of matter and density. Perhaps he could have resolved the problem as follows.

Descartes defines the concept of matter in terms of extension. Being extended is a necessary and sufficient condition of materiality, so that there is only a distinction of reason between the two concepts.12 Since a piece of matter has some size, it cannot be indivisible in principle; since an extended entity has some matter in it, there cannot be a pure vacuum. This kind of conceptual analysis, however, is inappropriate for discovering how many kinds of fundamental particles one must postulate in order to explain the variety of properties which matter exhibits. One is justified in postulating whatever particles are necessary to explain our experience of nature, and in this project "we cannot decide by reason alone." " The concept of matter does not provide any a priori insight into the kinds of particles one needs for the success of a scientific theory. Nor does the concept of matter determine in advance what it means for a particle to occupy a given space (or have a determinate extension).

It is possible for a very small particle of matter to occupy more or less space (to have a greater or smaller extension) by moving more or less fast within a determinate place. If it is objected that there must be a vacuum in such a place to permit inovement, Descartes could respond that since the space is occupied by the particle of matter, there is no sense in describing the particle's immediate environment as a vacuum. Therefore, one can conceive of more or less dense bodies in terms of the relative proportion of small interstices in their structures which are occupied by small imperceptible particles.14 This apparently is what Descartes had in mind when he defined solidity thus: "By solidity, in this context, I understand the quantity of matter of the

?0See Vol. VI, p. 43. "E.g., Vol. VIII, Pt. 1, p. 43. '2Cf. ibid., p. 31, Vol. X, pp. 443-445, Vol. XI, p. 36. '3Vol. VIII, Pt. 1, pp. 10 101; cf. Entretien avec Burman, ed. C. Adam (Paris: Boivin, 1937),

p. 112. 14Cf. Vol. XI, pp. 10-11.




third element . compared to its bulk and surface area."'5 Since different bodies may be more or less solid, one body may contain a greater quantity of matter than another either because it is larger in size or because it is of equal size but more solid than the other.'6

Thus the amount of matter in a given body is proportional to its volume and density. The weight of a body is the force it exerts in its motion toward the center of the earth and is a function of its quantity of matter, its size, and the resistance of the surrounding medium.'7 Thus the force of gravity of a body is relative to its solidity, and one could not determine the quantity of matter it contains from its weight alone. "Hence, from the weight alone one cannot easily estimate the amount of terrestrial matter in each body."' 8 A less dense body might contain more matter than another more dense body and still exert less force of gravity because of the greater resistance of the medium to its larger surface area.

Descartes' analysis of the force of a body in motion applies equally to gravitational and impressed motion, because the former is, for him, a special case of the latter. Consequently the solidity of a particular body, its size, its speed, and the resistance of other bodies it encounters determine the kind of force it can exert on impact with other bodies.

Size and speed are evidently significant factors in determining the force of a body in motion. Descartes writes, in Le monde, of "the more forceful [bodies], that is to say the larger ones among those which were equally moved, and the more moved ones among those which are equally large."'9 Relative solidity is also a factor, because Descartes supposes that every body on earth moves in an environment which more or less impedes its motion. This is borne out by the experience of moving more or less solid objects in a liquid:

When two bodies move equally fast it is true to say that if one contains twice as much matter as the other, it also has twice as much motion; that is not the same as saying that it has twice as much force to continue to move in a straight line. But it would have exactly twice as much force if its surface were also exactly twice as large, because it would always encounter twice as many other bodies which resist it.20

In other words, the product of quantity of matter and speed determines the measure of the motion of a body, but the relation between the motion and

15Vol. VIII, Pt. 1, p. 170. The French version is more explicit: "I understand here by the solidity of this star the quantity of the matter of the third element . . . insofar as it is compared with the extension of [its] surface and the size of the space which the star occupies" (Vol. IX, Pt. 2, p. 174).

'6"The one which contains the most matter, either because it is more solid, or because it is larger. Descartes to Mersenne, Feb. 23, 1643, Correspondance, ed. C. Adam and G. Milhaud (8 vols.; Paris: I, II, Alcan; III-VIII, Presses Universitaires de France, 1936-1963), Vol. V, pp. 268-269. All references to Descartes' correspondence are to this Adam-Milhaud edition, designated AM hereafter.

'7Cf. Oeuvres, Vol. VI, p. 291: "the pieces of ice . . . having a very large surface in proportion to the quantity of their matter, the resistance of the air that they would have to divide could easily have more force to stop them than their weight has force to cause them to descend."

I8bid., Vol. VIII, Pt. 1, p. 215. '9Ibid., Vol. XI, p. 50. Cf. Entretien avec Burman, p. 106: "To the extent to which a body

is larger, it continues its motion more easily, and resists other bodies." See also Vol. VI, p. 235. 20Vol. XI, pp. 66-67. See also the example of ships moving with more or less force, in Le

monde, ibid., p. 58.




force of a body depends on its surface area (among other things). "Consider that this force does not depend only on the quantity of matter of each body, but also on the extension of its surface." 2'

Descartes is evidently trying to come to terms with our experience of moving very dense objects and the exertion this requires, and also our experience of moving objects with more or less facility in fluids of differing viscosities. But details of his theory are never sufficiently developed, and so one finds the same rather vague references to size, surface area, resisting media, and speed in that section of the Principia between the laws and the rules in which he explains the theory underlying the derivation of the rules of impact.

This force [of a body to continue in motion] should be determined both from the size of the body in which it inheres and the surface by means of which it is separated from other bodies, and also from the speed of its motion, and the nature and degree of contrariety of the way in which other bodies mutually encounter it.22

Since Descartes' physics was much too undeveloped to consider all these factors in his theory of colliding bodies, he proposes the construction of an ideal situation in which two perfectly hard bodies collide, without interference from other bodies or the medium in which they move. The impact rules are applicable "if only two bodies collide, if they are perfectly hard, and if they are so separated from everything else that their motion is neither hindered nor assisted by anything in their environment."23 In this situation the only relevant factors which remain for calculating the relative forces of the two bodies are their speed and quantity of matter. If Q is the quantity of matter of a body and V its speed, then QV is the measure of its quantity of motion. The relative forces of two colliding bodies in the ideal conditions proposed by Descartes are proportional to their quantities of motion, QV.

To determine what happens when bodies of differing speed and quantity of matter collide, one need only consult L3. However, to apply this law of nature one needs to measure the force with which stationary bodies resist motion. Once a method is provided to make this kind of measurement, it is relatively simple to determine which of two colliding bodies has the greater force, whether the force in question is one which tends to cause motion or resist motion.

Descartes' concept of inertia is a corollary of his understanding of the force of a moving body. Assuming that one wishes to move two bodies, M and N, with a given speed S, then more force is required to move a body with a greater quantity of matter or a larger surface area. The reason for this is obvious, given the Cartesian concept of matter. If M and N both move with speed S, and M has a greater quantity of matter than N, then M has a greater motive force than N (all other things being equal). Since M can only have acquired this force when it began to move, and according to Li it tends to conserve its force intact, then a greater force must have been initially necessary to move M relative to N.24

21 Ibid., p. 66. 22Ibid., Vol. VIII, Pt. 1, p. 67. 23 Ibid. 24Cf. Descartes to Mersenne, Feb. 8, 1643, AM, Vol. V, p. 269.




On the other hand, if M has a larger surface area than N, then it will encounter more resistance from the medium in which they move, and to compensate for this M needs a greater impressed force to maintain the same speed as N. Since the first kind of resistance to motion is a characteristic of a body apart from the medium in which it moves, Descartes calls it the natural inertia of a body. The second type of resistance to impressed motion depends more directly on the medium in which a body moves, but it still represents a part of the inertia which must be overcome in changing the condition of a body at rest or in motion.

In either case, there is no such thing as a kind of active resistance of a body to motion. Rather, the force required to impart a certain motion to a body by impact is equal to the force of the motion of the same body after impact. Because it requires a given measure of force to achieve a certain degree of motion, a moved body is considered to resist motion with that same degree of force:

. . . if two unequal bodies receive the same amount of motion, [and] this equal quantity of motion does not give as much speed to the larger as to the smaller body, one could say, in this sense, that the more matter a body contains, the more natural inertia it has. To this one might add that a large body can more easily transfer its motion to other bodies than a small one can, and that it can be less easily moved by others. So there is one kind of inertia which depends on the quantity of matter and another which depends on the extension of their surfaces.25

In the idealized conditions in which Descartes' impact rules should apply, only the so-called natural inertia of bodies is relevant in determining the results of different collisions. For this special case also L3 could be re-written in the form of two theorems, the first of which is:

Theorem I: When two bodies collide, the body with the greater force of motion or natural inertia predominates.26

The actual manner in which the more "forceful" body predominates depends on the conditions of impact, and especially on the direction in which the bodies are moving relative to each other before impact. If the more forceful body must transfer some of its motion to the other body in order for its motion to predominate, then the total quantity of motion of the system remains unchanged. If V and V' represent the speed of a body before and after impact, respectively, and the subscripts b and c denote the two bodies in question, then the conservation of motion in the system of two colliding bodies is expressed by the following theorem:

Theorem II: Qb Vb + QC Vc = Qb Vb + Qc VC

Using these two theorems, we can derive the rules from Descartes' description of the seven different sets of conditions he considers.

25Descartes to [Debeaune], [Apr. 30, 16391, AM, Vol. III, pp. 214-215. 26When bodies collide, neither one is completely unchanged as a result of impact. In predicting

the outcome of such collisions Descartes thinks that the more "forceful" body imposes the direction or speed of its motion on that of the other body. Thus if two bodies A and B collide, A is said to predominate over B if: (1) A causes B to change the direction of its motion while the direction of A's motion is unchanged or (2) A causes B to increase the speed of its motion (or to begin to move if B is initially at rest).




R1: In this situation neither B nor C has more force, since they are exactly equal in size and speed. Hence, neither one can predominate according to Theorem I. Also, by Theorem II they must retain their original speeds if the total system is not to lose any motion in the collision. Since they are moving in opposite directions before impact, there is no other possiblity available except that the two bodies be reflected on impact and move in opposite directions with the same speed.27

R2: The conditions are similar to those in R1, except that B has a greater quantity of matter and, as a result, has a larger measure of motion than C. Consequently, when they collide B continues to move in the same direction as before impact, and C is forced to move in the same direction as B. Once C is reflected, it moves in its new direction with its original speed, V,. Because B and C are then moving in the same direction with equal speeds, C does not obstruct the passage of B. Therefore, no transfer of motion from B to C is required, and the final conditions are as described in R2.

R3: In this case, Vb is larger than V, and both bodies have the same quantity of matter. Hence B has more force than C and causes C to be reflected. If this were the only effect of impact, C would then be moving immediately in front of and with a lower speed than B. By applying Theorem I we find that B must transfer part of 'its motion to C in order to predominate in the collision. Otherwise C would effectively be impeding the motion of B. The minimum amount of motion transferred from B, to C enables both bodies to move in B's initial direction with the same post-impact speed.28

R4 and R6: Descartes gave two different versions of these rules, and one can only speculate about the source of the inconsistency in the two versions. One account of these rules is found in the Principia, in the same place as the other impact rules; the other version is found in his correspondence. Since the latter is apparently more consistent with our experience of collisions between a moving and a stationary body, it is best to examine it first and then investigate why Descartes changed his mind in formulating R4 in the Principia.

R4 applies when C is at rest and B moves toward C with an initial speed of Vb. Here Qb is less than QC. If Vb is sufficiently high, and if the difference in quantity of matter between B and C is not very large, experience seems to indicate that the two bodies would move after impact in the same direction as B moved initially. Following the line of argument involved in R3, the two bodies would have the same terminal speed of (Qb + QC) Vb/Qb-

In the situation to which R6 applies, the conditions are the same as in R4 except that Qb QC- In this case B would transfer half of its initial quantity of motion to C, and both B and C would move in B's initial direction, after impact, with a speed of 1/2 Vb. Descartes summarizes this version of R4 and R6 in a letter to Mersenne, December 25, 1639:

27Descartes distinguishes between what he calls the determination of a body's motion and the force of the motion. He regards the determination, or direction, of motion in such a way that no expenditure of force is required to change it, although it does require a cause, according to L2. The result is that two perfectly hard bodies can be reflected on impact without any loss or transfer of force.

28Descartes describes the resolution of this collision in a letter to Mersenne, Oct. 8, 1640: "I understand that this ball [B], making contact with the other [C], pushed it in front of itself in such a way that they moved together after the collision" (AM, Vol. IV, p. 181).




As regards inertia, I think that I have already written that in a completely frictionless space, if a body of a given size and speed collides with another. which is equal in size and without motion, it will transfer to it (the second body) half the initial speed of the first body. However, if it collides with another which is twice as large as itself, it will transfer two thirds of its motion and both will move afterwards with only one third of the initial speed of the first body. And in general, the greater the body, the slower it will move when pushed by the same force.29

This version of R4 and R6 represents one possible interpretation of L3, for Theorem I suggests that in deciding the consequences of this type of collision one need only consider the force of the moving body and the inertial force of the body at rest. Thus, no matter how large the body at rest is, and no matter how small or how slow the moving body is, the latter can always transfer a sufficiently large portion of its initial motion to the former so that both move, after impact, with the same velocity. However, it is also possible that at a certain critical point the inertia of C relative to the moving force of B is such that B is reflected on impact with C. It was apparently a consideration of this latter alternative which gave rise to the version of R4 and R6 which is found in the Principia. The following conjectured reasoning on Descartes' part explains the derivation of these two controversial rules.

R4: Assume that B does transfer some of its motion to C and that both bodies move, after impact, in B's initial direction. As in previously considered similar situations, B must transfer enough of its motion to C to make their final speeds equal; otherwise, C would be impeding B and this contradicts Theorem I. We can then assume that

Qb Vb = Qb Vb + QC Vc (from Theorem II) (1)

V = V (2)

and, from (1) and (2),

Qb Vb = Qb VC + QC Vc' (3)

And since

Qc > Qb (4)

it follows, from (2) and (4), that

Qb VC < QC Vc (5)

and, from (2), (3), and (5),

Qb Vb < I

Qb Vb (6)

In other words, if B transfers the minimum amount of its motion necessary to move C, the residual quantity of motion of B is less than half its initial value.

If this were to happen, Descartes considers that such a collision would contravene Theorem I. For if B must lose more than half of its initial motion in order to predominate in a collision, then C has a greater natural inertia to resist motion than B has force to cause C's movement. This situation obtains

29Ibid., Vol. III, pp. 299-300. Cf. Descartes to Mersenne, Oct. 28, 1640, for a similar resolution of the conditions to which R4 applies (Vol. IV, pp. 181-182).




no matter what initial speed B has as long as C has a greater quantity of matter than B, because the natural inertia of C increases in proportion to the increase in the speed of B. If one simply measures the moving force of B and the inertial force of C, then the former is greater than the latter. However, B has two alternatives: either to force C to move or to be completely reflected. As soon as the first of these alternatives involves overcoming an inertial force which is more than half of its own moving force then B is reflected on impact with C and retains its motion intact. At this critical point, although the force of B is still greater than the inertial force of C, it is not large enough, relative to the force of C, to cause the first rather than the second of the two possible results. This is the version of R4 which is found in the Principia.

Descartes suggests this kind of reasoning in defense of R4 in a letter to Clerselier, February 17, 1645:

The reason which prompted me to say that a body which is stationary could never be moved by another body which is smaller than it, no matter how fast this smaller one moved, is that it is a law of nature . . . if B is to C as 4 is to 5, B could not move C unless it transferred five of its nine degrees of motion to C, which is more than half [the motion] it has. Consequently, C resists this [imposed motion] more than B has force to act; that is why B must be reflected in the opposite direction rather than move C.30

Before applying L3 to the conditions considered by R6, it is useful to explain the derivation of R5. In this case Vc = 0 and Qb is greater than QC. Therefore B moves C by transferring enough of its motion to C to cause both bodies to move after impact with the same final speed. Since QC is less than Qb,

the terminal quantity of motion of C is less than half the initial quantity of motion of B. Hence C has a lower inertial force to resist motion than B has force to cause C's motion. The final speed of B and C is determined by applying Theorem II.

R6: This is a limiting case of the conditions which obtain in applying R4 and R5. If B is even slightly less than C in its quantity of matter, it is reflected with its speed unchanged; if B is slightly larger than C, it moves C so that the two bodies have a final speed of Qb Vb /(Qb + QC). As QC tends to equal Qb9 then Qb Vb /(Qb + QC) tends to 1/2 Vb. Descartes appears to resolve the question as follows: if B is slightly smaller than C, it transfers none of its motion to C. If B is slightly larger than C, it transfers almost half of its motion to C. If B and C have the same quantity of matter the result of impact is the mean of the two previous cases. Thus B transfers only one quarter of its motion to C and is reflected with three-quarters of its initial speed. This is consistent with Theorem II and agrees with the Principia version of R6.

R7: The first two parts of this rule are derived from Theorems I and II as in previous cases. If QC /Qb is less than Vc / Vb, then Qb Vb iS greater than

30"Ibid., Vol. VI, pp. 211-212. In the Entretien avec Burman, p. 88, Descartes claims that he has explained and elucidated the obscurity of the laws of nature in the French edition of the Principia. The explanation of R4 in Les principes adds the following comment to the original Latin: "Because to the extent that B cannot move C without making it go as fast as it moves itself after impact, it is certain that C should resist in proportion as B comes more quickly toward it; and its resistance should prevail over the action of B, because it is larger than B" (Oeuvres, Vol. IX, Pt. 2, pp. 90-91).




QC VC, and consequently B transfers enough motion to C to enable both bodies to move in B's initial direction, with equal final speeds. If B has a smaller quantity of matter than C, it cannot change C's motion, according to Theorem I. The only other possibility available is for B to be reflected with its speed unchanged.

The third part of R7 is unsatisfactory because it gives no measure for the amount of motion transferred from B to C. Since both B and C have equal force, there should be no transfer of motion according to Theorem I, and B should be reflected on impact with C. The justification of this part of R7 is apparently derived from two sources. On the one hand, it conforms to our experience of such collisions. On the other, it represents a limiting case between R7 (a) and R7 (b), where B has a greater or lesser quantity of motion than C. When their quantities of motion are equal, Descartes suggests that part of the effects of R7 (a) and R7 (b) occur.

In attempting to derive the impact rules from Descartes' laws of nature, it is apparent that R4 constitutes the principal stumbling block to a simple application of L3 to the different situations to which the rules apply. Alan Gabbey has tried to show that R4 conforms to Descartes' contest view of forces as expressed in L3.3' He argues that if a body B transfers more than half of its motion to a body C which is initially at rest, then C resists the impressed motion with a force equal to this motion.32 Since this is greater than the residual quantity of motion of B, the contest between the resistance of C and the residual force of B must be resolved in favor of C.

However, this argument hardly explains Descartes' use of L3 under these conditions. Consider the situation discussed by Descartes in the letter to Clerselier quoted above. If B transfers five of its nine degrees of motion to C, then C moves in the same direction as B and there is no longer any contest between the two bodies. The contest only occurs when B collides with C, and at that point there is a contest between the motive force of B and the inertia of C which are in the ratio of 9: 5. Therefore, according to L3, B can predominate because it has a greater force, but it can do so only at the expense of losing more than half of its motion. It could also be reflected when it collides with C, and there would be no loss of motion involved in this alternative. It is evident that Descartes has recourse to a specific interpretation of L3 when confronted with this choice, and he formulates this interpretation in the letter to Clerselier. If the force of one body can only predominate over that of another by losing more than half of its own force, then it is reflected instead with its quantity of motion unchanged.33 Once this corollary is added to L3 it is possible to derive the impact rules from the Cartesian laws of nature.

3'Alan Gabbey, "Force and Inertia in Seventeenth-Century Dynamics, I," Studies in History and Philosophy of Science, 1971, 2:20-28.

32"But C at rest resists the reception of the quantity of motion . . . with a force of equal size, and this force is contested by the remaining motive force of B, . . . Hence (by the Third Law) B will not move C." Ibid., p. 27.

33Descartes formulates this principle in the letter to Clerselier, Feb. 17, 1645: "When two bodies which have incompatible modes collide there must truly be some change in these modes to make them compatible, but this change should be always the least possible . . ." (AM, Vol. VI, p. 212). In his discussion of the impact rules Dubarle extends this principle to apply to all collision situations. However, he admits that it is difficult to see how it applies in some cases. See "Remarques," p. 332.




The corollary which stipulates that in resolving the incompatible motions of two bodies the minimal change of conditions possible should be expected, is not understood by Descartes as a distinct law of nature or an independent auxiliary hypothesis. In fact, if it were understood as such and applied in all seven sets of conditions, then some of the other rules would require modification to minimize the changes involved. On the other hand, Descartes can hardly intend this corollary as the most natural or literal application of L3 to the situation envisaged by R4. For the motive force of B is related to the inertial force of C as 9: 5, and therefore B could move C and lose as much of its motion as it transfers to C on impact. Presumably Descartes' minimal change hypothesis is a qualification of the second part of L3, which reads: "If however it has a greater force, then it moves the other body with itself...."34 In those cases where the transfer of motion from B to C would be more than half the initial motion of B, we have gone beyond a critical point at which B is reflected rather than lose more than half its motion.

The second question which arises concerning Descartes' impact rules is to what extent do these rules indicate the nonempirical character of Descartes' scientific method? The impact rules are especially significant for Descartes' method because of their close dependence on the general laws of nature and, at the same time, their obvious incompatibility with our experience of colliding bodies. They seem to point to a fundamental flaw in Cartesian physics or methodology; Descartes' elaboration of the justification of the rules (in the French edition of the Principia) suggests that the flaw is a methodological one. Responding to earlier objections that the rules were counter-experiential, the later edition of the Principia reads: "And the demonstrations of all this [i.e., the rules] are so certain that although experience seems to indicate the contrary, we ought to place more trust in our reason than in our senses."35

On closer examination it becomes apparent, however, that the impact rules do not contradict our experience and that Descartes was aware of this. For this reason he wrote that experience seems to indicate the contrary. When confronted with experiments which do not correspond with the predictions of the rules, Descartes concedes that all the bodies which we experience are somewhat elastic, and therefore the rules do not apply to them.36 Thus the most obviously counterfactual rule, R4, must be balanced with Descartes' admission that the whole earth can be imperceptibly moved by the impact of something falling to the ground under the weight of gravity.37 Besides, there are no actual collisions in a Cartesian world in which the viscosity of the medium and interaction with other bodies do not play a significant role in determining the results of collisions between bodies.

It was precisely because of the admitted complexity of real collisions that

34Oeuvres, Vol. VIII, Pt. 1, p. 65. 35Ibid., Vol. IX, Pt. 2, p. 93. 36See Descartes to Mersenne, Feb. 23, 1643 (AM, Vol. V, p. 267). On Apr. 26 of the same

year Descartes wrote to Mersenne: "I add also that these bodies must be perfectly hard; for if they are made of wood, or some other elastic material, as are all those we find on earth, then . . ." (ibid., Vol. VI, p. 281).

37VDescartes to [Debeaune], [Apr. 30, 1639]: "When a stone falls from a high altitude to the ground, if it does not rebound but is stopped instead, I believe that this results from the fact that it shakes the earth and thereby transfers its motion to the earth" (ibid., Vol. III, p. 214).




Descartes proposed the consideration of an ideal situation in which elasticity, viscosity of the medium, and other relevant factors would be eliminated from calculations. Thus, to make the problem of colliding bodies more tractable, he considers perfectly hard bodies in a frictionless medium. At the same time Descartes introduced the principle of conservation of motion to account for the transfer of motion from one body to another. Unwittingly Descartes was thereby involved in a dilemma which permeated rational mechanics in the seventeenth century until Newton realized that perfectly hard bodies do not transfer motion or reflect on impact, as Descartes suggests. The assumption of the conservation law and the consideration of perfectly hard bodies together was the theoretical source of Descartes' counterfactual rules of impact.

Although Cartesian commentators often refer to the impact rules as a symptom of the rationalism of Descartes' scientific method, there is no evidence to show that Descartes would defend the rules if they could be shown to be empirically disconfirmed. Descartes realized that the rules of the Principia did not apply to actual collisions of physical objects and that they were not even open to empirical testing. Hence, we may not conclude anything about the empirical or nonempirical nature of Descartes' scientific method from an examination of the impact rules. The nonexperiential character of the impact rules results from the idealization of the conditions to which they might apply. Their real failure is due to the theoretical inconsistency which is implicit in their formulation. As a result, it was a change in the concept of force and a development in theoretical mechanics, and not a change in scientific method, which heralded the overthrow of Cartesian dynamics.



the impact rules of descartes' physics

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