In Newtonian physics the law of the conservation of momentum is deduced from the three laws - and it is specific and quantitative; which follows from the specificity and quantitiveness of the three laws.

Given the impact of the atomism of antiquity on early modern physics - through; is it possible to deduce such a law is a neccessary consequence of its purely qualitative form?

A suggestion might be when two atoms collide whatever their motion; the motion of the first after impact is determined by the second, as by symmetry the second by the first; some quantity might then be supposed for each, which when added, cancels.

Descarte specifically suggested that a 'motion is conserved; what is his account or justification for such a law?

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    From the Principles of Philosophy: God is the primary cause of motion; and he always preserves the same quantity of motion in the universe.
    – JosEduSol
    Sep 11, 2015 at 5:17
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    You can see Descartes' Physics for details and references. Sep 11, 2015 at 6:09
  • Its a consequence of invariance of the laws of physics by Galilean boost, that is, that the good notion of change is given not by velocity, but by acceleration. As simple as that.
    – sure
    Sep 12, 2015 at 11:05
  • @sure: uniform velocity is also rest ie no change; acceleration is change; force is that which creates change; thus force implies change; when put like this, it's a tautology; it's the tautological content of Newton's second law - when quantification is dropped. Sep 12, 2015 at 11:11
  • Yes, but also change implies force... and this is the non-tautological aspect of N's second law; evry time we "perceive" or measure change (in motion) we are licensed to deduce that there is a force acting, and this drives him to deduce universal gravitation ! Sep 12, 2015 at 11:17

2 Answers 2


Descartes seems to invoke the following four arguments, when he aims, in The Principles of Philosophy part II, to justify the conservation of the "amount of motion" during collisions:

  1. Motion must be conserved, because it is "simple". And simple things are destructible only by an external cause.

Anything that is not composite but simple, as motion is, always stays in existence and in the same intrinsic state as long as it isn’t destroyed ·or altered· by an external cause. (Principles II 41)

  1. Since motion must be conserved, and since the direction of movement is not always conserved, after a collision, it must be the amount of motion which is conserved.

[The conservation of motion] is proved by the fact that there is a difference between how much a thing is moving and •in which particular direction it is moving; because the direction can be altered while the motion remains constant. (ibid)

  1. Conserving motion is part of God's preserving the world (which may indicate that the world would not exist if motion were not conserved).

[The conservation of motion] is proved from the unchangingness of God’s ways of operating, •keeping the world in existence by the very same kind of· action through which he •brought it into existence in the first place. (Principles II 42)

  1. Since the amount of motion must be preserved, and since this amount is not always preserved for individual bodies, it remains that the sum of the motions of the individual bodies is the preserved amount. And when bodies collide, some amount is transferred from one body to another.

Thus, since God preserves the world by the same ·kind of· action and in accordance with the same laws as when he created it, the motion that he preserves is not permanently fixed in each piece of matter but transferred from one piece to another when collisions occur. (ibid)


The context useful to interpret it is the Aristotelian philosophy; much of A's metaphysical theories are a response to the ancient "riddle" of becoming (originated with the Eleatic school).

According to Aristotle's Metaphysics, intelligibility of "becoming" derive from the concept of substance as a union of matter and form and from the principle that everything which becomes has an efficient cause which is the starting point for becoming.

Substance is therefore an act, the goal of becoming : the union of matter and form into substance thus acquires a dynamic value. In this way, becoming is no longer a nullification of being, but is instead the union of what is possible (the power : the capacity to produce or undergo change) and what really is (the act : an object's existence).

The passage from potentiality (or power) to act always requires a cause. Something must make it happen. From this derives the axiom which governs the last two books of Aristotle's Physics :

If a thing is in motion, it is of necessity being kept in motion by something". (see also Aristotle on Causality and Aristotle's Natural Philosophy.)

Following Galileo, in Descartes' Physics (uniform) motion is nor more becoming; it is a "state", and thus it requires no cause.

The same holds for Newton's Physics :

Law I is the law of inertia : the state of rest, or of uniform motion in a straight line is preserved "unless it is compelled to change that state by forces".

What requires a cause (an "explanation") is the "alteration of motion", and Law II quantify the "capability" to generate motion of a force.

In conclusion, uniform motion, like rest, is not "becoming", and thus it does not require a cause.

This is the context for Descartes' leges naturae; see Principia philosophiae, Pars II, art.XXXVII :

Atque ex hac eadem immutabilitate Dei, regulae quaedam sive leges naturae cognosci possunt, quae sunt causae secondariae ac particulares diversorum motuum [...]. Harum prima est, unamquanque rem, quatenus est simplex et indivisa, manere, quantum in se est, in eodem semper statu, nec unquam mutari nisi a causis externis.

This is the first clear and complete statement of the law of inertia : "each thing, as far as is in its power, always remains in the same state; and that consequently, when it is once moved, it always continues to move". It is supported by law two [art.XXXIX] : "all movement is, of itself, along straight lines; and consequently, bodies which are moving in a circle always tend to move away from the center of the circle which they are describing."

Causa hujus regulae eadem est quae praecedentis, nempe immutabilitas et simplicitas operationis, per quam Deus motum in materia conservat.

Thus, conservation of movement is justified by God immutability, which means that the world acts according to laws established by God, that are simple and stable.

Nitpicking comment : the first law speaks of "unamquanque rem [...] semper in eodem statu perseveret", followed by an example regarding the shape of a body, while the second law speaks of "omnis motus". We can see in this wording an allusion to a more general principle of "immutability", form which it is deduced the more specific law of inertia.

[art.XL] Tertia lex naturae haec est: ubi corpus quod movetur alteri occurrit, si minorem habet vim ad pergendum secundum lineam rectam, quam hoc alterium ad ei resistendum, tunc deflectitur in aliam partem, et motum suum retinendo solam motus determinationem amittit; si vero habeat majorem, tunc alterum corpus secum movet, ac quantum ei dat de suo motu, tantundem perdit. [“a body, on coming in contact with a stronger one, loses none of its motion; but that, upon coming in contact with a weaker one, it loses as much as it transfers to that weaker body.”]

The (wrong) first rule of impact is justified, for the first part simply by the conservation of movement (by first law) and, for the second part :

[art.XLII] Demonstratur etiam pars altera ex immutabilitate operationis Dei, mundum eadem actione, qua olim creavit, continuo jam conservantis. [...] Deum ab initio, mundum creando, [...] ipsum conservando eadem action, ac cum iisdem legibus cum quibus creavit, motum, non iisdem materiae partibus semper infixum, sed ex unis in alias prout sibi mutuo occurrunt transeuntem, conservet. Sicque haec ipsa creaturarum continua mutatio immutabilitatis Dei est argumentum.

Thus, the world moves and changes without end, following the immutable rules [leges or regulae] established form the beginning by God : the overall "amount" of motion given ab initio to the world will not change, and this is the ultimate source of the behaviour of bodies in motion.

  • The word motion has different senses; in Aristotle, it's almost a synonym for change; when Aristotle says that 'if a thing is in motion, it is by neccessity, kept in motion by something'; it is implied by his theory of change - a change requires a changer; it's also what lies behind Newton's second law; But it's often misinterpreted. Sep 12, 2015 at 11:20
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    Hello. This is interesting. However, the question does not seem to be about conservation of an uninterrupted uniform motion, but about conservation during collision. Sep 12, 2015 at 22:29
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    @MauroALLEGRANZA Right. But the question was not about the correctness of Descartes' position, but about the justification that he offered. Sep 13, 2015 at 12:22
  • Tobolski is right on both counts - that your answer is interesting (which is why I up-voted it), but it's answering a different question to the one I'm asking; as he suggests I am specifically asking how Descarte justifies his conservation law on collision; the usual physical derivation of the conservation of (linear) momentum from Newtons Laws of Motion analyses collisions. Sep 14, 2015 at 16:23
  • But I'm assuming that this isn't what Descarte is doing, otherwise we'd be talking about Descartes Laws of Motion rather than Newtons. Sep 14, 2015 at 16:25

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