Leibniz' Letter on his Law of Continuity and his "Critical Thoughts on the General Part of the Principles of Descartes," regarding Descartes' errors

by Corry Shores
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[The following is summary, my commentary is in brackets]

Gottfried Leibniz

Letter of Mr. Leibniz on a General Principle

Useful in Explaining the Laws of Nature

through a Consideration of the Divine Wisdom;

to Serve as a Reply to the Response

of the Rev. Father Malebranche

Nouvelles de la république des lettres, July, 1687

in

Philosophical Papers and Letters

Leibniz criticizes Malebranche for violating what Leibniz calls a "principle of general order," namely, his Law of Continuity. This principle begins with the notion of the infinite and it is absolutely necessary for geometry. And because God "acts as a perfect geometrician" by creating the world in a perfect harmony, this law should apply effectively as well in physics. Leibniz offers the following formulation for his principle, which he elsewhere refers to as the law of continuity:

When the difference between two instances in a given series or that which is presupposed can be diminished until it becomes smaller than any given quantity whatever, the corresponding difference in what is sought or in their results must of necessity also be diminished or become less than any given quantity whatever. Or to put it more commonly, when two instances or data approach each other continuously, so that one at last passes over into the other, it is necessary for their consequences or results (or the unknown) to do so also. This depends on a more general principle: that, as the data are ordered, so the unknowns are ordered also. [Datis ordinatis etiam quaesita sunt ordinata.]

(351d)

Leibniz then offers a mathematical and a physical example.

He explains that an ellipse can approach a parabola by extending one of the ellipse's foci out far from the other one. We may say that the one focus is infinitely far from the other. Or if we want to avoid the notion of infinity, we can say that the one focus is so far away that we may consider the resulting ellipse as different from a parabola by less than any given or givable difference. In this way, all geometrical theorems applying to an ellipse can be applied to parabola when the parabola are considered as indefinitely elongated ellipses.

(352a)

Likewise in physics, rest may be considered as an infinitely small velocity or an infinite slowness. In this way, laws applying to velocity or slowness may be applied to rest taken as infinite slowness.

Moreover, "equality can be considered as an infinitely small inequality, and inequality can be made to approach equality as closely as we wish."

(352b)

By neglecting these facts, Descartes committed errors when formulating his laws of nature. For example, according to Descartes' Second Law of Nature

if two bodies B and C collide in a straight line and with equal velocities, but B is but the least amount greater than C, C will be reflected with its former velocity, but B will continue its motion.

(352bc)

But, Leibniz notes, Descartes stated in his First Law of Nature that if B and C share the same mass and collide directly into each other, both will be reflected in opposite directions at the same velocity as their approach.

On the one hand, we might imagine why Descartes makes this claim. We can imagine that when two bodies moving toward each other at equal speeds encounter, both reflect equally in opposing directions. And if one's mass is less than the other, it would not have enough force to change the direction of the other. Hence we might imagine that the lesser body reflects back, just as it did when they were equal, while the larger body continues on its way.

But Leibniz here applies his law of continuity. So we might consider one body being only different by an indefinitely small amount, such that the difference between their values when equal and when barely different is less than any given or givable difference. In this case, the resulting effects of their collision should only be different by a negligible amount. In other words, we should still see both the objects retracting backwards after collision, just as they would when equal, only this time one of them moving slower by a negligible amount.

But Descartes' Second Law of Nature would suggest that any difference in mass between the two bodies would result in only one reflecting backward. So just the slightest possible change produces the greatest possible effect.

And this is an enormous leap from one extreme to another, whereas the body B should be reflected only a little less in this case, and the body C a little more, than in the case of their equality, from which this one can hardly be distinguished.

(352c)

[The remainder of the letter has little bearing on Leibniz' Law of Continuity, and so it will be passed over.]

From Leibniz' "Critical Thoughts on the General Part of the Principles of Descartes," 1692:

On Article 45:

Leibniz will critique Descartes' special rules of motion (Laws of Nature) by means of his Law of Continuity:

when two hypothetical conditions or two different data continuously approach each other until the one at last passes into the other, then the results sought for must also approach each other continuously until one at last passes over into the other, and vice versa.

(397d)

Leibniz again offers the example of the ellipse made into a parabola when one focus continually recedes further from the other so that the new ellipses created thereby continuously approach a parabola. In this way the properties of ellipses gradually approach those of parabola "until at last they pass over into them, and the parabola can be considered as an ellipse whose second focus is infinitely distant." (398a)

Likewise for physics and even for the notion of equality:

Thus gradually decreasing motion finally disappears in rest, and gradually diminishing inequality passes into exact equality, so that rest can be considered as infinitely small motion or as infinite slowness, and equality as infinitely small inequality.

Hence whatever holds for motion or equality should also hold for rest and inequality. "So the rules for rest or equality can in a sense be considered as special cases of the rules for motion or inequality." (398b)

On Article 46:

Rule 1: If two equal bodies B and C, with equal velocities, collide directly, both will be deflected with the velocities of their approach.

(398c)

Leibniz claims that this is the only one of Descartes' Laws of Nature that are true.

On Article 47:

Rule 2: If B and C collide with equal velocities, but B is the greater, then only C is deflected, and B continues; both with their earlier velocities, and so both moving in the original direction of B.

(398d)

Rule 2, Leibniz says, contradicts Rule 1:

For if the inequality, or the excess of B over C, is gradually diminished until it passes into full equality, the effects of inequality should also pass over continuously into the effects of equality. So if we assume that B, striking C, overcomes it with so excessive a force that it continues to advance after the collision, it will be necessary, if B is gradually diminished, for its advance also to diminish continuously until, when a certain ratio is reached between B and C, B will at length come to rest and then, by a continuous diminution, be turned into contrary motion; this will gradually increase until finally, when all inequality between B and C is removed, the motions end in the rule for equality, in which the regressive motion of each body after collision is equal to its progressive motion before collision, as the first rule states.

(398-399)

However, Descartes' Second Rule of Motion seems to imply that:

At the point, that is, where the excess of B over C finally disappears completely, and the very small difference between them is further decreased, the motion must pass over from a definite progression to a definite regression, with all the intervening degrees omitted in a single leap, as it were. The result will be that two instances which have an infinitely small variation in the hypotheses or given conditions (that is, a difference smaller than any given amount) will nevertheless have the greatest and most noticeable difference in their results, so that it must be in the very last moment only that the two bodies both begin and end their mutual approach and that they both approach each other and break apart in the moment they coincide, which is absurd.

(399b)

Leibniz, Gottfried. Philosophical Papers and Letters. Ed. & Transl. Leroy E. Loemker. Dordrecht: D. Reidel Publishing Company, 1956.