The law of continuity is a principle that we owe to Mr. Leibniz, which informs us that nothing jumps in nature and that one thing cannot pass from one state to another without passing through all the other states that can be conceived of between them. This law issues, according to Mr. Leibniz, from the axiom of sufficient reason. Here is the deduction. Every state in which a being finds itself must possess sufficient reason why this body finds itself in this state rather than in any other state; and this reason can only be found in its prior state. The prior state therefore contained something which gave birth to the actual state which it followed, and in such a way that these two states are so bound that it is impossible to place another in between them, for if there was a state between the actual state and that which immediately preceded it, nature would have left the first state even before it had been determined by the second to abandon the first; thus there would be no sufficient reason why it would sooner proceed to this state than to another. Thus no being passes from one state to another without passing through the intermediate states, in the same way that one cannot go from one town to the next without traveling along the road which is between them. This law is observed in Geometry with an exact precision. All of these changes occur in all lines which are one. That is to say, they occur in one line that remains the same or those lines which altogether make only one and uniquely one; all of these changes, I say, are made only after the figure has passed through all possible changes that lead it to the state that it acquires. The involutes or points of reflection that are in several curves appear to violate this law of continuity , since the line appears to end at that point and to re-track suddenly in the opposite direction. However, there is no violation. One can see at these points where the re-tracking occurs that there are knots which form in which one sees evidence that the law of continuity is followed since these knots, being infinitely small, assume the form of a sole and unique point of a bird’s beak. Thus in figure 104 of the Geometry, if the knot AD disappears, it will transform into the point of the involute at T. See Knot and Involute.

The same thing arrives in nature. It is not without reason that Plato called the Creator, the eternal Geometer. Properly said, there are no fixed angles in nature, neither inflexions nor any sudden involutes; however there is gradation in everything and everything is prepared at length for the changes that will test it and it will, little by little, proceed to the state to which it must submit. Thus a ray of light which is bent on a mirror, does not suddenly change direction and does not create a pointed angle at the point of reflection; however it engages in a new direction but it assumes the new direction which it takes by bending itself by a small curve, which leads it imperceptibly through all of the possible degrees which are between the two extreme points of incidence and reflection. It is the same for refraction: the ray of light is not broken at the point which separates the medium which it penetrates and which it leaves, however it begins to be subjected to an inflection before having penetrated into the new medium; the beginning of its refraction is a small curve which separates the two straight lines which it describes while traveling the two heterogeneous and continuous media.

The supporters of this principle pretend that we can use it to find the laws of motion. A body, they say, which moves in any random direction, would not be able to move in the opposite direction without going through its first movement at rest by all the degrees of intermediate slowing in order to pass once again through the imperceptible degrees of acceleration of rest to the new movement which it must achieve. Practically all of Descartes’ laws of motion are wrong, according to the Leibnizians, since they violate the principle of continuity . Such is, for example, the one that claims that if two bodies B and C enter into contact with one another with equal speeds, but that body B is larger than body C; then only body C will go backwards and body B will continue its path, each of which will continue at the same speed they had prior to the collision. This rule is contradicted by experimentation and is not in accordance with the principle of continuity , to which one must be completely attentive and by so doing, imitate nature, which also never breaks any of its own operations. Read chap. J. of the instit. of Physics of Madame du Châtelet, from § 13 to the end.

By this principle, one could further demonstrate that there are no perfectly elastic bodies in nature. The gradation that is expressed by the law of continuity would not take place in the collision of perfectly elastic bodies since these bodies would all of a sudden go from rest to motion and from motion in one sense to motion in an opposite sense. Therefore all bodies possess a degree of elasticity which allows them to satisfy the law of continuity that nature never violates. After which see Percussion. We owe this article to Mr. Formey .