Theorems

¹ If a body strikes into another resting body, or one occurring directly more slowly, or one preceding more slowly, it carries it away with itself (that is, it moves into the same direction) with a difference of speeds.

² If running in centrally, it will move more slowly than the one passing by; the one passing by [carries] with itself…

[16]**

…carries it away with itself by the difference of speeds.

³ If a striking thing moving centrally is faster than the receiving thing, the striking thing carries away the whole receiving thing by the difference of speeds.

⁴ If a striking thing and a passing thing move with equal speed, both will move just as in an equal-speed concurrence making an angle, about which see theorem 7.

⁵ If, however, the passing thing and the receiving thing are moved around their own axes [whether slower or faster], both the striking and receiving things will simultaneously retain their own motion and accept the motion of the other.

⁶ A thing striking eccentrically [whether faster or slower] into a coherent body at the point where it coheres, will continue its own motion and leave the prior motion to the receiver, and add to the same a motion around its own axis at the point of impact equal in speed to the motion of impact.

⁷ If 2 bodies concur with equal speed [or even if one runs in and the other passes by, see Theorem 4] and an angle is made [which always happens in an accursus, and never in a direct occursus] and it is bisected, the two bodies will simultaneously move in a straight line [unless the motion of one is uniform and the other accelerated, in which case parabolas and other kinds of lines arise, as seen by Hobbes, concerning which see elsewhere] out from the angle of concurrence [or incursion] along the bisector [unless two endeavors can be mutually added to each other, as the rectilineal and circular into a spiral, with the speed of the angles preserved; see foundation 19] with the prior speed. 8. From this it follows that the angle of incidence…

[17]**

…and of reflection are not always equal; but in our case [where each of the concurring things is mutually incident, and both are reflected as a composite into one], the rectilinear angle of incidence and reflection, whichever is smaller, is the double supplement to the right angle of the other. The cause of the equality in sensible bodies is given in the Theory of Concrete Motion, section 22.

⁹ From this it follows that only a rectilinear angle of incidence of 30 degrees has an equal angle of reflection according to the laws of abstract motion.

¹⁰ Incidence and reflection are not to be estimated from the surface into which it strikes, but from a straight line passing through the point of concurrence, perpendicular to the measure of the line of motion of the receiver, and parallel to the side of its coming.

¹¹ It follows also from theorem 4 that if two concur with equal speed along arcs of similar and equal curvilinear lines, both will proceed in a straight line.

¹² If an angle is not given which is bisectable [but an angle is not given at all in a direct occursus, nor is a bisectable angle given in another impact if the line of motion strikes between a straight and a curve, or a curve and a curve of a dissimilar or unequal figure] and the impact is of equal speed, both will come to rest [unless indeed there is no need for bisection, as in the concurrence of a uniform and an accelerated motion, or an accelerator of different forms; or there is no need for the section of an angle at all, as in the case of compatible endeavors, see theorem 7].

The striking thing and that into which it has struck, as far as…

[18]**

⁴ If a body strikes centrally into another which is resting, it moves it with its own speed, but it also remains in its place.

⁵ If it strikes eccentrically, it will move the other with less speed than its own, for a part of the endeavor (conatus) is diverted into a rotation or a grazing motion.

⁶ Every eccentric impact produces a circular motion**, or at least an endeavor toward it; for the part which is struck is impelled, while the part not struck resists by its own inertia, thus the body is turned around its own center.

⁷ If a body is moved by two equal and direct endeavors in opposite directions, it rests; but if they are not direct (that is, not in the same line), it is rotated. 8. Hence the variety of all “complex” motions in nature—such as the swaying of trees, the ripples of water, and the orbits of planets—can be reduced to the simple laws of central and eccentric impact.

[19]**

⁹ If a body strikes another while passing by**, it only grazes it, and the loss of speed is proportional to the portion of the surface touched.

¹⁰ In a plenum, there is no such thing as a “simple” grazing, for the surrounding aether immediately fills the space and mediates the pressure between the two bodies.

¹¹ If 2 bodies of the same magnitude and speed concur obliquely, they will proceed together in a third direction, which is the diagonal of the parallelogram formed by their individual endeavors.

¹² This is the most certain law of the composition of motion**, which Geometers have long used but which we here derive from the first principles of the Theory of Abstract Motion.

¹³ No motion is “broken” or “bent” in nature; rather, every change of direction is a new motion composed of the old endeavor and a new impulse from an obstacle or the aether.

¹⁴ Therefore, the path of any body is always a straight line in each of its unassignable parts (indivisibles), even if the whole path appears as a curve.

[20]**

…the same direction, with the prior speed of the striking body being preserved. 3. If a body strikes another which is moving more slowly in the same direction, it increases the speed of the other so that they both proceed with the speed of the striking body. 4. If a body strikes another resting body of the same magnitude, it gives its whole motion to it and itself rests, provided they are both in a plenum and there is no other obstacle. This is the reason why motion is transferred through a series of contiguous globes.

⁵ Reflection is not a motion through itself, but is composed of a new impulse**. For when a body strikes an obstacle that it cannot move, the aether, which is compressed between the body and the obstacle, restores itself with a swifter motion and repels the body in the opposite direction. Hence it follows that in the absolute geometry of motion, there is no reflection without an external elastic cause. 6. Every body which is moved in a plenum carries with it a part of the surrounding matter, and thus forms a certain “vortex” or “sphere of activity” around itself.

[21]**

⁷ If 2 bodies concur directly with equal speed and magnitude, they both rest** at the moment of impact. For their contrary endeavors (conatus) are mutually extinguished. However, because of the surrounding aether, they are immediately repelled from each other, which we call reflection. 8. If they are of unequal magnitude, the larger is less impeded by the smaller, and carries the smaller with it, with a speed proportional to the excess of its magnitude.

⁹ From this we see that no motion is ever lost in the universe**, but is only transferred from one part of the plenum to another. For even if a body seems to rest after impact, its motion has been distributed among the insensible parts of the obstacle or the surrounding aether.

10. This is the foundation of the conservation of force, which is the constant law of the Divine Creator in the administration of the world. For just as the sum of matter remains the same, so also the sum of the “endeavor” or “vital force” remains constant.

11. All variety in things is therefore but a variety of the directions and compositions of this single, universal motion.

[22]**

…carries with it the slower one, so that they both proceed with the speed of the difference of speeds. 3. If a body strikes into another which is resting but larger, it is reflected; if into a smaller one, it continues.

4. If a body is moved, it is either a single thing or an aggregate. If it is a single thing (which is a point of a body), it cannot be moved in many directions at once; for a point is that which is moved. 5. If it is an aggregate, it can be moved in many directions at once, for its different parts can have different endeavors (conatus). 6. An aggregate is moved by that motion which is the result of the endeavors of all its parts.

7. If the endeavors of the parts are contrary and equal, the whole aggregate rests, even if the parts are in the greatest motion. This is the reason for the cohesion of the universe and why the whole can be seen to rest while the parts are most agitated. 8. If the endeavors are unequal, the whole is moved in the direction of the stronger endeavor, with a speed that is the difference of the endeavors. 9. Any body in a plenum is moved by the pressure of the surrounding bodies; for since there is no vacuum, no body can be moved unless another gives way or pushes.