Principia 1687: Laws of Motion

ReferenceA Logical Approach to Theoretical Physics

This paper presents the chapter on LAWS OF MOTION from the English translation of NEWTON’S PRINCIPIA, American edition, 1846.

The paragraphs of original material are accompanied by brief comments in color based on present understanding. The heading below links to the original materials.


Laws of Motion

LAW I:  Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve then motions both progressive and circular for a much longer time.

The uniform motion of a body is characterized by constant velocity. This velocity changes only when the body is being pushed around by forces. The greater is the mass of a body, the lesser are the fluctuations in its velocity. This is seen as the uniform motion of the body being maintained by the inertia of matter (see Definition III).


LAW II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

The greater is the push the bigger is the change in the resulting velocity of a body. The magnitude of the push is determined by its force and duration. It is noted that for a push to continue, the body pushing must attempt to move faster than the body being pushed. The overall alteration of motion is proportional to the push.


LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinges upon another and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.

A push (impressed force) requires a contact. Here actions are considered to be reciprocal because they are balanced for the duration of the contact, for example, in the case of a person standing on the floor, or in an elastic collision. But, when the push is continuous over a long duration, it is balanced by the mass (inertia) coming into play through acceleration.


These three laws of Newton, when examined closely, lead to the conclusion that the natural uniform velocity of a body in space shall ultimately depend on its mass or inertia. The higher is the mass the lesser is the velocity and vice versa. You cannot keep increasing the velocity of a body without decreasing its mass (inertia). This is not the same consideration as given by relativity.

This explains why the velocity of light (that has no mass) is many degrees of magnitude higher than the velocity of matter (that has mass).


COROLLARY I: A body by two forces conjoined will describe the diagonal of a parallelogram, in the same time that it would describe the sides, by those forces apart.

When a body is pushed in two different directions simultaneously, it is equivalent to a single push along the diagonal of the parallelogram formed as above by the two pushes M and N. This geometry is based on the uniform velocity due to inertia.


COROLLARY II: And hence is explained the composition of any one direct force AD, out of any two oblique forces AC and CD ; and, on the contrary, the resolution of any one direct force AD into two oblique forces AC and CD : which composition and resolution are abundantly confirmed from mechanics.

Thus, two forces AC and CD may be shown as equivalent to a single force AD. Similarly, a single force AD may be resolved in any two directions AC and CD.


COROLLARY III: The quantity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves.

This is the principle of conservation of momentum of a system. In a collision, motion may transfer from one body to another as velocity. This change in mass is so small that it is ignored. If there is a change in the mass of a body it is in the inertia (quantization) of each particle, and not in the number of particles.


COROLLARY IV: The common centre of gravity of two or more bodies does not alter its state of motion or rest by the actions of the bodies among themselves; and therefore the common centre of gravity of all bodies acting upon each other (excluding outward actions and impediments) is either at rest, or moves uniformly in a right line.

This geometry is based on the rigidity of matter, and on the fact that the effect of a force is inversely proportional to the distance from the point of effect.


COROLLARY V: The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion.

Newton is assuming that the same relative velocities shall be maintained by the objects of a system if the same absolute velocity is added to all of them. This may be so if the velocity added is small. But when that velocity added is high the mass starts to reduce significantly (see the CRITICAL COMMENT under Law III).  Objects of different masses may respond differently, and the relative speeds may not be maintained.


COROLLARY VI: If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had been urged by no such forces.

Again, the objects of different masses may behave differently when the push is very large and/or applied for a long duration. The relative velocities among objects may not be maintained.



The experiments described by Newton in this scholium deal with low velocities only. Hence deviations from these conclusions at high velocities shall not be detectable.


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