Article 36. Kinematics and kinetics
We have hitherto been considering the motion of a system in its purely geometrical aspect. We have shown how to study and describe the motion of such a system, however arbitrary, without taking into account any of the conditions of motion which arise from the mutual action between the bodies.
The theory of motion treated in this way is called kinematics. When the mutual action between bodies is taken into account, the science of motion is called kinetics, and when special attention is paid to force as the cause of motion, it is called dynamics.

Article 37. Mutual action between two bodies—stress
The mutual action between two portions of matter receives different names according to the aspect under which it is studied, and this aspect depends on the extent of the material system which forms the subject of our attention.
If we take into account the whole phenomenon of the action between the two portions of matter, we call it stress. This stress, according to the mode in which it acts, may be described as attraction, repulsion, tension, pressure, shearing stress, torsion, etc.

Article 38. External force
But if, as in Article 2, we confine our attention to one of the portions of matter, we see, as it were, only one side of the transaction—namely, that which affects the portion of matter under our consideration—and we call this aspect of the phenomenon, with reference to its effect, an external force acting on that portion of matter, and with reference to its cause we call it the action of the other portion of matter. The opposite aspect of the stress is called the reaction on the other portion of matter.

Article 39. Different aspects of the same phenomenon
In commercial affairs the same transaction between two parties is called buying when we consider one party, selling when we consider the other, and trade when we take both parties into consideration.
The accountant who examines the records of the transaction finds that the two parties have entered it on opposite sides of their respective ledgers, and in comparing the books he must in every case bear in mind in whose interest each book is made up.
For similar reasons in dynamical investigations we must always remember which of the two bodies we are dealing with, so that we may state the forces in the interest of that body and not set down any of the forces on the wrong side of the account.

Article 40. Newton’s laws of motion
External or ‘‘impressed’’ force considered with reference to its effect—namely, the alteration of the motions of bodies—is completely defined and described in Newton’s three laws of motion.
The first law tells us under what conditions there is no external force.
The second shows us how to measure the force when it exists.
The third compares the two aspects of the action between two bodies, as it affects the one body or the other.

Article 41. The first law or motion
Law I. Every body perseveres in its state of rest or of moving uniformly in a straight line, except insofar as it is made to change that state by external forces.
The experimental argument for the truth of this law is that in every case in which we find an alteration of the state of motion of a body, we can trace this alteration to some action between that body and another, that is to say, to an external force. The existence of this action is indicated by its effect on the other body when the motion of that body can be observed. Thus the motion of a cannonball is retarded, but this arises from an action between the projectile and the air which surrounds it, whereby the ball experiences a force in the direction opposite to its relative motion, while the air, pushed forward by an equal force, is itself set in motion and constitutes what is called the wind of the cannonball.
But our conviction of the truth of this law may be greatly strengthened by considering what is involved in a denial of it. Given a body in motion. At a given instant let it be left to itself and not acted on by any force. What will happen? According to Newton’s law it will persevere in moving uniformly in a straight line, that is, its velocity will remain constant both in direction and magnitude.
If the velocity does not remain constant let us suppose it to vary. The change of velocity, as we saw in Article 31, must have a definite direction and magnitude. By the maxim of Article 19, this variation must be the same whatever be the time or place of the experiment. The direction of the change of motion must therefore be determined either by the direction of the motion itself or by some direction fixed in the body.
Let us, in the first place, suppose the law to be that the velocity diminishes at a certain rate, which for the sake of the argument we may suppose so slow that by no experiments on moving bodies could we have detected the diminution of velocity in hundreds of years.
The velocity referred to in this hypothetical law can only be the velocity referred to a point absolutely at rest. For if it is a relative velocity its direction as well as its magnitude depends on the velocity of the point of reference.
If, when referred to a certain point, the body appears to be moving northward with diminishing velocity, we have only to refer it to another point moving northward with a uniform velocity greater than that of the body, and it will appear to be moving southward with increasing velocity.
Hence the hypothetical law is without meaning, unless we admit the possibility of defining absolute rest and absolute velocity.
Even if we admit this as a possibility, the hypothetical law, if found to be true, might be interpreted, not as a contradiction of Newton’s law, but as evidence of the resisting action of some medium in space.
To take another case. Suppose the law to be that a body, not acted on by any force, ceases at once to move. This is not only contradicted by experience, but it leads to a definition of absolute rest as the state which a body assumes as soon as it is freed from the action of external forces.
It may thus be shown that the denial of Newton’s law is in contradiction to the only system of consistent doctrine about space and time which the human mind has been able to form.

Article 42. On the equilibrium of forces
If a body moves with constant velocity in a straight line, the external forces, if any, which act on it, balance each other or are in equilibrium.
Thus if a carriage in a railway train moves with constant velocity in a straight line, the external forces which act on it—such as the traction of the carriage in front of it pulling it forward, the drag of that behind it, the friction of the rails, the resistance of the air acting backward, the weight of the carriage acting downward, and the pressure of the rails acting upward—must exactly balance each other.
Bodies at rest with respect to the surface of the earth are really in motion, and their motion is not constant nor in a straight line. Hence the forces which act on them are not exactly balanced. The apparent weight of bodies is estimated by the upward force required to keep them at rest relatively to the earth. The apparent weight is therefore rather less than the attraction of the earth, and makes a smaller angle with the axis of the earth, so that the combined effect of the supporting force and the earth’s attraction is a force perpendicular to the earth’s axis just sufficient to cause the body to keep to the circular path which it must describe if resting on the earth.

Article 43. Definition of equal times
The first law of motion, by stating under what circumstances the velocity of a moving body remains constant, supplies us with a method of defining equal intervals of time. Let the material system consist of two bodies which do not act on one another, and which are not acted on by any body external to the system. If one of these bodies is in motion with respect to the other, the relative velocity will, by the first law of motion, be constant and in a straight line.
Hence intervals of time are equal when the relative displacements during those intervals are equal.
This might at first sight appear to be nothing more than a definition of what we mean by equal intervals of time, an expression which we have not hitherto defined at all.
But if we suppose another moving system of two bodies to exist, each of which is not acted upon by any body whatever, this second system will give us an independent method of comparing intervals of time.
The statement that equal intervals of time are those during which equal displacements occur in any such system is therefore equivalent to the assertion that the comparison of intervals of time leads to the same result whether we use the first system of two bodies or the second system as our timepiece.
We thus see the theoretical possibility of comparing intervals of time however distant, though it is hardly necessary to remark that the method cannot be put in practice in the neighbourhood of the earth, or any other large mass of gravitating matter.

Article 44. The second law of motion
Law II. Change of motion is proportional to the impressed force and takes place in the direction in which the force is impressed.
By motion Newton means what in modern scientific language is called momentum, in which the quantity of matter moved is taken into account as well as the rate at which it travels.
By impressed force he means what is now called impulse, in which the time during which the force acts is taken into account as well as the intensity of the force.

Article 45. Definition of equal masses and of equal forces
An exposition of the law therefore involves a definition of equal quantities of matter and of equal forces.
We shall assume that it is possible to cause the force with which one body acts on another to be of the same intensity on different occasions.
If we admit the permanency of the properties of bodies this can be done. We know that a thread of caoutchouc when stretched beyond a certain length exerts a tension which increases the more the thread is elongated. On account of this property the thread is said to be elastic. When the same thread is drawn out to the same length it will, if its properties remain constant, exert the same tension. Now let one end of the thread be fastened to a body, M, not acted on by any other force than the tension of the thread, and let the other end be held in the hand and pulled in a constant direction with a force just sufficient to elongate the thread to a given length. The force acting on the body will then be of a given intensity, F. The body will acquire velocity, and at the end of a unit of time this velocity will have a certain value, V.
If the same string be fastened to another body, N, and pulled as in the former case, so that the elongation is the same as before, the force acting on the body will be the same, and if the velocity communicated to N in a unit of time is also the same, namely V, then we say of the two bodies M and N that they consist of equal quantities of matter, or, in modern language, they are equal in mass. In this way, by the use of an elastic string, we might adjust the masses of a number of bodies so as to be each equal to a standard unit of mass, such as a pound avoirdupois, which is the standard of mass in Britain.

Article 46. Measurement of mass
The scientific value of the dynamical method of comparing quantities of matter is best seen by comparing it with other methods in actual use.
As long as we have to do with bodies of exactly the same kind, there is no difficulty in understanding how the quantity of matter is to be measured. If equal quantities of the substance produce equal effects of any kind, we may employ these effects as measures of the quantity of the substance.
For instance, if we are dealing with sulfuric acid of uniform strength, we may estimate the quantity of a given portion of it in several different ways. We may weigh it, we may pour it into a graduated vessel and so measure its volume, or we may ascertain how much of a standard solution of potash it will neutralize.
We might use the same methods to estimate a quantity of nitric acid if we were dealing only with nitric acid; but if we wished to compare a quantity of nitric acid with a quantity of sulfuric acid we should obtain different results by weighing, by measuring, and by testing with an alkaline solution.
Of these three methods, that of weighing depends on the attraction between the acid and the earth, that of measuring depends on the volume which the acid occupies, and that of titration depends on its power of combining with potash.
In abstract dynamics, however, matter is considered under no other aspect than as that which can have its motion changed by the application of force. Hence any two bodies are of equal mass if equal forces applied to these bodies produce, in equal times, equal changes of velocity. This is the only definition of equal masses which can be admitted in dynamics, and it is applicable to all material bodies, whatever they may be made of.
It is an observed fact that bodies of equal mass, placed in the same position relative to the earth, are attracted equally toward the earth, whatever they are made of; but this is not a doctrine of abstract dynamics, founded on axiomatic principles, but a fact discovered by observation and verified by the careful experiments of Newton, on the times of oscillation of hollow wooden balls suspended by strings of the same length and containing gold, silver, lead, glass, sand, common salt, wood, water, and wheat.
The fact, however, that in the same geographical position the weights of equal masses are equal is so well established that no other mode of comparing masses than that of comparing their weights is ever made use of, either in commerce or in science, except in researches undertaken for the special purpose of determining in absolute measure the weight of unit of mass at different parts of the earth’s surface. The method employed in these researches is essentially the same as that of Newton, namely, by measuring the length of a pendulum which swings seconds.
The unit of mass in this country is defined by the Act of Parliament (18 and 19 Vict. c. 72, July 30, 1855) to be a piece of platinum marked ‘‘P. S., 1844, 1 lb.’’ deposited in the office of the Exchequer, which ‘‘shall be and be denominated the Imperial Standard Pound Avoirdupois.’’ One seven-thousandth part of this pound is a grain. The French standard of mass is the ‘‘Kilogramme des Archives,’’ made of platinum by Borda. Professor Miller finds the kilogram equal to 15432.34874 grains.

Article 47. Numerical measurement of force
The unit of force is that force which, acting on the unit of mass for the unit of time, generates a unit of velocity.
Thus the weight of a gram—that is to say, the force which causes it to fall—may be ascertained by letting it fall freely. At the end of one second its velocity will be about 981 centimetres per second if the experiment be in Britain. Hence the weight of a gram is represented by the number 981 if the centimetre, the gram, and the second are taken as the fundamental units.
It is sometimes convenient to compare forces with the weight of a body and to speak of a force of so many pounds weight or gram weight. This is called gravitation measure. We must remember, however, that though a pound or a gram is the same all over the world, the weight of a pound or a gram is greater in high latitudes than near the equator, and therefore a measurement of force in gravitation measure is of no scientific value unless it is stated in what part of the world the measurement was made.
If, as in Britain, the units of length, mass, and time are one foot, one pound, and one second, the unit of force is that which, in one second, would communicate to one pound a velocity of one foot per second. This unit of force is called a poundal. In the French metric system the units are one centimetre, one gram, and one second. The force which in one second would communicate to one gram a velocity of one centimetre per second is called a dyne.
Since the foot is 30.4797 centimetres and the pound is 453.59 grams, the poundal is 13825.38 dynes.

Article 48. Simultaneous action of forces on a body
Now let a unit of force act for a unit of time upon a unit of mass. The velocity of the mass will be changed, and the total acceleration will be unity in the direction of the force.
The magnitude and direction of this total acceleration will be the same whether the body is originally at rest or in motion. For the expression ‘‘at rest’’ has no scientific meaning, and the expression ‘‘in motion,’’ if it refers to relative motion, may mean anything, and if it refers to absolute motion can only refer to some medium fixed in space. To discover the existence of a medium, and to determine our velocity with respect to it by observation on the motion of bodies, is a legitimate scientific inquiry, but supposing all this done we should have discovered, not an error in the laws of motion, but a new fact in science.
Hence the effect of a given force on a body does not depend on the motion of that body.
Neither is it affected by the simultaneous action of other forces on the body. For the effect of these forces on the body is only to produce motion in the body, and this does not affect the acceleration produced by the first force.
Hence we arrive at the following form of the law. When any number of forces act on a body, the acceleration due to each force is the same in direction and magnitude as if the others had not been in action.
When a force, constant in direction and magnitude, acts on a body, the total acceleration is proportional to the interval of time during which the force acts.
For if the force produces a certain total acceleration in a given interval of time, it will produce an equal total acceleration in the next because the effect of the force does not depend upon the velocity which the body has when the force acts on it. Hence in every equal interval of time there will be an equal change of the velocity, and the total change of velocity from the beginning of the motion will be proportional to the time of action of the force.
The total acceleration in a given time is proportional to the force.
For if several equal forces act in the same direction on the same body in the same direction, each produces its effect independently of the others. Hence the total acceleration is proportional to the number of the equal forces.

Article 49. On impulse
The total effect of a force in communicating velocity to a body is therefore proportional to the force and to the time during which it acts conjointly.
The product of the time of action of a force into its intensity if it is constant, or its mean intensity if it is variable, is called the impulse of the force.
There are certain cases in which a force acts for so short a time that it is difficult to estimate either its intensity or the time during which it acts. But it is comparatively easy to measure the effect of the force in altering the motion of the body on which it acts, which, as we have seen, depends on the impulse.
The word impulse was originally used to denote the effect of a force of short duration, such as that of a hammer striking a nail. There is no essential difference, however, between this case and any other case of the action of force. We shall therefore use the word impulse as above defined, without restricting it to cases in which the action is of an exceptionally transient character.

Article 50. Relation between force and mass
If a force acts on a unit of mass for a certain interval of time, the impulse, as we have seen, is measured by the velocity generated.
If a number of equal forces act in the same direction, each on a unit of mass, the different masses will all move in the same manner and may be joined together into one body without altering the phenomenon. The velocity of the whole body is equal to that produced by one of the forces acting on a unit of mass.
Hence the force required to produce a given change of velocity in a given time is proportional to the number of units of mass of which the body consists.

Article 51. On momentum
The numerical value of the momentum of a body is the product of the number of units of mass in the body into the number of units of velocity with which it is moving.
The momentum of any body is thus measured in terms of the momentum of a unit of mass moving with a unit of velocity, which is taken as the unit of momentum.
The direction of the momentum is the same as that of the velocity, and as the velocity can only be estimated with respect to some point of reference, so the particular value of the momentum depends on the point of reference which we assume. The momentum of the moon, for example, will be very different according as we take the earth or the sun for the point of reference.

Article 52. Statement of the second law of motion in terms of impulse and momentum
The change of momentum of a body is numerically equal to the impulse which produces it and is in the same direction.

Article 53. Addition of forces
If any number of forces act simultaneously on a body, each force produces an acceleration proportional to its own magnitude (Art. 48). Hence if in the diagram of accelerations (Art. 34) we draw from any origin a line representing in direction and magnitude the acceleration due to one of the forces, and from the end of this line another representing the acceleration due to another force, and so on, drawing lines for each of the forces taken in any order, then the line drawn from the origin to the extremity of the last of the lines will represent the acceleration due to the combined action of all the forces.
Since in this diagram lines which represent the accelerations are in the same proportion as the forces to which these accelerations are due, we may consider the lines as representing these forces themselves. The diagram, thus understood, may be called a diagram of forces, and the line from the origin to the extremity of the series represents the resultant force.
An important case is that in which the set of lines representing the forces terminate at the origin so as to form a closed figure. In this case there is no resultant force, and no acceleration. The effects of the forces are exactly balanced, and the case is one of equilibrium. The discussion of cases of equilibrium forms the subject of the science of statics.
It is manifest that since the system of forces is exactly balanced, and is equivalent to no force at all, the forces will also be balanced if they act in the same way on any other material system, whatever be the mass of that system. This is the reason why the consideration of mass does not enter into statical investigations.

Article 54. The third law of motion
Law III. Reaction is always equal and opposite to action, that is to say, the actions of two bodies upon each other are always equal and in opposite directions.
When the bodies between which the action takes place are not acted on by any other force, the changes in their respective momenta produced by the action are equal and in opposite directions.
The changes in the velocities of the two bodies are also in opposite directions but not equal, except in the case of equal masses. In other cases the changes of velocity are in the inverse ratio of the masses.

Article 55. Action and reaction are the partial aspects of a stress.
We have already (Art. 37) used the word stress to denote the mutual action between two portions of matter. This word was borrowed from common language and invested with a precise scientific meaning by the late Professor Rankine, to whom we are indebted for several other valuable scientific terms.
As soon as we have formed for ourselves the idea of a stress, such as the tension of a rope or the pressure between two bodies, and have recognized its double aspect as it affects the two portions of matter between which it acts, the third law of motion is seen to be equivalent to the statement that all force is of the nature of stress, that stress exists only between two portions of matter, and that its effects on these portions of matter (measured by the momentum generated in a given time) are equal and opposite.
The stress is measured numerically by the force exerted on either of the two portions of matter. It is distinguished as a tension when the force acting on either portion is toward the other, and as a pressure when the force acting on either portion is away from the other.
When the force is inclined to the surface which separates the two portions of matter the stress cannot be distinguished by any term in ordinary language but must be defined by technical mathematical terms.
When a tension is exerted between two bodies by the medium of a string, the stress, properly speaking, is between any two parts into which the string may be supposed to be divided by an imaginary section or transverse interface. If, however, we neglect the weight of the string, each portion of the string is in equilibrium under the action of the tensions at its extremities, so that the tensions at any two transverse interfaces of the string must be the same. For this reason we often speak of the tension of the string as a whole, without specifying any particular section of it, and also the tension between the two bodies, without considering the nature of the string through which the tension is exerted.

Article 56. Attraction and repulsion
There are other cases in which two bodies at a distance appear mutually to act on each other, though we are not able to detect any intermediate body, like the string in the former example, through which the action takes place. For instance, two magnets or two electrified bodies appear to act on each other when placed at considerable distances apart, and the motions of the heavenly bodies are observed to be affected in a manner which depends on their relative position.
This mutual action between distant bodies is called attraction when it tends to bring them nearer, and repulsion when it tends to separate them.
In all cases, however, the action and reaction between the bodies are equal and opposite.

Article 57. The third law true of action at a distance
The fact that a magnet draws iron toward it was noticed by the ancients, but no attention was paid to the force with which the iron attracts the magnet. Newton, however, by placing the magnet in one vessel and the iron in another, and floating both vessels in water so as to touch each other, showed experimentally that as neither vessel was able to propel the other along with itself through the water, the attraction of the iron on the magnet must be equal and opposite to that of the magnet on the iron, both being equal to the pressure between the two vessels.
Having given this experimental illustration, Newton goes on to point out the consequence of denying the truth of this law. For instance, if the attraction of any part of the earth, say a mountain, upon the remainder of the earth were greater or less than that of the remainder of the earth upon the mountain, there would be a residual force, acting upon the system of the earth and the mountain as a whole, which would cause it to move off, with an ever-increasing velocity, through infinite space.

Article 58. Newton’s proof not experimental
This is contrary to the first law of motion, which asserts that a body does not change its state of motion unless acted on by external force. It cannot be affirmed to be contrary to experience, for the effect of an inequality between the attraction of the earth on the mountain and the mountain on the earth would be the same as that of a force equal to the difference of these attractions acting in the direction of the line joining the centre of the earth with the mountain.
If the mountain were at the equator, the earth would be made to rotate about an axis parallel to the axis about which it would otherwise rotate, but not passing exactly through the centre of the earth’s mass.
If the mountain were at one of the poles, the constant force parallel to the earth’s axis would cause the orbit of the earth about the sun to be slightly shifted to the north or south of a plane passing through the centre of the sun’s mass.
If the mountain were at any other part of the earth’s surface, its effect would be partly of the one kind and partly of the other.
Neither of these effects, unless they were very large, could be detected by direct astronomical observations, and the indirect method of detecting small forces, by their effect in slowly altering the elements of a planet’s orbit, presupposes that the law of gravitation is known to be true. To prove the laws of motion by the law of gravitation would be an inversion of scientific order. We might as well prove the law of addition of numbers by the differential calculus.
We cannot, therefore, regard Newton’s statement as an appeal to experience and observation, but rather as a deduction of the third law of motion from the first.