Lex III. Actioni contrariam semper et aequalem esse reactionem; sive corporum duorum actiones in se mutuo tsemper esse aequales et in partes contrarias dirigi.

To every action there is always an equal and contrary reaction; or the mutual actions of any two bodies are always equal and oppositely directed.

§ 2. In 1863 Thomson and Tait (upon whose Treatise on Natural Philosophy much of what follows is based) called attention to the fact that, as regards Lex III., Newton gives in a scholium a second sense in which the words may be interpreted. In the first sense the action and reaction are mere forces, in the second they are the rates at which forces do work. Hence, and for another reason which will appear later, the word “activity” has been introduced as the English equivalent of the word actio in Newton’s second sense. Here is the passage :—

Si aestimetur agentis actio ex ejus vi et velocitate con-junctim; et similiter resistentis reactio aestimetur con-junctim ex ejus partium singularum velocitatibus et viribus resistendi ab earum attritione, cohaesione, pondere, et acceleratione oriundis; erunt actio et reactio, in omni instrumentorum usu, sibi invicem semper aequales.

If the activity of an agent be measured by its amount and its velocity conjointly; and if, similarly, the counter-activity of the resistance be measured by the velocities of its several parts and their several amounts conjointly, ιvhether these arise from friction, molecular forces, ιveight, or acceleration; .—activity and counter-activity, in all combinations of machines, will be equal and opposite.

This may be looked upon as a Fourth Law. But, in strict logic, the First Law is superfluous, because its consequences are all implied (by negation) in the statement of the Second. (See § 8 below.) Hence there are, virtually, only three laws, so far as Newton’s system is concerned.

§ 3. These laws are to be considered as deductions from observation and experiment, and in no sense as having an α priori foundation. Their proof, so far as rigorous proof is attainable in physical matters, is commonly looked on as being furnished in the most conclusive form by observational astronomy. The Nautical Almanac, published usually about four years in advance, contains the predicted places of the sun, moon, and principal planets from day to day, in some cases from hour to hour, throughout the year. The predictions are entirely based upon the laws of motion (along with the law of gravitation), and could not possibly be accurate unless these laws are true. So thoroughly satisfactory has hitherto been the coincidence between prediction and observation that, when a deviation occurs, no one dreams of a defect in the principles of the reasoning. On the contrary, such deviations are utilized for the purpose of correcting our knowledge of the “elements” of the orbits of the moon and planets, or our estimates of the masses of these bodies; and, as in the brilliant investigations of Adams and Leverrier, they sometimes enable us to discover the existence and even assign the position of a planet never before seen.

§ 4. It is not clear in what order, or by whom, these laws were first discovered. Galileo was undoubtedly acquainted with the first two; and Huygens, Wren, Hooke, and others were acquainted with the Third Law in some of its many applications. But they were first systematized and, as we have seen, extended in a most important manner by Newton. Though they were sadly disfigured in Britain during the fifty years which elapsed after the revival of mathematics in the early part of this century, they have of late been restored to the form in which Newton gave them. This readoption of Newton’s simple but comprehensive system has of itself aided in no small degree the recent rapid advance of science.

One peculiarity of Newton’s language must be noticed here, though very briefly, as we will return to the subject towards the end of the article. A force is said to “compel” a change of state in a body; bodies are said mutually to “act” on one another, &c. Such language is, of course, in its literal acceptation, of an anthropomorphic character; but, if one thinks of the habitual use even in scientific books of such expressions as “the sun rises,” “the wind blows,” &c., it cannot be construed into an assertion that force has real objective existence.

Comments on the Lares of Motion.

§ 5. Law I. First of all this law tells us what happens to a piece of matter which is left to itself, i.e., not acted on by forces. It preserves its “state,” whether of rest or of uniform motion in a straight line. This property (which, as we shall presently show, § 7, is considerably extended by Newton himself) is commonly called the “inertia” of matter, in virtue of which it is incapable of varying in any way its state of rest or motion. It may be the sport of forces for any length of time, but so soon as they cease to act it remains in the state in which it was left until they recommence their action on it. Hence, whenever we find the state of a piece of matter changing, we conclude that it is under the action of a force or forces. Thus, for the present, we have the definition of “force” as part of this First Law :—

Force is whatever changes the state of rest or uniform motion of a body.

When a body, originally at rest, begins to move, we conclude that force is acting on it. And when a moving body is seen to change either the speed or the direction of its motion, we conclude that this is due to force.

§ 6. But there is much more than this, even in the First Law. What is “rest”? The answer must be that the term is relative. Absolute rest and absolute motion are terms to which we find it impossible to assign a meaning. Maxwell has well said (in his Matter and Motion)’.—

“All our knowledge, both of time and place, is essentially relative. When a man has acquired the habit of putting words together, without troubling himself to form the thoughts which ought to correspond to them, it is easy for him to frame an antithesis between this relative knowledge and a so-called absolute knowledge, and to point out our ignorance of the absolute position of a point as an instance of the limitation of our faculties. Any one, however, who will try to imagine the state of a mind conscious of knowing the absolute position of a point will ever after be content with our relative knowledge.”

As will be seen later, the First Law gives us also a physical definition of “time,” and physical modes of measuring it.

§ 7. Newton’s own comment on this law is as follows :—

Projectilia perseverant in motibus suis, nisi quatenus a resistentia aeris retardantur, et vi gravitatis impelluntur deorsum. Trochus, cujus partes cohaerendo perpetuo retrahunt sese a motibus rectilineis, non cessat rotari, nisi quatenus ab aere retardatur. Majora autem planetarum et cometarum corpora motus suos et progressivos et circulares, in spatiis minus resistentibus factos, conservant diutius.

It is particularly worthy of notice that we have here the undisturbed rotation of a body about an axis introduced as another of those “states” in which it will continue, in virtue of the First Law, until force acts to compel it to change that state. Also it is to be noticed that Newton adduces a hoop, whose axis is fixed in direction both in the body and in space, as an example of this new form of state maintained in virtue of inertia. Later, it will be seen that the same thing is true of a body free in space and rotating about the principal axis of greatest or of least moment of inertia through its centre of mass.

§ 8. Law II. What Newton designates by the word motus is, as he has clearly pointed out, the same as is expressed by quantitas motus, that for which we now usually employ the term “momentum.” Its numerical value depends not only on the rate of motion, but also on the amount of matter, or “mass,” of the moving body, and is directly proportional to either of these when the other is unaltered. But it is regarded by Newton as having direction as well as magnitude. It is, in fact, what in the language of quaternions is called a “vector.” The change of such a quantity may be either in numerical magnitude, or in direction, alone, or simultaneously in both. We now see what this Second Law enables us to do. For (a) Given the mass of a body, the force acting on it, and the time during which it acts, we can calculate the change of motion. This is the direct problem of dynamics of a particle.

(6) Given the mass, and the change of velocity, we can calculate the magnitude and direction of the force acting. This is the inverse problem.

(c) We can compare, and so measure, forces by the changes of motion they produce in one and the same body.

(d) We can compare the masses of different bodies by finding what changes of velocity one and the same force produces in them.

(e) We can find the one force which is equivalent, in its action, to any given set of forces. For, however many changes of motion may be produced by the separate forces, they must obviously be capable of being compounded into a single change, and we can calculate what force would produce that.

§ 9. Hitherto, we have spoken of the motion of a body,— thus implying (except, of course, in the case of Newton’s hoop or that of the earth) that all its parts are moving in exactly the same way. From this point of view every body, however large, may be treated as if it were a single particle. But when the parts of a body have different velocities, as when a rigid body is rotating, or as when a non-rigid body is suffering a change of form, the-question becomes much more complex. We cannot at this stage enter into a full explanation, but will take a couple of very simple cases to »show the nature of the new difficulties, and thence the necessity for an additional law.

Suppose a bullet to be thrown in any direction. If we know with what force the earth attracts it, the calculation of the path it will pursue depends on the Second Law, which gives all the necessary preliminary information. But let two bullets be tied together by a string: we know by trial that each moves, in general, in a manner very different from that in which it would move if free. The path of each is now, usually, a tortuous curve, while its free path would be plane. It is no longer subject to gravity alone but also to what is called the “tension” of the string. If we knew the amount of this tension on either of the bullets and its direction, we could calculate, by the help of the Second Law alone, all the circumstances of the motion of that bullet. But how are we to find this tension Ί Is it even the same for each bullet? This, if answered in the affirmative, would simplify matters considerably, but we should still require to know the amount and direction of the tension. It is clear that, without a further axiom, we cannot advance to a solution of the question.

§ 10. Law HI. Furnished with this, in addition to our previous information, we can attack the question with more hope. We see by this law that, whatever force be exerted by the string on one of the bullets, an equal and opposite force, which must therefore be in the direction of the string, is exerted on the other. Still, the magnitude of these equal forces remains to be found. But the string in no way interferes with the motion of either bullet unless it is tight, i.e., unless the distance between the bullets is equal to the length of the string. Hence, whenever the unknown force comes into play, at the same time there comes in a geometrical relation of relative position between the two bullets. This supplies the additional equation necessary for the determination of the new unknown quantity.

§ 11. As an additional illustration, suppose the string to be made of india-rubber. The Third Law tells us that the tensions it exerts on the bullets are still equal and opposite. But we no longer have the geometrical condition we had before. We have, however, what is quite sufficient, a knowledge of how the tension of the string depends on its length. Thus the tension can be calculated from the relative position of the bullets.

§ 12. Scholium to Law III. On this we will, for the present, remark only that it furnishes us with the means of studying directly the transference of energy from one body or system to another. Experiment, however, was required to complete the application of this part of Newton’s systematic treatment of the subject. What was wanted, and how it has been obtainéd, will be treated of later. The first words of the scholium, however, claim for Newton the discovery of the clause we have extracted from it. For they run thus :—Hactenus principia tradidi a mathematicis recepta, et experientia multiplici confirmata.

§ 13. What has now been said enables us to see the order in which the fundamental ideas should be taken up, so that the necessities of each should be provided for before its turn comes. An indispensable preliminary is the study of motion in the abstract, i.e., without any reference to what is moving. This is demanded in order that we may be able to apply the Second Law. The science of pure motion, without reference to matter or force, is an extension of geometry by the introduction of the idea of time and the consequent idea of velocity. Ampère suggested for it the term Cinématique, or, as we shall write it, Kinematics. We include under it all changes of form and grouping which can occur in geometrical figures or in groups of points.

We shall then be prepared to deal with the action of force on a single particle of matter, or on a body which may be treated as if it were a mere particle. Thus we have the Dynamics of a Particle. This, again, splits into two heads, Statics and Kinetics of a Particle. But all this requires the Second Law only. When we have two or more connected particles, or two particles attracting one another or impinging on one another, the Third Law is required. Next in order of simplicity come the Statics and Kinetics of a Rigid Solid. Then we have to deal with bodies whose form, <kc., are altered by forces—flexible bodies, elastic solids, fluids, <fcc. Finally, we must briefly consider the general principles, such as “conservation of energy,” “least action,” <fcc., which are deducible by proper mathematical methods from Newton’s Laws, and of which some at least, if we could more clearly realize their intrinsic nature, would probably be found to express even more simply than do Newton’s Laws the true fundamental principles of abstract dynamics.

We will not restrict ourselves to one uniform course in the application of mathematical methods. Rather, as considerations of space require to be attended to, we will vary our methods from one part of the subject to another, so as to exhibit, each at least once, all the more usual processes. And we will endeavour to make the large-type portions of the article, in which only the most elementary mathematics will be introduced, a self-contained treatise which may be read by students of very moderate mathematical knowledge.

KINEMATICS.

Position and the Means of Assigning it.

§ 14. Motion (or displacement) consists simply in “change of position.” Hence, to describe motion, we must have the means of assigning position. This is, of course, a question of Geometry {q.vf. See also Quaternions.

From these articles it appears that the position of one free point with reference to another (all these space relations are relative, as we have already said) depends on three numbers, of which one at least must involve the unit of length. In Cartesian rectangular coordinates, we denote these by x, y, z, which indicate respectively the distance of the point from each of three planes at right angles to each other, and all passing through the origin (or reference point). From another point of view they may be called “degrees of freedom.” When the value of one is assigned, say by x=a, the point is said to have lost one degree of freedom, or to have had imposed upon it one “degree of constraint.” It must now lie in a plane parallel to the first of the reference planes, and at a distance a from it. When a second degree of constraint is applied, say by y=b, another degree of freedom is lost. The point’s position is limited to lie in a second plane in a given position at right angles to the first. It must therefore lie somewhere on the straight line which consists of the series of points common to the two planes. A third degree of constraint z=c takes away its one remaining degree of freedom; and its position is now definitely assigned as the single point of intersection of three given planes.

§ 15. But constraint may be applied in other ways. Thus if we assign the condition x 2 + y 2 + z 2 = a 2 t we deprive the point of one degree of freedom by compelling it to remain at a distance a from the origin. It is now limited to the surface of a sphere, but its latitude and longitude on that sphere may be any whatever. Here again the imposition of one degree of constraint has taken away one degree of freedom.

§ 16 In general, one degree of constraint may be expressed as fx, y, *)=f∙

This, when ξ has an assigned value, is the equation of a definite surface on which the point must lie. Three such conditions determine the position of the point, and may therefore be looked upon as introducing ξ, η, ζ, another set of coordinates, which may be used in place of x, y, z. The number of such systems is, of course, unlimited; but it is often possible to choose one in which the conditions of a problem are much more simply expressed than they were when expressed in x, y, z. The whole question belongs to what is called “change of variables.” To give an elementary instance of its use,—suppose we take the ordinary simple pendulum, a pellet supported by a fine thread or wire, and oscillating in a vertical plane. If the origin be placed at the point of suspension, and the axis of z be vertical, we have two conditions :—

x 2 + y i +...