force. virtual moment of the force (§ 237), or is the product of the Work of a resolved part of the force in the direction of the displacement into the displacement. From this it appears, that if the motion of the point of application be always perpendicular to the direction in which a force acts, such a force does no work. Thus the mutual normal pressure between a fixed and moving body, as the tension of the cord to which a pendulum bob is attached, or the attraction of the sun on a planet if the planet describe a circle with the sun in the centre, is a case in which no work is done by the force. couple. 240. The work done by a force, or by a couple, upon a body Work of a turning about an axis, is the product of the moment of the force or couple into the angle (in radians, or fraction of a radian) through which the body acted on turns, if the moment remains the same in all positions of the body. If the moment be variable, the statement is only valid for infinitely small displacements, but may be made accurate by employing the proper average moment of the force or of the couple. The proof is obvious. If Q be the moment of the force or couple for a position of the body given by the angle 0, Q(0, − 0.) if Q is constant, or ["2d0=9(0,-0.) where q is the proper average value of Q when variable, is the work done by the couple during the rotation 0 ation of 241. Work done on a body by a force is always shown by a Transformcorresponding increase of vis viva, or kinetic energy, if no other work. forces act on the body which can do work or have work done against them. If work be done against any forces, the increase of kinetic energy is less than in the former case by the amount of work so done. In virtue of this, however, the body possesses an equivalent in the form of Potential Energy (§ 273), if its Potential physical conditions are such that these forces will act equally, and in the same directions, if the motion of the system is reversed. Thus there may be no change of kinetic energy pro energy. Potential duced, and the work done may be wholly stored up as potential energy. Newton's Axiom. energy. Thus a weight requires work to raise it to a height, a spring requires work to bend it, air requires work to compress it, etc.; but a raised weight, a bent spring, compressed air, etc., are stores of energy which can be made use of at pleasure. 242. In what precedes we have given some of Newton's Definitiones nearly in his own words; others have been enunciated in a form more suitable to modern methods; and some terms have been introduced which were invented subsequent to the publication of the Principia. But the Axiomata, sive Leges Motûs, to which we now proceed, are given in Newton's own words; the two centuries which have nearly elapsed since he first gave them have not shown a necessity for any addition or modification. The first two, indeed, were discovered by Galileo, and the third, in some of its many forms, was known to Hooke, Huyghens, Wallis, Wren, and others; before the publication of the Principia. Of late there has been a tendency to split the second law into two, called respectively the second and third, and to ignore the third entirely, though using it directly in every dynamical problem; but all who have done so have been forced indirectly to acknowledge the completeness of Newton's system, by introducing as an axiom what is called D'Alembert's principle, which is really Newton's rejected third law in another form. Newton's own interpretation of his third law directly points out not only D'Alembert's principle, but also the modern principles of Work and Energy. 243. An Axiom is a proposition, the truth of which must be admitted as soon as the terms in which it is expressed are clearly understood. But, as we shall show in our chapter on Experience," physical axioms are axiomatic to those only who have sufficient knowledge of the action of physical causes to enable them to see their truth. Without further remark we shall give Newton's Three Laws; it being remembered that, as the properties of matter might have been such as to render a totally different set of laws axiomatic, these laws must be con sidered as resting on convictions drawn from observation and experiment, not on intuitive perception. first law. 244. LEX I. Corpus omne perseverare in statu suo quiescendi Newton's vel movendi uniformiter in directum, nisi quatenus illud à viribus impressis cogitur statum suum mutare. Every body continues in its state of rest or of uniform motion in a straight line, except in so far as it may be compelled by force to change that state. 245. The meaning of the term Rest, in physical science Rest. is essentially relative. Absolute rest is undefinable. If the universe of matter were finite, its centre of inertia might fairly be considered as absolutely at rest; or it might be imagined to be moving with any uniform velocity in any direction whatever through infinite space. But it is remarkable that the first law of motion enables us (§ 249, below) to explain what may be called directional rest. As will soon be shown, § 267, the plane in which the moment of momentum of the universe (if finite) round its centre of inertia is the greatest, which is clearly determinable from the actual motions at any instant, is fixed in direction in space. 246. We may logically convert the assertion of the first law of motion as to velocity into the following statements :— The times during which any particular body, not compelled by force to alter the speed of its motion, passes through equal spaces, are equal. And, again-Every other body in the universe, not compelled by force to alter the speed of its motion, moves over equal spaces in successive intervals, during which the particular chosen body moves over equal spaces. 247. The first part merely expresses the convention uni- Time. versally adopted for the measurement of Time. The earth, in its rotation about its axis, presents us with a case of motion in which the condition, of not being compelled by force to alter its speed, is more nearly fulfilled than in any other which we can easily or accurately observe. And the numerical measurement of time practically rests on defining equal intervals of time, as times during which the earth turns through equal VOL. I. 16 the law. angles. This is, of course, a mere convention, and not a law of nature; and, as we now see it, is a part of Newton's first law. Examples of 248. The remainder of the law is not a convention, but a great truth of nature, which we may illustrate by referring to small and trivial cases as well as to the grandest phenomena we can conceive. Directional fixedness. A curling-stone, projected along a horizontal surface of ice, travels equal distances, except in so far as it is retarded by friction and by the resistance of the air, in successive intervals of time during which the earth turns through equal angles. The sun moves through equal portions of interstellar space in times during which the earth turns through equal angles, except in so far as the resistance of interstellar matter, and the attraction of other bodies in the universe, alter his speed and that of the earth's rotation. 249. If two material points be projected from one position, A, at the same instant with any velocities in any directions, and each left to move uninfluenced by force, the line joining them will be always parallel to a fixed direction. For the law asserts, as we have seen, that AP: AP' :: AQ:AQ', if P, Q, and again P', Q' are simultaneous positions; and therefore PQ is parallel to P'Q'. Hence if four material points O, P, Q, R are all projected at one instant from one position, OP, OQ, OR The "Inva- are fixed directions of reference ever after. But, practically, the determination of fixed directions in space, § 267, is made to depend upon the rotation of groups of particles exerting forces on each other, and thus involves the Third Law of Motion. riable Plane" of the solar system. 250. The whole law is singularly at variance with the tenets of the ancient philosophers who maintained that circular motion is perfect. The last clause, “nisi quatenus," etc., admirably prepares for the introduction of the second law, by conveying the idea that it is force alone which can produce a change of motion. How, we naturally inquire, does the change of motion produced depend on the magnitude and direction of the force which produces it? And the answer is second law 251. LEX II. Mutationem motus proportionalem esse vi Newton's motrici impressæ, et fieri secundum lineam rectam quâ vis illa imprimitur. Change of motion is proportional to force applied, and takes place in the direction of the straight line in which the force acts. 252. If any force generates motion, a double force will generate double motion, and so on, whether simultaneously or successively, instantaneously, or gradually applied. And this motion, if the body was moving beforehand, is either added to the previous motion if directly conspiring with it; or is subtracted if directly opposed; or is geometrically compounded with it, according to the kinematical principles already explained, if the line of previous motion and the direction of the force are inclined to each other at an angle. (This is a paraphrase of Newton's own comments on the second law.) 253. In Chapter I. we have considered change of velocity, or acceleration, as a purely geometrical element, and have seen how it may be at once inferred from the given initial and final velocities of a body. By the definition of quantity of motion (§ 210), we see that, if we multiply the change of velocity, thus geometrically determined, by the mass of the body, we have the change of motion referred to in Newton's law as the measure of the force which produces it. It is to be particularly noticed, that in this statement there is nothing said about the actual motion of the body before it was acted on by the force: it is only the change of motion that concerns us. Thus the same force will produce precisely the same change of motion in a body, whether the body be at rest, or in motion with any velocity whatever. 254. Again, it is to be noticed that nothing is said as to the body being under the action of one force only; so that we may logically put a part of the second law in the following (apparently) amplified form : When any forces whatever act on a body, then, whether the body be originally at rest or moving with any velocity and in any direction, each force produces in the body the exact change of |