Newton's Space and Time

5. Newton’s absolute and relative motion.

If the earth is affected with an absolute rotation around its axis, centrifugal forces are set up in the earth it assumes an oblate form, the acceleration of gravity is diminished at the equator, the plane of Foucault’s pendulum rotates, and so on.

All these phenomena disappear if the earth is at rest and the other heavenly bodies are affected with absolute motion round it, such that the same relative rotation is produced.

This is the case, if we start ab initio from the idea of absolute space.

Relatively, not considering the unknown and neglected medium of space, the motions of the universe are the same whether we adopt the Ptolemaic or the Copernican mode of view.

It is, accordingly, not permitted us to say how things would be if the earth did not rotate.

We may interpret the one case that is given us, in different ways. If, however, we so interpret it that we come into conflict with experience, our interpretation is simply wrong. The principles of mechanics can, indeed, be so conceived, that even for relative rotations centrifugal forces arise.

Newton’s experiment with the rotating vessel of water shows that the relative rotation of the water:

• with respect to the vessel’s sides produces no noticeable centrifugal forces
• with respect to the earth’s mass and the other celestial bodies produces such forces

No one can say how the experiment would be if the thickness of the vessel’s sides increased to be several leagues thick.

6. The water’s motion is from inertia.

Most terrestrial motions are so short term that it is unnecessary to take into account the earth’s rotation and the changes of its progressive velocity with respect to the celestial bodies.

This is necessary only in projectiles that travel very far in the case of the vibrations of Foucault’s pendulum, and in similar instances.

When now Newton sought to apply the mechanical principles dis- covered since Galileo’s time to the planetary system, he found that, so far as it is possible to form any estimate at all thereof, the planets, irrespectively of dynamic effects, appear to preserve their direction and velocity with respect to bodies of the universe that are very remote and as regards each other apparently fixed, the same as bodies moving on the earth do with re- spect to the fixed objects of the earth.

The comportment of terrestrial bodies with respect to the earth is reducible to the comportment of the earth with respect to the remote heavenly bodies. If we were to assert that we knew more of moving objects than this their last-mentioned, experimentally-given comportment with respect to the celestial bodies, we should render ourselves culpable of a falsity.

A body [at rest] preserves its direction and velocity unchanged.

• This is in reference to the entire universe.

7. Instead of a body K moving in space or a system of coördinates, it moves in relation to the bodies of the universe.

• These relations create a system of coördinates

Bodies very remote from each other, mov- ing with constant direction and velocity with respect to other distant fixed bodies, change their mutual distances proportionately to the time.

We may also say, All very remote bodies-all mutual or other forces ne- glected-alter their mutual distances proportionately to those distances.

Two bodies, which, situated at a short distance from one another, move with constant direction and velocity with respect to other fixed bod- ies, exhibit more complicated relations. If we should regard the two bodies as dependent on one another, and call the distance, t the time, and a a constant dependent on the directions and velocities, the formula would be obtained: der/dt = (1/r) [a2 — (dr/dt)2]. It is manifestly much simpler and clearer to regard the two bodies as independent of each other and to con- sider the constancy of their direction and velocity with respect to other bodies.

Instead of saying, the direction and velocity of a mass μ in space remain constant, we may also employ the expression, the mean acceleration of the mass μ with respect to the masses m, m’, m". . . . at the dis- tances r, r’, qu’’.. is 0, or d2(mr/Zm)/dt2. = 0. The latter expression is equivalent to the former, as soon as we take into consideration a sufficient number of sufficiently distant and sufficiently large masses. The mutual influence of more proximate small masses, which are apparently not concerned about each other, is eliminated of itself. That the constancy of direction and velocity is given by the condition adduced, will be seen at once if we construct through μ as vertex cones that cut out different portions of space, and set up the condition with respect to the masses of these separate portions.

We may put, indeed, for the entire space encompassing μ, d2 (Emr/Zm) /d t2 = 0.

But the equation in this case asserts nothing with respect to the motion of μ, since it holds good for all species of mo- tion where μ is uniformly surrounded by an infinite number of masses.

If two masses μ1, M2 exert on each other a force which is dependent on their distance r, then der/dt2 (μ2+μ2)f(r). But, at the same time, the acceleration of the centre of gravity of the two masses or the mean acceleration of the mass-system with respect to the masses of the universe (by the prin- ciple of reaction) remains = 0; that is to say,

When we reflect that the time-factor that enters into the acceleration is nothing more than a quantity that is the measure of the distances (or angles of rotation) of the bodies of the universe, we see that even in the simplest case, in which apparently we deal with the mutual action of only two masses, the neglecting of the rest of the world is impossible.

Nature does not begin with elements, as we are obliged to begin with them.

It is certainly fortunate for us, that we can, from time to time, turn aside our eyes from the overpowering unity of the All, and allow them to rest on individual details.

But we should not omit, ultimately to complete and correct our views by a thorough con- sideration of the things which for the time being we left out of account.

8. The law of inertia does not come from a special absolute space.

The masses that exert forces on each other as well as those that exert none, stand with respect to acceleration in quite similar relations.

All masses are related to each other.

Accelerations play a prominent part in the relations of the masses.

…

quantities u play a de- terminative rôle. Differ- ences of temperature, of potential function, and so forth, induce the natural processes, which consist in the equalisation of The familiar expressions d2u/dx2, d2u/dy2, d2u/dz2, which are determinative of the character of the equalisation, may be regarded as the measure of the departure of the condition of any point from the mean of the conditions of its environment— to which mean the point tends. The accelerations of masses may be analogously conceived. The great dis- tances between masses that stand in no especial force- relation to one another, change proportionately to each other. If we lay off, therefore, a certain distance p as abscissa, and another as ordinate, we obtain a straight line. (Fig. 143.) Every r-ordinate corresponding to a definite p-value represents, accordingly, the mean of the adjacent ordinates.

If a force-relation exists be- tween the bodies, some value der/dt is determined by it which conformably to the remarks above we may replace by an expression of the form der/dp2. By the force-relation, therefore, a departure of the r-ordinate from the mean of the adjacent ordinates is produced, which would not exist if the supposed force-relation did not obtain. This intimation will suffice here.

9. We have attempted in the foregoing to give the ́law of inertia a different expression from that in ordi- nary use. This expression will, so long as a suffi- cient number of bodies are apparently fixed in space, accomplish the same as the ordinary one.

It is as easily applied, and it encounters the same difficulties. In the one case we are unable to come at an absolute space, in the other a limited number of masses only is within the reach of our knowledge, and the summation indicated can consequently not be fully carried out. It is impossible to say whether the new expression would still represent the true condition of things if the stars were to perform rapid movements among one another. The general experience cannot be constructed from the particular case given us. We must, on the contrary, wait until such an experience presents itself. Perhaps when our physico-astronomical knowledge has been extended, it will be offered somewhere in celestial space, where more violent and complicated motions take place than in our environment. The most impor– tant result of our reflexions is, however, that precisely the apparently simplest mechanical principles are of a very complicated character, that these principles are founded on uncompleted experiences, nay on experiences that never can be fully completed, that practically, indeed, they are suf- ficiently secured, in view of the tolerable stability of our environment, to serve as the foundation of mathematical deduction, but that they can by no means themselves be regarded as mathematically established truths but only as principles that not only admit of constant control by expe- rience but actually require it. This perception is valu- able in that it is propitious to the advancement of science. (Compare Appendix, XX., p. 542.)

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VII. SYNOPTICAL CRITIQUE OF THE NEWTONIAN ENUNCIATIONS

1. Now that we have discussed the details with sufficient particularity, we may pass again under re- view the form and the disposition of the Newtonian enunciations. Newton premises to his work several definitions, following which he gives the laws of mo- tion. We shall take up the former first.

Definition I.

The quantity of any matter is the “measure of it by its density and volume conjointly. This quantity is what I shall understand by the “term mass or body in the discussions to follow. It is “ascertainable from the weight of the body in ques- “tion. For I have found, by pendulum-experiments “of high precision, that the mass of a body is propor- “tional to its weight; as will hereafter be shown.

“Definition II. Quantity of motion is the measure “of it by the velocity and quantity of matter con- ‘jointly.

“Definition III. The resident force [vis insita, i. e. “the inertia] of matter is a power of resisting, by “which every body, so far as in it lies, perseveres in “its state of rest or of uniform motion in a straight line.

Definition IV. An impressed force is any action “upon a body which changes, or tends to change, its “state of rest, or of uniform motion in a straight line.

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2. Definition I is, as has already been set forth, a pseudo-definition. The concept of mass is not made clearer by describing mass as the product of the volume into the density, as density itself denotes simply the mass of unit of volume. The true definition of mass can be deduced only from the dynamical relations of bodies. To Definition II, which simply enunciates a mode of computation, no objection is to be made. Defini- tion (inertia), however, is rendered superfluous by Definitions IV-VIII of force, inertia being included and given in the fact that forces are accelerative. Definition IV defines force as the cause of the accel- eration, or tendency to acceleration, of a body. The latter part of this is justified by the fact that in the cases also in which accelerations cannot take place, other attractions that answer thereto, as the compres- sion and distension etc. of bodies occur. The cause of an acceleration towards a definite centre is defined in Definition v as centripetal force, and is distinguished in VI, VII, and VIII as absolute, accelerative, and mo- tive. It is, we may say, a matter of taste and of form whether we shall embody the explication of the idea of force in one or in several definitions. In point of principle the Newtonian definitions are open to no ob- jections.
3. The Axioms or Laws of Motion then follow, of which Newton enunciates three: “Law I. Every body perseveres in its state of rest “or of uniform motion in a straight line, except in so “far as it is compelled to change that state by im- “pressed forces.” “Law II. Change of motion [i. e. of momentum] is “proportional to the moving force impressed, and takes

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