Kepler’s third law of planetary motion follows readily from a
consideration of the character of the acceleration of a point
moving uniformly in a circle. Newton tells us that
Gravitation.
this agreement led him to adopt the law of the inverse
square of the distance about 1665–1666, before
Huygens’s results as to circular motion had been published. At
the same time he thought of the possibility of terrestrial gravity
extending to the moon, and made a calculation with regard to it.
Some years later he succeeded in showing that Kepler’s elliptic
orbit for planetary motion agreed with the assumed law of attraction;
he also completed the co-ordination with terrestrial gravity
by his investigation of the attractions of homogeneous spherical
bodies. Finally, he made substantial progress with more exact
calculations of the motions of the solar system, especially for the
case of the moon. The work of translating the law of gravitation
into the form of astronomical tables, and the comparison of these
with observations, has been in progress ever since. The discovery
of Neptune (1846), due to the influence of this planet on
the motion of Uranus, may be mentioned as its most dramatic
achievement. The verification is sufficiently exact to establish
the law of gravitation, as providing a statement of the motions
of the bodies composing the solar system which is correct to a
high degree of accuracy. In the meantime some confirmation
of the law has been obtained from terrestrial experiments, and
observations of double stars tend to indicate for it a wider if not
universal range. It should be noticed that the verification was
begun without any data as to the masses of the celestial bodies,
these being selected and adjusted to fit the observations.
The case of electro-magnetic forces between two conductors carrying electric currents affords an example of a statement of motion in terms of force of a highly artificial kind. It can only be contrived by means of complicated mathematical analysis. In this connexion a statement in terms of force is apt to be displaced by more direct and more comprehensive methods, and the attention of physicists is directed to the intervention of the ether. The study of such cases suggests that the statement in terms of force of the relations between the motions of bodies may be only a provisional one, which, though it may summarize the effect of the actual connexions between them sufficiently for some practical purposes, is not to be regarded as representing them completely. There are indications of this having been Newton’s own view.
The Newtonian base deserves some further consideration. It is defined by the property that relative to it all accelerations of particles correspond to forces. This test involves only changes of velocity, and so does not distinguish between two bases, each of which moves relatively to the other with uniform velocity without rotation. The establishment Newtonian base. of a true Newtonian base presumes knowledge of the motions of all bodies. But practically we are always dealing with limited systems, so any actual determination must always be regarded as to some extent provisional. In the treatment of the relative motions of a limited system, we may use a confessedly provisional base, though it may be necessary to introduce corrections, either exact or approximate, to take account either of the existence of bodies outside the system, or of the rotation of the base employed relative to a more correct one. Such corrections may be made by the device of applying additional unpaired, or what we may call external, forces to particles of the system. These are needed only so far as they introduce differences of accelerations of the several particles. The earth, which is commonly employed as a base for terrestrial motions, is not a very close approximation to being a Newtonian base. Differences of acceleration due to the attractions of the sun and moon are not important for terrestrial systems on a small scale, and can usually be ignored, but their effect (in combination with the rotation of the earth) is very apparent in the case of the ocean tides. A more considerable defect is due to the earth having a diurnal rotation relative to a Newtonian base, and this is never wholly ignored. Take a base attached to the centre of the earth, but without this diurnal rotation. A small body hanging by a string, at rest relatively to the earth, moves relatively to this base uniformly in a circle; that is to say, with constant acceleration directed towards the earth’s axis. What is done is to divide the resultant force due to gravitation into two components, one of which corresponds to this acceleration, while the other one is what is called the “weight” of the body. Weight is in fact not purely a combination of forces, in the sense in which that term is defined in connexion with the laws of motion, but corresponds to the Galileo acceleration with which the body would begin to move relatively to the earth if the string were cut. Another way of stating the same thing is to say that we introduce, as a correction for the earth’s rotation, a force called “centrifugal force,” which combined with gravitation gives the weight of the body. It is not, however, a true force in the sense of corresponding to any mutual relation between two portions of matter. The effect of centrifugal force at the equator is to make the weight of a body there about 35% less than the value it would have if due to gravitation alone. This represents about two-thirds of the total variation of Galileo’s acceleration between the equator and the poles, the balance being due to the ellipticity of the figure of the earth. In the case of a body moving relatively to the earth, the introduction of centrifugal force only partially corrects the effect of the earth’s rotation. Newton called attention to the fact that a falling body moves in a curve, diverging slightly from the plumb-line vertical. The divergence in a fall of 100 ft. in the latitude of Greenwich is about 111 in. Foucault’s pendulum is another example of motion relative to the earth which exhibits the fact that the earth is not a Newtonian base.
For the study of the relative motions of the solar system, a provisional base established for that system by itself, bodies outside it being disregarded, is a very good one. No correction for any defect in it has been found necessary; moreover, no rotation of the base relative to the directions of the stars without proper motion has been detected. This is not inconsistent with the law of gravitation, for such estimates as have been made of planetary perturbations due to stars give results which are insignificant in comparison with quantities at present measurable.
For the measurement of motion it must be presumed that we have a method of measuring time. The question of the standard to be employed for the scientific measurement of time accordingly demands attention. A definition of the measurement dependent on dynamical theory has been a characteristic of the subject as presented by some writers, Measurement of Time. and may possibly be justifiable; but it is neither necessary nor in accordance with the historical development of science. Galileo measured time for the purpose of his experiments by the flow of water through a small hole under approximately constant conditions, which was of course a very old method. He had, however, some years before, when he was a medical student, noticed the apparent regularity of successive swings of a pendulum, and devised an instrument for measuring, by means of a pendulum, such short periods of time as sufficed for testing the pulse of a patient. The use of the pendulum clock in its present form appears to date from the construction of such a clock by Huygens in 1657. Newton dealt with the question at the beginning of the Principia, distinguishing what he called “absolute time” from such measures of time as would be afforded by any particular examples of motion; but he did not give any clear definition. The selection of a standard may be regarded as a matter of arbitrary choice; that is to say, it would be possible to use any continuous time-measurer, and to adapt all scientific results to it. It is of the utmost importance, however, to make, if possible, such a choice of a standard as shall render it unnecessary to date all results which have any relation to time. Such a choice is practically made. It can be put into the form of a definition by saying that two periods of time are equal in which two physical operations, of whatever character, take place, which are identical in all respects except as regards lapse of time. The validity of this definition depends on the assumption that operations of different kinds all agree in giving the same measure of time, such allowances as experience dictates being made for changing conditions. This assumption has successfully stood all
