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VIII; Dynamics


The mechanical theory of natural phenomena consists of a formulation of the idea that all such phenomena can be viewed as essentially consisting in changes in the motions of parts of material systems. A scheme in which under given conditions all these motions can be numerically calculated in accordance with a set of fundamental laws is known as a dynamical scheme. The first postulate of such a scheme is that matter can be regarded as being of such a character that certain aggregations of it or certain parts of such aggregations remain unchanged through all changes of distribution and configuration in space; thus retaining a certain identity. The shape which this idea of conservation of matter through all changes has taken in modern times is formulated in the principle of the conservation of mass. Of the gradual emergence of this principle from less definite attempts to fix the character of the unchanging elements in the mechanical theory I shall give an account in the next lecture. In the present lecture I propose to give an account to some extent critical of the Classical Mechanics which is associated with the names of Galileo and Newton in which the fundamental concepts employed are those of Force Mass Time and Space. The Classical Mechanics as originally conceived is the Mechanics of gross bodies or Molar Mechanics; it is non-atomic in that it does not assume matter to have an atomic or a molecular constitution although as we shall see it requires the assumption that matter is indefinitely divisible into parts. The notions of force and mass which influenced the building of the Classical Mechanics were for the most part realistically and concretely conceived; and the mode in which they were employed was largely directed by the notion of efficient causation. Consequently a considerable amount of alteration of what have been the traditional modes of presenting the theory is requisite in order that it may be stated in the form of a conceptual scheme in which it is made clear what parts of that scheme consist of definitions and how far the postulations made in it have been derived from direct observations of the behaviour of actual bodies in motion under certain conditions.

The ideas as to the motion of heavy bodies which prevailed before the time of Galileo were so confused and conflicting that we may regard the foundation of the Science of Dynamics as due to Galileo (1564–1642). It should however be observed that a scientific treatment of Statics which is concerned with the conditions under which bodies remain at rest had been initiated by Archimedes and possessed some considerable body of doctrine at the time of Galileo. The first problem which Galileo set himself to solve was that of describing the mode in which heavy bodies actually fall; and this independently of any attempt to answer the question why they fall. The first attempt which he made to describe the motion of a falling body consisted of a guess that the velocity attained by a body falling freely from a state of rest is proportional to the height through which it has fallen. This hypothesis Galileo abandoned not because of a failure to verify it experimentally but because he convinced himself that it led to contradiction. The idea that there is anything self-contradictory in such a motion is however erroneous; such motion is conceivable although it is not that of a falling body. Galileo then made the hypothesis that the velocity acquired is proportional to the time of the fall and he correctly deduced as a consequence of this hypothesis that the height fallen would be proportional to the square of the time. In order to verify that this is the actual law of the motion of falling bodies he first experimented on a smooth ball rolling down an inclined plane. As no pendulum clock then existed he measured the time of the motion by weighing the water which flowed through a small orifice at the bottom of a very large vessel full of water during the motion of the ball. In this manner he confirmed his surmise that the height moved by the rolling sphere would be proportional to the square of the time of the motion from rest.
He then made the assumption confirmed by an experiment in which he employed a simple filar pendulum with a heavy ball attached to it that the velocity acquired depends only on the vertical height through which a body has moved from rest. He was then able to connect the acceleration of a freely falling body with that of a body moving along an inclined plane. In this manner Galileo ascertained the law of the fall of a body independently of any theory; thus obtaining a genuine scientific law by observation. A most important new conception introduced into Dynamics by Galileo is that of acceleration the gradient of velocity with respect to time. He perceived that when a body is in such circumstances that it is set in motion or has its motion altered it is the acceleration that characterizes the immediate manner in which these circumstances exhibit themselves. Before Galileo the immediate condition of the production of motion was recognized in pressure on the body due to contact with another body but it was quite unknown that it is acceleration and neither velocity nor position that is determined by the pressure. Thus Galileo's discovery paved the way for reaching the modern conception of force as determining acceleration. Of great importance are Galileo's investigations of the motion of projectiles which led him to the conception that the projectile has two independent motions a horizontal uniform motion and a vertical uniformly accelerated motion. Thus he introduced in this case the principle of compounding motions in accordance with the parallelogram law.
The next great contributor to Dynamical Science after Galileo was Christian Huygens (1629–1695) who invented the pendulum clock. One of his greatest discoveries is that of the existence and magnitude of the acceleration of a point which describes a circle uniformly directed towards the centre of the circle; its so-called centripetal acceleration. This is a case of acceleration in which the magnitude of the velocity remains unchanged the acceleration representing the effect of the change in its direction. Huygens was the first to ascertain the magnitude of the acceleration due to gravity by means of pendulum observations. In connection with his determination of the centre of oscillation of a compound pendulum Huygens was led to a particular case of the principle that work is what determines velocity in fact to a particular case of the modern principle that the change of kinetic energy in a system is equal to the work done upon the system. What was afterwards called the moment of inertia was also introduced by Huygens in this connection.
Apart from Newton's supremely important discovery of the universal law of gravitation but in close connection with that discovery and his deductions from it he laid down in a complete form the essential principles of Dynamics as they have been accepted by succeeding generations although as I have already stated the form in which abstract Dynamics as a conceptual scheme is now presented differs in some important respects from Newton's own formulation in his laws of motion and the accompanying scholia. Although Newton had by no means emancipated himself from the idea that a complete scientific theory must provide an explanation of phenomena in accordance with the law of efficient causation as is shown by his refusal to accept the notion of so-called action at a distance his method of procedure is that of the ascertainment of actual facts which he then employs for the formulation of scientific laws. His view of scientific method is formulated in a set of rules1 for the conduct of natural inquiry (the Regulae Philosophandi).
Rule I. No more causes of natural things are to be admitted than such as truly exist and are sufficient to explain the phenomena of these things.
Rule II. Therefore to natural effects of the same kind we must as far as possible assign the same causes; e.g. to respiration in man and animals; to the descent of stones in Europe and America; to the light of our kitchen fire and of the sun; to the reflection of light on the earth and on the planets.
Rule III. Those qualities of bodies that can be neither increased nor diminished and which are found to belong to all bodies within the reach of our experiments are to be regarded as the universal qualities of all bodies.
If it universally appear by experiments and astronomical observations that all bodies in the vicinity of the earth are heavy with respect to the earth and this in proportion to the quantity of matter which they severally contain that the moon is heavy with respect to the earth in the proportion of its mass and our seas with respect to the moon; and all the planets with respect to one another and the comets also with respect to the sun; we must in conformity with this rule declare that all bodies are heavy with respect to one another.
Rule IV. In experimental physics propositions collected by induction from phenomena are to be regarded either as accurately true or very nearly true notwithstanding any contrary hypotheses till other phenomena occur by which they are made more accurate or are rendered subject to exceptions. This rule must be adhered to that the results of induction may not be annulled by hypotheses.
I have already in an earlier lecture pointed out the defect in Newton's definition of absolute time and urged the view that in a purely conceptual scheme an independent variable which takes as its values the numbers of the arithmetic continuum must be employed; in the application of the scheme to percepts an interval of this continuum must be taken to correspond to the duration of some standard process an interval of public time.
Newton's views concerning space and motion he has stated substantially in the following form1:
Absolute space in its own nature and without regard to anything external always remains similar and unmovable.
Relative space is some movable dimension or measure of absolute space which our senses determine by its position with respect to other bodies and which is commonly taken for immovable space.
Absolute motion is the translation of a body from one absolute place to another absolute place: and relative motion the translation from one relative place to another relative place. And thus we use in common affairs instead of absolute places and motions relative ones; and that without any inconvenience. But in physical disquisitions we should abstract from the senses. For it may be that there is no body really at rest to which the places and motions of others can be referred.
In these statements we can recognize in a somewhat involved form the distinction which we now make between conceptual space that of abstract geometry and physical space the space of perceptual bodies. The conceptual space of ordinary Dynamics that which Newton calls absolute space is a three-fold ordered aggregate in which metrical relations of the Euclidean type are employed. As basis of the system of measurement in this space we postulate the existence of a definite frame usually conceived as a set of coordinate axes. In the conceptual scheme a particle is regarded as being at rest or in motion according as its coordinates remain unaltered or not when the independent time-variable takes up varying values. This is then the conceptual definition of rest and motion.
The frame of reference being once for all fixed conceptual motion may then be regarded as absolute. As to what we are to understand by a particle or a body in this absolute conceptual space I shall speak later. Abstract Dynamics consists of a scheme of rules by which having given certain specifications it is possible to calculate the positions of all the conceptual bodies of a system of such bodies in this purely conceptual space. The whole set of such rules is so devised that these calculated motions may be employed for the description and approximate determination of the actual motions of a system of perceptual bodies in physical space. In order that the conceptual scheme may be applicable in this manner it is necessary to define a mode in which the positions of conceptual bodies in conceptual space are to be made to correspond to positions of actual bodies in physical space.
The only meaning which we can attach to the motion or rest of a body in physical space is that it is in motion or at rest relatively to some other body taken as a standard; in other words motion and rest in physical space are purely relative. In order then to employ the conceptual scheme for the description of actual motions it is necessary to fix some frame in physical space which shall be taken to correspond to the conceptual frame or coordinate axes the existence of which has been postulated. This can only be done by taking some particular body regarded as unchangeable in its shape and dimensions to correspond to a conceptual body at rest and defining the frame as fixed in this body. Or some more complicated process may be employed for fixing upon a frame of reference the basis of which however always rests upon a choice of actual material bodies. The success of the conceptual scheme of abstract Dynamics consists in the fact that it is possible to determine such a frame of reference in physical space that when the positions of the bodies whose motions are under investigation are measured with reference to the frame their actual positions at different instants of public time are found to correspond with a sufficient degree of approximation to the calculated positions of the corresponding conceptual bodies in the conceptual scheme. What frame of reference is taken in any particular case depends upon the particular motions to be investigated and upon the degree of precision that is requisite in the determination of those motions. For the purposes of representing motions of bodies in this room it will often be sufficient to take a frame fixed relatively to the walls of the room say a vertical line and two perpendicular lines fixed on the floor. For more delicate observations this will give an insufficient determination of motions; when for example the fact of the rotation of the earth must be taken into account we take a frame determined by the directions of the so-called fixed stars. For the determination of the motions of the planets we take a frame of reference in the sun fixed relatively to the stars to correspond to the conceptual frame at rest in conceptual space. The fact that such actual motions are sufficiently described in this manner by the corresponding motions in the conceptual scheme of Newtonian abstract Dynamics is the only ground upon which that scheme can be accepted as adequate for its purpose. There is no a priori reason why that scheme may not have to be superseded in whole or in part by some different scheme in case it tails to describe with sufficient degree of approximation any actually observed set of motions. In particular there is no a priori reason for assuming that Newtonian Dynamics is sufficient for the purpose of describing the motions of the sub-molecular parts of bodies as all the direct verifications of the adequacy of the scheme have reference to the motions of molar bodies. Any extension of the scheme to the representation of motions in the microcosmic region in which the motions cannot be directly observed is a hypothesis the value of which must be judged by those deductions from it which are capable of direct verification by actual measurements.
In the Newtonian Dynamics there is embodied a principle that of inertia which is closely related to the Newtonian conception of Force. Newton has stated this principle in one of the definitions which precede his laws of motion and has further stated it in the first of his laws of motion. His definition takes the form that: “The resident force of matter is a power of resisting by which every body so tar as in it lies perseveres in its state of rest or of uniform motion in a straight line.” This definition may be regarded as rendered superfluous by his later definitions of force as they include the notion that all accelerations are dependent upon impressed forces.
Before we examine the precise meaning that can be attached to the principle of inertia and its position in the scheme of abstract Newtonian Dynamics it is advisable to glance briefly at the history of the principle; bound up as it is with the conception of force. The conception of a body persisting indefinitely in uniform rectilineal motion was quite unknown to the ancients. For example Aristotle employs the impossibility of such motion in an argument of redactio ad absurdum. He believed that a body can only be moved by the action of another body which is continually in contact with it. Under ordinary conditions this continuous contact is veiled from our eyes because when we project a body we at the same time impart a certain motion to the air and this continues to act upon the projectile; in a vacuum this would not take place. In accordance with another theory advanced by Hipparchus a body that is set in motion has received from another body an impulse which continues to reside in the projected body after the contact with the other body has ceased. This impulse keeps it in motion in a straight line although that motion is not uniform but gradually diminishes and finally ceases. There was no idea at that time that the velocity of the projectile could maintain itself without action from without. The same conception was expressed by Themistius a commentator on Aristotle who compared the impulse received by the body to the communication of heat to a body which remains in the body for some time whilst the body gradually cools down to its original temperature. Another notion which was commonly held by ancient physicists is that uniform circular motion is a natural kind of movement which persists unchanged when not interfered with.
The idea of the relativity of motion which by later Physicists was connected with the principle of inertia was distinctly conceived by Cardinal Nicolas de Cusa (1401–1464) in the first half of the fifteenth century. De Cusa who tried to demonstrate that the earth can move without our perceiving it uses as an illustration the fact that a boat in rapid motion may appear to be at rest to persons in it who do not see the banks; but he failed to connect the idea of relative motion with the principle of inertia. Although he affirmed the possibility of indefinite motion in a straight line he still accepted the Aristotelian doctrine of natural circular motion. He explained the fact that a smooth ball started in motion on a smooth floor continues in motion by the persistence of the tendency of the ball to rotate in its rolling motion. It is the perfection of rotundity that causes the perpetuity of the motion. The persistence of the motion of the ball is with him persistence of the rotation and not that of the translation.
Copernicus like Aristotle attributed to the celestial bodies a natural circular motion and denied that this gives rise to the appearance of a centrifugal force. On the other hand Kepler believed that the movements of the planets were due to a material emanation from the sun and that each of them would stop dead in its orbit if the sun ceased to act. Benedetti a precursor of Galileo still believed that the impulse communicated to a projectile decreases continually with the time but he regarded the motion of a projectile as compounded of the motion due to the original impulse which started it and of the natural motion due to its weight although he did not understand the cumulative effect of the weight. Even Galileo in his earlier utterances seems not to have rid himself of the idea of the gradual diminution of the impressed impulse. He never ceased to regard the circular motion of the heavenly bodies as a natural motion just as Copernicus and the Greek Philosophers had done but he distinctly affirmed the perpetuity of rectilineal motion in a horizontal direction and appealed for confirmation to the example of a ball rolling on a plane. The principle of inertia although implied in his works was not stated by him in its general form.
The first such statement in a complete form of the conservation of velocity in a straight line is due to Descartes who emphasized the fundamental importance of the principle in the general theory of motion. It is not unlikely that he was influenced by the writings of Galileo which had already been published when he formulated the principle for the first time. He divided the principle into two statements the first referring to a body at rest and the second to a body in motion. Thus he says:
A body when it is at rest has the power of remaining at rest and of resisting everything which could make it change. Similarly when it is in motion it has the power of continuing in motion with the same velocity and in the same direction.
A similar statement of the principle in two parts was given by D'Alembert who attempted to demonstrate it by means of the principle of sufficient reason without any appeal to experience except as regards the mere existence of motion.
Other attempts to give an a priori proof of the principle have been made by Kant and by Maxwell. The latter advanced what has at least the appearance of being a proof1 of the principle by a reductio ad absurdum. He supposes that the movement of a body left to itself might gradually cease in which case it would have a negative acceleration. This would be changed into a positive acceleration if we considered the motion relative to some body to which an appropriate motion was assigned. Maxwell infers that the law has no meaning unless the possibility of defining absolute rest and velocity be admitted and argues that the denial of the law contradicts the only doctrine of time and space which we can form. The same argument might however be applied to prove the impossibility of uniform motion. The fact is that in physical space there is no meaning in the assertion that a body moves uniformly in a straight line unless some material frame is specified with respect to which the motion is measured. However a definite meaning exists when the body is taken to be a conceptual body moving in conceptual space in which all positions are assigned absolutely by numerical specifications. The reluctance which Natural Philosophers have shown sharply to disentangle the conceptual statement of scientific laws and theories from statements relating to percepts has not only obscured the real nature of Science but has introduced much confusion into the formulation of results. It is essential for clarity of meaning in scientific theories to distinguish quite clearly between statements of facts of observation and postulations of the conceptual scheme which is designed to summarize and describe those facts.
An example of the inconvenience I have referred to is exhibited in proposals which have been made by C. Neumann Streintz and others to do away with the difficulties which arise in making statements about the motion of actual bodies in physical space by assuming the existence of some standard body. C. Neumann assumes1 the existence somewhere in physical space of a body called the body Alpha which is completely immovable; and all motions of other bodies are referred to this body Alpha. Such a body is a mere figment of the imagination and can certainly serve no purpose in actual measurements of motion. This notion of a body Alpha occurred to both Newton and Euler but was rejected by them as unsatisfactory. Streintz gives2 the name “fundamental body” to some body which can be regarded as independent of the bodies that surround it and is ascertained by the aid of pendulum experiments not to be in rotation. He then regards the principle of inertia as affirming that when any body is not subject to external influence it describes uniformly a straight line when referred to “fundamental coordinates” fixed in the standard body. It seems preferable in every way to regard the “fundamental body” or “the body Alpha” merely as the postulated ideal frame with reference to which rest and motion in conceptual space are reckoned instead of regarding it as an actual body in physical space. If luminiferous ether could be regarded as an actual substance filling all physical space it would be natural to regard it as affording the means of describing the motion of actual bodies; a body at rest would then be one which was at rest relatively to the ether. But in accordance with the view which I have maintained in these lectures the ether is if indeed it continues to retain any place in the Science of the future a concept introduced for the purposes of scientific theory and not a percept. Moreover as I shall explain in a later lecture all attempts to detect the supposed velocity of a body relatively to a material ether have proved fruitless.
The question arises as to what is the true nature of the principle of inertia. The view that it is an á priori truth is untenable unless the principle of efficient causation or that of sufficient reason employed; and the employment of such principles is not necessary for the purposes of Natural Science. It is hardly possible to maintain that the principled of inertia is a direct description of observed facts. We are not acquainted with any actual bodies which satisfy the conditions of validity of the principle. No actual body is or can be so isolated from other bodies as to be removed from conditions dependent or those other bodies; and as we have already seen when an actual body is said to be at rest that means relatively to some other material system. Uniform motion in a straight line relatively to say the earth would neither be uniform nor rectilinear relatively say to the sun. The observation of a ball moving on a smooth floor which is usually appealed to as an experimental proof of the principle does not really establish it as an empirical law or even as an approximate fact of observation partly on the ground I have just mentioned but also because the ball having weight and being in contact with the floor and the surrounding air does not satisfy the condition of being under no forces. That apart from friction the weight and the pressure of the ground taken together do not affect the horizontal motion can only be inferred when other parts of the dynamical scheme are taken into account. Thus the law of inertia is not a direct result of facts of observation; and this probably accounts to some extent at least for the length of the period during which it was not discovered or was not generally accepted. It follows then that the law of inertia cannot be taken in isolation but must necessarily be regarded as forming apart of a complete conceptual scheme of Dynamics. The exact position of the principle can only be determined when the whole of Newtonian Dynamics is taken into account including the doctrine of the relation of motion with force. We shall see that the principle appears in this conceptual scheme in the form of a definition. On the impossibility of regarding the principle of inertia either as an a priori truth or as embodying a direct result of observation Poincaré has remarked1:
If it be said that the velocity of a body cannot change if there is no reason for it to change may we not just as legitimately maintain that the position of a body cannot change or that the curvature of its path cannot change without the agency of an external cause? Is then the principle of inertia which is not an a priori truth an experimental fact? Have there ever been experiments on bodies acted on by no forces? and if so how did we know that no forces were acting?
One of the main points in which Newton's theory of Dynamics contains an advance beyond the conceptions of Galileo or Huygens is in the generalization of the conception of force. For the origin of the notion of force we must go back to the feeling of exertion which we have when our muscles are employed in changing the position of actual bodies. The notion was extended to cover the case of the interaction of two bodies in contact with one another; the force then acting on one of the bodies being regarded as the efficient cause of the motion produced in it. The explanation of motion as due to contact action appeared to explain the phenomenon in the sense to which I have referred in an earlier lecture that it reduced it to the familiar case of contact of the human body with another object. That a precise examination of what occurs in case of contact between two bodies must be undertaken in order to explain the effects of pressure was not recognized until modern times. We now know that such an examination involves conceptions as to the relations between the smallest parts of the bodies in contact and that this reduction to the corpuscular domain gives rise to problems as difficult as the original one which led to the whole investigation. The futility of the attempt to indicate the point at which efficient causation is to be found is exhibited by the endless regress to which we are led when we attempt to account for the interactions of the smallest parts of the bodies. If we refuse to regard a body as consisting of corpuscles of any kind but take it to be absolutely homogeneous we are unable to begin to understand the nature of contact action.
Involved in the discovery by Newton of the law of gravitation is the extension of the notion of force to such cases as the so-called forces of attraction between the sun and the planets. Newton himself under the influence of the ancient idea that all forces must be due to contact was as I have already observed never satisfied with the bare fact of the existence of these forces but thought that they were inconceivable without some mechanism such as a homogeneous ether which should condition the forces by ultimate reduction to contact action. However the existence of forces regarded as due to the action of one body on another body at a distance came to be accepted as part of the apparatus of Dynamics and was accepted by Newton himself with the reservation just indicated.
Newton's mode of measuring a force depends upon the notion of the mass of a body as a measurable quantity. The formulation by Newton of the definition of mass as the quantity of matter in a body measured by the product of its volume and density has the defect that it is circular since the conception of density is meaningless unless that of mass has been previously defined. Newton ascertained by means of experiments that a body has a property the mass distinct from its weight whereby the quantity of motion of the body is determined. He also showed that the mass of a body may in one and the same place be measured by its weight and that the ratio of the weight to the mass is independent of the chemical composition of bodies. There is a lack of clarity in the deductions from these experiments owing to the want of a satisfactory definition of the mass or quantity of matter which a body contains.
Newton's first two laws of motion contain a statement of the relation of an impressed force to the change of motion of the body which is really equivalent to a definition that he had previously given; and this we now express in the form that the force is proportional to the acceleration it produces in its own direction. As according to this definition there is no acceleration without force to produce it the first law is really contained in the second. Newton's third law of motion that the forces between two bodies which influence one another are equal and in opposite directions only attains a precise meaning when the notion of mass as having numerical measure has been made precise by means of a definition. If we consider the action and reaction between two bodies as equal the ratio of their relative masses may be taken to be the inverse ratio of their accelerations in opposite directions. This only amounts to the definition of the mass-ratios of the two bodies. If now we consider a third body we can then define the mass-ratio of any pair of them. It was pointed out by Mach that observation is required to establish the fact that to each one of the bodies a number may be assigned such that for each pair of the bodies their relative masses are given by the ratio of the two numbers assigned to the bodies of the pair. Thus a regular and consistent mode of assignment of mass-numbers can be set up such that the law of action and reaction between any pair of bodies will hold good. That the mass-number assigned to a body is independent of change in its form chemical and thermal conditions is a fact which must be verified by observation.
The rule that the force acting on a body is equal to the product of its mass and its acceleration depends upon the possibility of measuring three magnitudes the force the mass and the acceleration. I have pointed out that mass is not capable of measurement independently of the notion of the equality of two forces. The impossibility of an independent measurement of force has been trenchantly signalized1 by Poincaré as follows:
When we say that force is the cause of motion we are talking metaphysics; and this definition if we had to be content with it would be absolutely fruitless would lead to absolutely nothing. For a definition to be of any use it must tell us how to measure force; and that is quite sufficient for it is by no means necessary to tell what force is in itself nor whether it is the cause or the effect of motion. We must therefore define what is meant by the inequality of two forces. When are two forces equal? We are told that it is when they give the same acceleration to the same mass or when acting in opposite directions they are in equilibrium. This definition is a sham. A force applied to a body cannot be uncoupled and applied to another body as an engine is uncoupled from one train and coupled to another. It is therefore impossible to say what acceleration such a force applied to such a body would give to another body if it were applied to it. It is impossible to tell how two forces which are not acting in exactly opposite directions would behave if they were acting in opposite directions. It is this definition which we try to materialise as it were when we measure a force with a dynamometer or with a balance.
The impossibility of measuring independently of one another the three magnitudes force mass and acceleration as they are employed in Newton's scheme and of then verifying experimentally that force is proportional to the product of mass and acceleration leads to the conclusion that force must be defined as the product of mass into acceleration and thus does not appear in the scheme as an independent conception. Thus the first two laws of Newton regarded as part of a conceptual scheme consist in reality of a definition of force as the product of the mass-number of a body into its acceleration in a particular direction which direction is taken to define the direction of the force. The fact of observation which leads to the formulation of Newton's third law of motion is that a consistent set of mass-numbers can be assigned to a set of bodies so that any two of the bodies have towards each other accelerations inversely proportional to the mass-numbers of the two bodies.
Newton gave a deduction of the principle known as the parallelogram of forces from his second law of motion. But in order that this deduction may be valid the assumption is required that the accelerations produced in a body by two other bodies are independent of each other; so that the acceleration which B produces in A is independent of the fact that C is also producing an acceleration in A. That this is the case in actual bodies is not self evident and can only be established by experience; in the conceptual scheme the assumption must appear as a postulate. If it be assumed that all the forces between bodies are central forces depending in magnitude only on the distances between the bodies Newton's deduction will be justified; but if we do not make this assumption we have no means of resolving the acceleration of A into components one of which we regard as due to B and the other as due to C; and in that case we have no means of assigning definite mass-numbers to the three bodies and therefore the principle of equality of action and reaction would no longer be valid.
It would thus appear that the validity of the Newtonian scheme is dependent on the assumption that the forces between any two bodies are central. But this assumption has really been implied in the statement of the laws of motion. For the acceleration of a body has no definite meaning unless the body be either of negligible dimensions or unless the acceleration be taken to mean that of some one particular point in the body; for in general a body can move not only translationally but also rotationally and thus different parts of a body may have different accelerations. In order that Newton's laws of motion may have a precise meaning it must be assumed either that the bodies referred to in them are considered as masses concentrated at points or as bodies of non-negligible size which are equivalent to such concentrated masses at their centroids; the forces between any pair of such bodies being along the straight lines joining their centroids. This was realized with sufficient degree of approximation in the cases of motion of bodies of the solar system to which Newton applied his Dynamics.
The extension of Newton's Dynamics to the general case of bodies of various shapes acting and reacting on one another can only be made by a process of integration in which the bodies are divided up into a large number of very small parts so that very approximately any two of such parts can be regarded as two points at which the masses of the respective parts are concentrated; and it is assumed that the forces which act between two such parts whether they belong or not to one and the same body are central that is along the line joining these parts. By means of the procedure of passing to a limit that is of integration the dynamical scheme can be built up in a form applicable to a system of bodies. This scheme takes the form of a set of differential equations called the equations of motion of the system. In the conceptual scheme the bodies consist of defined portions of geometrical space which are movable in that space retaining in the case of a rigid body their dimensions unaltered; the mass of each such body appears in the equations of motion only as a coefficient to which in any special case a numerical value is assigned. The position for each value of the time variable of any ideal rigid body is denoted by a set of numbers which represent coordinates relatively to the frame of reference with respect to which all positions are referred. The motion of a point is denoted by a functional relation between its coordinates and the time-variable. As we have seen the principle of inertia and the conception of force appear in the scheme as definitions and the masses of bodies or of parts of bodies appear as coefficients to which numerical values may be assigned. The hypothesis of central forces between elementary parts of bodies of magnitudes dependent only on the distance is also an essential part of the scheme. The equations of motion of the conceptual bodies suffice to determine their motions and the results obtained suffice for the approximate determination of the motions of the perceptual bodies in an approximately isolated system provided a correlation between measurements in physical space and in geometrical space can be set up.
To do this it is requisite to assign for each special problem a frame of measurement in physical space which may be taken to correspond with the absolute conceptual frame by means of which all positions in the geometrical space are measured. This frame of reference must be fixed relatively to some actual body and when it consists of rectangular axes it may be spoken of as a Newtonian set of axes. The utility of the conceptual dynamical scheme for its purpose of describing the motions with sufficient accuracy of an approximately isolated material system depends upon the verifiable fact that it is possible to determine with sufficient degree of approximation for the purpose on hand a Newtonian set of axes1 suitable to the particular system. If such a Newtonian set of axes has been determined in any particular case any other set of axes parallel to the former will also be a Newtonian set provided its origin moves with uniform velocity relatively to the origin of the first set. This principle is subject to the test of experience but it is an essential principle of Newtonian Dynamics. But a set of axes in rotation with regard to Newtonian set will not be itself a Newtonian set unless additional forces including the so-called centrifugal forces are introduced into the system.
In this connection it has frequently been maintained that in physical space the rotation of a body is absolute whilst its motion of translation is relative; and thus that absolute directions unlike absolute positions can be defined. Newton's celebrated experiment with the water in a bucket is frequently appealed to in support of this contention. As long as the bucket is at rest the water in it has a flat surface but if the bucket be made to rotate about its vertical axis a motion of rotation is gradually set up in the water when its surface is no longer plane but has the form of a concave surface of revolution. It is then suggested that there is an absolute difference between axes fixed relatively to the bucket when it was at rest and axes fixed relatively to it when it is in motion. The only relevant fact would however appear to be that it is possible to determine with sufficient approximation in any concrete case the directions of a Newtonian set of axes; and other axes in rotation relatively to these do not form a Newtonian set. It can however be shown that the directions of the axes of a Newtonian set have no valid claim to be regarded as absolute.
In the first place a Newtonian set of axes determined for a particular isolated system need no longer be a Newtonian set for an enlarged system. For example in this room a vertical line and two horizontal lines fixed relatively to the walls would form a Newtonian set of axes for the purpose of describing the motions of ordinary bodies in this room; and in particular the rotation of Newton's bucket would mean rotation relatively to these axes. But for the purpose of describing the motion of Foucault's pendulum these axes would no longer form a Newtonian set. For that purpose we should have to take the axis of the earth and two other axes fixed in direction relatively to the stars. Again if we had to take account of the slow precessional motion of the earth's axis these last would no longer form a set of Newtonian axes. However far we may proceed in this manner to fix Newtonian axes which shall be sufficient for the motions of ever more extensive or complicated systems of bodies we are unable to assert that the axes obtained will necessarily be a Newtonian set for the purpose of measuring the motions of every system of bodies whatever which we may at any time have to consider. Moreover the approximate determination of a Newtonian set of axes is insufficient to determine even approximately absolute directions because the smallest error in the determination will be of a cumulative character involving the existence of a large and no longer negligible error after a sufficiently long time. The rotation of a set of axes however slow that rotation may be will in a sufficiently long time produce a large deviation of the axes from their original position. There is thus no warrant for the assertion that it is possible to assign directions in physical space which may for every possible purpose be regarded as Newtonian axes. On this ground and also because the Newtonian system of Dynamics although the simplest system for the descriptive purposes for which it was devised is not the only possible system that might be employed and may not even suffice for all purposes it is not correct to assert that absolute directions in physical space can be determined by any means at our disposal; nor are we compelled to conceive absolute directions in physical space as existent.
To regard the earth as rotating round its axis is convenient for the purposes of Descriptive Astronomy and it is also convenient for the purposes of the dynamical description of the motions of bodies in the solar system. But however convenient is the assertion that the earth rotates it still remains a convention although the opposite assertion would vastly complicate both the astronomical and the dynamical schemes which would then have to be employed. In connection with the new general theory of relativity both spatial relations and dynamical theory have to be considered from a point of view which necessitates a rediscussion of such matters as the introduction of centrifugal and other forces when non-Newtonian sets of axes are employed in connection with the motions of a material system.