R F Muirhead's Laws of Motion

Robert Franklin Muirhead studied at the University of Glasgow before matriculating at St Catharine's College, Cambridge in 1881. There he studied the mathematical tripos, was nineteenth wrangler in 1884, was classed Division I in Part III, in 1885, and was awarded a Smith's Prize in 1886 for his essay on Newton's Laws of Motion. We give a version of his interesting essay below.

The Laws of Motion. By Robert Franklin Muirhead, B.A., of St Catharine's College, Cambridge.

Communicated by Professor James Thomson; being the Essay to which the second Smith's Prize was awarded in 1886.

Preface.

The aim of this Essay is to state in the clearest manner possible the best existing conception of dynamical science. The writer believes that the statement of dynamical principles here given is to be found implicitly in He reasonings of the best modern masters of the science, but that it has never hitherto been stated explicitly. The general statement indeed has sometimes been made that the proof of a hypothesis or theory is its agreement with the facts, or that the whole Principia is the proof of the Laws of Motion. But I have pointed out in detail that the very conceptions and definitions of Dynamics are unintelligible when taken singly. I have endeavoured to free the science of Dynamics from survivals from its childhood, in the shape of extra-kinetic definitions of dynamical concepts, and A priori assumptions.

The Laws of Motion.

In view of the enormous development to which the science of Dynamics has attained in modern times, of the simplicity of its fundamental conceptions, and of the unquestioned validity oi its processes and results, it may appear somewhat strange that much difficulty has been found-in stating its principles in a satisfactory form.

In the preface to the second edition of Tait and Steele's Dynamics of a Particle we read (referring to the chapter on the Laws of Motion): -

These five pages, faulty and even erroneous as I have since seen them to be, cost me almost as much labour and thought as the utterly disproportionate remainder of my contributions to the volume; and I cannot but ascribe this result in part, at least, to the vicious system of the present day, which ignores Newton's Third Law, etc.

And when we read Clerk Maxwell's notice of the 2nd edition of Thomson and Tait's treatise in Nature, we feel that the reform introduced by Thomson and Tait, in "returning to Newton", still leaves something to be desired. This feeling is strengthened when we learn from the late Prof Clifford, that "no mathematician can attach any meaning to the language about force, mass, inertia, etc. used in current text-books of Mechanics".

It will then be worth while to clear up the logic of the science, and, if possible, to state the laws of motion in a form that shall be free from all ambiguity and confusion.

Let us cast a brief and partial glance over the history of the development of dynamical first principles.

Though one region of the science of Dynamics, namely Statics, was cultivated by the ancients, it was left for Galileo Galilei to become the pioneer of dynamical science in its full extent.

Before Galileo, the idea of force as something measurable was attained to. The causes tending to disturb rest were perceived to have a common kind of effect, so that for the purposes of Statics they could be represented by the tension of cords produced by suspending from them weights of determinate magnitude. Galileo paved the way for the introduction of the kinetic idea of force, i.e. that of the cause of the acceleration of the motion of bodies. It is noteworthy, however, that he approached the subject from a kinematical standpoint. In his Dialogues, he treats of the science of "Local Motion", not of the science of Force; and in his investigations on the motion of Projectiles in that work, his aim is to describe the properties of their motion, not to speculate on causes.

Another stage was reached when Newton published the Principia. The Definitions and Axioms therein propounded include all the principles underlying the modern science of Dynamics. Subsequent progress has been either in the direction of mathematical development or application to special problems, or in attempts to improve the form of statement.

Let us now inquire whether Newton's scheme of Definitions and Axioms is satisfactory.

We are struck at once by the fact (noticed by many writers) that the First Law of Motion is previously stated or implied in the Definition of Inertia. This, however, may be passed over as a mere awkwardness of arrangement.

Another defect which has been pointed out by several writers, is the absence of any definition of equal times, which renders the expression "uniformiter" in Law I. perfectly indefinite.

Of course the law implies that all bodies un-acted on by force pass through spaces in any interval of time whatever, which are in the same proportion, so that taking any one such body as chronometer, the First Law of Newton may be affirmed of all the rest. We may, however, object to a form of statement which does not directly state, but implies the physical fact.

Again, "uniform rectilinear motion" has no meaning unless with reference to some base of measurement. And the Law is not true except with reference to bases of a certain type. For instance, the "fixed stars describe not straight lines, but circles, taking the Earth as base of measurement".

Newton's own statement is that the Laws of Motion are to be understood with reference to absolute position and absolute time.

The only explanation given of absolute time, is that in itself and of its own nature, without reference to anything else, it flows uniformly.

In explanation of the expressions "absolute motion" and "absolute position," we have the statement that "Absolute and relative motion and rest are distinguished from one another by their properties, causes, and effects. It is a property of rest that bodies truly at rest are at rest among themselves, but true rest cannot be defined by the relative positions of bodies we observe ... The causes by which true and relative motion are distinguished from one another are the forces impressed on the bodies to produce motion. True motion cannot change except by forces impressed.

The effects by which absolute motion is distinguished from relative are the centrifugal forces of rotation. For merely relative rotation these forces are zero; in true rotation they exist in greater or less degree.

Thereafter comes the well-known experiment of the rotating vessel of water.

Now the first criterion helps us only in a negative way, by enabling us to deny the attribution of true rest to both of two systems when they are moving relatively to each other.

The second criterion involves reasoning in a circle. Force is defined as that which produces change of motion; hence to define unchanged or uniform motion as that which takes place when no force acts does not carry us beyond the previous definition, and is nugatory.

The third criterion, taken along with the first, implies a physical fact, viz. that when two bodies severally show no centrifugal force, they have no rotation relative to one another.

Consider now Law II. It amounts merely to a definition of force, specifying how it is to be measured.

This has been recognised by several writers. Some, however, have expanded it into the further assertion that when two forces act simultaneously on a body, each produces its own effect independently of the other, in accelerating the body's motion. But such a statement is entirely nugatory if we keep by the kinetic definition of force. It is then simply an identical proposition like "A is A," as will be seen by substituting in the statement "acceleration of mass" for "force."

We now perceive that even the residuum of meaning which remained after our criticism of Law I. and the statements regarding Absolute Motion seems to disappear. For we were supposed to recognise a body absolutely at rest by the absence of centrifugal force. But force is recognisable only by its accelerative effect, while the acceleration must be reckoned relative to a body absolutely at rest, which rest, again, we cannot recognise until we know absolute motions. We are thus reasoning in a circle.

Law III. This law at first sight undoubtedly seems to express an experimental fact. We may therefore be surprised to find that Newton deduces one case of it (viz. that of two mutually attracting bodies) from Law I. (see Scholium to the Axiomata).

This seeming paradox arises from the fact that in this Scholium Newton makes Law I. apply to a body or system of finite size, and not necessarily without rotation. This assumes that there is some one point (centre of Inertia) whose motion may be taken to represent that of the system, which implies that the 3rd Law is true so far as the parts of such a system are concerned. Now it seems difficult to draw a valid distinction between such a system and any mass-system whatever; in fact it seems quite as legitimate to assume that every mass-system has a centre of Inertia.

But if this assumption were made, then clearly the first Law could be deduced from the third in all its generality, and vice versa.

We see that in this respect again Newton's arrangement is defective. We find that the experimental fact is not stated directly, but implied in the assumption of the existence of a mass-centre. In fact, strictly read, Newton's Definitions and Axioms abound in logical circles, nugatory statements, and illusory definitions; and what real meaning they imply is not at all explicit.

The need for the removal of many obscurities which pertain to the science of Dynamics as set forth in the Principia of Newton, and in the writings of his successors, has been clearly perceived by Professor James Thomson. In his paper on the Law of Inertia, etc. he propounds the following Law of Inertia:-

For any set of bodies acted on each by any force, a Reference-Frame and a Reference Dial-traveller are kinematically possible, such that relatively to them conjointly the motion of the mass-centre of each body undergoes change simultaneously with any infinitely short element of the dial-traveller progress, or with any element during which the force on the body does not alter in direction nor in magnitude, which change is proportional to the intensity of the force acting on that body, and to the simultaneous progress of the dial-traveller, and is made in the direction of the force.

For explanations of the terms used I refer to the paper itself. At the end of this paper we have the assertion:

The Law of Inertia here enunciated sets forth all the truth which is either explicitly stated, or is suggested by the First and Second Laws in Sir Isaac Newton's arrangement.

Professor Thomson's Law is doubtless, so far as order and logic are concerned, an immense advance on the Newtonian arrangement. Let us inquire whether it can be accepted as absolutely satisfactory.

How are we to measure the "forces" referred to? If kinematically, then we are again involved in a logical circle, as may be seen by substituting in the Law, for the words "force acting on that body" the words "rate of change of motion of that body", and for the words "direction of force" the words "direction of change of motion". And we cannot entertain any other measure of force, for reasons which will be adduced later on.

Again, Prof Thomson, by not restricting his statement to infinitesimal particles, has to assume the existence of mass-centres. How is a mass-centre to be defined? We shall give reasons later for rejecting any but a kinetic definition of mass and mass-centre. But it is impossible to arrive at a kinetic definition when we start by assuming a knowledge of the measurement of mass in the Fundamental Law of Motion, as is done by Professor Thomson.

While noting therefore that Professor Thomson has adopted the right method of defining chronometry and "true rests" we cannot accept his Law as a satisfactory statement of the fundamental principle of Dynamical science.

Let us endeavour to frame, after the manner of Professor Thomson, a statement which shall be satisfactory. Taking the definitions of dial-traveller and reference-frame, as given in the paper referred to, let us proceed thus:-

Let a material system be conceived divided into an infinite number of particles whose greatest linear dimensions are all infinitesimal. To each particle let us attribute a certain value called its provisional-mass. Let us adopt a reference-frame and dial-traveller. Let the acceleration of any particle multiplied into its provisional-mass be called the apparent-force on the particle. Then it is possible so to choose the provisional-masses, the dial-traveller, and the reference-frame, so that the provisional-masses and the apparent-forces shall, within the limits of error of observation, have relations expressible by the laws of physical science, i.e. the law of the Indestructibility of Matter, the law of Equality of Action and Reaction the law of Universal Gravitation, the laws of electric, magnetic, elastic, and capillary action, etc., etc. Such a system being chosen, the provisional-masses in it are masses and the apparent-forces, forces. The dial-traveller indicates "absolute time," and the reference-frame is absolutely without rotation or acceleration.

We have thus kinetic definitions of force, mass, absolute time-measurement, and absolute rest so far as that is possible.

It is evident kinematically that any other reference-frame which has no rotation or acceleration relatively to one chosen as above would lead to exactly the same results; and that this would not be the case if any reference-frame not fulfilling this condition were chosen.

The above statement includes all in the First and Second Laws of Newton that can conceivably be tested by experiment or observation.

We observe that Newton's Third Law appears classed along with other laws of physics, and along with that of the Indestructibility of Matter, which must be assumed as a preliminary to the ordinary statement of Dynamical Laws before the measurement of matter has received its definition. In our statement of the fundamental principle of Dynamics, neither of these Laws is assumed, and it could be modified so as to be equally definite and intelligible were they untrue.

By dealing with infinitesimal particles, we have avoided the necessity of assuming à priori the existence of mass-centres; for on the supposition that the angular motion of no element is infinite (or, more generally, that there is no finite relative acceleration or velocity between the parts of any particle), the motion of any point of a particle might be taken to represent the motion of that particle.

To define the expression force acting on a body, used in Dynamics, we would require simply to define the centre of mass by the usual analytical equations of the type

\bar{x} =\Large \frac{\sum {mx}}{\sum m}

where the summation extends over all the particles of the body, and then to define the mass of the body by

\Sigma m

, and the force on the body as that acting on its whole mass supposed concentrated at its centre of mass.

What would be the meaning of "a force acting on a body at a certain point"? This expression is appropriate only to rigid bodies, or at least to such as retain their shape unaltered. while under consideration. The meaning would be that this force, acting on the particle at the point referred to, together with the forces between particles determined by the kinematical conditions of rigidity, are the actual forces on the body.

One objection might be raised to the fundamental Law of Dynamics, as above stated by us; it seems awkward to imply a knowledge of the whole of physical science in stating that fundamental principle.

This objection leads us to cast aside Prof James Thomson's type of statement, and to adopt another, which states exactly the same thought in a different form. We shall propound as preliminary a science of Abstract Dynamics, which shall be a pure science to the same extent as Kinematics is a pure science.

It is as follows:-

In a dynamical system, each particle is credited with a certain mass, and by coordinates with reference to a system of coordinate axes its position and motion are determined. When a particle is accelerated, it is said to have a force acting upon it in the direction of the acceleration and of magnitude proportional to the acceleration and mass conjointly.

The system of chronometry is arbitrary, as well as the system of coordinate axes.

The expressions, mass of a body, centre of mass of a body, force on a body, and force acting, on a body at a point, are defined in the same way as before.

This forms the subject of "Abstract Dynamics," which deals only with mental conceptions, and which is a sort of Kinematics, but Kinematics enriched by the conceptions of force and mass.

This being premised, then, in place of Newton's Definitions and his First and Second Laws of Motion, we have the Physical Law or Theory that we can so choose the masses to be assigned to our material particles, our coordinate axes, and our system of chronometry, that the forces may be resolved by the parallelogram of forces into such as are expressed by our Physical Laws.

Perhaps we should keep more faithfully to the historical conception of Dynamics were we to state our Law of Experimental Dynamics as follows:-

It is possible to choose the masses of the solar system, the axis, and the chronometry, so that the masses shall correspond with those of Astronomy, and the forces shall be resolvable into such as will be expressed by the Law of Universal Gravitation, and conformable to Newton's 3rd Law of Motion and to the Law of the Indestructibility of Matter (Conservation of Mass).

Then true time, absolute velocity, and mass-measurement being defined from this system, there would be the further Law of Physics, that the forces on the various particles composing the different members of the solar system and others are expressible by our various Physical Laws or Theories.

We have now arrived at the conclusion that the attempt to state the Laws of Motion by means of a set of detached definitions and axioms is futile. We have found that Newton's First Law of Motion cannot be stated until we have the conception of a certain system of reference, whose definition involves the knowledge of the First Law, as well as the definition of force, etc. We have therefore seen that the Experimental Principle of Dynamics should be stated as an organic theory or hypothesis. We have found it convenient to formulate a science of Abstract Dynamics, which is an extended Kinematics,. depending only on space and time-measurements, but including the ideas of force and mass (abstract).

By means of this we can state in a succinct form the experimental Law or Hypothesis of Dynamics (applied), which enables us to give to time-measurement such a specification that durations of time, as well as other dynamical magnitudes, are made to depend ultimately for their measurement solely on space-measurement and observations of coincidences in time.

These conclusions we have arrived at by assuming that only kinetic specifications for the measurement of force, mass, and time, and only a kinetic definition of "true rest" are admissible.

Before attempting to justify these assumptions, it may be expedient to devote a few paragraphs to a general consideration of the idea of our method. A theory is an attempt to dominate our experience; it is a conception which may enable us, with as little expenditure of thought as possible, to remember the past and forecast the future.

The theory of Universal Gravitation is an example of a very successful attempt, perfectly successful so far as it has been tested. So with the Euclidean Geometry.

On the other hand, we have theories which have been found useful to enable us to dominate one region of experience, while they break down in certain directions. The Newtonian Emission Theory of Light is an example. There are others which, if they do not break down absolutely, involve the mind in difficulties hitherto unsolved; e.g. the "elastic-solid" Wave Theory of Light.

Theories which are found to break down when applied to their full extent, as well as theories which have not been sufficiently tested, are often called "working hypotheses".

The only merits or demerits a theory can have arise from these two desiderata: (1) it must not be contradicted by any part of our experience; (2) it must be as simple as possible.

Thus, for example, consider tile two rival theories: (1) that the earth has a certain amount of rotation about its axis; (2) that it has no rotation. The latter will be found to agree perfectly with our experience, provided we assume as a new physical law that there is a repulsive force of magnitude

\omega^{2}r

away from the Earth's axis at every point of space, and also at every point a force

2 \omega \Large \frac {dr}{dt}

at right angles to the axis of the Earth, and to the shortest distance of the point from the axis, where

\omega

is the angular velocity in the first theory, and

r

is the distance of any point from the axis. But we reject the latter theory on account of its greater complexity. It is incorrect to say that the one is true and the other false.

It follows that there is no essential difference between a hypothesis and a theory, or what is called a law of nature. One may be less exact than another, or less simple, or less sufficiently tested, but the difference is one of degree.

Now there are two opposite methods of stating dynamical principles; the one employing independent definitions of the various conceptions, the other that adopted in this Essay. Both, so far as observation has tested them, correspond equally to the facts. The question is, then, Which is the simpler? Which comprehends the various relations with the least expenditure of mental energy?

According to the former method, force, mass, time measurement, and "true rest" would be defined as preliminaries to the science of Dynamics, and independently of that science. According to the latter, these conceptions are defined by means of one Law or Hypothesis.

Probably to learners unaccustomed to abstract reasoning, who do not probe the processes of proof employed to the bottom, the former method may be preferable because its conceptions are more concrete; but to one who has mastered the essential relations of the subject, the latter will be found superior.

Let us discuss the idea of force. What are the alternatives to the kinetic definition of force and force-measurement? We might take some arbitrary standard, such as a spring-balance having a graduated scale. This would obviously have the disadvantage of want of permanence, or, to speak more accurately, that of liability to invalidate all our other methods of reckoning force, by reason of some physical change occurring in the standard balance. Further, such a method would be incapable of accuracy sufficient for many of our physical problems, where we deal with forces so small as to be insensible to our present observing powers on such a standard; forces whose magnitude, therefore, we could not define, even theoretically. And, besides, any such arbitrary definition of force would be contrary to our whole tendency in modern science. Suppose, for instance, experiment were to disclose that Newton's Second Law was untrue, the forces being thus measured, should we hesitate between rejecting the law or rejecting the method of force-measurement? And it is certain that we cannot find a spring-balance which would render this event unlikely to happen.

A more promising method would be the definition of unit of force as the weight of a certain piece of matter at a certain place on the Earth's surface. The force

F

would then be defined as being equal to the weight of a body whose mass was

F

times the standard mass. This would involve an independent method of mass-measurement, which we shall consider later. In treating questions of the secular changes of the Earth such a definition would be useless, unless we were also to specify the date as well as the place of the weighing supposed to be at the base of force-measurement; and this could not be brought into connexion with measurements at any other date without employing the whole science of Dynamics, which would thus involve reasoning in a circle.

A modification of this method would be one in which force-measurement would be made to depend on the gravitational or astronomical unit of mass, as well as the theory of the force of gravitation. But this also would be a system of force-measurement, involving for its conception the whole science of Dynamics, of which it would not be independent.

When Statics is treated as a science, independent of Kinetics, force is sometimes left undefined at first, while the mode of procedure is as follows:- We are supposed to have a certain idea of the nature of force, partly based on the sensations we experience when our body forms one of the two bodies which exert force on one another, and starting from this, by the aid of á priori reasoning the idea of the measurement of force is evolved. Then, with the help of certain physical axioms and constructions ("transmissibility of force," "superposition of forces in equilibrium," etc.), the parallelogram of forces is proved.

All this has a very artificial character, and would lead us to prefer the simpler kinetic conception of force; but still further argument is required before we get to Kinetics. The "Second Law of Motion" is proved by means of experiments which could not be accurately performed, and whose interpretation generally involves a knowledge of the science whose foundations we are laying. Then the proportionality of force to mass is thus proved:-

Suppose two equal masses acted on by equal and parallel forces; they have the same acceleration. Next, suppose they form parts of a single body; the acceleration will "evidently" be the same as before, etc. (Third Law of Motion assumed.) Hence accelerations being equal, force varies as mass.

This method has been discredited of late, chiefly through the influence of Thomson and Tait's 'Natural Philosophy,' so that we may omit further discussion upon it.

It may be remarked, however, that those who have most emphatically declared against the statical measure of force do not seem to perceive what is logically implied in that course. (Cf. Professor Tait's Lecture on Force.)

Consider next the idea of mass.

The definition based on the weight of bodies is open to the same objections as the corresponding method in the case of force.

If we define mass by reference to chemical affinity, or to volumetric observations, we in the first place lose the simplicity of the kinetic method, and secondly we adopt a conception of mass which is different from the actual conception of modern science. This is demonstrated if we ask ourselves: Supposing experiment to show a discrepancy between the mass as measured kinetically and as measured otherwise, which method should we call inexact? If the former, Kinetics could no longer be considered an exact science.

Consider next the reference system, and the idea of true rest. The most obvious arbitrary definition of the system to which the motions of bodies in Dynamics are to be referred is to look on the centre of gravity of the Solar system as the fixed point, and the directions of certain fixed stars as fixed directions. The objections are, first, this would be a very inconvenient system in discussing the cosmical Dynamics; second, it is not the actual conception of the science of the present day. If one of the stars chosen were found to have a motion compared with the average position of neighbouring stars, we should certainly conclude that its direction was not "fixed" in the dynamical sense.

It has been suggested to take as a fixed direction that of the perpendicular to the "invariable plane of the Solar system." This really is not an independent definition, and is open to the objections we previously urged against such, when isolated from the fundamental law of Experimental Dynamics.

The foregoing methods have been well criticised by Heinrich Streintz, who propounds in their stead a method of reference to a "Fundamental Körper," which is any body not acted on by external forces and having no rotation. The absence of rotation is to be determined by observations of centrifugal force (as in Newton's experiment of the rotating bucket of water). Now as Streintz takes the kinetic definition of force, it involves reasoning in a circle to speak at this stage of a body "not acted on by forces." Further, if the observations of centrifugal force are to be made with the whole resources of Dynamics, and our knowledge of the laws of nature, this is virtually the kinetic definition of force, but stated in a form which involves reasoning in a circle. If, on the other hand, want of rotation is to be defined as existing when the surface of a bucket of water does not appear to deviate from planeness, then our stock objections to such definitions of dynamical ideas reappear.

A most instructive discussion relating to this subject is given by Professor Mach in his book Die Mechanik in ihrer Entwickelung, historisch-kritisch dargestellt, pp. 214-222. Let us quote a sentence on p. 218:-

Instead of saying 'the direction and velocity of a mass in space remain constant,' we can say 'the mean acceleration of the mass $\mu$ with reference to the masses $m, m', m'', ...$ at the distances $r, r', r'', ...$ is = 0, or $\Large \frac {d^2}{dt^2}\frac{\sum {mx}}{\sum m}$ '. The latter expression is equivalent to the former, so soon as we take into consideration masses which are great enough, numerous enough, and distant enough.

On the previous page, referring to Newton's bucket experiment, he remarks that no one can say how the experiment would come out were we to increase the mass of the bucket continually; and, further, that we should be guilty of dishonesty were we to maintain that we know more of the motion of bodies than that their motion relative to the very distant stars appears to follow the same laws as Galileo formulated for terrestrial bodies relative to the Earth.

Of course this charge of dishonesty cannot be urged against the method of this Essay, as explained in our paragraphs on the nature of theories. And our definition of "true rest" being based entirely on experiment and observation, is not affected by Prof Mach's strictures on the use of the terms absolute rest, absolute space, etc.

Though on the principles of this Essay no exception in principle can be taken to Prof Mach's "Substitute for the "First Law of Motion above quoted," we reject it because it is not the actual conception which has been historically evolved in Dynamics.

Lastly, let us consider the conception of time-measurement.

The only rival definition of equal times that need be considered is that adopted by Streintz, and ascribed by him to D'Alembert and Poisson, viz. "Times are equal in which identical processes take place." The difficulty here would be to distinguish when we have identical processes going on. We find that practically this will reduce to assuming each rotation of the Earth with reference to the fixed stars a process identical with all the others. For the "processes" must consist in movements of matter, of which the Earth's rotations are the most "identical" we have experience of.

But even these we know are not absolutely identical, so that our definition is not practicable. With this definition, what should we mean by saying that the rotation period of the Earth is altering? We should mean that if identical processes happened at different dates, their durations measured by sidereal time would differ. But the only identical processes actually available are wrapped up in the general dynamical theory of the Solar system; so that this theoretically independent definition of time turns out to involve all our Dynamics implicitly when we try to give it physical meaning.

In seeking to justify our preference of kinetic definitions over non-kinetic definitions of our fundamental dynamical conceptions, we have found that the latter, besides being theoretically inconvenient, very often have only an illusory independence of Dynamics.

In fact no one has ever built up a science of Dynamics from independently formed conceptions; and to do so in a strictly logical manner would require expositions whose length would render them tedious in the extreme.

We have hitherto made no reference to any scheme of dynamical principles apart from that of Newton, and those various modifications of it proposed by later writers. This course has been adopted in order to concentrate attention upon the principle at issue.

Systems of Dynamics founded on such principles as Maupertius's "Principle of Least Action," or Gauss's " Principle of Least Coercion" (Kleinsten Zwanges), may be treated from exactly the same point of view, and will not he further referred to.

Note A. - On Theories and Hypotheses.

In the preceding Essay we have assumed as known the science of Geometry; but of course the views put forward in this Essay concerning the nature of physical theories apply equally to geometrical theories. This is the standpoint adopted by Riemann in his epoch-making paper, Über die Hypothesen welche der Geometrie zu Grunde liegen. That space is infinite and that one and only one parallel to a straight line can be drawn through any point, are, it is true, the simplest hypotheses which serve to express our experience; but, as Helmholtz points out in his tract Über die Erhaltung der Kraft, at page 7, the task of theoretical science is only completed when we have proved that our theories are the only ones by which the phenomena can be explained.

Dann wäre dieselbe als die nothwendige Begriffsform der Naturauffassung erwiesen; es würde derselben alsdann also auch objective Wahrheit zuzuschreiben sein.

[Then this would be proved to be the necessary conceptual form of the conception of nature; it would then also be ascribable to objective truth.]

In his critique of the second edition of Thomson and Tait's treatise on Natural Philosophy ('Nature,' vol. xx. p. 213), Clerk Maxwell clearly indicates the hypothetical nature of abstract Dynamics. On p. 214 we read:-

Why, then, should we have any change of method when we pass on from Kinematics to abstract Dynamics? Why should we find it more difficult to endow moving figures with mass than to endow stationary figures with motion? The bodies we deal with in abstract Dynamics are just as completely known to us as the figures in Euclid. They have no properties whatever, except those which we explicitly assign to them ... We have thus vindicated for figures with mass, and, therefore, for force and stress, impulse and momentum, work and energy, their place in abstract science beside form and motion.

The phenomena of real bodies are found to correspond so exactly with the necessary laws of dynamical systems that we cannot help applying the language of Dynamics to real bodies, ...

It will be seen that, so far as they go, the above extracts are in complete harmony with the views in this Essay. It is to be regretted that these views are not consistently followed out in Clerk Maxwell's book Matter and Motion. In that book, while there are very many clear expositions of particular points, the arrangement is in many parts highly illogical. This has been pointed out to a certain extent by Streintz in his aforementioned book, and the reader of the foregoing Essay will have little difficulty in making further criticisms.

One point in Maxwell's book (Matter and Motion) calls for special notice, viz., his à priori proof of the first law of Motion. This proof rests on the assumption of the impossibility of defining absolute rest. "Hence," he says, "the hypothetical law is without meaning unless we admit the possibility of defining absolute rest and absolute velocity." But it is obvious that if the "hypothetical law" spoken of (velocity diminishing at a certain rate) corresponded with experience, we should then have, by that very fact, a conception of absolute rest and absolute velocity which would be perfectly intelligible, so that the assumption "absolute rest unintelligible" would not be justified. Thus, Maxwell's conclusion, "It may thus be shown that the denial of Newton's law is in contradiction to the only system of consistent doctrine about space and time which the mind has been able to form" is unwarranted.

Kirchhoff in his Mechanik appears to adopt a view somewhat similar to that set forth in this Essay. In his preface we find him stating as the problem of Mechanik,

die in der Natur vor sich gehenden Bewegungen vollständig und zwar auf die einfachste Weise zu beschreiben.

[to describe the movements that take place in nature completely and in the simplest way.]

This author uses the term force only as a convenient means of expressing equations shortly in words. Mass appears as a coefficient in the equations of motion, and thus receives a kinetic definition. But no explanations are given as to time-measurement, or as to the axes of reference.

Note B. - Newton's Absolute Space and Time.

My criticisms of the Newtonian scheme of Definitions and Axioms have been directed not so much against what I suppose to be Newton's meaning, as against the form in which it is put, especially as against that form on the supposition that force is to be measured kinetically.

Thus, instead of looking on the Second Law as a mere definition of force-measurement, we might suppose that Newton had in his mind some non-kinetic conception of force-measurement; in which ease the Second Law would be a real and not an illusory statement of physical fact, though imperfect through the want of any specification of how force was to be measured.

Again, take the question of absolute space and time, with respect to which Newton's laws are stated.

There are three ways of looking at it. Some characterise these terms as mere metaphysical nonsense (Much, p. 209). Streintz quotes the Hypothesis I. from the third Book of Newton's Principia to show that by absolute rest Newton means rest relative to the centre of gravity of the universe. But Newton evidently places this Hypothesis in a different category from his laws of motion.

I think the meaning of the terms amounts simply to this, that Newton looked on Dynamics as an abstract science. "In rebus philosophicis abstrahendum est a sensibus", "loca primaria moveri absurdum est". And an abstract science is one which deals with a certain body of conceptions, every relation in which holds with absolute exactness. The point at which considerations as to degree of exactitude may arise, is its application to experience.

If this be the correct view of Newton's meaning, then the foregoing Essay has been simply the explicit and developed statement of that meaning.

Thomson and Tait, while in various ways improving the form in which they state the Newtonian theory, entirely ignore his idea of "absolute space and time," which, as I have tried to show, is the germ of the true theory.

The late Carl Neumann, in his pamphlet Über die Principien der Galilei-Newtonschen Theorie (Leipzig, 1870), like Newton, postulates an "absolute rest." He does so by assuming that there is a "Körper Alpha," an ideally existing body which is absolutely at rest and absolutely rigid, with respect to which the first Law of Newton holds good.

Streintz criticises this rather unintelligently, I think, for it is evident in reading Neumann's essay that this is merely an awkward and metaphorical way of stating the theory of an "Abstract Dynamics."

Note C. - The Parallelogram of Force.

Force being defined kinetically, it is hardly necessary to demonstrate this proposition. It follows as easily from the parallelogram of accelerations as that does from the parallelogram of velocities, or the parallelogram of velocities from the parallelogram of steps. This applies primarily to forces acting on a particle, but it is easy to extend the theorem to "forces acting on a body," as defined in the Essay.

Last Updated March 2021