¹ Before Galileo force is known solely as pressure, by the principle: cessante causa cessat et effectus.

² “…when this friction is drastically reduced, as on an ice rink, their velocity is maintained for a long time. This makes Galileo's discovery seem plausible.” (Sciama, The Unity of the Universe, p. 85). And compare Mach, Science of Mechanics, pp. 330 ff.

³ Two New Sciences, IIIrd Day, 242–246.

⁴ As whenever the First Law is characterized as being but a limiting case of the Second Law, i.e., when Σ F = 0 it follows that a = 0 (whether m > 0 or = 0). That is, d ²x/dt ² = 0; d ²y/dt ² = 0; d ²z/dt ² = 0: the second time derivatives of co-ordinates x, y, z vanish when a body is force-free. From this much, however, one can infer the rectilinearity of an inertial path only by building that concept into “force-free”.

“But along a horizontal plane the motion is uniform since here it experiences neither acceleration nor retardation:…”

“Furthermore we may remark that any velocity once imparted to a moving body will be rigidly maintained as long as the external causes of acceleration or retardation are removed, a condition which is found only on horizontal planes; for in the case of planes which slope downwards there is already present a cause of acceleration, while on planes sloping upward there is retardation; from this it follows that motion along a horizontal plane is perpetual; for, if the velocity be uniform, it cannot be diminished or slackened, much less destroyed. Further, although any velocity which a body may have acquired through natural fall is permanently maintained so far as its own nature (suapte natura) is concerned, yet it must be remembered that if, after descent along a plane inclined downwards the body is deflected to a plane inclined upwards, there is already existing in this latter plane a cause of retardation; for in any such plane this same body is subject to a natural acceleration downwards. Accordingly here we have the supposition of two different states, namely, the velocity acquired during the preceding which if acting alone would carry the body at a uniform rate to infinity, and the velocity which results from a natural acceleration downwards common to all bodies. It seems altogether reasonable, therefore, if we wish to trace the future history of a body which has descended along some inclined plane and has been deflected along some plane inclined upwards, for us to assume that the maximum speed acquired during descent is permanently maintained during the ascent. In the ascent, however, there supervenes a natural inclination downwards, namely, a notion which, starting from rest, is accelerated at the usual rate.” Galileo, op. cit., pp. 215–16.

⁵ Compare Mach, op. cit., pp. 168–169.

⁶ This very supposition is internally inconsistent for anyone who accepts Mach's kinetic definition of mass. Since ex hypothesi a single particle cannot interact with other particles, it is idle to discuss its mass in Machian terms. But are there any other really meaningful terms?

⁷ “If every place is relative then every motion is relative and as motion cannot be understood without a determination of its direction which in its turn cannot be understood except in relation to our or some other body. Up, down, right, left, all directions and places are based on some relation and it is necessary to suppose another body distinct from the moving one… so that motion is given in relation to which it exists, or generally there cannot be any relation, if there are no terms to be related.

Therefore if we suppose that everything is annihilated except one globe, it would be impossible to imagine any movement of that globe.

Let us imagine two globes and that besides them nothing else material exists, then the motion in a circle of these two globes round their common centre cannot be imagined. But suppose that the heaven of fixed stars was suddenly created and we shall be in a position to imagine the motion of the globes by their relative position to the different parts of heaven.” From Berkeley, De Motu, written thirty years after the publication of the Principia of Newton.

⁸ “Definition: By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal.” Galileo, Two New Sciences, IIIrd Day, 191.

⁹ Compare a different, but somewhat analogous claim by Keynes:

“The law of the Uniformity of Nature appears to me to amount to an assertion that an anlogy which is perfect, except that mere differences of position in time and space are treated as irrelevant, is a valid basis for a generalisation, two total causes being regarded as the same if they only differ in their positions in time or space. This, I think, is the whole of the importance which this law has for the theory of inductive argument. It involves the assertion of a generalised judgmentof irrelevance, namely, of the irrelevance of mere position in time and space to generalisations which have no reference to particular position in time and space. It is in respect of such position in time or space that ‘nature’ is supposed ‘uniform’.” A Treatise on Probability, p. 226.

¹⁰ A summary of Newton's interpretation of his “bucket experiment” (Principia, “Scholium on Space on Time”) is that absolute rotation has nothing whatever to do with the relative rotations which are directly observed (e.g., the spinning bucket), and that nevertheless we can determine experimentally the amount of absolute rotation possessed by a body. All we need do is measure the curvature of a water surface rotating with the body. This determination of absolute rotation consists in measuring centrifugal force. Foucault's pendulum-experiment reveals the motion of the pendulum-plane to be acted upon by a Coriolis force.

Compare Mach:

“For me only relative motions exist… When a body rotates relatively to the fixed stars, centrifugal forces are produced; whenit rotates relatively to some different body and not relative to the fixed stars, no centrifugal forces are produced. I have no objection to just calling the first rotation so long as it be remembered that nothing is meant except relative rotation with respect to the fixed stars.”

“Obviously it does not matter if we think of the earth as turning round on its axis, or at rest while the fixed stars revolve round it. Geometrically these are exactly the same case of a relative rotation of the earth and the fixed stars with respect to one another. But if we think of the earth at rest and the fixed stars revolving round it, there is no flattening of the earth, no Foucault's experiment, and so on — at least according to our usual conception of the law of inertia. Now one can solve the difficulty in two ways. Either all motion is absolute, or our law of inertia is wrongly expressed. I prefer the second way. The law of inertia must be so conceived that exactly the same thing results from the second supposition as from the first. By this it will be evident that in its expression, regard must be paid to the masses of the universe.”

Compare also Newton's Principia Mathematica Philosophiae Naturalis, Fifth Corollary (p. 19, 1st edition).

¹¹ Consider the conceptual difficulties Hertz encountered when he laid it down that:

“Every unfree system we conceive to be a portion of a more extended free system; from our point of view there are no unfree systems for which this assumption does not obtain. If, however, we wish to emphasise this relation, we shall denote the unfree system as a partial system, and the free system of which it forms a part, as the complete system.”

This statement is continuous, of course, with Hertz’ own “Fundamental Law” — namely, The Law of Inertia:

“Fundamental law. Every free system persists in its state of rest or of uniform motion in a straightest path. Systerna omne liberum perseverare in statu suo quiescendi vel movendi uniformiter in directissima.” The Principles of Mechanics, pp. 144, 178.

¹² These discussions all turn on the fact that, according to Newton's theory the only way rotation relative to absolute space can be detected is from the existence of centrifugal forces (the “bucket”) and Coriolis forces (the Foucault pendulum). But Absolute Space in the Principia was “invented” precisely to account for these forces — so it adds nothing to what we knew before it was “invented”. Whenever an ‘explanatory’ hypothesis has as its consequences only those anomalous phenomena for which we originally sought an explanation, that hypothesis may quite rightly be regarded as being but a ‘quasi-explanatory’ restatement of the very descriptions with which enquiry began. As such it is not an explanation at all — or, at best, an ad hoc explanation. Leibniz, Huygens, Bernoulli and Poleni originally reacted thus to F α γ Mm/r ², noting that its only consequences were descriptions of planetary purturbations, which latter set the initial problem! Further consequences of the law were soon drawn, making its claims to be a physical explanation thereby more plausible.

¹³ Compare Russell: “It seems evident that the question whether one body is at rest or in motion must have as good a meaning as the same question concerning any other body; and this seems sufficient to condemn Neumann's suggested escape from absolute motion.” (The Principles of Mathematics, 2nd edition, p. 464.)

¹⁴ Notice that this counters Mach's Principle, to wit, that other matter in the universe will make a small contribution to a body's total inertia, rather than no contribution at all — because, according to Mach, a body has inertia only because it interacts (in some way) with all the matter in the universe. This directly opposes the views of Galileo and Newton, for whom inertia was an intrinsic property of matter. Einstein, who was greatly influenced by this position, is responsible for naming it “Mach's Principle”. The view was held also by Stallo, J. J. Thomson, Lange, Kleinpeter, J. G. MacGregor and Pearson.

¹⁵ Thus Einstein noted the similarities between gravitational and inertial forces — both are proportional to the mass of the body acted on — and the dissimilarity of both vis-a-vis electric and magnetic forces (which do not induce the same accelerations in all bodies: consider neutral bodies). This similarity so impressed Einstein that he came to claim that it was impossible to distinguish gravitational from inertial forces. If challenged to discover whether a field of force is gravitational or electrical we just measure the accelerations of a neutral and a charged body. The acceleration of the former discloses the strength of the gravitational part of the field. The additional acceleration of the latter gives the strength of the electrical force. The same experiment with “inertial force” substituted for “electrical force” will permit of no comparable distinction in measurement: the total strength of the field is determinable, but not the individual contributions of the gravitational and inertial components. A man's trousers may fall either when he hits the ground after jumping from a ledge or when he is lifted quickly from the ground in a rapidly accelerating rocket. From the man's point of view the situations are mechanically indistinguishable. There is thus no general criterion for distinguishing inertial from gravitational forces: this is Einstein's Principle of Equivalence. It is “resolved” by treating all inertial forces as being fundamentally gravitational. Our observed inertial forces result from stellar accelerations. Without the latter, the former would not exist. Cf. Bergmann P. G., Theory of Relativity, pp. 153 ff.

It is interesting to note that Euler defended the necessity of Absolute Space by pointing out that, since (according to the Law of Universal Gravitation) all material reference frames are accelerated, inertial motion could exist only relatively to a non-material reference frame, i.e., absolute space. This constitutes a preservation of the Euclidean geometrical meaning of “rectilinearity” at the complete sacrifice of its physical meaning.

¹⁶ Another way of making this point is to say that not all frames of reference are equivalent for the formulation of the laws of motion. There does exist in nature a special set of reference frames in which Newton's Laws hold without additional inertial forces (e.g., Coriolis forces) having to be introduced. But not all phenomena in fact occur within that special set.

Mach argued (op. cit., 286 ff.) that inertial reference frames were just those which were unaccelerated relative to the “fixed stars” — i.e., some suitably defined average of all the matter in the universe. For him matter has inertia only because there is other matter in the universe. Redistribute that other matter, and our conception of inertia might have to be changed at once. Bertrand Russell denies this:

“Mach has a very curious argument by which he attempts to refute the grounds in favour of absolute rotation. He remarks that, in the actual world, the earth rotates relative to the fixed stars, and that the universe is not given twice over in different shapes, but only once, and as we find it. Hence any argument that the rotation of earth could be inferred if there were no heavenly bodies is futile. This argument contains the very essence of empiricism, in a sense in which empiricism is radically opposed to the philosophy advocated in the present work. The logical basis of the argument is that all propositions are essentially concerned with actual existents, not with entities which may or may not exist. For if, as has been held throughout our previous discussion, the whole dynamical world with its laws can be considered without regard to existence, than it can be no part of the meaning of these laws to assert that the matter to which they apply exists, and therefore they can be applied to universes which do not exist. Apart from general arguments, it is evident that the laws are so applied throughout rational Dynamics, and that, in all exact calculations, the distribution of matter which is assumed is not that of the actual world. It seems impossible to deny significance to such calculations; and yet, if true or false, then it can be of no necessary part of their meaning to assert the existence of the matter to which they are applied. This being so, the universe is given, as an entity, not only twice, but as many times as there are possible distributions of matter, and Mach's argument falls to the ground.” op. cit.

And compare Cassirer:

“We need only apply these considerations to the discovery and expression of the principle of inertia in order to recognize that the real validity of this principle of inertia is not bound to any definite material system of reference. Even if we have found the law at first verified with respect to the fixed stars, there would be nothing to hinder us from freeing it from this condition by calling to mind that we can allow the original substratum to vary arbitrarily without the meaning and content of the law itself being thereby affected.”

¹⁷ Compare Patterns of Discovery (N. R. Hanson) Ch. V, with a much less flexible approach, to wit:

“What is the status of this law ? One possible approach to the subject would be to regard Newton's theory as axiomatic, and to work out as an exercise in abstract mathematics the consequences of the theory, without claiming any relationship between the results of the theory and the events of the actual world. But this would be a sadly mistaken policy. It would rob the theory of the greater part of its interest, and it would deprive us of the immensely valuable help provided by our knowledge of what does in fact happen in the real world. It is therefore altogether preferable to regard Newtonian mechanics as a science based upon experiment.” L. S. Pars, Introduction to Dynamics, p. 33.

This is reminiscent of Mach (op. cit.) and C. D. Broad (Perception, Physics, and Reality (1913), both of whom characterize the Second Law of Motion as being an abstract induction from experience.

¹⁸ In contemporary physics the Second Law is usually amended to read: force is mass times absolute acceleration. Were the earth at rest and the Sun moving around it, F = Ma would not be satisfied since the Sun's gravitational force would produce no acceleration on the earth at all, ex hypothesi. Only if the accelerations of bodies are measured in a special way, therefore, will the Second Law hold: the accelerations must be measured relative to an inertial frame of reference, i.e., a collection of bodies on which no forces act, and which have no absolute acceleration. Plenty of objects on earth are not accelerating for us, although they all have an absolute acceleration. Additional inertial forces — like the Coriolis force — must sometimes be introduced in order to cope with unusual reference frames, e.g., that in which two rockets are launched at exactly the earth's rotational + (plus) revolutionary velocity; one rocket is shot “forward” with the earth's velocities, the other opposing them.

Compare Russell (writing in 1924):

“Nevertheless Newtonian dynamics will explain Foucault's pendulum and the flattening of the earth at the poles if the earth rotates, but not if the heavens revolve. This shows a defect in Newtonian dynamics, since the empirical science ought not to contain a metaphysical assumption which can never be proved or disproved by observation — and no observation can distinguish the rotation of the earth from the revolution of the heavens. This philosophical principle, that distinctions which make no difference to observable phenomena must play no part in physics, has inspired a good deal of the work on relativity, and is advocated by many writers.”