Mach’s Science of Mechanics Ch.V








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1. Purely mechanical phenomena do not exist. The production of mutual accelerations in masses is, to all appearances, a purely dynamical phenomenon. But with these dynamical results are always associated thermal, magnetic, electrical, and chemical phenomena, and the former are always modified in proportion as the latter are asserted. On the other hand, thermal, magnetic, electrical, and chemical conditions also can produce motions. Purely mechanical phenomena, accordingly, are abstractions, made, either intentionally or from necessity, for facilitating our comprehension of things. The same thing is true of the other classes of physical phenomena. Every event belongs, in a strict sense, to all the departments of physics, the latter being separated only by an artificial classification, which is partly conventional, partly physiological, and partly historical.


2. The view that makes mechanics the basis of the remaining branches of physics, and explains all physical phenomena by mechanical ideas, is in our judgment a prejudice. Knowledge which is historically first, is not necessarily the foundation of all that is subsequently [496] gained.


 As more and more facts are discovered and classified, entirely new ideas of general scope can be formed. We have no means of knowing, as yet, which physical phenomena go deepest, whether the mechanical phenomena are perhaps not the most superficial of all, or whether all do not go equally deep. Even in mechanics we no longer regard the oldest law, the law of the lever, as the foundation of all the other principles.

The mechanical theory of nature, is, undoubtedly, in an historical view, both intelligible and pardonable : and it may also, for a time, have been of much value.  But, upon the whole, it is an artificial conception.



Faithful adherence to the method that led the greatest investigators of nature, Galileo, Newton, Sadi Carnot, Faraday, and J. R. Mayer, to their great results, restricts physics to the expression of actual facts, and forbids the construction of hypotheses behind the facts, where nothing tangible and verifiable is found. *If this is done, only the simple connection of the motions of masses, of changes of temperature, of changes in the values of the potential function, of chemical changes, and so forth is to be ascertained, and nothing is to be imagined along with these elements except the physical attributes or characteristics directly or indirectly given by observation.

 This idea was elsewhere * [* Mach, Die Geschichte und die Wurzel des Satzes von der Erhaltung der Arbeit] developed by the author with respect to the phenomena of heat, and indicated, in the same place, with respect to electricity. All hypotheses of fluids or media are eliminated from the theory of electricity as entirely superfluous, when we reflect that electrical conditions are all given by the [497] values of the potential function V and the dielectric constants. If we assume the differences of the values of V to be measured (on the electrometer) by the forces, and regard V and not the quantity of electricity Q as the primary notion, or measurable physical attribute, we shall have, for any simple insulator, for our quantity of electricity



(where x, y, z denote the coordinates and dv the element of volume,) and for our potential*


Here Q and W appear as derived notions, in which no conception of fluid or medium is contained. If we work over in a similar manner the entire domain of physics, we shall restrict ourselves wholly to the quantitative conceptual expression of actual facts. All superfluous and futile notions are eliminated, and the imaginary problems to which they have given rise forestalled. (See Appendix XXVIII, p. 583.)


The removal of notions whose foundations are historical, conventional, or accidental, can best be furthered by a comparison of the conceptions obtaining in the different departments, and by finding for the conceptions of every department the corresponding conceptions of others. We discover, thus, that temperatures and potential functions correspond to the velocities of mass-motions. A single velocity-value, a single temperature-value, or a single value of potential function, never changes alone. But whilst in the case of velocities and potential functions, so far as we yet [498]


 _________ footnote _____________


* Using the terminology of Clausius.



know, only differences come into consideration, the significance of temperature is not only contained in its difference with respect to other temperatures. Thermal capacities correspond to masses, the potential of an electric charge to quantity of heat, quantity of electricity to entropy, and so on. The pursuit of such resemblances and differences lays the foundation of a comparative physics, which shall ultimately render possible the concise expression of extensive groups of facts, without arbitrary additions. We shall then possess a homogeneous physics, unmingled with artificial atomic theories.


It will also be perceived, that a real economy of scientific thought cannot be attained by mechanical hypotheses. Even if an hypothesis were fully competent to reproduce a given department of natural phenomena, say, the phenomena of heat, we should, by accepting it, only substitute for the actual relations between the mechanical and thermal processes, the hypothesis.  The real fundamental facts are replaced by an equally large number of hypotheses, which is certainly no gain. Once an hypothesis has facilitated, as best it can, our view of new facts, by the substitution of more familiar ideas, its powers are exhausted.  We err when we expect more enlightenment from an hypothesis than from the facts themselves.




3. The development of the mechanical view was favored by many circumstances. In the first place, a connection of all natural events with mechanical processes is unmistakable, and it is natural, therefore, that we should be led to explain less known phenomena by better known mechanical events. Then again, it was first in the department of mechanics that laws of general and extensive scope were discovered. A law of [499] this kind is the principle of vis viva ∑ (U1 – U0) = ∑ ½ m(v1 2 – v0 2), which states that the increase of the vis viva of a system in its passage from one position to another is equal to the increment of the force-function, or work, which is expressed as a function of the final and initial positions. If we fix our attention on the work a system can perform and call it with Helmholtz the Spannkraft, S, * then the work actually performed, U, will appear as a diminution of the Spannkraft, K, initially present; accordingly, S=K –U, and the principle of vis viva takes the form


S + ½ mv2 = const,




that is to say. every diminution of the Spannkraft, is compensated for by an increase of the vis viva. In this form the principle is also called the law of the Conservation of Energy, in that the sum of the Spannkraft (the potential energy) and the vis viva (the kinetic energy) remains constant in the system. But since, in nature, it is possible that not only vis viva should appear as the consequence of work performed, but also quantities of heat, or the potential of an electric charge, and so forth, scientists saw in this law the expression of a mechanical action as the basis of all natural actions. However, nothing is contained in the expression but the fact of an invariable quantitative connection between mechanical and other kinds of phenomena.


4. It would be a mistake to suppose that a wide and extensive view of things was first introduced into physical science by mechanics. On the contrary, this [500]






 ____ footnote _____________


Helmholtz used this term in 1847; but it is not found in his subsequent papers; and in 1882 (Wissenschaftliche Abhandlungen, II, 965) he expressly discards it in favor of the English “potential energy”.  He even (p. 968) prefers Clausius’s word Ergal to Spannkraft, which is quite out of agreement with modern terminology. — Trans.






insight was possessed at all times by the foremost of inquirers and even entered into the construction of mechanics itself, and was, accordingly, not first created by the latter. Galileo and Huygens constantly alternated the consideration of particular details with the consideration of universal aspects, and reached their results only by a persistent effort after a simple and consistent view. The fact that the velocities of individual bodies and systems are dependent on the spaces descended through, was perceived by Galileo and Huygens only by a very detailed investigation of the motion of descent in particular cases, combined with the consideration of the circumstance that bodies generally, of their own accord, only sink. Huygens especially speaks, on the occasion of this inquiry, of the impossibility of a mechanical perpetual motion ; he possessed, therefore, the modern point of view. He felt the incompatibility of the idea of a perpetual motion with the notions of the natural mechanical processes with which he was familiar.


Take the fictions of Stevinus say, that of the endless chain on the prism. Here, too, a deep, broad insight is displayed. We have here a mind, disciplined by a multitude of experiences, brought to bear on an individual case. The moving endless chain is to Stevinus a motion of descent that Is not a descent, a motion without a purpose, an intentional act that does not answer to the intention, an endeavor for a change which does not produce the change. If motion, generally, is the result of descent, then in the particular case descent is the result of motion. It is a sense of the mutual Interdependence of v and h in the equation v = 2gh that is here displayed, though of course in not so definite a form. A contradiction exists in this [501] fiction for Stevinus’s exquisite investigative sense that would escape less profound thinkers.


This same breadth of view, which alternates the individual with the universal, is also displayed, only in this instance not restricted to mechanics, in the performances of Sadi Carnot. When Carnot finds that the quantity of heat Q which, for a given amount of work L, has flowed from a higher temperature t to a lower temperature t’, can only depend on the temperatures and not on the material constitution of the bodies, he reasons in exact conformity with the method of Galileo. Similarly does J. R. Mayer proceed in the enunciation of the principle of the equivalence of heat and work. In this achievement the mechanical view was quite remote from Mayer’s mind ; nor had he need of it. They who require the crutch of the mechanical philosophy to understand the doctrine of the equivalence of heat and work, have only half comprehended the progress which it signalises. Yet, high as we may place Mayer’s original achievement, it is not on that account necessary to depreciate the merits of the professional physicists Joule, Helmholtz, Clausius, and Thomson, who have done very much, perhaps all, towards the detailed establishment and perfection of the new view. The assumption of a plagiarism of Mayer’s ideas is in our opinion gratuitous. They who advance it, are under the obligation to prove it. The repeated appearance of the same idea is not new in history. We shall not take up here the discussion of purely personal questions, which thirty years from now will no longer Interest students. But It is unfair, from a pretense of justice, to insult men, who If they had accomplished but a third of their actual services, would have lived highly honored and unmolested lives, (See p. 584.)


5. We shall now attempt to show that the broad view expressed in the principle of the conservation of energy, is not peculiar to mechanics, but is a condition of logical and sound scientific thought generally. The business of physical science is the reconstruction of facts in thought, or the abstract quantitative expression of facts. The rules which we form for these reconstructions are the laws of nature. In the conviction that such rules are possible lies the law of causality. The law of causality simply asserts that the phenomena of nature are dependent on one another. The special emphasis put on space and time in the expression of the law of causality is unnecessary, since the relations of space and time themselves implicitly express that phenomena are dependent on one another.


The laws of nature are equations between the measurable elements α,β,γ,δ . . . . ω of phenomena. As nature is variable, the number of these equations is always less than the number of the elements.


If we know all the values of α,β,γ,δ . . . . by which, for example, the values of  λ,μ,ν . . . are given, we may call the group α,β,γ,δ . . . . the cause and the group λ,μ,ν . . . the effect.  In this sense we may say that the effect is uniquely determined by the cause. The principle of sufficient reason, in the form, for instance, in which Archimedes employed it in the development of the laws of the lever, consequently asserts nothing more than that the effect cannot by any given set of circumstances be at once determined and undetermined. If two circumstances α and λ are connected, then, supposing all others are constant, a change of λ will be accompanied by a change of α, and as a general rule a change of α by a change of λ. The constant observance of this mutual interdependence is met with [503]



in Stevinus, Galileo, Huygens, and other great inquirers. The idea is also at the basis of the discovery of counter-phenomena. Thus, a change in the volume of a gas due to a change of temperature is supplemented by the counter-phenomenon of a change of temperature on an alteration of volume ; Seebeck’s phenomenon by Peltier’s effect, and so forth.





Care must, of course, be exercised, in such inversions, respecting the form of the dependence. Figure 235 will render clear how a perceptible alteration of α may always be produced by λ , but a change of  λ not necessarily by a change of α. The relations between electromagnetic and induction phenomena, discovered by Faraday, are a good instance of this truth.


If a set of circumstances  α,β,γ,δ . . . . , by which a second set λ,μ,ν . . . is determined, be made to pass from its initial values to the terminal values  α’,β’,γ’,δ’ . . . .then λ,μ,ν . . . also will pass into λ’,μ’,ν’ . . . If the first set be brought back to its initial state, also the second set will be brought back to its initial state.  This is the meaning of the “equivalence of cause and effect,’ which Mayer again and again emphasizes.


If the first group suffer only periodical changes, the second group also can suffer only periodical changes, not continuous permanent ones. The fertile methods of thought of Galileo, Huygens, S. Carnot, Mayer, and their peers, are all reducible to the simple but significant perception, that purely periodical alterations of one set of circumstances can only constitute the source of similarly periodical alterations of a second set of circumstances, not of continuous and permanent alterations. Such maxims, as “the effect is equivalent to the cause,” [504] “work cannot be created out of nothing,” “a perpetual motion is impossible,” are  particular, less definite, and less evident forms of this perception, which in itself is not especially concerned with mechanics, but is a constituent of scientific thought generally. With the perception of this truth, any metaphysical mysticism that may still adhere to the principle of the conservation of energy* is dissipated. (See p. 585.)



All ideas of conservation, like the notion of substance, have a solid foundation in the economy of thought.  A mere unrelated change, without fixed point of support, or reference, is not comprehensible, not mentally reconstructible. We always inquire, accordingly, what idea can be retained amid all variations as permanent, what law prevails, what equation remains fulfilled, what quantitative values remain constant ?  When we say the refractive index remains constant in all cases of refraction, g remains = 9.810 m in all cases of the motion of heavy bodies, the energy remains constant in every isolated system, all our assertions have one and the same economical function, namely that of facilitating our mental reconstruction of facts.






All science has its origin in the needs of life.  However minutely it may be subdivided by particular vocations or by the restricted tempers and capacities of those who foster it, each branch can attain its full and best development only by a living connection with the whole. Through such a union alone can it approach [505] its true maturity, and be insured against lop-sided and monstrous growths.




___ footnote ____________


* When we reflect that the principles of science are all abstractions that presuppose repetitions of similar cases, the absurd applications of the law of the conservation of forces to the universe as a whole fall to the ground.





The division of labor, the restriction of individual inquirers to limited provinces, the investigation of those provinces as a life-work, are the fundamental conditions of a fruitful development of science. Only by such specialisation and restriction of work can the economical instruments of thought requisite for the mastery of a special field be perfected. But just here lies a danger the danger of our overestimating the instruments, with which we are so constantly employed, or even of regarding them as the objective point of science.



2. Now, such a state of affairs has, in our opinion, actually been produced by the disproportionate formal development of physics. The majority of natural inquirers ascribe to the intellectual implements of physics, to the concepts mass, force, atom, and so forth, whose sole office is to revive economically arranged experiences, a reality beyond and independent of thought.  Not only so, but it has even been held that these forces and masses are the real objects of inquiry, and, if once they were fully explored, all the rest would follow from the equilibrium and motion of these masses. A person who knew the world only through the theatre, if brought behind the scenes and permitted to view the mechanism of the stage’s action, might possibly believe that the real world also was in need of a machine-room, and that if this were once thoroughly explored, we should know all. Similarly, we, too, should beware lest the intellectual machinery, employed in the representation of the world on the stage of thought, be regarded as the basis of the real world.


3. A philosophy is involved in any correct view of [506]


the relations of special knowledge to the great body of knowledge at large, a philosophy that must be demanded of every special investigator. The lack of it is asserted in the formulation of imaginary problems, in the very enunciation of which, whether regarded as soluble or insoluble, flagrant absurdity is involved.  Such an overestimation of physics, in contrast to physiology, such a mistaken conception of the true relations of the two sciences, is displayed in the inquiry whether it is possible to explain feelings by the motions of atoms?



Let us seek the conditions that could have impelled the mind to formulate so curious a question. We find in the first place that greater confidence is placed in our experiences concerning relations of time and space ; that we attribute to them a more objective, a more real character than to our experiences of colors, sounds, temperatures, and so forth. Yet, if we investigate the matter accurately, we must surely admit that our sensations of time and space are just as much sensations as are our sensations of colors, sounds, and odors, only that in our knowledge of the former we are surer and clearer than in that of the latter. Space and time are well-ordered systems of sets of sensations. The quantities stated in mechanical equations are simply ordinal symbols, representing those members of these sets that are to be mentally isolated and emphasised. The equations express the form of interdependence of these ordinal symbols.


A body is a relatively constant sum of touch and sight sensations associated with the same space and time sensations.   Mechanical principles, like that, for instance, of the mutually induced accelerations of two masses, give, either directly or indirectly, only some[507] combination of touch, sight, light, and time sensations.  They possess intelligible meaning only by virtue of the sensations they involve, the contents of which may of course be very complicated.



It would be equivalent, accordingly, to explaining the more simple and immediate by the more complicated and remote, if we were to attempt to derive sensations from the motions of masses, wholly aside from the consideration that the notions of mechanics are economical implements or expedients perfected to represent mechanical and not physiological or psychological facts. If the means and aims of research were properly distinguished, and our expositions were restricted to the presentation of actual facts, false problems of this kind could not arise.


4. All physical knowledge can only mentally represent and anticipate compounds of those elements we call sensations. It is concerned with the connection of these elements. Such an element, say the heat of a body A, is connected, not only with other elements, say with such whose aggregate makes up the flame B, but also with the aggregate of certain elements of our body, say with the aggregate of the elements of a nerve, N.  As simple object and element N is not essentially, but only conventionally, different from A and B, The connection of A and B is a problem of physics, that of A and N a problem of physiology. Neither is alone existent; both exist at once. Only provisionally can we neglect either. Processes, thus, that in appearance are purely mechanical, are, in addition to their evident mechanical features, always physiological, and, consequently, also electrical, chemical, and so forth. The science of mechanics does not comprise the foundations, no, nor even a part of the world, but only an aspect of it.


_____________  end of chapter V ____________________






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