Figure 44:

                     Ernst Mach: The Science of Mechanics

 

Ernst Mach: Die Mechanik in ihrer Entwickelung. 1883.

[First English translation 1893]

 

English translation by Thomas J .McCormack.

Ernst Mach: The Science of Mechanics. A Critical and Historical Account of Its Development.

Sixth edition. With revisions through the ninth German edition [1933].

La Salle, Ill.: Open Court 1960.

 

 

pp. 516-517

CHAPTER IV.

 

THE FORMAL DEVELOPMENT OF MECHANICS.

I.

THE ISOPERIMETRICAL PROBLEMS.

 

1. When the chief facts of a physical science have once been fixed by observation, a new period of its development begins - the deductive, which we treated in the previous chapter.

In this period, the facts are reproducible in the mind [… die Thatsachen in Gedanken nachzubilden] without constant recourse to observation. Facts of a more general and complex character are copied in thought on the theory that they are made up of simpler and more familiar observational elements. But even after we have deduced from our expressions for the most elementary facts (the principles) expressions for more common and more complex facts (the theorems) and have discovered in all phenomena the same elements, the developmental process of the science is not yet completed.

The deductive development of the science is followed by its formal development. Here it is sought to put in a clear compendious form, or system, the facts to be reproduced, so that each can be reached and mentally pictured with the least intellectual effort. Into our rules for the mental reconstruction of facts we strive to incorporate the greatest possible uniformity, so that these rules shall be easy of acquisition.

 

It is to be remarked, that the three periods distinguished are not sharply separated from one another, but that the processes of development referred to frequently go hand in hand, although an the whole the order designated is unmistakable.

 

 

pp. 577-579

 

IV.

THE ECONOMY OF SCIENCE.

 

1. It is the object of science to replace, or save, experiences, by the reproduction and anticipation of facts in thought. Memory is handier than experience, and often answers the same purpose. This economical office of science, which fills its whole life, is apparent at first glance; and with its full recognition all mysticism in science disappears. Science is communicated by instruction, in order that one man may profit by the experience of another and be spared the trouble of accumulating it for himself; and thus, to spare posterity, the experiences of whole generations are stored up in libraries.

 

Language, the instrument of this communication, is itself an economical contrivance. Experiences are analyzed, or broken up, into simpler and more familiar experiences, and then symbolized at some sacrifice of precision. The symbols of speech are as yet restricted in their use within national boundaries, and doubtless will long remain so.

But written language is gradually being metamorphosed into an ideal universal character. It is certainly no longer a mere transcript of speech. Numerals, algebraic signs, chemical symbols, musical notes, phonetic alphabets, may be regarded as parts already formed of this universal character of the future; they are, to some extent, decidedly conceptual, and of almost general international use. The analysis of colors, physical and physiological, is already far enough advanced to render an international system of color-signs perfectly practical.

In Chinese writing, we have an actual example of a true ideographic language, pronounced diversely in different provinces, yet everywhere carrying the same meaning. Were the system and its signs only of a simpler character, the use of Chinese writing might become universal. The dropping of unmeaning and needless accidents of grammar, as English mostly drops them, would be quite requisite to the adoption of such a system. But universality would not be the sole merit of such a character; since to read it would be to understand it. Our children often read what they do not understand; but that which a Chinaman cannot understand, he is precluded from reading.

 

2. In the reproduction of facts in thought, we never reproduce the facts in full, but only that side of them which is important to us, moved to this directly or indirectly by a practical interest. Our reproductions are invariably abstractions. Here again is an economical tendency.

 

Nature is composed of sensations as its elements. Primitive man, however, first picks out certain compounds of these elements - those namely that are relatively permanent and of greater importance to him. The first and oldest words are names of "things." Even here, there is an abstractive process, an abstraction from the surroundings of the things, and from the continual small changes which these compound sensations undergo, which being practically unimportant are not noticed. No inalterable thing exists. The thing is an abstraction, the name a symbol, for a compound [Complex] of elements from whose changes we abstract. The reason we assign a single word to a whole compound is that we need to suggest all the constituent sensations [Eindrücke] at once.

When, later, we come to remark the changeableness, we cannot at the same time hold fast to the idea of the thing's permanence, unless we have recourse to the conception of a thing-in-itself, or other such like absurdity. Sensations [Empfindungen] are not signs of things; but, on the contrary, a thing is a thought-symbol for a compound sensation of relative fixedness. Properly speaking the world is not composed of "things" as its elements, but of colors, tones, pressures, spaces, times, in short what we ordinarily call individual sensations.

 

The whole operation is a mere affair of economy. In the reproduction of facts, we begin with the more durable and familiar compounds, and supplement these later with the unusual by way of corrections. Thus, we speak of a perforated cylinder, of a cube with beveled edges, expressions involving contradictions, unless we accept the view here taken.

All judgments are such amplifications and corrections of ideas already admitted.

 

 

pp. 586-589

 

6. The science of physics also furnishes examples of this economy of thought, altogether similar to those we have just examined. A brief reference here will suffice. The moment of inertia saves us the separate consideration of the individual particles of masses. By the force-function we dispense with the separate investigation of individual force-components. The simplicity of reasonings involving force functions springs from the fact that a great amount of mental work had to be performed before the discovery of the properties of the force-functions was possible. Gauss's dioptrics dispenses us from the separate consideration of the single refracting surfaces of a dioptrical system and substitutes for it the principal and nodal points. But a careful consideration of the single surfaces had to precede the discovery of the principal and nodal points. Gauss's dioptrics simply saves us the necessity of often repeating this consideration.

We must admit, therefore, that there is no result of science which in point of principle could not have been arrived at wholly without methods. But, as a matter of fact, within the short span of a human life and with man's limited powers of memory, any stock of knowledge worthy of the name is unattainable except by the greatest mental economy. Science itself, therefore, may be regarded as a minimal problem, consisting of the completest possible presentment of facts with the least possible expenditure of thought.

 

7. The function of science, as we take it, is to re-place experience. Thus, on the one hand, science must remain in the province of experience, but, on the other, must hasten beyond it, constantly expecting confirmation, constantly expecting the reverse. Where neither confirmation nor refutation is possible, science is not concerned. Science acts and acts only in the domain of uncompleted experience. Exemplars of such branches of science are the theories of elasticity and of the conduction of heat, both of which ascribe to the smallest particles of matter only such properties as Observation supplies in the study of the larger portions. The comparison of theory and experience may be farther and farther extended, as our means of observation increase in refinement.

 

Experience alone, without the ideas that are associated with it, would forever remain strange to us. Those ideas that hold good throughout the widest domains of research and that supplement the greatest amount of experience, are the most scientific. The principle of continuity, the use of which everywhere pervades modern inquiry, simply prescribes a mode of conception which conduces in the highest degree to the economy of thought.

 

8. If a long elastic rod be fastened in a vise, the rod may be made to execute slow vibrations. These are directly observable, can be seen, touched, and graphically recorded. If the rod be shortened, the vibrations will increase in rapidity and cannot be directly seen; the rod will present to the sight a blurred image. This is a new phenomenon. But the sensation of touch is still like that of the previous case; we can still make the rod record its movements; and if we mentally retain the conception of vibrations, we can still anticipate the results of experiments. On further shortening the rod the sensation of touch is altered; the rod begins to sound; again a new phenomenon is presented. But the phenomena do not all change at once; only this or that phenomenon changes; consequently the accompanying notion of vibration, which is not confined to any single one, is still serviceable, still economical. Even when the Sound has reached so high a pitch and the vibrations have become so small that the previous means of observation are not of avail, we still advantageously imagine the sounding rod to perform vibrations, and can predict the vibrations of the dark lines in the spectrum of the polarized light of a rod of glass. If on the rod being further shortened all the phenomena suddenly passed into new phenomena, the conception of vibration would no longer be serviceable because it would no longer afford us a means of supplementing the new experiences by the previous ones.

 

When we mentally add to those actions of a human being which we can perceive, sensations and ideas like our own which we cannot perceive, the object of the idea we so form is economical. The idea makes experience intelligible to us; it supplements and supplants experience.

This idea is not regarded as a great scientific discovery, only because its formation is so natural that every child conceives it. Now, this is exactly what we do when we imagine a moving body which has just disappeared behind a pillar, or a comet at the moment invisible, as continuing its motion and retaining its previously observed properties. We do this that we may not be surprised by its reappearance. We fill out the gaps in experience by the ideas that experience suggests.

 

9. Yet not all the prevalent scientific theories originated so naturally and artlessly. Thus chemical, electrical, and optical phenomena are explained by atoms. But the mental artifice atom was not formed by the principle of continuity; on the contrary, it is a product especially devised for the purpose in view. Atoms cannot be perceived by the senses; like all substances, they are things of thought. Furthermore, the atoms are invested with properties that absolutely contradict the attributes hitherto observed in bodies. However well fitted atomic theories may be to reproduce certain groups of facts, the physical inquirer who has laid to heart Newton's rules will only admit those theories as provisional helps, and will strive to attain, in some more natural way, a satisfactory substitute.

 

The atomic theory plays a part in physics similar to that of certain auxiliary concepts in mathematics; it is a mathematical model for facilitating the mental reproduction of facts [ein mathematisches Modell zur Darstellung der Thatsachen]. Although we represent vibrations by the harmonic formula, the phenomena of cooling by exponentials, falls by squares of times, etc., no one will fancy that vibrations in themselves have any-thing to do with the circular functions, or the motion of falling bodies with squares. It has simply been observed that the relations between the quantities investigated were similar to certain relations obtaining between familiar mathematical functions, and these more familiar ideas are employed as an easy means of supplementing experience. Natural phenomena whose relations are not similar to those of functions with which we are familiar, are at present very difficult to reconstruct. But the progress of mathematics may facilitate the matter.

As mathematical helps of this kind, spaces of more than three dimensions may be used, as I have elsewhere shown. But it is not necessary to regard these, on this account, as anything more than mental artifices.

 

 

pp. 610-613

 

A person who knew the world only through the theater, 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 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 emphasized. 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 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 one exists alone; 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.

 

At the beginning of this book, the view was expressed that the doctrines of mechanics have developed out of the collected experiences of handicraft by an intellectual process of refinement. In fact, if we consider the matter without prejudice, we see that the savage discoverers of bow and arrows, of the sling, and of the javelin, set up the most important law of modern dynamics - the law of inertia - long before it was misunderstood with thorough-going perversity by Aristotle and his learned commentators. And although first ancient machines for throwing projectiles and catapults and then modern firearms brought this law daily be-fore our eyes, many centuries were needed before the correct theoretical idealization was discovered by the genius of Galileo and Newton. It lay in exactly the opposite direction to that in which the great majority of human beings expected it to lie. Not the conservation, but the decrease of the velocity of projection was to be theoretically explained and justified.

 

The simple machines - the five mechanical powers - as they are described by Hero of Alexandria in the work of which an Arabian translation came down to the Middle Ages, are without question a product of handicraft. If, now, a child busies himself with mechanical work with quite simple and primitive means - as was the case with my son, Ludwig Mach - the dynamical sensations observed in this connection and the dynamical experiences obtained when adaptive motions are made, make a powerful and lasting impression. If we pay attention to these sensations, we come closer, intellectually speaking, to the instinctive origin of machines. We understand why one prefers the longer lever yielding to the lesser pressure, and why a hammer swung round by its handle through a longer distance can transmit more work or vis viva.

We understand at once by experiment the transport of loads an rollers, and also how the wheel - the fixed roller - arose. The making of rollers must have gained a great technical importance and have led to the discovery of the lathe. In possession of this, mankind easily discovered the wheel, the wheel and axle, and the pulley. But the primitive lathe is the very ancient fire-drill of savages, which had a bow and cord, though of course this primitive lathe is only fitted for small objects.

The Arabians still use it, and, up to quite recent times, it was almost universally in use with our watchmakers. The potter's wheel of the ancient Egyptians was also a kind of lathe. Perhaps these forms served as models for the larger lathe, whose discovery, as well as that of the plumb-line and theodolite, is ascribed to Theodorus of Samos. On it pillars of stone may well have been turned (532 B. C.).

Not all knowledge finds immediate use; often it lies fallow for a long time.