Figure 47:

                     Heinrich Hertz: The Principles of Mechanics

 

Heinrich Hertz: Die Prinzipien der Mechanik. 1894.

 

First English translation by D. E. Jones and J. T. Walley:

Heinrich Hertz: The Principles of Mechanics Presented in a New Form. New York: Dover Publications 1956.

 

 

Introduction, pp. 1-3

 

The most direct, and in a sense the most important, problem which our conscious knowledge of nature should enable us to solve is the anticipation of future events, so that we may arrange our present affairs in accordance with such anticipation.

As a basis for the solution of this problem we always make use of our knowledge of events which have already occurred, obtained by chance observation or by prearranged experiment. In endeavouring thus to draw inferences as to the future from the past, we always adopt the following process. We form for ourselves images [innere Scheinbilder] or symbols of external objects; and the form which we give them is such that the necessary consequents of the images in thought are always the images of the necessary consequents in nature of the things pictured.

In order that this requirement may be satisfied, there must be a certain conformity between nature and our thought. Experience teaches us that the requirement can be satisfied, and hence that such a conformity does in fact exist. When from our accumulated previous experience we have once succeeded in deducing images of the desired nature, we can then in a short time develop by means of them, as by means of models, the consequences which in the external world only arise in a comparatively long time, or as the result of our own interposition. We are thus enabled to be in advance of the facts, and to decide as to present affairs in accordance with the insight so obtained.

The images which we here speak of are our conceptions of things. With the things themselves they are in conformity in one important respect, namely, in satisfying the above-mentioned, requirement. For our purpose it is not necessary that they should be in conformity with the things in any other respect whatever. As a matter of fact, we do not know, nor have we any means of knowing, whether our conceptions of things are in conformity with them in any other than this one fundamental respect.

 

The images which we may form of things are not determined without ambiguity by the requirement that the consequents of the images must be the images of the consequents. Various images of the same objects are possible, and these images may differ in various respects. We should at once denote as inadmissible all images which implicitly contradict the laws of our thought.

Hence we postulate in the first place that all our images shall be logically permissible - or, briefly, that they shall be permissible. We shall denote as incorrect any permissible images, if their essential relations contradict the relations of external things, i.e. if they do not satisfy our first fundamental requirement.

Hence we postulate in the second place that our images shall be correct. But two permissible and correct images of the same external objects may yet differ in respect of appropriateness. Of two images of the same object that is the more appropriate which pictures more of the essential relations of the object, - the one which we may call the more distinct. Of two images of equal distinctness the more appropriate is the one which contains, in addition to the essential characteristics, the smaller number of superfluous or empty relations, - the simpler of the two. Empty relations cannot be altogether avoided: they enter into the images because they are simply images, - images produced by our mind and necessarily affected by the characteristics of its mode of portrayal.

 

The postulates already mentioned are those which we assign to the images themselves: to a scientific representation of the images we assign different postulates. We require of this that it should lead us to a clear conception of what properties are to be ascribed to the images for the sake of permissibility, what for correctness, and what for appropriateness. Only thus can we attain the possibility of modifying and improving our images.

What is ascribed to the images for the sake of appropriateness is contained in the notations, definitions, abbreviations, and, in short, all that we can arbitrarily add or take away. What enters into the images for the sake of correctness is contained in the results of experience, from which the images are built up. What enters into the images, in order that they may be permissible, is given by the nature of our mind. To the question whether an image is permissible or not, we can without ambiguity answer yes or no; and our decision will hold good for all time. And equally without ambiguity we can decide whether an image is correct or not; but only according to the state of our present experience, and permitting an appeal to later and riper experience. But we cannot decide without ambiguity whether an image is appropriate or not; as to this differences of opinion may arise. One image may be more suitable for one purpose, another for another; only by gradually testing many images can we finally succeed in obtaining the most appropriate.

 

Those are, in my opinion, the standpoints from which we must estimate the value of physical theories and the value of the representations of physical theories. They are the standpoints from which we shall here consider the representations which have been given of the Principles of Mechanics.

 

 

 

Dynamical Models, pp. 175-177

 

418. Definition. A material system is said to be a dynamical model of a second system when the connections of the first can be expressed by such coordinates as to satisfy the following conditions: -

(1) That the number of coordinates of the first System is equal to the number of the second.

(2) That with a suitable arrangement of the coordinates for both systems the same equations of condition exist.

(3) That by this arrangement of the coordinates the expression for the magnitude of a displacement agrees in both Systems.

Any two of the coordinates so related to one another in the two systems are called corresponding coordinates. Corresponding positions, displacements, etc., are those positions, displacements, etc., in the two systems which involve similar values of the corresponding coordinates and their changes.

 

419. Corollary 1. If one system is a model of a second, then, conversely, the second is also a model of the first. If two systems are models of a third system, then each of these systems is also a model of the other. The model of the model of a system is also a model of the original system.

All systems which are models of one another are said to be dynamically similar.

 

420. Corollary 2. The property which one system possesses of being a model of another, is independent of the choice of the coordinates of one or the other system, although it is only clearly exhibited by a particular choice of coordinates.

 

421. Corollary 3. A system is not completely determined by the fact that it is a model of a given System. An infinite number of systems, quite different physically, can be models of one and the same system. Any given System is a model of an infinite number of totally different systems.

For the coordinates of the masses of the two systems which are models of one another can be quite different in number and can be totally different functions of the corresponding coordinates.

 

422. Corollary 4. The models of holonomous systems are themselves holonomous. The models of non-holonomous systems are themselves non-holonomous.

 

423. Observation. In order that a holonomous system may be a model of another, it is sufficient that both should have such free coordinates that the expression for the magnitude of the displacements of both systems should be the same.

 

424. Proposition. If two systems, each of which is a model of the other, have corresponding conditions at a definite time, then they have corresponding conditions at all times.

For by the equations of condition of a system, the expression for the magnitude of the displacement (§ 164) and the initial values of the coordinates and their change (§ 332), the course of these coordinates is determined for all times, - this being true whatever function of these coordinates the position of the masses of the system is.

 

425. Corollary 1. In order to determine beforehand the course of the natural motion of a material system, it is sufficient to have a model of that system. The model may be much simpler than the system whose motion it represents.

 

426. Corollary 2. If the same quantities are corresponding coordinates of a number of material systems which are models of one another, and if these corresponding coordinates alone are accessible to observation, then, so far as this limited observation is concerned, all these systems are not different from one another; they appear as like systems, however different in reality they may be in the number and the connection of their material points.

Thus it is impossible, from observation alone of the natural motions of a free system, i. e. without direct determination of its masses (§ 300), to obtain any wider knowledge of the connection of the system than that one could specify a model of the system.

 

427. Observation 1. If we admit generally and without limitation that hypothetical masses (§ 301) can exist in nature in addition to those which can be directly determined by the balance, then it is impossible to carry our knowledge of the connection of natural systems further than is involved in specifying models of the actual systems. We can then, in fact, have no knowledge as to whether the systems which we consider in mechanics agree in any other respect with the actual systems of nature which we intend to consider, than in this alone, - that the one set of systems are models of the other.

 

428. Observation 2. The relation of a dynamical model to the system of which it is regarded as the model, is precisely the same as the relation of the images which our mind forms of things to the things themselves. For if we regard the condition of the model as the representation of the condition of the system, then the consequents of this representation, which according to the laws of this representation must appear, are also the representation of the consequents which must proceed from the original object according to the laws of this original object. The agreement between mind and nature may therefore be likened to the agreement between two systems which are models of one another, and we can even account for this agreement by assuming that the mind is capable of making actual dynamical models of things, and of working with them.