As our further studies will show, the criticism to be applied here does not only leave the validity of measurement and the mathematical treatment of the data thus obtained fully intact, but by giving them their appropriate place in a wider conception of nature it opens the way to an ever more firmly grounded and, at the same time, enhanced application of both.

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Expressed in another way, a force of this magnitude working in the reverse direction (R') will establish an equilibrium with the other two forces. In technical practice, as is well known, this theorem is used for countless calculations, in both statics and dynamics, and indeed more frequently not in the form given here but in the converse manner, when a single known force is resolved into two component forces. (Distribution of a pressure along frameworks, of air pressure along moving surfaces, etc.)

As we see, in this experience of force that of mass is at once implied. Still, we can strengthen the latter by experimenting with some outer physical object. Take a fairly heavy object in your hand, stretch out your arm lightly and move it slowly up and down, watching intently the sensation this operation rouses in you.² Evidently the experience of mass outside ourselves, as with that of our own body, comes to us through the experience of the force which we ourselves must exert in order to overcome some resisting force occasioned by the mass. Already this simple observation - as such made by means of the sense of movement and therefore outside the frontiers of the onlooker-consciousness - tells us that mass is nothing but a particular manifestation of force.

Let us, therefore, transpose ourselves once more into the condition of the child who is still entirely volition, and thus experiences himself as one with the world. Let us consider, from the point of view of this condition, the process of lifting the body into the vertical position and the acquisition of the faculty of maintaining it in this position; and let us ask what the soul, though with no consciousness of itself, experiences in all this. It is the child's will which wrestles in this act with the dynamic structure of external space, and what his will experiences is accompanied by corresponding perceptions through the sense of movement and other related bodily senses. In this way the parallelogram of forces becomes an inner experience of our organism at the beginning of our earthly life. What we thus carry in the body's will-region in the form of experienced geometry - this, together with the freeing and crystallizing of part of our will-substance into our conceptual capacity, is transformed into our faculty of forming geometrical concepts, and among them the concept of the parallelogram of movements.

Our primary knowledge of the existence of something we call 'warmth' or 'heat' is due to a particular sense of warmth which modern research has recognized as a clearly definable sense. Naturally, seen from the spectator-standpoint, the experiences of this sense appear to be of purely subjective value and therefore useless for obtaining an objective insight into the nature of warmth and its effects in the physical world. In order to learn about these, resort is had to certain instruments which, through the change of the spatial position of a point, allow the onlooker-observer to register changes in the thermal condition of a physical object. An instrument of this kind is the thermometer. In the following way an indubitable proof seems to be given of the correctness of the view concerning the subjectivity of the impressions obtained through the sense of warmth, and of the objectivity of thermometrical measurement. A description of it is frequently given in physical textbooks as an introduction to the chapter on Heat.

It will now be our task to examine the logical link which is believed to connect one theorem with the other. This link is found in the well-known definition of physical force as a product of 'mass' and 'acceleration' - in algebraic symbols F=ma. We will discuss the implications of this definition in more detail later on. Let us first see how it is used as a foundation for the above assertion.

Seen in the light of this experience, the equation F=ma requires to be interpreted in a manner quite different from that to which scientific logic has submitted it. For if we have to ascribe to F and m the same quality, then the rule of multiplication allows us to ascribe to a nothing but the character of a pure number. This implies that there is no such thing as acceleration as a self-contained entity, merely attached to mass in an external way.

Looked at in this way, the true relationship between the two parallelogram-theorems is seen to be the very opposite of the one held with conviction by scientific thinking up to now. Instead of the parallelogram of forces following from the parallelogram of movements, and the entire science of dynamics from that of kinematics, our very faculty of thinking in kinematic concepts is the evolutionary product of our previously acquired intuitive experience of the dynamic order of the world.

To begin with, the well-known fact is cited that if one plunges one's hands first into two different bowls, one filled with hot water and the other with cold, and then plunges them together into a bowl of tepid water, this will feel cold to the hand coming from the hot water and warm to the hand coming from the cold. Next, it is pointed out that two thermometers which are put through the same procedure will register an equal degree of temperature for the tepid water. In this way the student is given a lasting impression of the superiority of the 'objective' recording of the instrument over the 'subjective' character of the experiences mediated by his sense of warmth.

The conception of 'force' as the product of 'mass' and 'acceleration' is based on the fact - easily experienced by anyone who cycles along a level road - that it is not velocity itself which requires the exertion of force, but the change of velocity - that is, acceleration or retardation ('negative acceleration' in the sense of mathematical physics); also that in the case of equal accelerations, the force depends upon the mass of the accelerated object. The more massive the object, the greater will be the force necessary for accelerating it. This mass, in turn, reveals itself in the resistance a particular object offers to any change of its state of motion. Where different accelerations and the same mass are considered, the factor m in the above formula remains constant, and force and acceleration are directly proportional to each other. Thus in the acceleration is discovered a measure for the magnitude of the force which thereby acts.