What we designate as acceleration, and measure as such, is nothing else than a numerical factor comparing two different conditions of force within the physical-material world.

If this is the truth concerning the origin of our knowledge of force and its behaviour on the one hand, and our capacity to conceive mathematical concepts in a purely ideal way on the other, what is it then that causes man to dwell in such illusion as regards the relationship between the two? From our account it follows that no illusion of this kind could arise if we were able to remember throughout life our experiences in early childhood. Now we know from our considerations in Chapter VI that in former times man had such a memory. In those times, therefore, he was under no illusion as to the reality of force in the world. In the working of outer forces he saw a manifestation of spiritual beings, just as in himself he experienced force as a manifestation of his own spiritual being. We have seen also that this form of memory had to fade away to enable man to find himself as a self-conscious personality between birth and death. As such a personality, Galileo was able to think the parallelogram of forces, but he was unable to comprehend the origin of his faculty of mathematical thinking, or of his intuitive knowledge of the mathematical behaviour of nature in that realm of hers where she sets physical forces into action.

Let us now test this procedure by carrying out the same experiment with the help of thermometrical instruments in their original form, that is, the form in which Galileo first applied them. By doing so we proceed in a truly Goethean manner, because we divest the experiment of all accessories which prevent the phenomenon from appearing in its primary form.

Now it is logically evident that the theorem of the parallelogram of velocities is equally valid for movements with constant or variable velocities. Even though it is somewhat more difficult to perceive mentally the movement of a point in two different directions with two differently accelerated motions, and to form an inner conception of the resulting movement, we are nevertheless still within a domain which may be fully embraced by thought. Thus accelerated movements and movements under constant velocity can be resolved and combined according to the law of the parallelogram of movements, a law which is fully attainable by means of logical thought.

Only when we give the three factors in our equation this meaning, does it express some concrete outer reality. At the same time it forbids the use of this equation for a logical derivation of the parallelogram of forces from that of pure velocities.

Deep below in Galileo's soul there lived, as it does in every human being, the intuitive knowledge, acquired in early childhood, that part of nature's order is recordable in the conceptual language of mathematics. In order that this intuition should rise sufficiently far into his conscious mind to guide him, as it did, in his observations, the veil of oblivion which otherwise separates our waking consciousness from the experiences of earliest childhood must have been momentarily lightened. Unaware of all this, Galileo was duly surprised when in the onlooker-part of his being the truth of his intuition was confirmed in a way accessible to it, namely through outer experiment. Yet with the veil immediately darkening again the onlooker soon became subject to the illusion that for his recognition of mathematics as a means of describing nature he was in need of nothing but what was accessible to him on the near side of the veil.

To turn a modern thermometer into a thermoscope we need only remove the figures from its scale. If we make the experiment with two such thermoscopes we at once become aware of something which usually escapes us, our attention being fixed on the figures recorded by the two instruments. For we now notice that the two instruments, when transferred from the hot and cold water into the tepid water, behave quite differently. In one the column will fall, in the other it will rise.

With the help of the definition of force as the product of mass and acceleration it seems possible, indeed, to derive the parallelogram of forces from that of accelerations in a purely logical manner. For it is necessary only to extend all sides of an a parallelogram by means of the same factor m in order to turn it into an F parallelogram. A single geometrical figure on paper can represent both cases, since only the scale needs to be altered in order that the same geometrical length should represent at one time the magnitude a and on another occasion ma. It is in this way that present-day scientific thought keeps itself convinced that the parallelogram of forces follows with logical evidence from the parallelogram of accelerations, and that the discovery of the former is therefore due to a purely mental process.

The same method which has enabled us to restore its true meaning to the formula connecting mass and force will serve to find the true source of man's knowledge of the parallelogram of forces. Accordingly, our procedure will be as follows.

Thus it became man's fate in the first phase of science, which fills the period from Galileo and his contemporaries up to the present time, that the very faculty which man needed for creating this science prevented him from recognizing its true foundations. Restricted as he was to the building of a purely kinematic world-picture, he had to persuade himself that the order of interdependence of the two parallelogram-theorems was the opposite of the one which it really is.