We now turn to Galileo's discovery known as the theorem of the Parallelogram of Forces. The illusion which has been woven round this theorem expresses itself in the way it is described as being connected ideally with another theorem, outwardly similar in character, known as the theorem of the Parallelogram of Movements (or Velocities), by stating that the former follows logically from the latter. This statement is to be found in every textbook on physics at the outset of the chapter on dynamics (kinetics), where it serves to establish the right to treat the dynamic occurrences in nature in a purely kinematic fashion, true to the requirements of the onlooker-consciousness.1
The following description will show that, directly we free ourselves from the onlooker-limitations of our consciousness in the way shown by Goethe - and, in respect of the present problem, in particular also by Reid - the ideal relationship between the two theorems is seen to be precisely the opposite to the one expressed in the above statement. The reason why we take pains to show this at the present point of our discussion is that only through replacing the fallacious conception by the correct one, do we open the way for forming a concrete concept of Force and thereby for establishing a truly dynamic conception of nature.
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Let us begin by describing briefly the content of the two theorems in question. In Fig. 1, a diagrammatical representation is given of the parallelogram of movements. It sets out to show that when a point moves with a certain velocity in the direction indicated by the arrow a, so that in a certain time it passes from P to A, and when it simultaneously moves with a second velocity in the direction indicated by
b, through which alone it would pass to B in the same time, its actual movement is indicated by c, the diagonal in the parallelogram formed by a and b. An example of the way in which this
theorem is practically applied is the well-known case of a rower who sets out from P in order to cross at right angles a river indicated by the parallel lines. He has to overcome the velocity a of the water of the river flowing to the right by steering obliquely left towards B in order to arrive finally at C.
It is essential to observe that the content of this theorem does not need the confirmation of any outer experience for its discovery, or to establish its truth. Even though the recognition of the fact which it expresses may have first come to men through practical observation, yet the content of this theorem can be discovered and proved by purely logical means. In this respect it resembles any purely geometrical statement such as, that the sum of the angles of a triangle is two right angles (180°). Even though this too may have first been learnt through outer observation, yet it remains true that for the discovery of the fact expressed by it - valid for all plane triangles - no outer experience is needed. In both cases we find ourselves in the domain of pure geometric conceptions (length and direction of straight lines, movement of a point along these), whose reciprocal relationships are ordered by the laws of pure geometric logic. So in the theorem of the Parallelogram of Velocities we have a strictly geometrical theorem, whose content is in the narrowest sense kinematic. In fact, it is the basic theorem of kinematics.
We now turn to the second theorem which speaks of an outwardly similar relationship between forces. As is well nown, this states that
two forces of different magnitude and direction, when they apply at the same point, act together in the manner of a single force whose magnitude and direction may be represented by the diagonal of a parallelogram whose sides express in extent and direction the first two forces. Thus in Fig. 2, R exercises upon P the same effect as F1 and F2 together.