Lecture I

And now I beg you to remember the so-called Parallelogram of forces, (Figure 1b). This time, the point a will signify a material thing—some little grain of material substance. I exert a force to draw it on from \(a\) to \(b\). Mark the difference between the way I am now speaking and the way I spoke before. Before, I spoke of movement as such; now I am saying that a force draws the little ball from \(a\) to \(b\). Suppose the measure of this force, pulling from \(a\) to \(b\), to be five grammes; you can denote it by a corresponding length in this direction. With a force of five grammes I am pulling the little ball from \(a\) to \(b\). Now I might also do it differently. Namely I might first pull with a certain force from \(a\) to \(c\). Pulling from \(a\) to \(c\) (with a force denoted by this length) I need a different force than when I pulled direct from \(a\) to \(b\). Then I might add a second pull, in the direction of the line from \(c\) to \(b\), and with a force denoted by the length of this line. Having pulled in the first instance from a towards \(b\) with a force of five grammes, I should have to calculate from this figure, how big the pull \(ac\) and also how big the pull \(cd\) would have to be. Then if I pulled simultaneously with forces represented by the lines \(ac\) and \(ad\) of the parallelogram, I should be pulling the object along in such a way that it eventually got to \(b\); thus I can calculate how strongly I must pull towards \(c\) and \(d\) respectively. Yet I cannot calculate this in the same way as I did the displacements in our previous example. What I found previously (as to the movement pure and simple), that I could calculate, purely in thought. Not so when a real pull, a real force is exercised. Here I must somehow measure the force; I must approach Nature herself; I must go on from thought to the world of facts. If once you realize this difference between the Parallelogram of Movements and that of Forces, you have a clear and sharp formulation of the essential difference between all those things that can be determined within the realm of thought, and those that lie beyond the range of thoughts and mental pictures. You can reach movements but not forces with your mental activity. Forces you have to measure in the outer world. The fact that when two pulls come into play—the one from \(a\) to \(c\), the other from \(a\) to \(d\),—the thing is actually pulled from \(a\) to \(b\) according to the Parallelogram of Forces, this you cannot make sure of in any other way than by an outer experiment. There is no proof by dint of thought, as for the Parallelogram of Movements. It must be measured and ascertained externally. Thus in conclusion we may say: while we derive the parallelogram of movements by pure reasoning, the parallelogram of forces must be derived empirically, by dint of outer experience. Distinguishing the parallelogram of movements and that of forces, you have the difference—clear and keen—between Phoronomy and Mechanics, or Kinematics and Mechanics. Mechanics has to do with forces, no mere movements; it is already a Natural Science. Mechanics is concerned with the way forces work in space and time. Arithmetic, Geometry and Kinematics are not yet Natural Sciences in the proper sense. To reach the first of the Natural Sciences, which is Mechanics, we have to go beyond the life of ideas and mental pictures.