Einstein, the searcher

by Alexander Moszkowski

Schopenhauer has, however, quite forgotten that he himself, some pages earlier, trumpeted forth Newton's imperishable fame with the words: "To form an estimate of the great value of the gravitational system which was at least completed and firmly established by Newton, we must remind ourselves how entirely nonplussed about the origin of the motion of celestial bodies thinkers had previously been for thousands of years." That bears the ring of truth. Newton's greatness can be grasped only if thousands of years are used as a measure.

Whereas Schopenhauer argued from grounds drawn from psychology and the principle of universal knowledge, his antagonist Hegel, who was still more vague in these fields, sought to dispense with both Newton and Kepler by calling to his aid the so-called pure intuition of the curved line. In an exposition of truly comical prolixity, such as would have delighted the hearts of scholiasts, he proves that the ellipse must represent the fundamental type of planetary motion, this being quite independent of Newton's laws, Kepler's observations, and resulting mathematical relationships. And Hegel actually succeeds, with a nebulous verbosity almost stultifying in its unmeaningness, in paraphrasing Kepler's second law in his own fashion. It reads like an extract from some carnival publication issued by scientists in a bibulous mood to make fun of themselves.

But these extravagances, too, serve to add lustre to Newton, for his genius shines out most brilliantly when it is a question of expressing clearly, and without assumptions, a phenomenon of cosmic motion. Here there are no forerunners, not even with regard to his own law of gravitation. Newton showed with truly triumphant logic that Kepler's second law belongs to those things that are really self-evident.

This law, taken alone, offers considerable difficulties to anyone who learns of it for the first time. Every planet describes an ellipse; that is accepted without demur. But the uninitiated will possibly or even probably deduce from this that the planet will pass over equal lengths of arc in equal times. By no means, says Kepler; the arcs traversed in equal times are unequal. But if we connect every point of the elliptic path with a definite point within the curve (the focus of the ellipse) by means of straight lines, each of which is called a radius vector, we get that the areas swept out by the radius vector in equal times (and not the arcs) are equally great.

Why is this so? This cannot be understood a priori. But one might argue that since the attraction of the sun is the governing force, this will probably have something to do with Newton's law of gravitation, in particular with the inverse square of the distance. And one might further infer that, if a different principle of gravitation existed, Kepler's law would assume a new form.

A fact amazing in its simplicity here comes to light. Newton states the proposition: "According to whatever law an accelerating force acts from a centre on a body moving freely, the radius vector will always sweep out equal areas in equal lengths of time."

Nothing is assumed except the law of inertia and a little elementary mathematics, namely, the theorem that triangles on the same base and of the same altitude are equal in area. The form in which this theorem occurs in Newton's simple drawing is certainly astonishing. One feels that there in a few strokes a cosmic problem is solved; the impression is ineffaceable.

This theorem together with its proof is contained in Newton's chief work, Philosophiæ naturalis principia mathematica. The interfusion of philosophy and mathematics furnished him with the natural principles of knowledge.*

Einstein made some illuminating remarks about Newton's famous phrase: "Hypotheses non fingo." I had said that Newton must have been aware that it is impossible to build up a science entirely free from hypotheses. Even geometry itself has arrived at that critical stage at which Gauss and Riemann discovered its hypothetical foundations.

Einstein replied: "Accentuate the words correctly and the true sense will reveal itself!" It is the last word that is to be stressed and not the first. Newton did not want to feel himself free from hypotheses, but rather from the assumption that he invented them, except when this was absolutely necessary. Newton, then, wished to express that he did not go further back in his analysis of causes than was absolutely inevitable.