4 9 4 D O C U M E N T 5 0 3 N E W T O N ’ S M E C H A N I C S These laws gave a complete answer to the question how the planets moved round the sun (elliptical orbit, equal areas described by the radius vector in equal periods, relation between semi-major axis and period of revolution). But these rules do not satisfy the requirement of causality. The three rules are logically inde- pendent of one another, and show no sign of any interconnection. The third law cannot be extended numerically as it stands from the sun to another central body there is for instance, no relation between a planet’s period of revolution round the sun and the period of revolution of a moon round its planet. But the principal thing is that these laws have reference to motion as a whole, and not to the question how there is developed from one condition of motion of a system that which immediately follows it in time. They are, in our phraseology of today, integral laws, and not differential laws. The differential law is the form which alone entirely satisfies the modern phys- icist’s requirement of causality. The clear conception of the differential law is one of the greatest of Newton’s intellectual achievements. What was needed was not only the idea but a formal mathematical method which was, indeed, extant in rudi- ment but had still to gain a systematic shape. This also Newton found in the differ- ential and integral calculus. It is unnecessary to consider whether Leibnitz arrived at these same mathematical methods independently of Newton or not in any case, their development was a necessity for Newton, as they were required in order to give Newton the means of expressing his thought. Galileo had already made a significant first step in the recognition of the law of motion. He discovered the law of inertia and the law of free falling in the earth’s field of gravitation a mass (or, more accurately, a material point) uninfluenced by other masses moves uniformly in a straight line the vertical velocity of a free body increases in the field of gravity in proportion to the time. It may seem to us today to be only a small step from Galileo’s observations to Newton’s laws of motion. But it has to be observed that the two propositions above, in the form in which they are given, relate to motion as a whole, while Newton’s law of motion gives an answer to the question: How does the condition of motion of a point-mass change in an in- finitely small period under the influence of an external force? Only after proceed- ing to consider the phenomenon during an infinitely short period (differential law) does Newton arrive at a formula which is applicable to all motions. He takes the conception of force from the already highly developed theory of statics. He is only able to connect force with acceleration by introducing the new conception of mass, which, indeed, is supported curiously enough by a pseudo-definition.[3] Today we are so accustomed to forming conceptions which correspond to differential quo- tients that we can hardly realize any longer how great a capacity for abstraction was [p. 274]