On the Philosophy of Discovery, Chapters Historical and Critical

by William Whewell

"In the circle, the arc or angle which is included by the two radii is independent of them. But in the motion [of a planet] as determined by the conception, the distance from the center and the arc run over in a certain time must be compounded in one determination, and must make out a whole. This whole is the sector, a space of two dimensions. And hence the arc is essentially a Function of the radius vector; and the former (the arc) being unequal, brings with it the inequality of the radii."

As was said in the former case, if we could regard this as reasoning, it would not prove the conclusion, but only, that the arc is some function or other of the radii.

Hegel indeed offers (t) a reason why there must be an arc involved. This arises, he says, from "the determinateness [of the nature of motion], at one while as time in the root, at another while as space in the square. But here the quadratic character of the space is, by the returning of the line of motion into itself, limited to a sector."

Probably my readers have had a sufficient specimen of Hegel's mode of dealing with these matters. I will however add his proof of Kepler's third law, that the cubes of the distances are as the squares of the times.

Hegel's proof in this case (u) has a reference to a previous doctrine concerning falling bodies, in which time and space have, he says, a relation to each other as root and square. Falling bodies however are the case of only half-free motion, and the determination is incomplete.

"But in the case of absolute motion, the domain of free masses, the determination attains its totality. The time as the root is a mere empirical magnitude: but as a component of the developed Totality, it is a Totality in itself: it produces itself, and therein has a reference to itself. And in this process, Time, being itself the dimensionless element, only comes to a formal identity with itself and reaches the square: Space, on the other hand, as a positive external relation, comes to the full dimensions of the conception of space, that is, the cube. The Realization of the two conceptions (space and time) preserves their original difference. This is the third Keplerian law, the relation of the Cubes of the distances to the squares of the times."

"And this," he adds, (v) with remarkable complacency, "represents simply and immediately the reason of the thing:—while on the contrary, the Newtonian Formula, by means of which the Law is changed into a Law for the Force of Gravity, shows the distortion and inversion of Reflexion, which stops half-way."

I am not able to assign any precise meaning to the Reflexion, which is here used as a term of condemnation, applicable especially to the Newtonian doctrine. It is repeatedly applied in the same manner by Hegel. Thus he says, (g) "that what Kepler expresses in a simple and sublime manner in the form of Laws of the Celestial Motions, Newton has metamorphosed into the Reflexion-Form of the Force of Gravitation."

Though Hegel thus denies Newton all merit with regard to the explanation of Kepler's laws by means of the gravitation of the planets to the sun, he allows that to the Keplerian Laws Newton added the Principle of Perturbations (k). This Principle he accepts to a certain extent, transforming the expression of it after his peculiar fashion. "It lies," he says, (l) "in this: that matter in general assigns a center for itself: the collective bodies of the system recognise a reference to their sun, and all the individual bodies, according to the relative positions into which they are brought by their motions, form a momentary relation of their gravity towards each other."

This must appear to us a very loose and insufficient way of stating the Principle of Perturbations, but loose as it is, it recognises that the Perturbations depend upon the gravity of the planets one to another, and to the sun. And if the Perturbations depend upon these forces, one can hardly suppose that any one who allows this will deny that the primary undisturbed motions depend upon these forces, and must be explained by means of them; yet this is what Hegel denies.