700 Dr. WIIEWELL, ON HEGEL'S CRITICISM 



into itself in which all the radii are equal; there is, for it, only one determining quantity, the 

 radius. 



"But in free motion, the determination according to space and to time come into view with 

 differences. There must be a difference in the spatial aspect in itself, and therefore the form requires 

 two determining quantities. Hence the form of the path returning into itself is an ellipse." 



Now even if we could regard this as reasoning, the conclusion does not in the smallest degree 

 follow. A curve returning into itself and determined by two quantities, may have innumerable 

 forms besides the ellipse ; for instance, any oval form whatever, besides that of the conic section. 



But why must the curve be a curve returning into itself ? Hegel has professed to prove this 

 previously (m) from "the determination of particularity and individuality of the bodies in general, 

 so that they have partly a center in themselves, and partly at the same time their center in another." 

 Without seeking to find any precise meaning in this, we may ask whether it proves the impossi- 

 bility of the orbits with moveable apses, (which do not return into themselves,) such as the planets 

 (affected by perturbations) really do describe, and such as we know that bodies must describe in all 

 cases, except when the force varies exactly as the square of the distance .i" It appears to do so : and 

 it proves this impossibility of known facts at least as much as it proves anything. 



Let us now look at Hegel's proof of Kepler's second law, that the elliptical sectors swept by 

 the radius vector are proportional to the time. It is this : (s). 



" 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 JCepler's third law, that the cubes of the distances are as 

 the squares of the times. 



Hegel's proof in this case (7/) 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 deve- 

 loped Totality, it is a Totality in itself: it produces itself, and therein has a reference to itself. 

 And in tliis process, Time, being itself the diniensionless element, only comes to a formal identity 

 with itself and reaches the square : Space, on the other hand, as a positive external relation, tomes 

 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, (y) with remarkable complacency, " represents simply and immediately the 

 reason of the fkhig :—vihile on the contrary, the Newtonian Formula, by means of which the Law 



