KEGELS CRITICISM OF NEWTON'S PRINCIPIA. 705 



(9) fugal Force it ends in a straight line. But now, if in place of the suddenness of the conversion, 

 we suppose a gradual increase of the Force in question, then, since rather the other Force ought 

 to be assumed as increasing, we lose the opposition wliich is assumed for the sake of the ex- 

 planation ; and if the increase of the one is assumed to be different from that of the other, 

 (which is the case in some representations,) then there is found at the mean distance between 

 the apsides a point in which the Forces are in equiUhrio And the transition of the Forces 

 out of Equilibrium is a thing just as little without any sufficient reason as the aforesaid 

 suddenness of inversion. And in the whole of this kind of explanation, we .see that the 

 mode of remedying a bad mode of dealing with a subject leads to newer and greater confu- 

 sion A similar confusion makes its appearance in the explanation of the phienomenon that 



the pendulum oscillates more slowly at the equator. This pha?nomenon is ascribed to the 

 Centrifugal Force, which it is asserted must then be greater ; but it is easy to see that we 

 may just as well ascribe it to the augmented gravity, inasmuch as that holds the pendulum 

 more strongly to the perpendicular line of rest. 



§ 240. 

 (r) And now first, as to what concerns the Form of the Path, the Circle only can be conceived 



as the path of an absulutely uniform motion Conceivable, as people express it, no doubt it 

 is, that an increasing and diminishing motion should take place in a circle. But this con- 

 ceivableness or possibility means only an abstract capability of being represented, which leaves 

 out of sight that Determinate Thing on which the question turns. 



The Circle is the line returning into itself in which all the radii are equal, that is, it is 

 completely determined by means of the radius. There is only 07ie Determination, and that 

 is the whole Determination. 



But in free motion, in which the Determinations according to space and according to time 

 come into view with Differences, in a qualitative relation to each other, this Relation appears 

 in the spatial aspect as a Difference thereof in itself, which therefore requires two Deter- 

 minations. Hereby the Form of the path returning into itself is essentially an Ellipse. 



(») The abstract Determinateness which produces the circle appears also in this way, that the 



arc or angle which is included by two Radii is independent of them, a magnitude with reo-ard 

 to them completely empirical. But since in the motion as determined by the Conception, the 

 distance from the center, and the arc which is run over in a certain time, must be compre- 

 hended in one determinateness, [a?iti] make out a whole, this is the sector, a space-deter- 

 mination of two dimensions : in this way, 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. 

 That the determination with regard to the space by means of the time appears as a Deter- 

 mination of two Dimensions, — as a Superficies-Determination, — agrees with what was said 



(/) before (^ 26(5) respecting Falling Bodies, with regard to the exposition of the same Deter- 

 minateness, at one while as Time in the root, at another while as Space in the square. Here, 

 however, the Quadratic character of the space is, by the returning of the Line of motion into 

 itself, limited to a Sector. These are, as may be seen, the general principles on which the 

 Keplerian Law, that in equal times equal sectors are cut off", rests. 



This Law becomes, as is clear, only the relation of the arc to the Radius Vector, and the 

 Time enters there as the abstract Unity, in which the different Sectors are compared, because 

 as Unity it is the Determining Element. But the further relation is that of the Time, not 

 as Ilmty, but as a Quantity in general, — as the time of Revolution — to tile magnitude of the 

 Path, or, what is tile same tiling, the distance from the center. As R( ot and Siiuarc, we saw 

 that Time and Space had a relation to each other, in the case of Falling Bodies, the case of 

 half-free motion — because that [inotion\ is determined on one side by the conception, on the 



