decrease and increase of the magnitude of the Centripetal Force and Centrifugal Force, the confusion which the assumption of such independent Forces produces is at its greatest (p) height. According to this explanation, in the motion of a Planet from the Aphelion to the Perihelion, the centrifugal is less than the centripetal force, and on the contrary, in the Perihelion itself, the centrifugal force is supposed to become greater than the centripetal. For the motion from the Perihelion to the Aphelion, this representation makes the forces pass into the opposite relation in the same manner. It is apparent that such a sudden conversion of the preponderance which a force has obtained over another, into an inferiority to the other, cannot be anything taken out of the nature of Forces. On the contrary it must be concluded, that a preponderance which one Force has obtained over another must not only be preserved, but must go onwards to the complete annihilation of the other Force, and the motion must either, by the Preponderance of the Centripetal Force, proceed till it ends in rest, that is, in the Collision of the Planet with the Central Body, or till by the Preponderance of the Centrifugal (9) 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 which is assumed for the sake of the explanation; 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 equilibrio. 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 confusion.-A similar confusion makes its appearance in the explanation of the phænomenon that the pendulum oscillates more slowly at the equator. This phænomenon 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 absolutely 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 conceivableness 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 one 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 Determinations. Hereby the Form of the path returning into itself is essentially an Ellipse.

(s) 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 regard 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 comprehended in one determinateness, [and] make out a whole, this is the sector, a space-determination 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 Determination of two Dimensions, as a Superficies-Determi(t) nation, agrees with what was said before (§ 266) respecting Falling Bodies, with regard to the exposition of the same Determinateness, 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 Unity, but as a Quantity in general,- -as the time of Revolution-to the magnitude of the Path, or, what is the same thing, the distance from the center. As Root and Square, 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 [motion] is determined on one side by the conception, on the other by external [conditions]. But in the case of absolute motion-the domain (u) of free masses the determination attains its Totality. The Time as the Root is a mere empirical magnitude; but as a component (moment) of the developed Totality, it is a Totality in itself,-it produces itself, and therein has a reference to itself; as the Dimensionless Element in itself, it only comes to a formal identity with itself, the Square; Space, on the other hand, as the positive Distribution (aussereinander) [comes] to the Dimension of the Conception, the CUBE. Their (v) Realization preserves their original difference. This is the third Keplerian Law, the relation of the Cubes of the Distances to the Squares of the Times ;-a Law which is so great on this account, that it represents so simply and immediately Reason as belonging to 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, Perversion and Inversion of Reflexion which stops half-way.

Additions to new Edition. § 269.

The center has no sense without the circumference, nor the circumference without the center. This makes all physical hypotheses vanish which sometimes proceed from the center, sometimes from the particular bodies, and sometimes assign this, sometimes that, as the original [cause of motion]...It is silly (läppisch) to suppose that the centrifugal force, as a tendency to fly off in a Tangent, has been produced by a lateral projection, a projectile force, an impulse which they have retained ever since they set out on their journey (von Haus aus). Such casualty of the motion produced by external causes belongs to inert matter; as when a stone fastened to a thread which is thrown transversely tries to fly from the thread. We are not to talk in this way of Forces. If we will speak of Force, there is one Force, whose elements

do not draw bodies to different sides as if they were two (w) Forces. The motion of the heavenly bodies is not a being pulled this way or that, such as is thus imagined; it is free motion: they go along, as the ancients said, as blessed Gods (sie gehen als selige Götter einher). The celestial corporeity is not such a one as has the principle of rest or motion external to itself. Because stone is inert, and all the earth consists of stones, and the other heavenly bodies are of the same nature,—is a conclusion which makes the properties of the whole the same as those of the part. Impulse, Pressure, Resistance, Friction, Pulling, and the like, are valid only for (r) an existence of matter other than the celestial. Doubtless

that which is common to the two is matter, as a good thought and a bad thought are both thoughts; but the bad one is not therefore good, because it is a thought.

APPENDIX K.

DEMONSTRATION THAT ALL MATTER IS HEAVY.

(Cam. Phil. Soc. FEB. 22, 1841.)

THE discussion of the nature of the grounds and proofs of the most general propositions which the physical sciences include, belongs rather to Metaphysics than to that course of experimental and mathematical investigation by which the sciences are formed. But such discussions seem by no means unfitted to occupy the attention of the cultivators of physical science. The ideal, as well as the experimental side of our knowledge must be carefully studied and scrutinized, in order that its true import may be seen; and this province of human speculation has been perhaps of late unjustly depreciated and neglected by men of science. Yet it can be prosecuted in the most advantageous manner by them only: for no one can speculate securely and rightly respecting the nature and proofs of the truths of science without a steady possession of some large and solid portions of such truths. A man must be a mathe. matician, a mechanical philosopher, a natural historian, in order that he may philosophize well concerning mathematics, and mechanics, and natural history; and the mere metaphysician who without such preparation and fitness sets himself to determine the grounds of mathematical or mechanical truths, or the principles of classification, will be liable to be led into error at every step. He must speculate by means of general terms, which he will not be able to use as instruments of discovering and conveying philosophical truth, because he cannot, in his own mind, habitually and familiarly, embody their import in special examples,

Acting upon such views, I have already laid before the Philosophical Society of Cambridge essays on such subjects as I here refer to; especially a memoir "On the Nature of the Truth of the Laws of Motion," which was printed by the Society in its Transactions.