OF NEWTON'S PRINCIPIA. 699 



This, which is put as another mode of explanation, is, in fact, the same mode; for, as I have 

 already said, the centrifugal force, which is less than the centripetal at the aphelion, becomes the 

 greater of the two before the perihelion ; and there is an intermediate position, at which the two forces 

 are equal. But at this point, is there no reason why, being equal, the forces should become unequal ? 

 Reason abundant : for the body, being there, moves in a line oblique to the distance, and so chanires 

 its distance ; and the centripetal and centrifugal force, depending upon the distance by different 

 laws, they forthwith become unequal. 



But these modes of explanation, by means of the centripetal and centrifugal forces and their 

 relation, are not necessary to Newton's doctrine, and are nowhere used by Newton , and undoubtedly 

 much confusion has been produced in other minds, as well as Hegel's, by speaking of the centrifugal 

 force, which is a mere intrinsic geometrical result of a body's curvilinear motion round a center, in 

 conjunction with centripetal force, which is an extrinsic force, acting upon the body and urgino- it to 

 the center. Neither Newton, nor any intelligent Newtonian, ever spoke of the centripetal and centri- 

 fugal force as two distinct forces both extrinsic to the motion, which Hegel accuses them of doino-. hi) 



I have spoken of the third and second of Kepler's laws ; of Newton's explanations of them, 

 and of Hegel's criticism. Let us now, in the same manner, consider the first law, that the planets 

 move in ellipses. Newton's proof that this was the result of a central force varyino- inversely as 

 the square of the distance, was the solution of a problem at which his contemporaries had laboured 

 in vain, and is commonly looked upon as an important step. " But," says Hegel, (rf) "the proof 

 gives a conic section generally, whereas the main point which ought to be proved is, tliat the path 

 of the body is an ellipse only, not a circle or any other conic section." Certainly if Newton had 

 proved that a planet cannot move in a circle, (which Hegel says he ought to have done), his 

 system would have perplexed astronomers, since there are planets which move in orbits hardly 

 distinguishable from circles, and the variation of the extremity from planet to planet shews that 

 there is notliing to prevent the excentricity vanishing and the orbit becoming a circle. 



" But," says Hegel again, (e) " the conditions which make the path to be an ellipse rather than 

 any other conic section, are empirical and extraneous ; — the supposed casual strength of the im- 

 pulsion originally received." Certainly the circumstances which determine the amount of excen- 

 tricity of a planet's orbit arc derived from experience, or rather, observation. It is not a part of 

 Newton's system to determine u priori what the excentricity of a planet's orbit must be. A system 

 that professes to do this will undoubtedly be one very different from his. And as our knowledge 

 of the excentricity is derived from observation, it is, in that sense, empirical and casual. The 

 strength of the original impulsion is a hypothetical and impartial way of expressing this result of 

 observation. And as we see no reason why the excentricity should be of any certain magnitude, 

 we see none why the fraction which expresses tlie excentricity should not become as large as unity, 

 that is, why the orbit should not become a parabola ; and accordingly, some of the bodies which 

 revolve about the same appear to move in orbits of this form : so little is the motion in an ellipse, 

 as Hegel .says, (/) " the only thing to be proved." 



But Hegel himself lias offered proof of Kepler's laws, to which, considering his objections to 

 Newton's proofs, we cannot help turning with some curiosity. 



And first, let us look at tlie proof of the Proposition which we have been considering, that the 

 path of a planet is necessarily an ellipse. I will translate Hegel's language as well as I can ; but 

 without answering for the correctness of my translation, since it docs not appear to me to conform 

 to the first condition of translation, of being intelligible. The translation however, such as it is, 

 may help us to form some opinion of the validity and value of Hegel's proofs as compared with 

 .Newton's, (r) 



" For absolutely uniform motion, the circle is tlie only palli . . .The circle is the line returning 



