698 Dr. WHEWELL, ON HEGEL'S CRITICISM 



This is apparently in order to shew that the "lines" of the Newtonian construction are superfluous. 

 We know very well that analysis does not always refer to visihle representations of such lines : hut 

 we know too, (and Francoeur would testify to this also,) that the analytical proofs contain equiva- 

 lents to the Newtonian lines. We, in this place, are too familiar with the substitution of analytical 

 for geometrical proofs, to be led to suppose that such a substitution affects the substance of the truth 

 proved. The conversion of Newton's geometrical proofs of his discoveries into analytical processes 

 by succeeding writers, has not made them cease to be discoveries : and accordingly, those who have 

 taken the most prominent share in such a conversion, have been the most ardent admirers of New- 

 ton's genius and good fortune. 



So much for Newton's comparison of the Forces in different circular orbits, and for Hegel's power 

 of understanding and criticising it. Now let us look at the motion in different parts of tlie same 

 elliptical orbit, as a further illustration of the value of Hegel's criticism. In an elliptical orbit the 

 velocity alternately increases and diminishes. This follows necessarily from Kepler's law of the 

 equal description of the areas, and so Newton explains it. Hegel, however, treats of this acceleration 

 and retardation as a separate fact, and talks of another explanation of it, founded upon Centripetal 

 and Centrifugal Force (o). Where he finds this explanation, I know not; certainly not in Newton, 

 who in the second and third section of the Pri7icipia explains the variation of the velocity in a quite 

 different manner, as I have said; and nowhere, I think, employs centrifugal force in his explana- 

 tions. However, the notion of centrifugal as acting along with centripetal force is introduced in 

 some treatises, and may undoubtedly be used with perfect truth and propriety. How far Hegel 

 can judge when it is so used, we may see from what he says of the confusion produced by such an 

 explanation, which is, he says, a maximum. In the first place, he speaks of the motion being titii- 

 fornily accelerated and retarded in an elliptical orbit, which, in any exact use of the word imiformly, 

 it is not. But passing by this, he proceeds to criticise an explanation, not of the variable velocity 

 of the body in its orbit, but of the alternate access and recess of the body to and from the center. 

 Let us overlook this confusion also, and see what is the value of his criticism on the explanation. 

 He says (p), "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 in the perihelion itself the centri- 

 petal force is supposed suddenly to become greater than the centrifugal;" and so, of course, the 

 body re-ascends to the aphelion. 



Now I will not say that this explanation has never been given in a book professing to be scien- 

 tific ; but I have never seen it given ; and it never can have been given but by a very ignorant and 

 foolish person. It goes upon the utterly unmechanical supposition that the approach of a body to the 

 center at any moment depends solely upon the excess of the centripetal over the centrifugal force ; and 

 reversely. But the most elementary knowledge of mechanics shews us that when a body is moving 

 obliquely to the distance from the center, it approaches to or recedes from the center in virtue of this 

 obliquity, even if no force at all act. And the total approach to the center is the approach due to 

 this cause, plus the approach due to the centripetal force, minus the recess due to the centrifugal 

 force. At the aphelion, the centripetal is greater than the centrifugal force; and hence the motion 

 becomes oblique; and then, tlie body approaches to the center on both accounts, and approaches on 

 account of the obliquity of the path even when the centrifugal has become greater than the centri- 

 petal force, which it becomes before the body reaches the perihelion. This reasoning is so elemen- 

 tary, that when a person who cannot sec this, writes on the subject with an air of authority, I do 

 not see what can be done but to point out the oversight and leave it. 



But there is, says Hegel ((/), another way of explaining the motion by means of centripetal 

 and centrifugal forces. The two forces are supposed to increase and decrease gradually, according 

 to different laws. In this case, there must be a point where they are equal, and in equilibrio ; and 

 this being the case, they will always continue equal, for there will be no reason for their going 

 out of equilibrium. 



