APPENDIX TO Dr. WHEWELLS MEMOIR ON HEGEL'S CRITICISM, ETC. 70S 



immortal fame. Kepler has proved these laws in this sense, that for the empirical data he 

 found tiieir general expression. Since then, it has become a common way of speaking to say 

 (a) that Newton first found out the proof of these Laws. It has rarely happened that fame has 

 been more unjustly transferred from the first discoverer to another person. On this subject I 

 make the following remarks. 



1. That it is allowed by Mathematicians that the Newtonian Formula may be derived 

 (6) from the Keplerian Laws. The completely immediate derivation is this : In the third Kep- 



(c) lerian Law, — is the constant quantity. This being put as -~^, and calling, with Newton, 

 — universal Gravitation, his expression of the effect of gravity in the reciprocal ratio of the 

 square of the distances is obvious. 



(d) 2. That the Newtonian proof of the Proposition that a body subjected to the Law of 

 Gravitation moves about the central body in an Ellipse, gives a Conic Section generally, 

 while the main Proposition which ought to be proved is that the fall of such a Body is not a 

 Circle or any other Conic Section, but an Ellipse only. Moreover, there are objections which 

 may be made against this proof in itself; (Princ. Math. 1. 1. Sect. ii. Prop, l.) and althouo-h 



(e) it is the foundation of the Newtonian Theory, analysis has no longer any need of it. The 

 conditions which in the sequel make the path of the Body to a determinate Conic Section, are 

 referred to an empirical circumstance, namely, a particular position of the Body at a deter- 

 mined moment of time, and the casual strength of an impulsion which it is supposed to have 



(/) received originally; so that the circumstance which makes the Curve be an Ellipse, which 

 alone ought to be the thing proved, is extraneous to the Formula. 



3. That the Newtonian Law of the so-called Force of Gravitation is in like manner only 

 proved from experience by Induction. 



(g) The sum of the difference is this, that what Kepler expressed in a simple and sublime 



manner in the Form of Laws of the Celestial Motions, Newton has metamorphosed into the 

 Reflexion- Form of the Force of Gravitation. If the Newtonian Form has not only its con- 

 venience but its necessity in reference to the analytical method, this is only a difference of the 



(Ji) mathematical formula; ; Analysis has long been able to derive the Newtonian expression, and 

 the Propositions therewith connected, out of the Form of the Keplerian Laws ; (on this subject 

 I refer to the elegant exposition in Francoeur's Traite Elem. de Mecanique, Liv. ii. Ch. xi. 



(j) n. iv.) — The old method of so-called proof is conspicuous as offering to us a tangled web, 

 formed of the Lines of the mere geometrical construction, to which a physical meaning of 

 independent Forces is given ; and of empty Reflexion-determinations of the already men- 

 tioned Accelerating Force and Vis InerticB, and especially of the relation of the so-called 

 gravitation itself to the centripetal force and centrifugal force, and so on. 



The remarks which are here made would undoubtedly have need of a further explica- 

 tion to shew how well founded they are: in a Comjjendiuni, projjositions of tliis kind which 

 do not agree with that which is assumed, can only have the shape of assertions. Indeed, 

 since they contradict such high authorities, they must appear as something worse, as pre- 

 sumptuous assertions. I will not, on this subject, support myself by saying, by the bye, 

 that an interest in these subjects has occupied me for S.") years ; but it is more precisely to 

 the purpose to remark, that the distinctions and determinations which Mathematical Analysis 

 introduces, and the course wiiicii it must take according to its nietliod, is nltogctiier did'crent 

 from that which a pliysical reality must have. Tlie Presuppositions, the (.'oursc, and tlie 

 Results, which the Analysis necessarily has and gives, remain quite extraneous to tlie considera- 

 tions which determine the physical value and the signification of those determinatiuns and of 



4. x'2 



