On the Philosophy of Discovery, Chapters Historical and Critical

But, says Hegel, as further showing the artificial nature of the Newtonian formulæ, (h) "Analysis has long been able to derive the Newtonian expression and the laws therewith connected out of the Form of the Keplerian Laws;" an assertion, to verify which he refers to Francœur's Mécanique. This is apparently in order to show that the "lines" of the Newtonian construction are superfluous. We know very well that analysis does not always refer to visible representations of such lines: but we know too, (and Francœur would testify to this also,) that the analytical proofs contain equivalents 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 Newton'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 the 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 Principia explains the variation of the velocity in a quite different manner, as I have said; and nowhere, I think, employs centrifugal force in his explanations. 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 uniformly accelerated and retarded in an elliptical orbit, which, in any exact use of the word uniformly, 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 centripetal 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 scientific; 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 shows 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, the 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 centripetal force, which it becomes before the body reaches the perihelion. This reasoning is so elementary, that when a person who cannot see 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 (q), 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.

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 changes 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 urging it to the center. Neither Newton, nor any intelligent Newtonian, ever spoke of the centripetal and centrifugal force as two distinct forces both extrinsic to the motion, which Hegel accuses them of doing. (n)

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 varying 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, (d) "the proof gives a conic section generally, whereas the main point which ought to be proved is, that 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 shows that there is nothing 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 impulsion originally received." Certainly the circumstances which determine the amount of excentricity of a planet's orbit are derived from experience, or rather, observation. It is not a part of Newton's system to determine à 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 the 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, (f) "the only thing to be proved."

But Hegel himself has 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 the 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 does 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 the only path.... The circle is the line returning into itself in which all the radii are equal; there is, for it, only one determining quantity, the radius.

"But in free motion, the determination according to space and to time come into view with differences. There must be a difference in the spatial aspect in itself, and therefore the form requires two determining quantities. Hence the form of the path returning into itself is an ellipse."

Now even if we could regard this as reasoning, the conclusion does not in the smallest degree follow. A curve returning into itself and determined by two quantities, may have innumerable forms besides the ellipse; for instance, any oval form whatever, besides that of the conic section.

But why must the curve be a curve returning into itself? Hegel has professed to prove this previously (m) from "the determination of particularity and individuality of the bodies in general, so that they have partly a center in themselves, and partly at the same time their center in another." Without seeking to find any precise meaning in this, we may ask whether it proves the impossibility of the orbits with moveable apses, (which do not return into themselves,) such as the planets (affected by perturbations) really do describe, and such as we know that bodies must describe in all cases, except when the force varies exactly as the square of the distance? It appears to do so: and it proves this impossibility of known facts at least as much as it proves anything.

Let us now look at Hegel's proof of Kepler's second law, that the elliptical sectors swept by the radius vector are proportional to the time. It is this: (s).

"In the circle, the arc or angle which is included by the two radii is independent of them. But in the motion [of a planet] as determined by the conception, the distance from the center and the arc run over in a certain time must be compounded in one determination, and must make out a whole. This whole is the sector, a space of two dimensions. And hence 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."

As was said in the former case, if we could regard this as reasoning, it would not prove the conclusion, but only, that the arc is some function or other of the radii.

Hegel indeed offers (t) a reason why there must be an arc involved. This arises, he says, from "the determinateness [of the nature of motion], at one while as time in the root, at another while as space in the square. But here the quadratic character of the space is, by the returning of the line of motion into itself, limited to a sector."

Probably my readers have had a sufficient specimen of Hegel's mode of dealing with these matters. I will however add his proof of Kepler's third law, that the cubes of the distances are as the squares of the times.

Hegel's proof in this case (u) has a reference to a previous doctrine concerning falling bodies, in which time and space have, he says, a relation to each other as root and square. Falling bodies however are the case of only half-free motion, and the determination is incomplete.

"But in the case of absolute motion, the domain of free masses, the determination attains its totality. The time as the root is a mere empirical magnitude: but as a component of the developed Totality, it is a Totality in itself: it produces itself, and therein has a reference to itself. And in this process, Time, being itself the dimensionless element, only comes to a formal identity with itself and reaches the square: Space, on the other hand, as a positive external relation, comes to the full dimensions of the conception of space, that is, the cube. The Realization of the two conceptions (space and time) preserves their original difference. This is the third Keplerian law, the relation of the Cubes of the distances to the squares of the times."

"And this," he adds, (v) with remarkable complacency, "represents simply and immediately the reason of 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 and inversion of Reflexion, which stops half-way."

I am not able to assign any precise meaning to the Reflexion, which is here used as a term of condemnation, applicable especially to the Newtonian doctrine. It is repeatedly applied in the same manner by Hegel. Thus he says, (g) "that what Kepler expresses 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."

Though Hegel thus denies Newton all merit with regard to the explanation of Kepler's laws by means of the gravitation of the planets to the sun, he allows that to the Keplerian Laws Newton added the Principle of Perturbations (k). This Principle he accepts to a certain extent, transforming the expression of it after his peculiar fashion. "It lies," he says, (l) "in this: that matter in general assigns a center for itself: the collective bodies of the system recognise a reference to their sun, and all the individual bodies, according to the relative positions into which they are brought by their motions, form a momentary relation of their gravity towards each other."

This must appear to us a very loose and insufficient way of stating the Principle of Perturbations, but loose as it is, it recognises that the Perturbations depend upon the gravity of the planets one to another, and to the sun. And if the Perturbations depend upon these forces, one can hardly suppose that any one who allows this will deny that the primary undisturbed motions depend upon these forces, and must be explained by means of them; yet this is what Hegel denies.

It is evident, on looking at Hegel's mode of reasoning on such subjects, that his views approach towards those of Aristotle and the Aristotelians; according to which motions were divided into natural and unnatural;—the celestial motions were circular and uniform in their nature;—and the like. Perhaps it may be worth while to show how completely Hegel adheres to these ancient views, by an extract from the additions to the Articles on Celestial Motions, made in the last edition of the Encyclopædia. He says (w),

"The motion of the heavenly bodies is not a being pulled this way and that, as is imagined (by the Newtonians). They go along, as the ancients said, like blessed gods. The celestial conformity is not such a one as has the principle of rest or motion external to itself. It is not right to say because a stone is inert, and the whole earth consists of stones, and the other heavenly bodies are of the same nature as the earth, therefore the heavenly bodies are inert. This conclusion 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 other than celestial matter."

There can be no doubt that this is a very different doctrine from that of Newton.

I will only add to these specimens of Hegel's physics, a specimen of the logic by which he refutes the Newtonian argument which has just been adduced; namely, that the celestial bodies are matter, and that matter, as we see in terrestrial matter, is inert. He says (x),

"Doubtless both are 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 TO THE MEMOIR ON HEGEL'S CRITICISM OF NEWTON'S PRINCIPIAHegel. Encyclopædia (2nd Ed. 1827), Part XI. p. 250C. Absolute Mechanics§ 269

Gravitation is the true and determinate conception of material Corporeity, which (Conception) is realized to the Idea (zur Idee). General Corporeity is separable essentially into particular Bodies, and connects itself with the Element of Individuality or subjectivity, as apparent (phenomenal) presence in the Motion, which by this means is immediately a system of several Bodies.

Universal gravitation must, as to itself, be recognised as a profound thought, although it was principally as apprehended in the sphere of Reflexion that it eminently attracted notice and confidence on account of the quantitative determinations therewith connected, and was supposed to find its confirmation in Experiments (Erfahrung) pursued from the Solar System down to the phenomena of Capillary Tubes.—But Gravitation contradicts immediately the Law of Inertia, for in virtue of it (Gravitation) matter tends out of itself to the other (matter).—In the Conception of Weight, there are, as has been shown, involved the two elements—Self-existence, and Continuity, which takes away self-existence. These elements of the Conception, however, experience a fate, as particular forces, corresponding to Attractive and Repulsive Force, and are thereby apprehended in nearer determination, as Centripetal and Centrifugal Force, which (Forces) like weight, act upon Bodies, independent of each other, and are supposed to come in contact accidentally in a third thing, Body. By this means, what there is of profound in the thought of universal weight is again reduced to nothing; and Conception and Reason cannot make their way into the doctrine of absolute motion, so long as the so highly-prized discoveries of Forces are dominant there. In the conclusion which contains the Idea of Weight, namely, [contains this Idea] as the Conception which, in the case of motion, enters into external Reality through the particularity of the Bodies, and at the same time into this [Reality] and into their Ideality and self-regarding Reflexion, (Reflexion-in-sich), the rational identity and inseparability of the elements is involved, which at other times are represented as independent. Motion itself, as such, has only its meaning and existence in a system of several bodies, and those, such as stand in relation to each other according to different determinations.

§ 270

As to what concerns bodies in which the conception of gravity (weight) is realized free by itself, we say that they have for the determinations of their different nature the elements (momente) of their conception. One [conception of this kind] is the universal center of the abstract reference [of a body] to itself. Opposite to this [conception] stands the immediate, extrinsic, centerless Individuality, appearing as Corporeity similarly independent. Those [Bodies] however which are particular, which stand in the determination of extrinsic, and at the same time of intrinsic relation, are centers for themselves, and [also] have a reference to the first as to their essential unity.

The Planetary Bodies, as the immediately concrete, are in their existence the most complete. Men are accustomed to take the Sun as the most excellent, inasmuch as the understanding prefers the abstract to the concrete, and in like manner the fixed stars are esteemed higher than the Bodies of the Solar System. Centerless Corporeity, as belonging to externality, naturally separates itself into the opposition of the lunar and the cometary Body. The laws of absolutely free motion, as is well known, were discovered by Kepler;—a discovery of immortal fame. Kepler has proved these laws in this sense, that for the empirical data he found their general expression.

|(a)| Since then, it has become a common way of speaking to say 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 Formulæ may be derived from the Keplerian Laws.

|(b)| The completely immediate derivation is this: In the third Keplerian Law, A3/T2 is the constant quantity.

|(c)| This being put as A.A2/T2 and calling, with Newton, A/T2 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. I. 1. Sect. II. Prop. 1); and although it is the foundation of the Newtonian Theory, analysis has no longer any need of it.

|(f)| 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 determined 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 Reflection-Form of the Force of Gravitation.

|(h)| If the Newtonian Form has not only its convenience but its necessity in reference to the analytical method, this is only a difference of the mathematical formulæ; Analysis has long been able to derive the Newtonian expression, and the Propositions therewith connected, out of the Form of the Keplerian Laws;

|(i)| (on this subject I refer to the elegant exposition in Francœur's Traité Elém. de Mécanique, Liv. II. Ch. xi. n. 4.)—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 mentioned Accelerating Force and Vis Inertiæ, 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 explication to show how well founded they are: in a Compendium, propositions of this 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 presumptuous assertions. I will not, on this subject, support myself by saying, by the bye, that an interest in these subjects has occupied me for 25 years; but it is more precisely to the purpose to remark, that the distinctions and determinations which Mathematical Analysis introduces, and the course which it must take according to its method, is altogether different from that which a physical reality must have. The Presuppositions, the Course, and the Results, which the Analysis necessarily has and gives, remain quite extraneous to the considerations which determine the physical value and the signification of those determinations and of that course. To this it is that attention should be directed. We have to do with a consciousness relative to the deluging of physical Mechanics with an inconceivable (unsäglichen) Metaphysic, which—contrary to experience and conception—has those mathematical determinations alone for its source.