Chapter XLIX: Appendix: To the Memoir on Hegel's Criticism of Newton's Principia

HEGEL. _Encyclopædia_ (2nd Ed. 1827), Part XI. p. 250.

C. _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
(_a_) general expression. Since then, (_a_) 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
(_b_) Formulæ may be derived from the Keplerian Laws. The
completely immediate derivation is this: In the third Keplerian
(_c_) Law, _A_^3/_T_^2 is the constant quantity. This being put
as _A.A_^2/_T_^2 and calling, with Newton, _A_/_T_^2 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.
(_e_) Prop. 1); and although 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 determined moment of time, and
(_f_) 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_. If the Newtonian
Form has not only its convenience but its necessity in reference
(_h_) 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; (on this subject I refer
(_i_) 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.

It is recognized that what Newton--besides the foundation of the
analytical treatment, the development of which, by the bye, has
of itself rendered superfluous, or indeed rejected much which
belonged to Newton's essential Principles and glory--has added
to the Keplerian Laws is the Principle of _Perturbations_,--a
Principle whose importance we may here accept thus far (hier in
(_k_) sofern anzuführen ist); namely, so far as it rests upon the
Proposition that the so-called attraction is an operation of all
(_l_) the individual parts of bodies, as being material. It lies
in this, that matter in general assigns a center for itself (sich
das centrum setzt), and the figure of the body is an element in the
determination of its place; that collective bodies of the system
recognize a reference to their Sun (sich ihre Sonne setzen), but
also the individual bodies themselves, according to the relative
position with regard to each other into which they come by their
general motion, form a momentary relation of their gravity
(schwere) _towards each other_, and are related to each other not
only in abstract spatial relations, but at the same time assign to
themselves a joint center, which however is again resolved [into
the general center] in the universal system.

(_m_) As to what concerns the features of the path, to show how
the fundamental determinations of Free Motion are connected _with
the Conception_, cannot here be undertaken in a satisfactory and
detailed manner, and must therefore be left to its fate. The proof
from reason of the quantitative determinations of free motion can
only rest upon the _determinations_ of _Conceptions_ of space and
time, the elements whose relation (intrinsic not extrinsic) motion
is.

That, _in the first place_, the motion in general is a motion
_returning into itself_, is founded on the determination of
particularity and individuality of the bodies in general (§ 269),
so that partly they have a center in themselves, and partly at the
same time their center in another. These are the determinations of
Conceptions which form the basis of the false representatives of
(_n_) Centripetal Force and Centrifugal Force, as if each of these
were self-existing, extraneous to the other, and independent of
it; and as if they only came in contact in their operations and
consequently _externally_. They are, as has already been mentioned,
the Lines which must be drawn for the mathematical determinations,
transformed into physical realities.

Further, this motion is _uniformly accelerated_, (and--as returning
into itself--in turn uniformly retarded). In motion as _free_,
Time and Space enter as _different_ things which are to make
(_o_) themselves effective in the determination of the motion
(§ 266, note). In the so-called _Explanation_ of the uniformly
accelerated and retarded motion, by means of the alternate
decrease and increase of the magnitude of the Centripetal Force
and Centrifugal Force, the _confusion_ which the assumption of
(_p_) such independent Forces produces is at its greatest 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
(_q_) Body, or till by the Preponderance of the Centrifugal 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 Determinations 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
(_t_) time appears as a Determination of two Dimensions,--as a
Superficies-Determination,--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
(_u_) the other by external [_conditions_]. 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 (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
(_w_) draw bodies to different sides as if they were two 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
(_x_) only for 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.

FOOTNOTES:

[Footnote 353: Cartes. _Princip._ iv. 23.]

[Footnote 354: Jac. Bernoulli, _Nouvelles Pensées sur le Système de M. Descartes_, op. t. i. p. 239 (1686).]

[Footnote 355: _De la Cause de la Pesanteur_ (1689), p. 135.]

[Footnote 356: _Journal des Savans_, 1703. Mém. Acad. Par. 1709.

Bulfinger, in 1726 (Acad. Petrop.), conceived that by making a sphere revolve at the same time about two axes at right angles to each other, every particle would describe a great circle; but this is not so.]

[Footnote 357: Acad. Par. 1714, _Hist._ p. 106.]

[Footnote 358: Acad. Par. 1733.]

[Footnote 359: Acad. Sc. 1709. If we abandon the clear principles of mechanics, the writer says, "toute la lumière que nous pouvons avoir est éteinte, et nous voilà replongés de nouveau dans les anciennes ténèbres du Peripatetisme, dont le Ciel nous veuille preserver!"

It was also objected to the Newtonian system, that it did not account for the remarkable facts, that all the motions of the primary planets, all the motions of the satellites, and all the motions of rotation, including that of the sun, are in the same direction, and nearly in the same plane; facts which have been urged by Laplace as so strongly recommending the Nebular Hypothesis; and that hypothesis is, in truth, a hypothesis of vortices respecting the _origin_ of the system of the world.]

[Footnote 360: _Nouvelle Physique Céleste_, Op. t. iii. p. 163.

The deviation of the orbits of the planets from the plane of the sun's equator was of course a difficulty in the system which supposed that they were carried round by the vortices which the sun's rotation caused, or at least rendered evident. Bernoulli's explanation consists in supposing the planets to have a sort of _leeway_ (_dérive des vaisseaux_) in the stream of the vortex.]

[Footnote 361: See _Hist. Sc. Ideas_, b. iii. c. ix. Art. 7.]

[Footnote 362: See Mill's _Logic_, vol. i. p. 311, 2nd ed.]

[Footnote 363: These letters refer to passages in the Translation annexed to this Memoir.]