For Hegel, gravitating matter is matter in motion. The solar system, which is produced by gravity, is thus a system of moving bodies. From the perspective of philosophical mechan
ics, indeed, nothing else about celestial bodies needs to be taken into account: what makes the sun and the planets different kinds of gravitating body is not the physical fact that the sun is a source of light whereas the planets are not, but the simple mechanical fact that they move in different ways. In Hegel’s view, the character of celestial motion and the laws governing it are determined by the inherent gravity of matter itself in much the same way as free fall and Galileo’s law.
Stars have their centres within themselves and do not (as a matter of logical necessity) explicidy seek their centres in another body. Accordingly, their essential nature is not to move, but to remain at rest, in relation to one another and to the planets. As material, grav
itational bodies, of course, they cannot be absolutely immobile. Their motion, for Hegel, is, however, restricted to rotation around their own axis. Planets, by contrast, are not so unambiguously self-contained. This finds expression in the fact that ‘they seek their centre’
outside themselves and so always ‘leave their place and occupy another one’. Planets, there
fore, are necessarily in motion in relation to their sun.149
Planets, however, not only actively seek their centre within the sun, but also remain irre- ducibly independent of the sun. This finds expression in the fact that in their motion they preserve their distance from the sun and do not fall towards it. Accordingly, Hegel argues, planets move in an endless orbit around the sun, for only in this way can they show that they are both bound to the sun and free: ‘in altering their position at various distances from one another they describe a curve and return into themselves. For it is only in this way that they express their independence in regard to the central body; just as their unity with the central body finds expression in their motion round it in this same curve.’150 Similarly, comets must orbit the sun and moons must orbit their respective planets, because they, too, are inde
pendent bodies that explicitly seek to unite with a central body outside themselves.151 The orbital motion of the planets is thus made necessary by the fact that they are attracted to yet also set themselves apart from, and so repel> their sun. Planets combine explicit attraction and repulsion in this way because they are the most perfect and concrete embodiment of gravitational motion. What causes planets to orbit their sun, therefore, is nothing but gravity itself. In Hegel’s view, the characteristic motion of the planets does not arise due to the impact of other bodies or the ‘pull’ of an external centripetal force; rather, matter determines itself - through its inherent gravity - to move in a closed orbit around a central body. Planetary motion is thus not only necessary but also ‘absolutely free motion.
It is the closed, curvilinear motion that is intrinsic to the planets themselves.152
Hegel maintains that the laws governing planetary motion are also determined by the inherent gravity of matter. These laws were discovered by Johannes Kepler in the seven
teenth century and, in Hegel’s words, concern ‘the shape of the orbit and the velocity of the motion’. They are usually stated as follows:
1st Law: each planet moves in its own elliptical orbit around the sun, with the sun at one o f the two foci.
2nd Law: an imaginary line joining the sun to a planet moving in its orbit around the sun sweeps out equal areas in equal times.
3rd Law: the squares of the orbital periods of any two planets are proportional to the cubes of their distances from the sun.153
Hegel notes that ‘Kepler discovered his laws empirically by induction, based on the inves
tigations of Tycho Brahe5, but he believes that they can also be derived a priori from the nature of gravitational matter. He admits, however, that ‘the difficulty of the task is such that this has not yet been fully accomplished’.154 Hegel’s proofs of Kepler’s laws - like his proof of Galileo’s law of fall - are, in my view, a priori proofs whose presentation is made possible by a prior empirical discovery. In this case, however, Hegel openly admits that he did not work out those proofs as fully as he would have liked to.
Kepler’s laws are to be derived from the very nature of gravitational matter. From the mechanical point of view, however, matter is essentially motion. Motion in turn is gener
ated by the unity of space and time; that is, by place negating itself into another place and then another in time. To derive the laws of motion from the nature of gravitational matter is, therefore, to derive them from the nature of space and time that ultimately give rise to matter. More specifically, philosophy must show how the distinctive qualities of space and time determine the quantitative relations between space and time in planetary motion.155 The derivation of Kepler’s first law proceeds as follows.
Planets move in a closed orbit around their sun, so their motion is either circular or elliptical. When a body moves in a circular orbit, Hegel claims, it covers equal arcs of the orbit in equal times. The magnitude of the space - that is, the arc - travelled in a given time thus always remains the same and, in this sense, the body orbits with a uniform velo
city. Hegel recognizes that a circular orbit could conceivably be produced by a body accel
erating and decelerating over the course of its orbit, but he regards such a possibility as a
‘superficial’ product o f ‘abstract imagination’.156 In Hegel’s view, there is nothing about cir
cular orbital motion as such that requires it to involve acceleration, and it is more rational to conceive it as motion generated by covering equal arcs in equal times.
Whether one considers circular motion to be the product of uniform or accelerated motion, one thing is clear: regardless of their length, the arcs covered in equal times all have the same curvature and in that sense are all of the same kind. This is because they are all spanned between radii vectores of equal length. The radius vector is the straight line con
necting the orbiting body to its central body, and a circular orbit is defined by the fact that the radius vector remains constant throughout the orbit.157
Now planetary motion, for Hegel, is free motion; that is, motion determined by gravity and, ultimately, by the nature of space and time alone. As we saw in the discussion of falling bodies, however, space and time play different roles in the generation of free, gravitational motion: in this sense they remain free and independent of one another in their very unity.
In Hegel’s view, this qualitative difference between space and time must determine, and find expression in, their quantitative relation in planetary motion: ‘in the motion which is free, space and time come to assert themselves as what they are - that is, as different - in determining the magnitude of the motion’.158 But what precisely does this mean? It means that the quantitative relation between space and time in planetary motion must itself be one in which the difference and independence of space from time is clearly evident. Space proves its independence from time by constantly changing the relation in which it stands to time in the course of the motion. A planet in orbit around the sun must, therefore,
cover unequal distances or arcs in equal times. Hegel claims that this inequality must be twofold.
First, an orbiting planet must cover arcs of different lengths in equal times. This dictates that its velocity cannot be simply uniform, but must be ‘uniformly accelerated (and, as returning into itself, in turn uniformly retarded)}59 Hegel assumes here that free motion will be uniformly - rather than erratically - accelerated or decelerated, because it is self
determining and so (largely) unaffected by contingent factors. He also assumes that this suffices to prove that planetary orbits cannot be circular. Yet by his own admission it is con
ceivable - if only by ‘abstract imagination’ - that circular orbits can be generated by accel
erating (and decelerating) bodies.
Hegel argues, however, that if arcs covered in equal times are to be genuinely unequal and different from one another, they cannot differ in length alone but must also differ ‘in their function’. That is to say, they must play different roles in the constitution of the orbit.
Arcs of the same orbit perform different functions when they have different curvature, and they have different curvature only when they span radii vectores of different lengths. It is this fact that proves definitively that the orbit followed by a planet covering unequal arcs in equal times cannot be circular: because the radii vectores in a circular orbit are all equal rather than different:
if different arcs are traversed in the same time, then they must differ not only empirically but in regard to their function.. . . The radius - the relation of the circumference to the centre - belongs essentially to the function of an arc. If the arcs differ, then so must the radii too, and thus the Concept of the circle is overcome [aufgehoben]}60
Since planets move freely and continuously around the sun, with uniform acceleration and deceleration rather than in an erratic manner, their radii vectores must themselves increase and decrease in a regular way. The orbit followed by any such planet must, therefore, be elliptical
The passage just quoted is taken from the addition to §270 of the Philosophy of Nature.
In §270 itself Hegel presents his argument in a much more abstract form. As we have just seen, in free planetary motion the magnitudes of space and time are differentiated from one another not only because unequal spaces are covered in equal times but because those unequal spaces are themselves the product of two different determinations, namely the arc and the radius vector. In §270 Hegel condenses this idea into the following claim: time and space are distinct because time is simply a given magnitude whereas the space covered in that time is the product of a difference within space itself, ‘in free motion, where the determinations of space and time come together in diversity, in a qualitative relation, this relation necessarily emerges in the element o f space itself, as a difference of it, which accordingly demands two determinations’. It is this, Hegel maintains, that requires planets to move in elliptical orbits.
It has to be admitted that, taken by itself, Hegel’s condensed argument in the lines just quoted is barely intelligible. If we read those lines in conjunction with those from the addi
tion, however, his overall point is clear. The spaces covered in equal times are unequal due to differences in the respective arcs that are connected to differences in the respective radii vectores. The fact that these two determinations together differentiate the spaces from one another - as arcs with different curvature - is what makes planets move in elliptical orbits.
The elliptical orbit is thus the orbit that freely moving celestial bodies are required to follow by space and time themselves. Accordingly, it is inherent in the very nature of grav
itating matter. This is not to say that planets can never follow circular orbits, but only that
such orbits would be contingent from the philosophical point o f view. Nor is it to say that planets will follow perfectly elliptical orbits: Hegel admits that ‘observation shows that even the ellipse does not exactly correspond to the path of the planets, and so other perturba
tions must be assumed’ The task of philosophy, however, is simply to show that an ellip
tical orbit is the rational one for planets to follow - the one that is made logically necessary by gravitating matter itself. Accordingly, Hegel declares, ‘it will be for future astronomy to decide whether the path has not functions more profound than the ellipse, whether it is not perhaps an oval, etc.’161
In contrast to his proof of Galileo’s law of fall, Hegel’s proof of Kepler’s first law does not make direct reference to the specific qualities of space and time; that is, to ‘self- externality5 and ‘negativity5. It rests on the general idea that there is a qualitative difference between space and time and on the claim that in free, planetary motion their respective magnitudes must also be genuinely differentiated. For this reason, in planetary motion unequal spaces (or arcs) must be traversed in equal times. Hegel’s proof of Kepler’s second law does, however, make reference to the particular quality of space, albeit indirectly.
This proof is founded on the idea that space in free motion must not only be differen
tiated from time but also united with time in a single determination. That is to say, space must be proportional to and determined by time. Hegel reminds us that this is also the case in free fall: in such motion, as we recall, the space travelled is proportional to and determined by the square of the time elapsed (so that a body falling y metres in 1 second will, in 2 seconds, fall 4y metres rather than 2y metres). Although the time has to be squared in free fall in order to calculate the distance travelled, it is actually the quality of space that requires there to be any squaring at all. Space is being that is external to itself and so con
stantly extends itself beyond itself; it thereby augments itself by itself If this is to be reflected in the magnitude of the space travelled in free fall, that magnitude must also increase itself
‘solely in accordance with its own determinateness’ and so be a square.162
In the free motion of the planets the space that is proportional to the time must also be a ‘square’ and Hegel’s reference back to the law of fall suggests that now, as then, this is due to the quality or logical structure of space. In the case of planetary motion, however, the space concerned will not be proportional to the square of the time elapsed but will be a
‘squared’ space in its own right. That is to say, it will be a space that is itself a geometrical
‘square’ But what exactly does this mean? It does not mean that the space proportional to and determined by time will have the particular shape of a square as opposed to a trian
gle or pentagon. It means simply that such a space will be an area or a plane - measured in, for example, square kilometres - rather than a mere line. What area will this be? It will be the area that corresponds to the line, or arc, that is covered by the freely orbiting planet.
We have already seen in the proof of Kepler’s first law that the arcs traversed in free orbital motion span radii vectores of differing lengths. Indeed, the different curvature of each arc is determined by the different radii vectores that it connects: ‘the arc is in this way essentially a function of the radius vector [Radiusvektor] \ Hegel points out that a given arc and its two principal radii vectores together constitute a whole) namely the space that they enclose. This space is ‘a space-determination of two dimensions - the sector’. It is this sector, Hegel claims, that must be proportional to the time in free orbital motion. This is because such sectors constitute precisely the ‘squared space’ that must be proportional to time, if space and time are to be united in free motion as qualitatively distinct:
That the spatial determination should appear here, by means of time, as a two-dimensional deter
mination - as a plane - is connected with what was said above (§267) about the exposition of the same determinateness in fall. . . as space in the square [Quadrat]. Here, however, the quadraticity or
squaredness [das Quadratische] of space is, through the return of the Jine of motion into itself, confined to the sector.
Since the sectors - as ‘squared’ space - are directly proportional to the time elapsed, ‘equal sectors are swept out in equal times]’ as stated in Kepler’s second law of motion.163
It might seem that there is no more than a superficial verbal connection between the idea of a ‘squared’ number and a ‘squared’ space and that Hegel’s proof of Kepler’s second law is thus spurious. In Hegel’s view, however, there is a logical connection between the two ideas that goes beyond the fact that the same adjective is used in each case. What connects them is the fact that the specific quality of space requires the space that is proportional to time in free motion to have the following logical structure: it must become other than itself but, ‘in becoming an Other, [be] related solely to itself’.164
Now in free fall the space traversed is one-dimensional: it is the straight line of descent.
That space is thus not itself, and cannot ever be, explicitly self-relating space, since such space does not arise until the line connects up to itself and forms a two-dimensional plane (as we learn at the start of the philosophy of nature). Accordingly, the space in free fall cannot - simply through being the space that it is - satisfy the logical demand to become
‘other’ than itself and thereby explicitly‘relate to itself’ in a given time. This logical demand can only be met if the space or distance covered increases by a quantum or number that
‘becomes other than itself’ and in so doing ‘relates solely to itself’. Such a number that changes into another number purely by itself is, as we have seen, one that is squared. Thus,
‘in the abstract motion of falling, the squares. . . are only numerical determinations; the square is not to be taken in a spatial sense, because in falling what is traversed is only a straight line. It is this that constitutes the formal element in falling.’165 Precisely because space is not itself explicitly self-relating in free fall, however, it is not the numerical value of the space or distance travelled in a given time that is squared. Rather, it is the time that is squared, and the distance travelled is thus proportional to the square of the time.
In free orbital motion, by contrast, the space traversed is not merely a straight line but an arc that, together with its radii vectores, encloses a whole sector of space. Such a sector
In free orbital motion, by contrast, the space traversed is not merely a straight line but an arc that, together with its radii vectores, encloses a whole sector of space. Such a sector