Dynamics of the celestial spheres

Classical and medieval scientists hypothesized about the dynamics of the celestial spheres, the various heavens nested one within the other with the earth at the center.

Inertia in the celestial spheres

However, the motions of the celestial spheres came to be seen as presenting a major anomaly for Aristotelian dynamics, and as even refuting its general law of motion v α F/R. According to this law all motion is the product of a motive force (F) and some resistance to motion (R), and whose ratio determines its average speed (v). And the ancestor of the central concept of Newtonian dynamics, the concept of the force of inertia as an inherent resistance to motion in all bodies, was born out of attempts to resolve it. This problem of celestial motion for Aristotelian dynamics arose as follows.

In Aristotle's sublunar dynamics all motion is either 'natural' or 'violent'. Natural motion is motion driven solely by the body's own internal 'nature' or gravity (or levity), that is, a centripetal tendency to move straight downward towards their natural place at the centre of the Earth (and universe) and to be at rest there. And its contrary, violent motion, is simply motion in any other direction whatever, including motion along the horizontal. Any such motion is resisted by the body's own 'nature' or gravity, thus being essentially anti-gravitational motion.

Hence gravity is the driver of natural motion, but a brake on violent motion, or as Aristotle put it, a 'principle of both motion and rest'. And gravitational resistance to motion is virtually omni-directional, whereby in effect bodies have horizontal 'weight' as well as vertically downward weight. The former consists of a tendency to be at rest and resist motion along the horizontal wherever the body may be on it (technically termed an inclinatio ad quietem in scholastic dynamics, as distinct from its tendency to centripetal motion as downwards weight that resists upward motion (technically termed an inclinatio ad contraria in scholastic dynamics).

The only two resistances to sublunar motion Aristotle identified were this gravitational internal resistance just to violent motion, measured by the body's weight, and more generally in both natural and violent motion also the external resistance of the medium of motion to being cleaved by the mobile in the sublunar plenum, measured by the density of the medium.

Thus Aristotle's general law of motion assumed two different interpretations for the two different dynamical cases of natural and violent sublunar motion. In the case of sublunar natural motion the general law v α F/R becomes v α W/R (because Weight is the measure of the motive force of gravity), with the body's motion driven by its weight and resisted by the medium.But in the case of violent motion the general law v α F/R then becomes v α F/W because the body's weight now acts as a resistance that resists the violent mover F, whatever that might be, such as a hand pulling a weight up from the floor or a gang of ship-hauliers hauling a ship along the shore or a canal.

However, in Aristotle's celestial physics, whilst the spheres have movers, each being 'pushed' around by its own soul seeking the love of its own god as its unmoved mover, whereby F > 0, there is no resistance to their motion whatever, since Aristotle's quintessence has neither gravity nor levity, whereby they have no internal resistance to their motion. And nor is there any external resistance such as any resistant medium to be cut through, whereby altogether R = 0. Yet in dynamically similar terrestrial motion, such as in the hypothetical case of gravitational fall in a vacuum,driven by gravity (i.e. F = W > 0), but without any resistant medium (i.e. R = 0), Aristotle's law of motion therefore predicts it would be infinitely fast or instantaneous, since then v α W/R = W/0 = infinite.

But in spite of these very same dynamical conditions of celestial bodies having movers but no resistance to them, in the heavens even the fastest sphere of all, the stellar sphere, apparently took 24 hours to rotate, rather than being infinitely fast or instantaneous as Aristotle's law predicted sublunar gravitational free-fall would be.

Thus when interpreted as a cosmologically universal law, Aristotle's basic law of motion was cosmologically refuted by his own dynamical model of celestial natural motion as a driven motion that has no resistance to it.

Hence in the 6th century, John Philoponus argued that the finite speed rotation of the celestial spheres empirically refuted Aristotle's thesis that natural motion would be instantaneous in a vacuum where there is no medium the mobile has to cut through, as follows:

"For if in general the reason why motion takes time were the physical [medium] that is cut through in the course of this motion, and for this reason things that moved through a vacuum would have to move without taking time because of there being nothing for them to cut through, this ought to happen all the more in the case of the fastest of all motions, I mean the [celestial] rotation. For what rotates does not cut through any physical [medium] either. But in fact this [timeless motion] does not happen. All rotation takes time, even without there being anything to cut through in the motion."

Consequently Philoponus sought to resolve this devastating celestial empirical refutation of Aristotelian mathematical dynamics by Aristotle's own rotating celestial spheres by rejecting Aristotle's core law of motion and replacing it with the alternative law v α F - R, whereby a finite force does not produce an infinite speed when R = 0. The essential logic of this refutation of Aristotle's law of motion can be reconstructed as follows. The prediction of the speed of the spheres' rotations in Aristotelian celestial dynamics is given by the following logical argument

[ (i) v α F/R & (ii) F > 0 & (iii) R = 0 ] entail v is infinite.

These premises comprise the conjunction of Aristotle's law of motion in premise (i) with his dynamical model of celestial motion expressed in premises (ii) & (iii). But the contrary observation v is not infinite entails at least one premise of this conjunction must be false. But which one ?

Philoponus decided to direct the falsifying logical arrow of modus tollens at the very first of the three theoretical premises of this prediction, namely Aristotle's law of motion, and replace it with his alternative law v α F - R. But logically premises (ii) or (iii) could have been rejected and replaced instead. And indeed some six centuries later premise (iii) was rejected and replaced.

For in the 12th century Averroes rejected Philoponus's 'anti-Aristotelian' solution to this refutation of Aristotelian celestial dynamics that had rejected its core law of motion v α F/R. Instead he restored Aristotle's law of motion as premise (i) by adopting the 'hidden variable' approach to resolving apparent refutations of parametric laws that posits a previously unaccounted variable and its value(s) for some parameter, thereby modifying the predicted value of the subject variable, in this case the average speed of motion v. For he posited there was a non-gravitational previously unaccounted inherent resistance to motion hidden within the celestial spheres. This was a non-gravitational inherent resistance to motion of superlunary quintessential matter, whereby R > 0 even when there is neither any gravitational nor any media resistance to motion.

Hence the alternative logic of Averroes' solution to the refutation of the prediction of Aristotelian celestial dynamics

[ (i) v α F/R & (ii) F > 0 & (iii) R = 0 ] entail v is infinite

was to reject its third premise R = 0 instead of rejecting its first premise as Philoponus had, and assert R > 0.

Thus Averroes most significantly revised Aristotle's law of motion v α F/R into v α F/M for the case of celestial motion with his auxiliary theory of what may be called celestial inertia M, whereby R = M > 0. But Averroes restricted inertia to celestial bodies and denied sublunar bodies have any inherent resistance to motion other than their gravitational (or levitational) inherent resistance to violent motion, just as in Aristotle's original sublunar physics.

However, Averroes’ 13th century follower Thomas Aquinas accepted Averroes' theory of celestial inertia, but rejected his denial of sublunar inertia, and extended Averroes' innovation in the celestial physics of the spheres to all sublunar bodies. He posited all bodies universally have a non-gravitational inherent resistance to motion constituted by their magnitude or mass. In his Systeme du Monde the pioneering historian of medieval science Pierre Duhem said of Aquinas' innovation:

"For the first time we have seen human reason distinguish two elements in a heavy body: the motive force, that is, in modern terms, the weight; and the moved thing, the corpus quantum, or as we say today, the mass. For the first time we have seen the notion of mass being introduced in mechanics, and being introduced as equivalent to what remains in a body when one has suppressed all forms in order to leave only the prime matter quantified by its determined dimensions. Saint Thomas Aquinas's analysis, completing Ibn Bajja's, came to distinguish three notions in a falling body: the weight, the mass, and the resistance of the medium, about which physics will reason during the modern era....This mass, this quantified body, resists the motor attempting to transport it from one place to another, stated Thomas Aquinas."

Aquinas thereby predicted this non-gravitational inherent resistance to motion of all bodies would also prevent an infinite speed of gravitational free-fall for sub-lunar bodies as otherwise predicted by Aristotle's law of motion applied to pre-inertial Aristotelian dynamics in Aristotle's famous Physics 4.8.215a25f argument for the impossibility of natural motion in a vacuum i.e. of gravitational free-fall. Thus by eliminating the prediction of its infinite speed, Aquinas made gravitational fall in a vacuum dynamically possible in an alternative way to that in which Philoponus had rendered it theoretically possible.

Another logical consequence of Aquinas's theory of inertia was that all bodies would fall with the same speed in a vacuum because the ratio between their weight, i.e. the motive force, and their mass which resists it, is always the same. Or in other words in the Aristotelian law of average speed v α W/m, W/m = 1 and so v = k, a constant. But it seems the first known published recognition of this consequence of the Thomist theory of inertia was in the early 15th century by Paul of Venice in his critical exposition on Aristotle's Physics, in which he argued equal speeds of unequal weights in natural motion in a vacuum was not an absurdity and thus a reductio ad absurdum against the very possibility of natural motion in a vacuum as follows:

"It is not absurd that two unequal weights move with equal speed in the void; there is, in fact, no resistance other than the intrinsic resistance due to the application of the motor to the mobile, in order that its natural movement be accomplished. And the proportion of the motor to the mobile, with respect to the heavier body and the lighter body, is the same. They would then move with the same speed in the void. In the plenum, on the other hand, they would move with unequal speed because the medium would prevent the mobile from taking its natural movement."

As Duhem commented, this "glimpses what we, from the time of Newton, have expressed as follows: Unequal weights fall with the same speed in the void because the proportion between their weight and their mass has the same value." But the first mention of a way of empirically testing this novel prediction of this Thomist revision of Aristotelian dynamics seems to be that detailed in the First Day of Galileo's 1638 Discorsi, namely by comparing the pendulum motions in air of two bobs of the same size but different weights.

However, yet another consequence of Aquinas's innovation in Aristotelian dynamics was that it contradicted its original law of interminable rest or locomotion in a void that an externally unforced body in motion in a void without gravity or any other resistance to motion would either remain at rest forever or if moving continue moving forever.For any such motion would now be terminated or prevented by the body's own internal resistance to motion posited by Aquinas, just as projectile violent motion against the countervailing resistance of gravity was impossible in a vacuum for Aristotle. Hence by the same token that Aquinas's theory of inertia predicted gravitational fall in a vacuum would not be infinitely fast, contra Aristotle's Physics 4.8.215a25f, so it also predicted there would not be interminable locomotion in a gravity-free void, in which any locomotion would terminate, contrary to Aristotle's Physics 4.8.215a19-22 and Newton's first law of motion.

Some five centuries after Averroes' and Aquinas's innovation, it was Kepler who first dubbed this non-gravitational inherent resistance to motion in all bodies universally 'inertia'. Hence the crucial notion of 17th century early classical mechanics of a resistant force of inertia inherent in all bodies was born in the heavens of medieval astrophysics, in the Aristotelian physics of the celestial spheres, rather than in terrestrial physics or in experiments.

This auxiliary theory of Aristotelian dynamics, originally devised to account for the otherwise anomalous finite speed rotations of the celestial spheres for Aristotle's law of motion, was a most important conceptual development in physics and Aristotelian dynamics in its second millennium of progress in the dialectical evolutionary transformation of its core law of motion into the basic law of motion of classical mechanics a α (F - R)/m. For it provided what was eventually to become that law's denominator, whereby when there is no other resistance to motion, the acceleration produced by a motive force is still not infinite by virtue of the inherent resistant force of inertia m. Its first millennium had seen Philoponus's 6th century innovation of net force in which those forces of resistance by which the motive force was to be divided in Aristotle's dynamics (e.g. media resistance and gravity) were rather to be subtracted instead to give the net motive force, thus providing what was eventually to become the numerator of net force F - R in the classical mechanics law of motion.

The first millennium had also seen the Hipparchan innovation in Aristotelian dynamics of its auxiliary theory of a self-dissipating impressed force or impetus to explain the sublunar phenomenon of detached violent motion such as projectile motion against gravity, which Philoponus had also applied to celestial motion. The second millennium then saw a radically different impetus theory of an essentially self-conserving impetus developed by Avicenna and Buridan which was also applied to celestial motion to provide what seems to have been the first non-animistic explanation of the continuing celestial motions once initiated by God.

Impetus in the celestial spheres

In the 14th century the logician and natural philosopher Jean Buridan, Rector of Paris University, subscribed to the Avicennan variant of Aristotelian impetus dynamics according to which impetus is conserved forever in the absence of any resistance to motion, rather than being evanescent and self-decaying as in the Hipparchan variant. In order to dispense with the need for positing continually moving intelligences or souls in the celestial spheres, which he pointed out are not posited by the Bible, Buridan applied the Avicennan self-conserving impetus theory to their endless rotation by extension of a terrestrial example of its application to rotary motion in the form of a rotating millwheel that continues rotating for a long time after the originally propelling hand is withdrawn, driven by the impetus impressed within it.

Earlier Franciscus de Marchia had given a 'part impetus dynamics - part animistic' account of celestial motion in the form of the sphere’s angel continually impressing impetus in its sphere whereby it was moved directly by impetus and only indirectly by its moving angel. This hybrid mechanico-animistic explanation was necessitated by the fact that de Marchia only subscribed to the Hipparchan-Philoponan impetus theory in which impetus is self-dissipating rather than self-conserving, and thus would not last forever but need constant renewal even in the absence of any resistance to motion.

But Buridan attributed the cause of the continuing motion of the spheres wholly to impetus as follows:

"God, when He created the world, moved each of the celestial orbs as He pleased, and in moving them he impressed in them impetuses which moved them without his having to move them any more...And those impetuses which he impressed in the celestial bodies were not decreased or corrupted afterwards, because there was no inclination of the celestial bodies for other movements. Nor was there resistance which would be corruptive or repressive of that impetus."

However, having discounted the possibility of any resistance due to a contrary inclination to move in any opposite direction or due to any external resistance, in concluding their impetus was therefore not corrupted by any resistance Buridan also discounted any inherent resistance to motion in the form of an inclination to rest within the spheres themselves, such as the inertia posited by Averroes and Aquinas. For otherwise that resistance would destroy their impetus, as the anti-Duhemian historian of science Annaliese Maier maintained the Parisian impetus dynamicists were forced to conclude because of their belief in an inherent inclinatio ad quietem (tendency to rest) or inertia in all bodies. But in fact contrary to that inertial variant of Aristotelian dynamics, according to Buridan prime matter does not resist motion.

But this then raised the question within Aristotelian dynamics of why the motive force of impetus does not therefore move the spheres with infinite speed. One impetus dynamics answer seemed to be that it was a secondary kind of motive force that produced uniform motion rather than infinite speed, just as it seemed Aristotle had supposed the spheres' moving souls do, or rather than uniformly accelerated motion like the primary force of gravity did by producing constantly increasing amounts of impetus.

However in his Treatise on the heavens and the world in which the heavens are moved by inanimate inherent mechanical forces, Buridan's pupil Oresme offered an alternative Thomist response to this problem in that he did posit a resistance to motion inherent in the heavens (i.e. in the spheres), but which is only a resistance to acceleration beyond their natural speed, rather than to motion itself, and was thus a tendency to preserve their natural speed. This analysis of the dynamics of the motions of the spheres seems to have been a first anticipation of Newton's subsequent more generally revised conception of inertia as resisting accelerated motion but not uniform motion.