On the Threshold of Inertial Mass?

Francesco d’Appignano on Resistance and Infinite Velocity

Chris Schabel*

Genealogists generally take little interest in people of the past who had no children. Much less justifiably, historians of physics, probably more than historians of any other subject, also tend to neglect dead ends. Despite our awareness that to understand the mind of the Middle Ages we often need to understand theories and ideas that later societies rejected, nevertheless we are most excited when we find something in medieval science that resembles something of Classical Mechanics, something that looks “modern.” Indeed, we can even speculate in print about the possible connection between, say, Peter Auriol and Isaac Newton, usually without doing much to test this speculation. Of course, we do not often do this for Quantum Mechanics and contemporary physics, either because medieval historians do not understand it or because they cannot imagine any direct link between medieval and quantum physics.[1]

                Francesco d’Appignano, or, as he is called in most medieval texts, Franciscus de Marchia, is a scholar whose ideas have been linked vaguely with the elements of the Newtonian System or with its forerunners. First, there is his role in the development of the theory of impetus to explain the motion of projectiles, about which we have heard so much ever since the works of Anneliese Maier. This theory modifies Aristotle in the direction of Galileo, and so draws our attention, even though we are not completely sure about the nature of the impact of Francesco’s input, the theme of Fabio Zanin’s contribution to this volume.[2] The subject of Russell Friedman’s paper, Francesco’s support of an actual infinite, entailed his conception “of a world of infinite space,” according to Graziella Vescovini, “a new concept in medieval cosmology,” she remarks, but easily recognizable as part of the Newtonian worldview.[3] Notker Schneider has shown how Francesco did away with the notion of the Quintessence of the supralunary realm, thus making possible the future unification of the cosmos into one system obeying one set of laws.[4] And as Maier pointed out, Francesco’s views on the status of natural laws are very much in line with those of Classical Mechanics: natural causation works necessarily.[5] Finally, we might mention Domenico Priori’s presentation of “Francesco’s razor,” another general scientific principle that Francesco shared with Copernicus, Galileo, Kepler, and Newton.

                Again leaving aside the question of direct influence on later Parisian thinkers like Jean Buridan and Nicole Oresme, I wish to ask here whether Francesco was also on the threshold of inertial mass. Included in the notion of inertia, i.e. that force equals mass times acceleration (F = MA), are various ideas: that a body “resists” change in velocity unless acted upon by an outside force, that a body has a specific “resisting” mass that can be mathematically determined by the force that can accelerate it, and that these things appear most clearly in a real void space. The short answer to the question is, of course, no, Francesco was not on the threshold of inertial mass. But the long answer is that he was dealing with questions in such a way that, had others done so, over a period of time, this background would have helped the likes of Galileo to see the pertinent factors in a non-Aristotelian light. Whether this in fact happened is not under present consideration.

                Francesco’s interesting remarks on this subject come in the traditional places where scholastics considered the possibility of motion in an instant, that is, of infinite or instantaneous velocity, which Aristotle rejects completely. For Aristotle, velocity varies as the ratio of force to the resistance of the medium (V ~ F/R). There are three factors: the mover, the medium, and the mobile. (This last, the mobile, was more a medieval addition to the equation than an Aristotelian element.) With the augmentation or diminution of the motive force, the velocity varies accordingly. Likewise, when the resistance of the medium changes by densification or rarification, the velocity also changes. Finally, the “size” of the mobile is a factor, for the bigger the mobile, the harder it is to move, and the smaller is it, the easier it is to move. Thus instantaneous velocity comes up for discussion in three contexts in Francesco’s works: first, when the possibility of the existence of a void is treated, because in a void there would be no medium to offer resistance and hence no ratio of force to resistance (F/R = infinity); second, when the motive force itself may be infinite, as in the case of God; third, when the mobile is incorporeal, and hence any force at all would move it at infinite speed. Here I will look at each of these contexts in turn.

Resistance and Void in the Physics Commentary

In book IV, chapter 8, of the Physics, Aristotle proves that a vacuum or void is impossible because it entails absurdities. Since motion varies with the ratio of force to resistance, with zero resistance, motion in a void would be infinitely fast. Moreover, two motions in the same medium are related according to the force, but with no resistance, unequal forces would both result in infinite velocity. Also, such motion would take place in no time, or in an instant, but Aristotle rejects instantaneous motion—being in two places at the same time—in book VIII, chapter 7, of the Physics.

In his introduction to Francesco’s Physics commentary, Father Mariani remarks that in general our Franciscan merely explains the Philosopher unless he contradicts the faith. In this way, Francesco’s commentary can be called a Christian expositio textus. In the case of Physics IV, chapter 8, however, it appears that Francesco corrects Aristotle for scientific reasons. First he dismisses Aristotle’s argument since it applies only to heavy and light things, but not to things that move themselves, like animals and celestial bodies. For these, motion does not take time solely on account of the resistance of the medium, since in the case of celestial bodies, “there is no medium that impedes or slows” motion. Even animals, says Francesco, would move in time if a void were posited here.[6]

A mere assertion that motion in a void would take time does not separate Francesco from the likes of Albertus Magnus and Thomas Aquinas. On this issue they followed the lead of Avempace, whose views were relayed by Averroes.[7] Of greater importance, then, is Francesco’s explanation for how finite motion in a void is possible:

Because celestial bodies are actually distinct from their motions, therefore one understands that there is in them a certain resistance on account of which time is required in motion there [in the heavens] (ibi). Animals also, because of the mobile’s resistance—that is the heaviness of their bodies— [have a resistance] to the desire that is moving them. For we see that animals’ limbs become tired when moving because of such resistance. Because of this they also move in a greater time when a full medium resists them.[8]

The reference to the animals is plain enough, and of course entails a rejection of the universal applicability of Aristotle’s theory of velocity being a mere function of force over resistance offered by the medium. Even if animals were in a vacuum, they would move in time. Notably, here it is the very weight of the animals’ bodies that acts as a resister, so we can see a vague link between mass and motion in a void. With the motion of animals, however, there are too many factors left out for us to draw any conclusions.

The celestial bodies example is where we want to go, especially since Francesco tends to apply the same natural rules to the celestial realm as he does to the terrestrial. Exactly what kind of resistance do the celestial bodies have that enables them to move in time without the resistance of the medium? And does this entail that, for Francesco, the celestial realm is an actual void space? If so, then we may have the beginnings of a theory of inertial mass. On the one hand, Francesco takes the resistance of the medium out of the equation in discussing an instance of real motion, not counterfactual, or secundum imaginationem. On the other, he says in the quotation above that you cannot simply move a body just like that, immediately, from one place to the next, because there is some internal resistance that prevents this. With the medium removed, the only possible source for resistance lies in the mobile itself. In this connection one could point out that Francesco’s notion of virtus derelicta, developed in the first question of book IV of his Sentences commentary, shows that a body can absorb a kind of force that keeps it in motion.[9] That is, there is some internal property of bodies that is related to motion. Admittedly, this is a temporary force, and only Buridan will make it permanent for the celestial bodies, but one can see that Francesco is juggling with some of the elements of inertial mass.

Motion and the Intensively Infinite in I Sentences

There are other conditions in which motion in an instant might be thought to arise, and in discussing these possibilities Francesco provides further clues about his notion, or lack of notion, of inertial mass. One way is with a power that is infinite in vigor or intensity. Father Mariani has edited some of the relevant material. In his Quodlibet, question 9, article 2, Francesco asks “Whether the infinite intensity of the First Mover can be concluded on the basis of the infinity of motion.”[10] The answer is negative if one takes ‘infinity of motion’ to refer to motion over an infinite amount of time. But if one means an infinitely fast motion, then that would again mean motion in an instant. Since this is impossible, Francesco’s imaginary opponent claims that so must a power infinite in intensity be impossible.[11]

                The difficulty with that response is that God appears to be just such an infinitely intense power, and so it seems that He could move things in an instant.[12] We are denied the promised clarification following the rubric “contra ista,” however, because Father Mariani’s manuscript trails off and even omits the last two articles of this final Quodlibetal question. Luckily, the answer is supplied in Mariani’s appendix in the corresponding question from Francesco’s Sentences commentary, book I, distinction 2, question 3, where the problem is not only expressed in terms of motion in an instant, but also in “non-time.”[13]

                In the first article, Francesco gives two possible ways to solve this dilemma, the first being that there is a “repugnance on the part of the mobile toward a motion of infinite velocity,” but, interestingly, both this and the other, less potent, explanation of Averroes are rejected.[14] He returns to the problem later in the second article, where again we are faced with the obvious difficulty with instantaneous motion: “The same body would be simultaneously in a place and not in that same place, and this is a contradiction; therefore positing a motive power [that is] infinite in vigor is impossible.”[15] Francesco again provides and refutes the two solutions he gave in article one, for those suggestions do not prove why the existence of such a power would not result in motion in an instant.[16]

                The reader, who is awaiting Francesco’s explanation of why instantaneous motion would not occur, is taken by surprise when he instead reads the conclusion that “according to the faith, which posits a God of intensively infinite power positively and not negatively, it seems that one must concede that God, by His absolute power, could move an infinite mobile even in an instant.”[17] In particular, as God’s absolute power applies to motion in an instant:

It does not entail a contradiction for change of place to occur in an instant, because if the cause does not entail a contradiction, then neither does the effect entail a contradiction; but the cause of instantaneous change does not entail a contradiction, because an infinite power does not entail a contradiction.[18]

But would not a body be in two places at once, or at once be in a place and also not be in that place? No, “this does not follow,”

Because God could release (absolvere) the moved body from every place, and once this is done, then the body that is moved in an instant would not be in several places either simultaneously or successively, because it would be in no place. Second, I say that, although the same body being in several places at once is impossible with respect to a finite power, it is not impossible with respect to an infinite power.[19]

In fact, in his Physics commentary, treating book VIII, chapter 10, where Aristotle discusses the possibility of an infinite force, Francesco hints at much the same thing, explicitly linking the question of the intensively infinite with that of the vacuum treated earlier in book IV. In a passage that further suggests that Francesco's Physics commentary is not as "literal" as one initially thinks, our Franciscan states:

It could also be said that there is no contradiction for some body to be transferred from place to place in an instant. Whence in the chapter "On the Vacuum" the Commentator says that in a vacuum heavy things move in an instant because there is no resistance to the mobile either of the medium or of the mover, and so [motion in an instant] will be able to exist if God is posited to be of infinite power. But such a transference is not motion strictly speaking, because it will not be anything that is successive, which is part of the nature of motion. Therefore an agent of infinite power can transfer a body from place to place in an instant, and accordingly it could appear to someone that the Philosopher's argument is [just] a probable discussion and does not reach a conclusion demonstratively, unless one handles it differently.[20]

My purpose is bringing up these passages is to show that Francesco is in fact not opposed to motion in an instant per se, so that when Francesco brings up the matter of internal resistance later on in his Sentences commentary, it is not because he has to defeat instantaneous motion at all costs. Nevertheless, Francesco does realize that what he has said is odd and that simply accepting instantaneous motion and maintaining that God can separate bodies from their places does not make things much easier. Francesco tries to talk about this infinite velocity in negative terms. In fact, his language resembles that of Peter Auriol in a similar context: Auriol prefers to describe God’s relationship to time and space as one of indistance, asserting that God is not positively present to every time and place, but negatively indistant from every time and place, thus denying any distance. Correspondingly, Francesco here claims that infinite velocity is the lack of all slowness (tarditas), so that instead of talking about velocity, we should really be talking about lack of tardity.[21]

                Perhaps this is an instance where our linking medieval thought with Quantum Mechanics would not be so ridiculous, for Francesco thinks that something really strange, really counter-intuitive occurs when we talk about infinite velocity, or complete lack of tardity. We have already seen that being in several places at once is possible, and even that God can somehow divorce things from every place at such speeds. Now Francesco also mentions something about the very acquisition of place being of a different mode in normal motion and in motion in an instant.[22] In Francesco’s overall conclusion he explains why velocity is no longer the “subject” of discussion with instantaneous motion:

Time is not preserved (salvatur) in an instant, nor a line in a point—thus the passions of a line are not preserved in a point, and so a point is neither straight nor curved, and likewise, the passions of time are not preserved in an instant, and so an instant is not long or short. [Therefore, I say] similarly that the passions of motion, which are fast and slow, are not preserved in change of place. Therefore this is not the subject of velocity because in the argument one proceeds from the subject of velocity to a non-subject of velocity, and consequently from positive velocity to negative velocity.[23]

So what happens with normal, continuous motion cannot be discussed in the same terms as what occurs in the “indivisible change” of instantaneous motion: they are alterius racionis, different things altogether.[24] A phase shift occurs when one gets to infinite speed. In Francesco’s little thought experiment, he resembles Einstein wondering about what happens at the speed of light. But this is not of our concern.

Angels and Inertia in II Sentences.

Thus the topic of infinite velocity, motion in an instant, comes up in the context of the possibility of a void and in treatments of a power of infinite intensity. Besides the medium and the motive force, the third factor in medieval Aristotelian discussions of velocity is the mobile. A final locus for treating infinite velocity is the motion of angels, an odd sort of mobile. The last text I would like to discuss is question 16, article 5, of Francesco’s commentary on the second book of the Sentences. Book II is usually the focus of discussion of natural philosophy, both for the medieval theologians themselves and, understandably, for modern historians: Anneliese Maier has examined questions 1 and 12, on creation and eternity, questions 33-36 on the elements, and question 48 on causation; Pierre Duhem has studied question 6, on time and motion; Notker Schneider has edited questions 29-32, on cosmology.[25] One of the interesting remaining questions is number 16, on the place of angels. Here Francesco talks about the place and local motion of angels in five articles, the last of which asks “whether an angel can move in an instant.”[26]

                Francesco immediately links his question to Aristotle’s arguments about the void in book four of the Physics. Because the resistance of the medium is a factor in motion, as we have seen, then for Aristotle a complete lack of resistance, as in a void, would produce infinite, instantaneous motion, because there is no proportion between the motion of an object in any medium and the motion of the same object in a void. Such instantaneous motion is impossible for Aristotle but not for Francesco. Our Franciscan then extrapolates from Aristotle’s principle that velocity varies inversely with the resistance, by saying that we can vary the mobile instead of the medium and talk about varying velocity.[27] Let us take an example of what seems to be Francesco’s point: If we take two things moving through a medium, for example a rock and an arrow, the medium will resist the rock more than the arrow, and the arrow will move faster. But if we compare the rock with an angel, since an angel is incorporeal and therefore cannot be resisted by the medium, there is no proportion between the motion of the rock and that of the angel: the angel will move in an instant.

                In fact, if an angel did move in time and not instantaneously, Francesco still does not see how on Aristotelian lines, the rock, for example, could move with the same velocity as the angel. No matter what force is applied to the rock, since the medium does not resist the angel at all, the rock can never move at a speed commensurate with the angel’s. This is the case even if we break the rock into ever smaller particles, since these particles can never reach the angelic state of incorporeality.[28]

                Unlike in the case of God, the intensively infinite power, Francesco does not think it possible for an angel to move in an instant by its own power. Therefore he has to explain why not. One way seems to be to hold to an atomistic view of space, but Francesco merely mentions this without refutation, saying that he maintains otherwise. Francesco’s explanation is that the angel resists the mover, not of course like a rock’s weight and natural inclination to fall to the ground resist a catapult’s efforts, because for one thing the accidents of violent motion do not apply to angels in the heavens. Rather the angel “does not have perfect obedience to [the mover],” and this is because “it cannot naturally be in several places at once,” so that “when it is moved from one place to another, it resists the mover, since it is not in perfect obedience to it such that it is moved or transferred from place to place in any measure [of time] whatever.”[29]

                Now if Francesco had left it at that, he would not have said anything substantially different from, say, what Thomas Aquinas had stated in denying the instantaneous motion of angels.[30] But Francesco tries to go further than this: it is because of a “privative resistance, such as in every local motion—caused by a finite power—of every finite thing whatever, whether it is corporeal or spiritual.”[31] Unlike a rock, an angel does not have a positive contrary resistance to violent motion, in that the rock actually tends to go in the opposite direction. But an angel does have a privative resistance. What this suggests is that all bodies have a privative resistance, so that even in a void space without a gravitational force (heaviness) or friction, they would still resist movement, and if Francesco means that they resist movement when they are at rest, then he is heading toward inertial mass.

                There is still a fundamental incommensurability between angels and rocks, because, as was mentioned, no matter how small a rock gets, it is still “alterius rationis.” That is, to use Francesco’s example, just like an acute angle can get wider and wider but can never become a right angle, a corpus quantum—a mass—can become less and less dense, but it can never reach the subtlety of an angel. So the resistance of an angel and the resistance of a corporeal body are alterius rationis, of a completely different nature, and can never be commensurable.[32]

Conclusion: Inertial Mass?

For Galileo, a constant force acting on a mass in a void will produce a constant acceleration, and the mass and acceleration vary inversely. Some of Francesco’s Oxford successors would discuss uniform acceleration, but Francesco does not even bring acceleration into the picture. What he does do is analyze motion in a void. Although unlike Aristotle Francesco is not against infinite velocity in principle, he does claim that natural motion in a void would not be of infinite velocity. The reason he disagrees with Aristotle about the possibility of such finite motion is the reason why his theory is interesting: internal resistance. In addition to the positive resistance offered by corporeal bodies, bodies have a privative resistance to being moved, even if their natural inclination is removed from the picture, whether in reality or in a thought experiment. Moreover, even incorporeal bodies like angels offer this privative resistance, which thus prevents the natural occurence of infinite velocity. But because privative resistance seems to vary with the mass of a corpus quantum, Francesco has to maintain that the resistance of a massless angel is simply incommensurate with that of a corpus quantum. Still, Francesco does accept the possibility of real infinite velocity in the case of God the infinitely powerful motive force, although this infinity velocity is quite odd and can only be described using negative language.

                One could only say that, with thinkers like Francesco, interesting things are happening, interesting possibilities are discussed, and Aristotelian physics can be modified as much as desired. Indeed, Francesco is really not that much of an Aristotelian. This climate is conducive to change, but Francesco probably did not influence Galileo directly. Nevertheless, one wonders whether Francesco’s thought on this subject had an impact on Buridan and Oresme, Francesco’s more famous followers in time, whose ideas, via Albert of Saxony and Marsilius of Inghen, were not forgotten. And as far as I know, Einstein did not read his Francesco, although Francesco’s musings about infinite velocity are remarkably “free” of the shackles of common sense and tradition.

Summario

Nei suoi commenti sulla Fisica e sulle Sentenze e nel suo Quolibet, Francesco d'Appignano presenta alcune teorie non-aristoteliche sul concetto della velocità infinita. Diversamente da Aristotele, Francesco sostiene che il movimento naturale in uno spazio vuoto é possibile perché, oltre alla resistenza positiva, i soggetti corporei possiedono una resistenza privativa allo spostamento. E inoltre, perfino i corpi spirituali, come gli angeli, emanano questa resistenza privativa, che così impedisce l'avvenire naturale della velocità infinita. Ma poiché la resistenza privativa sembra variare secondo la massa del corpus quantum, Francesco deve pur sempre sostenere che la resistenza di un angelo senza materia é semplicemente incommensurabile rispetto a quella di un corpus quantum. Eppure, Francesco accetta la possibilità di una velocità infinita reale nel caso di Dio, la forza motrice infinitamente potente, anche se poi questa velocità infinita é abbastanza difficile a definire e può essere descritta soltanto usando un linguaggio negativo.