Aristotelian Physics, Impetus Theory, and the Mean Speed Theorem
Aristotelian Physics, Impetus Theory, and the Mean Speed Theorem
Prior to the seventeenth century, many of the most fundamental problems of physics concerned difficulties associated with local motion—changes in place or position. Medieval attempts to explain how and why such changes took place were developed within an Aristotelian framework. These efforts progressively undermined and eventually led to the rejection of certain tenets of Aristotle's (384-322 b.c.) doctrine of motion. They also culminated in what many consider the single most important contribution of medieval scholars to physics—the mean speed theorem.
Aristotle dichotomized the universe into a terrestrial or sublunar region, encompassing Earth and extending to the sphere of the Moon, and a celestial or supralunar region, extending from the sphere of the Moon to the fixed stars. All matter in the terrestrial region was thought to be composed of four elements—earth, water, air, and fire—while the celestial region was assumed to be filled with the fifth or divine element, ether. The ether was believed to be immune to all changes except local motion. Ordinary matter was subject not only to local motion but other types of change as well.
Aristotle distinguished between two types of local motion—natural and violent. Natural motions are those that a body exhibits when unimpeded. Violent motions occur when a body is displaced from its natural resting place. According to Aristotle, celestial bodies naturally move in circles or combinations thereof. Since it was thought that the ether in no way hindered this motion, it was concluded that these bodies do not exhibit violent motions. Bodies composed of ordinary matter behave quite differently and in a manner intimately tied to the structure of the sublunar world.
The terrestrial region of Aristotle's universe was composed of four concentric areas, each the natural place for one of the four elements. When displaced, it was believed that each element naturally moves rectilinearly toward its concentric ring (if unimpeded). The outermost ring was the natural place of fire, below that the ring of air, below that the ring of water, and below that the ring of earth. Aristotle attributed different degrees of heaviness or lightness to the basic elements to explain their tendency to seek their natural places. The element earth was deemed absolutely heavy. As such, it naturally moved downward toward Earth's center from the regions above. Similarly, the absolute lightness of fire caused it to rise from below to its natural place above the ring of air.
Explaining violent motion required a different mechanism. Aristotle believed the force responsible for motion had to be in constant physical contact with the moved body. When violent motion was initiated, the originating motive impulse was easily identified—the bow string for a shooting arrow, the hand for a thrown rock, etc. Aristotle maintained that the original mover not only sets the arrow or the stone in motion, but also activates the surrounding medium—in this case air. The air parts before the arrow or rock and circles back to maintain a continuous motive force behind the object. This force gradually diminishes due to the resistance of the medium. When it completely dissipates, the arrow or stone falls downward according to its natural motion.
Aristotle formulated specific rules to describe the consequences of this doctrine. He stated that the speed of a body in violent motion is directly proportional to the motive force and inversely proportional to resistance. The latter included the resistive power of the body in motion (a concept left undefined) and the resistance offered by the external medium. Accordingly, the speed of a body could be doubled by doubling the applied force or halving the resistance.
This doctrine of motion also led Aristotle to deny the existence of a vacuum. Since speed was proportional to the medium density, an indefinite rarefaction of the medium would produce a corresponding indefinite increase in speed. But if the medium were to vanish completely, then the speed of a body would be infinite. This was clearly absurd. Even more serious, motion in a void violated Aristotle's claim that violent motion necessarily occurred in a medium. Furthermore, Aristotle took it as axiomatic that bodies of different weights necessarily fall at different speeds—their speeds being directly proportional to their weight. However, he realized that without a material medium, lighter bodies would move just as swiftly as heavier bodies. To avoid these conclusions, Aristotle rejected the void and postulated a universe filled everywhere with matter.
John Philoponus (sixth century a.d.) strongly objected to Aristotle's theory of motion. He was especially critical of the role Aristotle assigned to the medium. He argued convincingly that it was not the agent or cause of violent motion. If Aristotle were correct, then it would be possible to move an arrow or rock by simply agitating the air behind them. This, however, was contrary to experience. He concluded that violent motion occurs by the mover transferring to the object of motion an incorporeal kinetic power—later known as impetus. By divorcing motion from the external medium, Philoponus held open the possibility that motion through a vacuum was possible, thus undermining one of the chief reasons for rejecting the void.
The idea of an impressed force was further developed by Islamic scholars who referred to it as mail. Ibn Sina (980-1037) propounded a version of mail theory that delineated three different types: psychic, natural, and violent. Only natural and violent mail are relevant to the present discussion. They were used by Ibn Sina to explain Aristotle's natural and violent motions. Ibn Sina argued that bodies in motion received mail directly in proportion to their weight. He further maintained that violent mail would endure in a body indefinitely in the absence of external resistance. A consequence of this was that motion in a void would continue without end since there would be nothing to stop it. Ibn Sina rejected the existence of a vacuum because such motion had never been observed.
Other anti-Aristotelian critiques were promulgated in the Islamic world. One of the most notable was advanced by Ibn Bajja (1095?-1138?). Possibly influenced by Philoponus, Ibn Bajja denied Aristotle's claim that speed was necessarily inversely proportional to the density of the medium. He pointed out that celestial bodies traveled with different speeds through the ether, which supposedly offered no resistance to motion. Thus, he concluded differences in speed were not the result of an external medium. In fact, Ibn Bajja argued, the only function of the medium was to retard motion.
Thomas Aquinas (1225-1274) elaborated upon Ibn Bajja's ideas. He attempted to demonstrate that motion in a void, or resistanceless media such as the ether, would not be instantaneous as Aristotle had feared. He argued that motion from one point to another required traversing the intervening points successively. This was only possible if motion were finite. Consequently, infinite speeds must be impossible in a vacuum. Aquinas, along with Roger Bacon (1214?-1294), rejected all attempts to explain violent motion by means of incorporeal forces imparted to moving bodies.
It was not until the fourteenth century that Philoponus's impressed-force theory gained general acceptance. Known as impetus theory, it became popular around 1320 and was taught at the University of Paris. Franciscus of Marchia (?-1344?) proposed one version in which a self-dissipating impetus was imparted not only to the body set in motion, but also to the surrounding medium. Thereafter, Jean Buridan (1295?-1358), who is perhaps responsible for the term "impetus," developed the theory so fully that he must be considered its main proponent.
Buridan conceived of impetus as a motive force transferred to bodies. He considered its magnitude directly proportional to body weight and velocity. Furthermore, he followed Ibn Sina in ascribing permanence to impetus, claiming it endured indefinitely unless diminished by external resistance. This clearly implied that, if all resistance were eliminated, a body in motion would continue in motion indefinitely at a constant speed. Buridan's formulation bears a striking resemblance to the concept of inertia and may have prepared the way for its development. Unfortunately, he denied the possibility of indefinite rectilinear motion, which is an essential feature of inertia. He did, however, propose that impetus might be the cause of the eternal circular motions of celestial bodies.
Buridan further applied impetus theory to the accelerated motion of free fall. Buridan identified a body's heaviness or gravitas as the cause of its natural uniform fall. A body's gravitas not only initiated the downward motion, it also augmented it through accumulated increments of impetus. Each increment generated a corresponding increase in velocity, which in turn generated a further increment and so on, resulting in continuously accelerated motion. Buridan's account is essentially Aristotelian since the cause of motion—impetus—is proportional to velocity and not acceleration as in Newtonian physics.
The most significant medieval results on local motion were achieved by members of Merton College, Oxford University. In 1328 Thomas Bradwardine (1290?-1349) developed a new law of motion that proved very influential. Richard Swineshead worked out the implications of Bradwardine's rule and together with William Heytesbury (1300?-1380), John Dumbleton, and other Mertonian scholars produced the correct definitions for uniform speed and uniformly accelerated motion. Applying these definitions, the Mertonians derived the single most important medieval contribution to physics—the mean speed theorem. The theorem relates uniformly accelerated motion to uniform velocity, making it possible to express the distance traveled by the former in terms of the distance traveled by the latter. Nicole Oresme (1320?-1382), in On the Configurations of Qualities (c. 1350), provided geometrical and arithmetical proofs of the theorem. These were widely disseminated throughout Europe in the fourteenth and fifteenth centuries and probably influenced Galileo (1564-1642), who made the mean speed theorem the foundation of his new science of motion.
STEPHEN D. NORTON
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