The theory of impetus was initially an auxiliary or secondary theory of Aristotelian dynamics, put forth initially to explain projectile motion against gravity. It was first introduced by John Philoponus in the 6th century AD. A radically different version was later developed by Avicenna (11th century) and Jean Buridan (14th century), which became an ancestor to the concepts of inertia, momentum and acceleration in classical mechanics. In the article, Philoponus' theory will be referred to as the Philoponan theory, while the Avicennan-Buridan theory will be referred to as the A-B theory.
The problem of projectile motion in Aristotelian dynamics
Aristotelian dynamics presupposes that all motion against resistance requires a conjoined mover continuously to supply motive force. In cases of projectile motion, however, there is no apparent mover to counteract gravity. To resolve the problem of continued motion after contact is lost with the original projector, Aristotle tentatively suggested the auxiliary theory that the propellant is the medium through which the projectile travels. The medium was postulated to be endowed with an incorporeal motive force impressed within its parts by the original projector. In the theories described below, the motive force or "impetus" is instead regarded to be impressed directly within the projectile itself by the original projector and is not mediated by the medium through which the projectile moves.
In the paradigm case of a projectile motion as a form of violent motion against gravity, that of a stone thrown vertically upwards against the downward force of gravity that then falls down again after reaching its zenith, according to Philoponan impetus dynamics the projector impresses an upward motive force within the stone which is greater than the downward force of gravity or its weight (I > W) whereby the stone moves upwards as its gravity is overcome by its greater upward impetus. But this force of impetus was held to be essentially evanescent and starts to decay away to nothing just of its own accord from the moment the stone is released, rather than its decay being due to any external resistance or to the downward resistance of gravity. Thus the stone’s upward motion decelerates as its impetus decays and its excess over gravity diminishes. Then at the turning point or moment of stasis from upward to downward motion, the stone was said to be in dynamical equilibrium where the upward force of impetus equals the downward force of gravity or the stone’s weight (I = W). After this in the third phase of this projectile motion where I < W, the stone accelerates downwards as its impetus decays even further until it is wholly exhausted (I = 0) and the stone assumes a characteristic constant speed of natural motion proportional to its excess natural weight over that of the medium. Thus this theory predicts all natural motion (i.e. gravitational fall) has a terminal velocity, even in a vacuum.
Thus in its dynamical account of projectile motion, the Philoponan impetus theory also offered an explanation of the acceleration of gravitational fall when it is the downward second stage of vertical projectile motion. Natural motion was already accepted by Aristotle as being swifter at the end than at the beginning. The Philoponan impetus theory explained the acceleration as a case of dynamical de-retardation, the continual erosion of the power of a brake on the natural speed of natural motion according to the body's excess natural weight in the medium.
However, it also offered an explanation of the case of gravitational fall just from a state of rest, as when a stone is dropped from the hand. In this case the initial situation of the stone at rest before release was analysed as a state of dynamical equilibrium between the weight of the stone acting downwards and a counterbalancing equal upward force impressed within the stone by the hand pressing upwards. Then when the stone is released from the hand its counterbalancing impetus that was continually refreshed by the hand then immediately starts to decay without further refreshment, and thus becomes increasingly less than its weight and the stone moves downward since I < W, accelerating as a case of de-retardation until I = 0.
In the 6th century, John Philoponus partly accepted Aristotle's theory that "continuation of motion depends on continued action of a force," but modified it to include his idea that the hurled body acquires a motive power or inclination for forced movement from the agent producing the initial motion and that this power secures the continuation of such motion. However, he argued that this impressed virtue was temporary; that it was a self-expending inclination, and thus the violent motion produced comes to an end, changing back into natural motion.
The Avicennan-Buridan impetus theory
In the 11th century, Avicenna developed an elaborate theory of motion in The Book of Healing, in which he made a distinction between the inclination and force of a projectile, and concluded that motion was a result of an inclination (mayl) transferred to the projectile by the thrower, and that projectile motion in a vacuum would not cease. The violent inclination he conceived was non-self-consuming, a permanent force whose effect is dissipated only as a result of external forces such as air resistance, making him "the first to conceive such a permanent type of impressed virtue for non-natural motion." Avicenna's concept of mayl is almost the opposite of the Aristotelian conception of violent motion and is reminiscent of the concept of inertia, known as Newton's first law of motion, roughly equivalent to Jean Buridan's concept of impetus. Avicenna's theory of mayl also attempted to provide a quantitive relation between the weight and velocity of a moving body, resembling the concept of momentum. Avicenna's theory also holds that the object is pushed along by the air it displaces.
In the 12th century, Hibat Allah Abu'l-Barakat al-Baghdaadi adopted and modified Avicenna's theory on projectile motion. In his Kitab al-Mu'tabar, Abu'l-Barakat stated that the mover imparts a violent inclination (mayl qasri) on the moved and that this diminishes as the moving object distances itself from the mover. He was also the first to reject Aristotle's law that a constant force produces uniform motion, as he realized that a force applied continuously produces acceleration, which is now considered "the fundamental law of classical mechanics." He also described acceleration as the rate of change of velocity. Jean Buridan and Albert of Saxony later refer to Abu'l-Barakat in explaining that the acceleration of a falling body is a result of its increasing impetus.
In the 14th century, Jean Buridan rejected the Philoponan notion that the motive force, which he named impetus, dissipated spontaneously, and adopted the Avicennan impetus theory in which (i) it is only corrupted by the resistances of the medium and of gravity in the case of anti-gravitational motion, but would otherwise be permanently conserved in the absence of any resistances to motion, and in which (ii) gravity is also a downward projector and creator of downward impetus, unlike in the radically different Philoponan theory in which gravity neither creates not destroys impetus. The assimilation of the role of gravity in natural motion to the role of a projector that creates impetus just as it is created by a thrower in anti-gravitational violent motion was explicitly stated by Buridan's pupil Dominicus de Clavasio in his 1357 De Caelo, as follows:
- "When something moves a stone by violence, in addition to imposing on it an actual force, it impresses in it a certain impetus. In the same way gravity not only gives motion itself to a moving body, but also gives it a motive power and an impetus, ...".
Buridan's position was that a moving object would only be arrested by the resistance of the air and the weight of the body which would oppose its impetus. Buridan also maintained that impetus was proportional to speed; thus, his initial idea of impetus was similar in many ways to the modern concept of momentum. Despite the obvious similarities to more modern idea of momentum, Buridan saw his theory as only a modification to Aristotle's basic philosophy, maintaining many other peripatetic views, including the belief that there was still a fundamental difference between an object in motion and an object at rest. Buridan also maintained that impetus could be not only linear, but also circular in nature, causing objects (such as celestial bodies) to move in a circle.
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, he applied 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. He wrote on the celestial impetus of the spheres 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 either 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 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 producing 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 inertial 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 revised conception of inertia as only resisting accelerated motion but not resisting uniform motion.
Buridan's thought was followed up by his pupil Albert of Saxony (1316–1390) and the Oxford Calculators. Their work in turn was elaborated by Nicole Oresme who pioneered the practice of demonstrating laws of motion in the form of graphs.
The tunnel experiment and oscillatory motion
The Avicennan-Buridan (A-B) self-conserving impetus theory developed one of the most important thought-experiments in the history of science, namely the so-called 'tunnel-experiment', so important because it brought oscillatory and pendulum motion within the pale of dynamical analysis and understanding in the science of motion for the very first time and thereby also established one of the important principles of classical mechanics. The pendulum was to play a crucially important role in the development of mechanics in the 17th century, and so more generally was the axiomatic principle of Galilean, Huygenian and Leibnizian dynamics to which the tunnel experiment also gave rise, namely that a body rises to the same height from which it has fallen, a principle of gravitational potential energy. As Galileo Galilei expressed this fundamental principle of his dynamics in his 1632 Dialogo:
"The heavy falling body acquires sufficient impetus [in falling from a given height] to carry it back to an equal height." 
This imaginary experiment predicted that a cannonball dropped down a tunnel going straight through the centre of the Earth and out the other side would go past the centre and rise on the opposite surface to the same height from which it had first fallen on the other side, driven upwards past the centre by the gravitationally created impetus it had continually accumulated in falling downwards to the centre. This impetus would require a violent motion correspondingly rising to the same height past the centre for the now opposing force of gravity to destroy it all in the same distance which it had previously required to create it, and whereupon at this turning point the ball would then descend again and oscillate back and forth between the two opposing surfaces about the centre ad infinitum in principle. Thus the tunnel experiment provided the first dynamical model of oscillatory motion, albeit a purely imaginary one in the first instance, and specifically in terms of A-B impetus dynamics.
However, this thought-experiment was then most cunningly applied to the dynamical explanation of a real world oscillatory motion, namely that of the pendulum, as follows. The oscillating motion of the cannonball was dynamically assimilated to that of a pendulum bob by imagining it to be attached to the end of an immensely cosmologically long cord suspended from the vault of the fixed stars centred on the Earth, whereby the relatively short arc of its path through the enormously distant Earth was practically a straight line along the tunnel. Real world pendula were then conceived of as just micro versions of this 'tunnel pendulum', the macro-cosmological paradigmatic dynamical model of the pendulum, but just with far shorter cords and with their bobs oscillating above the Earth's surface in arcs corresponding to the tunnel inasmuch as their oscillatory mid-point was dynamically assimilated to the centre of the tunnel as the centre of the Earth.
Hence by means of such impressive literally 'lateral thinking', rather than the dynamics of pendulum motion being conceived of as the bob inexplicably somehow falling downwards compared to the vertical to a gravitationally lowest point and then inexplicably being pulled back up again on the same upper side of that point, rather it was its lateral horizontal motion that was conceived of as a case of gravitational free-fall followed by violent motion in a recurring cycle, with the bob repeatedly travelling through and beyond the motion's vertically lowest but horizontally middle point that stood proxy for the centre of the Earth in the tunnel pendulum. So on this imaginative lateral gravitational thinking outside the box the lateral motions of the bob first towards and then away from the normal in the downswing and upswing become lateral downward and upward motions in relation to the horizontal rather than to the vertical.
Thus whereas the orthodox Aristotelians could only see pendulum motion as a dynamical anomaly, as inexplicably somehow 'falling to rest with difficulty' as historian and philosopher of science Thomas Kuhn put it in his 1962 The Structure of Scientific Revolutions, on the A-B impetus theory's novel analysis it was not falling with any dynamical difficulty at all in principle, but was rather falling in repeated and potentially endless cycles of alternating downward gravitationally natural motion and upward gravitationally violent motion. Hence, for example, Galileo was eventually to appeal to pendulum motion to demonstrate that the speed of gravitational free-fall is the same for all unequal weights precisely by virtue of dynamically modelling pendulum motion in this manner as a case of cyclically repeated gravitational free-fall along the horizontal in principle.
In fact the tunnel experiment, and hence pendulum motion, was an imaginary crucial experiment in favour of A-B impetus dynamics against both orthodox Aristotelian dynamics without any auxiliary impetus theory, and also against Aristotelian dynamics with its Philoponan variant. For according to the latter two theories the bob cannot possibly pass beyond the normal. In orthodox Aristotelian dynamics there is no force to carry the bob upwards beyond the centre in violent motion against its own gravity that carries it to the centre, where it stops. And when conjoined with the Philoponan auxiliary theory, in the case where the cannonball is released from rest, again there is no such force because either all the initial upward force of impetus originally impressed within it to hold it in static dynamical equilibrium has been exhausted, or else if any remained it would be acting in the opposite direction and combine with gravity to prevent motion through and beyond the centre. Nor were the cannonball to be positively hurled downwards, and thus with a downward initial impetus, could it possibly result in an oscillatory motion. For although it could then possibly pass beyond the centre, it could never return to pass through it and rise back up again. For dynamically in this case although it would be logically possible for it to pass beyond the centre if when it reached it some of the constantly decaying downward impetus remained and still sufficiently much to be stronger than gravity to push it beyond the centre and upwards again, nevertheless when it eventually then became weaker than gravity, whereupon the ball would then be pulled back towards the centre by its gravity, it could not then pass beyond the centre to rise up again, because it would have no force directed against gravity to overcome it. For any possibly remaining impetus would be directed 'downwards' towards the centre, that is, in the same direction in which it was originally created.
Thus pendulum motion was dynamically impossible for both orthodox Aristotelian dynamics and also for Philoponan impetus dynamics on this 'tunnel model' analogical reasoning. But it was predicted by the A-B impetus theory's tunnel prediction precisely because that theory posited that a continually accumulating downwards force of impetus directed towards the centre is acquired in natural motion, sufficient to then carry it upwards beyond the centre against gravity, and rather than only having an initially upwards force of impetus away from the centre as in the Philoponan theory of natural motion. So the tunnel experiment constituted a crucial experiment between three alternative theories of natural motion.
On this analysis then A-B impetus dynamics was to be preferred if the Aristotelian science of motion was to incorporate a dynamical explanation of pendulum motion. And indeed it was also to be preferred more generally if it was to explain other oscillatory motions, such as the to and fro vibrations around the normal of musical strings in tension, such as those of a zither, lute or guitar. For here the analogy made with the gravitational tunnel experiment was that the tension in the string pulling it towards the normal played the role of gravity, and thus when plucked i.e. pulled away from the normal and then released, this was the equivalent of pulling the cannonball to the Earth's surface and then releasing it. Thus the musical string vibrated in a continual cycle of the alternating creation of impetus towards the normal and its destruction after passing through the normal until this process starts again with the creation of fresh 'downward' impetus once all the 'upward' impetus has been destroyed.
This positing of a dynamical family resemblance of the motions of pendula and vibrating strings with the paradigmatic tunnel-experiment, the original mother of all oscillations in the history of dynamics, was one of the greatest imaginative developments of medieval Aristotelian dynamics in its increasing repertoire of dynamical models of different kinds of motion.
Shortly before Galileo's theory of impetus, Giambattista Benedetti modified the growing theory of impetus to involve linear motion alone:
"…[Any] portion of corporeal matter which moves by itself when an impetus has been impressed on it by any external motive force has a natural tendency to move on a rectilinear, not a curved, path."
Benedetti cites the motion of a rock in a sling as an example of the inherent linear motion of objects, forced into circular motion.
References and footnotes
- Aristotle's Physics 4.8.215a15-19
- Aydin Sayili (1987), "Ibn Sīnā and Buridan on the Motion of the Projectile", Annals of the New York Academy of Sciences 500 (1): 477–482 
- Fernando Espinoza (2005). "An analysis of the historical development of ideas about motion and its implications for teaching", Physics Education 40 (2), p. 141.
- Aydin Sayili (1987), "Ibn Sīnā and Buridan on the Motion of the Projectile", Annals of the New York Academy of Sciences 500 (1), p. 477–482 :
"It was a permanent force whose effect got dissipated only as a result of external agents such as air resistance. He is apparently the first to conceive such a permanent type of impressed virtue for non-natural motion."
- Aydin Sayili (1987), "Ibn Sīnā and Buridan on the Motion of the Projectile", Annals of the New York Academy of Sciences 500 (1), p. 477–482 
- Aydin Sayili (1987), "Ibn Sīnā and Buridan on the Motion of the Projectile", Annals of the New York Academy of Sciences 500 (1): 477–482 : "Indeed, self-motion of the type conceived by Ibn Sina is almost the opposite of the Aristotelian conception of violent motion of the projectile type, and it is rather reminiscent of the principle of inertia, i.e., Newton's first law of motion."
- Seyyed Hossein Nasr & Mehdi Amin Razavi (1996), The Islamic intellectual tradition in Persia, Routledge, p. 72, ISBN 0700703144
- Classical Arabic philosophy: an anthology of sources Jon McGinnis, David C. Reisman Hackett Publishing, 2007 ISBN 0872208710, 9780872208711
- Gutman, Oliver (2003), Pseudo-Avicenna, Liber Celi Et Mundi: A Critical Edition, Brill Publishers, p. 193, ISBN 9004132287
- Shlomo Pines (1970). "Abu'l-Barakāt al-Baghdādī , Hibat Allah". Dictionary of Scientific Biography 1. New York: Charles Scribner's Sons. 26–28. ISBN 0684101149.
(cf. Abel B. Franco (October 2003), "Avempace, Projectile Motion, and Impetus Theory", Journal of the History of Ideas 64 (4): 521-546 )
- A. C. Crombie, Augustine to Galileo 2, p. 67.
- "Jean Buridan: Quaestiones on Aristotle's Physics".
- According to Buridan's theory impetus acts in the same direction or manner in which it was created, and thus a circularly or rotationally created impetus acts circularly thereafter.
- Questions on the Eight Books of the Physics of Aristotle: Book VIII Question 12 English translation in Clagett's 1959 Science of Mechanics in the Middle Ages p536
- See e.g. Moody's statement "What I have found in Buridan's writings...is the repeated assertion that "prime matter" does not resist motion..." in footnote 7, p32 of his essay Galileo and his precursors in Galileo Reappraised Golino (ed) University of California Press 1966
- The distinction between primary motive forces and secondary motive forces such as impetus was expressed by Oresme, for example, in his De Caelo Bk2 Qu13, which said of impetus, "it is a certain quality of the second species...; it is generated by the motor by means of motion,.." [See p552 Clagett 1959]. And in 1494 Thomas Bricot of Paris also spoke of impetus as a second quality, and as an instrument which begins motion under the influence of a principal particular agent but which continues it alone. [See p639 Clagett 1959].
- "For the resistance that is in the heavens does not tend to some other motion or to rest, but only to not being moved any faster." Bk2 Ch 3 Treatise on the heavens and the world
- See p22-3 & p227 Dialogo, Stillman Drake (tr) University of California Press 1953, where the tunnel experiment is discussed. Also see Galileo's Discorsi, p206-8 on p162-4 Drake 1974 where Salviati presents 'experimental proof' of this postulate by pendulum motions.
- For statements of the relationship between pendulum motion and the tunnel prediction, see for example Oresme's discussion in his Treatise on the Heavens and the World translated on p.570 of Clagett's 1959, and Benedetti's discussion on p235 of Drake & Drabkin 1959. For Buridan's discussion of pendulum motion in his Questiones see p.537-8 of Clagett 1959
- See pp117-125 of 1962 edition and pp118-26 of its 1970 second edition.
- See p128-131 of his 1638 Discorsi, translated on p86-90 of Drake's 1974 English edition.
- Giovanni Benedetti, selection from Speculationum, in Stillman Drake and I.E. Drabkin, Mechanics in Sixteenth Century Italy (The University of Wisconsin Press, 1969), p. 156.
- Clagett, Marshall Science of Mechanics in the Middle Ages University of Wisconsin Press 1959
- Duhem, Pierre. [1906-13]: Etudes sur Leonard de Vinci
- Duhem, Pierre History of Physics Section IX, XVI & XVII in The Catholic Encyclopedia
- Drake, Stillman & Drabkin, I.E. Mechanics in Sixteenth Century Italy The University of Wisconsin Press, 1969
- Galileo De Motu 1590, translated in On Motion and On Mechanics Drabkin & Drake
- Galilei, Galileo Dialogo, Stillman Drake (tr), University of California Press 1953
- Galilei, Galileo Discorsi, Stillman Drake (tr), 1974
- Grant, Edward The Foundations of Modern Science in the Middle Ages 1996
- Koyré, Alexander Galilean Studies
- Kuhn, Thomas The Copernican Revolution 1957
- Kuhn, Thomas The Structure of Scientific Revolutions 1962/70
- Moody, E.A. Galileo and his precursors in Galileo Reappraised Golino (ed) University of California Press 1966
- Moody, E. A. Galileo and Avempace: The Dynamics of the Leaning Tower Experiment published in Journal of the History of Ideas, Vol. 12 1951.