Theory of impetus

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.[5]

In the 11th century, Avicenna (Ibn Sīnā) discussed Philoponus' theory in The Book of Healing, in Physics IV.14 he says:[6]

When we independently verify the issue (of projectile motion), we find the most correct doctrine is the doctrine of those who think that the moved object acquires an inclination from the mover

Ibn Sīnā agreed that an impetus is imparted to a projectile by the thrower, but unlike Philoponus, who believed that it was a temporary virtue that would decline even in a vacuum, he viewed it as persistent, requiring external forces such as air resistance to dissipate it.[7][8][9] Ibn Sina made distinction between 'force' and 'inclination' (called "mayl"), and argued that an object gained mayl when the object is in opposition to its natural motion. Therefore, he concluded that continuation of motion is attributed to the inclination that is transferred to the object, and that object will be in motion until the mayl is spent. He also claimed that a projectile in a vacuum would not stop unless it is acted upon, which is consistent with Newton's concept of inertia.[10] This idea (which dissented from the Aristotelian view) was later described as "impetus" by Jean Buridan, who may have been influenced by Ibn Sina.[11][12]

In the 12th century, Hibat Allah Abu'l-Barakat al-Baghdaadi adopted Philoponus' theory of impetus. 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.[13] Like Philoponus, and unlike Ibn Sina, al-Baghdaadi believed that the mayl self-extinguishes itself.[14]

He also proposed an explanation of the acceleration of falling bodies where "one mayl after another" is successively applied, because it is the falling body itself which provides the mayl, as opposed to shooting a bow, where only one violent mayl is applied.[14] According to Shlomo Pines, al-Baghdaadi's theory was

the oldest negation of Aristotle's fundamental dynamic law [namely, that a constant force produces a uniform motion], [and is thus an] anticipation in a vague fashion of the fundamental law of classical mechanics [namely, that a force applied continuously produces acceleration].[14]

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.[13]

In the 14th century, Jean Buridan postulated the notion of motive force, which he named impetus.

When a mover sets a body in motion he implants into it a certain impetus, that is, a certain force enabling a body to move in the direction in which the mover starts it, be it upwards, downwards, sidewards, or in a circle. The implanted impetus increases in the same ratio as the velocity. It is because of this impetus that a stone moves on after the thrower has ceased moving it. But because of the resistance of the air (and also because of the gravity of the stone) which strives to move it in the opposite direction to the motion caused by the impetus, the latter will weaken all the time. Therefore the motion of the stone will be gradually slower, and finally the impetus is so diminished or destroyed that the gravity of the stone prevails and moves the stone towards its natural place. In my opinion one can accept this explanation because the other explanations prove to be false whereas all phenomena agree with this one.[15]

Buridan gives his theory a mathematical value: impetus = weight x velocity

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.[16] 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. 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.

Buridan pointed out that neither Aristotle's unmoved movers nor Plato's souls are in the Bible, so he applied impetus theory to the eternal rotation of the celestial spheres 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.[17] 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."[18]

However, by discounting the possibility of any resistance either due to a contrary inclination to move in any opposite direction or due to any external resistance, he concluded 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.

This raised the question 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,[19] rather than producing uniformly accelerated motion like the primary force 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. His response was to 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.[20]

Buridan's thought was followed up by his pupil Albert of Saxony (1316–1390), by writers in Poland such as John Cantius, 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 Buridan impetus theory developed one of the most important thought experiments in the history of science, namely the so-called 'tunnel-experiment'. This experiment was important because it incorporated oscillatory and pendulum motion into dynamical analysis and the science of motion for the very first time. It thereby also established one of the important principles of classical mechanics. The pendulum was crucially important to the development of mechanics in the 17th century. The tunnel experiment also gave rise to the more generally important axiomatic principle of Galilean, Huygenian and Leibnizian dynamics, 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.[21]

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.[22]

This thought-experiment was then 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 midpoint 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,[23] on the 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 eventually appealed 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.[24]

In fact the tunnel experiment, and hence pendulum motion, was an imaginary crucial experiment in favour of impetus dynamics against both orthodox Aristotelian dynamics without any auxiliary impetus theory, and also against Aristotelian dynamics with its H-P 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 Philoponus 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 H-P impetus dynamics on this 'tunnel model' analogical reasoning. But it was predicted by the 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 theory of natural motion. So the tunnel experiment constituted a crucial experiment between three alternative theories of natural motion.

On this analysis then 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."[25]

Benedetti cites the motion of a rock in a sling as an example of the inherent linear motion of objects, forced into circular motion.

• Conatus
• Inertia
• Momentum
• Physics in medieval Islam
• Science in the Middle Ages

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