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According to the SEP Buridan was the first to develop the physical motion of impetus: a force applied for a certain time; it is silent, however on just where and how Buridan took this notion - suggesting Avicennas doctrine of mayl (inclination) or Philoponus on the Stoic notion of horme (impetus).

Given that Buridan was famous as an expositor of Aristotle - as was Avicenna - it's worth looking at the most plausible candidate there: this is, I suggest, section VII.5 of Aristotle's Physics, titled 'power acting is to weight moved as distance moved is to time taken'

Given the lack of an efficient algebraic notation in Aristotle's time this is a little confusing, but he explains it as follows:

an agent of movement always moves something, does so in something, and does so to some extent; by 'in something' I mean 'in some time'; and by 'to some extent' I mean 'over a certain amount of distance'.

Redacted, this says: an agent of movement always moves something, and does so in some time, and over a certain distance.

Aristotle then designates F as the agent of movement on an object m; and d the distance moved and t the time taken (in Sorabjis translation the letters chosen are different - they are just alphabetically chosen, and this is presumably aligned to Aristotles own choice); he then states:

The ratios will be preserved.

And

So that in an equal time t, an equal power F will move half m double the distance d.

And in half the time t, it will move half m, in the distance d.

The question, is how to translate this into modern physical notation: given that he introduces weight, distance and time; and we are talking about motion and power, the obvious possibility to try is linear momentum mv - but this introduces the new notion of velocity; but noting that Aristotle speaks of a fixed length of time, we ought to write this then as md/t.

The question is does this work? Let us write a given power F moves an object m a distance d in a time t as:

F <--> md/t

Where the RHS is to be understood as shorthand for the preceding paragraph; and also algebraically.

Then his first example is:

(1/2 m) (2d) / t = md/t

And this is exactly what we said F would accomplish.

And his second example is:

(1/2 m) (d) / (1/2 t) = md/t

And again this is what F would accomplish.

Aristotle, then adds as a third example:

Also, if the same power moves the same object just such a distance on just such a time, and half the distance in half the time, then half the power will take an equal amount of time to move an object half the weight over an equal distance.

1/2 F <-> (1/2 m) d / t

This, again is correct.

Question: Is this analysis a fair reflection of Aristotle's prose?

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The notion of velocity (rather speed) was not new to Aristotle, it was defined by Eudoxus and used in astronomy, including by Aristotle himself. Aristotle roughly thought that the speed of forced motion is proportional to the "power" (force) causing it, and inverse proportional to the resistance of the medium, so mv=F/R is roughly right.

This applies with caveat that the body is actually moving, because the rest is a fundamentally different state for Aristotle. This also leads to infinite velocity in the vacuum (zero resistance), which is one of the reasons why he thought that "nature abhors vacuum". So there is some inescapable distortion in translating Aistotle's mechanics into modern mathematical notation. Rovelli makes a systematic attempt, and shows that his mechanics is largely reproduced in the limit of highly resistant medium.

Aristotle's account of projectile motion however can only be interpreted very artificially as a forced motion (because of low air resistance), and was criticized already in antiquity, by Philoponus and possibly Hipparchus earlier. Philoponus pointed out that on Aristotle's account of force moving air moving arrow one should be able to make the arrow move by waiving hands behind it. This convinced Avicenna, Avempace, and later Buridan, to adopt the impetus theory, where forced motion can be maintained without constant impact from a force. See Franco's paper.

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