EXCURSE TO THE HISTORY OF UNDERSTANDING OF MOTION – DE MOTU

Igal Galili and Michael Tseitlin

The Hebrew University of Jerusalem

February 2010

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Excurse to the History of Understanding of Motion – De Motu

Igal Galili and Michael Tseitlin

The Hebrew University of Jerusalem

igal@vms.huji.ac.il

Abstract

This excursus furnishes students' understanding of the conceptual basis of the classical theory of motion through displaying the older theories, the one by Aristotle in Hellenic science, the first scientific account of motion, and the subsequent theory known as the theory of impetus, that reigned in physics until the scientific revolution of the 17th century. The theory of impetus emerged in the critique of the Aristotelian theory of motion and was refuted in the transition to the modern science of classical mechanics. The concept of impetus became central in physics after Aristotle and served as a mediator between the Aristotelian and the Newtonian mechanics. The relevance of students' familiarizing with the old theories of motion draws on the fact that the contemporary students of physics continuously rediscover the impetus idea as a spontaneous idea and thus erroneously account for motion prior, during and after their learning of physics. Familiarizing with older theories, provide the considered knowledge with both conceptual as well as in cultural perspective. This experience is educationally valuable because it encourages students to organize there knowledge in a format of scientific theory, to argue pro- and contra- in advocating for certain ideas in science. The revision of pertinent conceptions and the conceptual change to the classical Newtonian view becomes easier. Classical mechanics emerged in the painstaking effort of critique with respect to the impetus theory. The excurse familiarizes students with the scientists of different cultural periods, from Aristotle to Newton, and their approaches to the scientific knowledge which changed greatly from period to period. The intention is to show that despite this variety scholars created the objective knowledge of motion. The pioneers of theory of motions were at the same time great philosophers and enthusiasts of exploration of reality in terms of objective knowledge regardless their errors and views that we do not share today. Our depiction and analysis expanded not only on the subject matter claims but also on the employed epistemology, the kind of evidence adopted, and the reasoning used. Acquaintance with their epistemologies, may bring the students to appreciate both the new theory and the new method, which drew on the experimental verification and refuted other metaphysical considerations. It is easier to understand the new physical ideas through comparison with older ones, such as impetus. It is instructive to acquaint with the intellectual products of the bright minds of the past who suggested other theories of motion and thus essentially contributed to the scientific progress. These scientists may serve examples of spiritual excellence, and encourage appreciation of our cultural debt to those scholars of the past and this way to understand the cumulative nature of science as a continuous human endeavor of constructing the objective knowledge of reality. .

Keywords: force, impetus, momentum, theory of motion, "charge of motion", rest-motion equivalence.

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UNDERSTANDING OF MOTION

Description of the Case Study And the Lord said unto Cain, Where is Abel thy brother? And he said, I know not: Am I my brother’s keeper?

Genesis 4: 9

And the Lord put a mark on Cain, lest any who came upon him should kill him. Then Cain went away from the presence of the Lord, and dwelt in the land of Nod ["the land of wandering"], east of Eden.

Genesis 4:14-16

Forgive us, our Father, for … we have failed to judge ourselves in this particular area. How many times we have glossed over our prejudices and treated them as unimportant trivialities, ... Thank you for … helping us to understand what is happening ... May we face it in realism and in truth. ...Amen.

From a prayer When we all learn the First Law of Newton, we read:1

Law I. Every body perseveres in its state of being at rest or of moving uniformly straight forward except insofar as it is compelled to change its state by forces impressed.

We see here Newton addressing a body which "perseveres in its being at rest or of moving uniformly straight forward". Newton placed the two states at rest and moving uniformly straight forward next to each other separated with or which signifies them as if being not exactly the same. In the rest of this Newton fundamental treatise of classical mechanics, as well as from any modern textbook of mechanics, we learn that in fact the two mentioned states of motion are practically equivalent: velocity is a relative entity. This implies that the state of rest is a special case of the uniform rectilinear motion with zero velocity, and thus Newton could just not mention it at all. Why did he?

The answer is in the history of physics prior to Newton. From the very beginning of physics in the fourth century B.C. and up to the scientific revolution of the 17th century, scholars distinguished between the rest and motion as fundamentally different concepts. Keeping with this background one may interpret the text of Newton as a vestige of certain mental commitment (or the "Mark of Cain") and/or as his saying in the ongoing discourse, never stopping in science regarding the nature of motion, that Newton saw himself as a part it. Newton talked other scholars using their concepts that was going to change.

In the following, we are going to retrieve in the major conceptual points this discourse in order to understand the conceptual fundamentals of the classical mechanics, which since then became the basic of any physics or science curriculum.

1 Newton, I. (1687/1999). The Principia. Mathematical Principles of Natural Philosophy. Translated

by B. Cohen & A. Whitman. University of California Press, Berkeley, CA, p. 416.

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I. THE FIRST THEORY OF MOTION – ARISTOTLE The first scientific theory of motion was produced by Aristotle in his Physics (IVc. B.C.). It was a great accomplishment and all further theories drew on that one this or other way, on this theory in which the subject matter and the conceptual agenda were determined. Aristotle defined change in general to be a subject for physics inquiry. Motion – the change of location in time, or locomotion, in terms of Aristotle – was one of possible ways of changes taking place in the world. The theory focuses on motion of bodies and leaves all other aspects out of interest. Yet, even under this assumption, it was not a simple task to describe motion and explain it.

Explanation in science Aristotle cared that any investigation should be organized within some general frame, a plan that will determine in general, what will be the goal of our effort. He asked himself, what does it mean to know something? 2

Knowledge is the object of our inquiry, and people do not think they know a thing till they have grasped the 'why' of (which is to grasp its primary cause). So, clearly, we too must do this as regards both coming to be and passing away and every kind of physical change, in order that, knowing their principles, we may try to refer to these principles each of our problems.

So, to explain a thing one needs to find out and demonstrate the causes for it from knowing general principles. Firstly, he mentioned that the material of the objects is important to understand it. For example, if the object is a statue and is made of marble, this explains its beauty – providing the material cause. Then, he proceeded, it is clear that the shape of things and events is also of importance. In the case of a statue, it is especially simple to see the formal cause – the shape of the statue.

In the next type, Aristotle pointed to the "primary source" of the subject or considered change. In the case of a statue this is the special skill of the artisan who knows how exactly to make the statue. This is the effective cause of the subject. This was the effective cause.

Finally, Aristotle mentioned that the goal for the subject is also important for its understanding since it may explain much of its nature. We then think about what is it for? How that it fits the considered goal? This is the teleological cause. In the case of a statue, if we know that the statue will be served to decorate, say, the sport complex, we may better understand the specific features of the states which were aimed to serve this goal.

Explanation of motion Following this framework Aristotle addressed the question of motion. Motion was opposed the state of rest, essentially different existence of things. Motion was perceived as a change, in particular the change of location. It was therefore called locomotion. The rest, being natural does not need any cause. Motion was considered

2 Aristotle, Physics. (-IVc. B.C.), Book II, part 3. http://classics.mit.edu/Aristotle/physics.2.ii.html

Aristotle

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not a state but a process of changing place (Fig. 1). The motion according to Aristotle was realization, actualization of some causes.

Aristotle distinguished two major types of locomotions natural and violent.

The natural motion The natural motion was related to the natural order of things. Firstly, all material objects were considered to be comprised of four elements, which naturally intend to arrange in order of their weight: from heavier down, closer to the center of the universe, to lighter – up, in the direction from the center of the universe. The elements were: earth, water air, and fire. Beyond the fire area, the celestial spheres start, comprised of the fifth element – ether (Fig. 2).

When this order is violated by a certain amount of element, a cause, and even more than one, appear for the natural motion to restore the order and for the element to return to the

natural place.

The motion takes place for material (such as air in water medium), formal (such as fire placed

underground) and teleological (for restoring the order) causes. Clearly, the natural motion could be upwards (lighter than the environment) and downwards (heavier than the environment). In the case of a body comprised of several elements, the net intention, the one resulting the summarized effect of all compounding elements in the body in comparison with the environment, determines the direction of the natural movement.

Weight concept was determined by Aristotle in this context – as the tendency to restore the natural order, the measure of the intention to occupies lower or higher radial location. Importantly, in accordance with the way the Greeks conceived the reality, weight was introduce with its opposite: levity – the tendency of the light things to ascend upwards. Thus, levity was ascribed to air and fire.

When motion starts the moving object may change the place with different speed. The swiftness of the body is proportional to its weight and inversely proportional to the resistance of the medium. Although Aristotle did not use the concept of velocity or speed as well as did not present the dependences in mathematical form, we may express his conception of natural motion in the form of the following formula:

RWv ∝ (1)

State 1

State 2

The process of motion

Figure 1. Motion as a process of transition between two states of resrt.

Figure 2. The natural order of elements in the cosmos in accordance to the Aristotelian view.

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with v – for velocity, W – for weight of the falling body and R – for the medium resistance. Natural falling was considered the paradigmatic case of natural motion. This dependence makes sense for our everyday experience. Indeed, a metallic ball falls faster than a plastic one, and the speed of falling decreases if one drops the metallic ball in water in comparison to its fall in air. All these situations support the dependence expressed in (1).

The violent motion Any movement different from the vertical natural spontaneous motion was considered violent. Unlike the natural motion, the violent motion required efficient cause – force. It was conceived as some agent or engine, external to the body serving as a mover (Fig. 3). The outside mover presents the principle of the Aristotelian dynamics.

Any lifting things was considered a violent motion. Suppose a man is pushing a heavy barrel along inclined plane (Fig. 4). The efficient cause of lifting (motion) is the man pushing. The man

is the source of this movement upwards. Let's emphasize the difference with the modern account. While the efficient cause of Aristotle, the man who is the creator of motion, would not be of our interest which is on physical force applied, regardless its origin.

As with regard to the natural motion, one may be interested in the rate of change the location by the moving body (what we name – velocity). In regarding the violent motion, Aristotle related the speed in this case in direct proportion to the power of the applied force and inverse proportion to the resistance of the medium or the weight of the body:

RFv ∝ or

WFv ∝ (2)

with v – for velocity, F – for the strength of the applied force, W – for weight of the falling body and R – for the medium resistance. In a sense, the two formulas say that weight of the object present a resistance to move it. Good to see that these formulars hold only when F is bigger than R(W). Only then the force may overcome the resistance and cause motion. If not (F ≤ R), the body remains at rest (v=0).

These formulas make sense in many regular situations. It is not difficult to illustrate them. If one pushes the object harder, it moves faster. If we put an object on the rough surface and then push or pull it, applying the same effort, our effort will cause less speed of the movement. Finally, if one applies the same force to a big, and then, to a smaller weight, their speeds will be in the inverse proportion to their weights. It is clear that to move smaller object is easier.

Despite of many simple situations that go well with the established regularity, this theory immediately matched with the challenging situation – the movement of projectiles. Indeed, what serves as an efficient cause when we throw a stone, after it

Figure 3. To move an object one needs a mover – the agent applying force

Figure 4. Any lifting presents a violent motion

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loses the contact with our hand? What serves as a mover of the stone? These questions created the most difficult challenge to the Aristotelian theory of motion.

Two worlds separation Observation of the world brought Aristotle to the idea that the world is separated into two areas of different laws with respect to motion. One area includes our regular environment and extends from the center of the world to the orbit of the Moon. In that area the natural motion is radial and vertical to the surface of the Earth. The second area is a celestial area of ether. There the only motion is natural and unlike the under-Moon world the natural motion of that world is uniform circular motion of planets.

The two figures presented here reflect already the medieval modification of the Aristotelian picture of the world. As the first change of the original theory of Aristotle, the medieval scholars suggested that the motion of the celestial bodies require some sort of the mover. It is represented with the mashinary kind of wheel depicted in the picture (Fig. 5). On Fig. 6, angels are shown in the role of providing circular motion to celestial spheres by means of a mechanical machine (Fig. 6).

The motion of projectiles Let's cosider the motion of a stone thrown by a hand. We need a force that could reason this violent motion after the stone leaves the hand. What could it be? Aristotle's answer was quite original – the air, this is the efficient cause of projectiles. How could it be? When the hand accelerates the stone, it moves the surrounding air too. Air in the layers in contact with the stone is set in motion and circulate around it. This because the stone splits the air, moves forward leaving a vacancy behind the stone. When the circulating air rushes into this area behind the stone, it applies force (push) on the rare side of the stone, propelling it in the direction of the motion. This

R F

Figure 7. Stream vortexes – antiperistasis – creating, in vew of Aristotle, the push forward, propagating the shell in the medium.

Figure 5. Two worlds separation on the old engraving. The mover mechanism is represented by a wheel (arrow).

Figure 6. Two angels (arrows) set in motion the celestial spheres, serving as a mover.

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mechanism was called antiperistasis – inverse, or reciprocal, circular pressure (Fig. 7).3

The result is rather strange: the air which might be seen as a creating the resisting force R on the shell obtains in addition the role of the mover and apply the pushing force F. It follows that without the air the motion of projectiles is impossible.

Addressing the stone thrown upwards Aristotle4 describes both tendencies: the natural (without force) acting downwards and the impressed upon the body from outside (by the force of the air):

But since 'nature' means a source of movement within the thing itself, while a force is a source of movement in something other than it or in itself qua other, and since movement is always due either to nature or to constraint, movement which is natural, as downward movement is to a stone, will be merely maintained by an external force, while an unnatural movement will be due to the force alone. In either case the air is as it were instrumental to the force.

Aristotle5 proceeded to explain the dual function of the air by it being at the same time light (in comparison with water and earth) and heavy (in comparison with fire):

For air is both light and heavy, and thus qua light produces upward motion, being propelled and set in motion by the force, and qua heavy produces a downward motion. In either case the force transmits the movement to the body by first, as it were, impregnating the air. That is why a body moved by constraint continues to move when that which gave the impulse ceases to accompany it. Otherwise, i.e. if the air were not endowed with this function, constrained movement would be impossible. And the natural movement of a body may be helped on in the same way. This discussion suffices to show (1) that all bodies are either light or heavy, and (2) how unnatural movement takes place.

Was it a convincing explanation? Seemingly, not since although consistent with the rest of the theory, it spurred a desperate critique not much time after Aristotle.

II. PHYSICS OF THE IMPETUS

The explanation of projectiles by means of the air flow both supporting and resisting the motion at the same time (antiperistasis) looked artificial already to the contemporaries of Aristotle. The complexity of explanation was at odds with the apparent simplicity of the projectile motion. What was perceived as problem was the fact that the thrown stone continues to move when without any apparent external mover which maintains the movement of the thrown body. Other ideas were seemingly required to explain projectiles.

3 In Greek, άντιπερίστασὶς, is formed of άντί ("against") and περίστασις ("standing around"), and terms the resistance to anything that surrounds or besets another. For the projectile this suggested the force against the air resistance.

4 Aristotle (1952), On the Heavens, Book III, Part 2. Great Books of the Western World. Encyclopeadia Britannica. Chicago.

5 Ibid.

Hipparchus

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Early history Among the first who provided an alternative explanation for the projectile motion was the Greek astronomer Hipparchus (the 2nd c. BC) – a brilliant mind of the Hellenistic culture that followed the Hellenic one. That was his explanation of the tossed stone. The hand who throws a stone at the same time, endows it with some capacity ("the throwing force"), which counteracts gravity. We may call it impetus – the name this force, or virtue, received much later, during the medieval time in Europe. While the "the throwing force" exceeds the gravity, the body is moving slowly upwards, stopping when they are equal (Aristotelian principle: motion of a body implies force exerted on it), and begins to move rapidly down when the gravity prevails the throwing force. A motion reaches the maximum speed when the impressed a sort of kinetic force (impetus), which exhausts itself in the course of movement. It continues until the impetus if fully dispensed. By our modern means, we can represent this scenario graphically (Fig. 8).

In the 6th century, John Philoponus (490 to 570) ["lover of toil"], the Early

Christian commentator of Aristotle considered the general case of projectile motion. His view practically coincided with that of Hipparchus. Philoponus proposed to explain the projectile motion, after the projector ceases to move it, by some "immaterial driving ability" imprinted in the movable body by the projector. It looks like a charging the projectile with the charge of motion.

The situation when a body is simply dropped from the height (zero initial velocity) apparently required a different treatment. In this case, Philoponus considered permanent influx of impetus through the supporting hand into the ball in accord with the required by the particular weight (and so the hand is tired). The influx stops when the body is dropped and the impetus spontaneously disappears explaining the growing velocity of the falling body.

Philoponus criticized Aristotelian theory of motion in two ways. First, he appealed to experience. When two heavy and unequal stones were dropped, and one was twice as heavy as the other, the two stones reached the ground almost together and not as Aristotle would predict: the heavier stone should be twice faster. Furthermore, why cannot we through light things (feather) farther than heavier (stone)?

Weight of the stone

Impetus

time

The stone reaches the highest point and stops for a moment: I = W

The stone ascends and decelerates: I > W

The stone descends and accelerates: I < W

Figure 8. The dependence of the impetus of the stone thrown upwards on time

0

The stone leaves the hand

The impetus is expended I=0 and the stone continues to fall thereafter at a maximum speed ("terminal velocity")

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The empirical discrepancy was not, however, perceived as decisive. After all, Aristotle could address the major order and the reality may add other factors, such as the complex dependence on the air resistance. A more puzzling was the appeal to the special cases of motion: that by a spear and the spinning top.

Beyond the question why an arrow continue to fly after it has left the bowstring, there is another puzzle. It is well known that a spear (as well as an arrow) sharp in both ends flies much faster and longer than a body whose rare side was flat, although one might think that that flat rare surface of a body would match more push due to the antiperistasis and therefore move faster and longer – a contradiction.

Even a more striking contradiction is provided by a spinning top. In the case of such motion, different parts of a moving body replace each other without any support of the pushing air. Why the tope set up in rotation continues to spin?

These questions could not be answered within the Aristotelian theory of motion, and they supported the new

alternative understanding: the virtue of motion is provided to the body set in motion. The agent of motion is internal and not external.

Furthermore, Philoponus criticized the major law of motion by Aristotle that we have expressed by formula 1. This was important also because through using this law Aristotle demonstrated the absence of vacuum: if there were no void (R=0) there would be an infinite speed – on obvious absurdity. Instead, Philoponus suggested a different dependence:

RFv −∝ (3)

This implies that motion happens when the force F overcomes resistance R or gravity (in throwing upwards), that is, F > R. Obviously, nothing dramatic happened when the resistance of the medium disappears (R=0). Although Philoponus did not stated the existence of void, he did not reject it: when bodies move, they exchange places and this suggests of an empty space that is filled by the bodies.

Philoponus further points to the principle of pipette (clepsydra – "water thief") – a device in ancient Greece, using for drawing liquids from vessels too large to pour (Fig. 9). For the long history that included Galileo in the 17th century, its work was erroneously explained by the ability of vacuum to suck liquids. We widely use this

device to suck light beverages from a bottle.

Philoponus, thus, changed the understanding of the role

of medium for motion: from the essential and vital for movement support – Aristotle, to the impediment to motion.

Figure 9. Pipetta works: After being smashed with pressure, the small balloon recovers, for being elastic, causing the liquid to rise and enter the balloon.

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Impetus in the medieval physics The scientific tradition of the Hellenic and Hellenistic cultures were kept by the following culture of the Muslim world. Scientific development drew on the knowledge preserved. Two scholars are known for their addressing the idea of impetus: Ibn Sīnā (Latinized name Avicenna) (981-1037), a Persian polymath and Abu'l-Barakat al-Baghdadi (1080-1165) a physicist and philosopher of Jewish-Arab descent from Baghdad. Aristotle continued to be a stimulator of critique and the innovative alternative idea was similar to the ideas of Hipparchus and Philoponus.

Avicenna introduced the entity of inclination (mayl) to motion that is transferred to the projectile by the thrower. He stated that projectile motion in a vacuum would not cease. The inclination is dissipated by the influence of the medium such as air resistance. Mayl was defined as being proportional to weight (W) times velocity (v), a clear precursor to the concept of momentum (mv):

vWM ⋅∝ (4)

Abu'l-Barakat al-Baghdadi also stated that the mover imparts a violent inclination (mayl qasri) on the moved, diminishing later, when the moving object distances itself from the mover. He stated that a constant force produced uniform motion quite in accord with Aristotle (2), but described motion in a much more mature way: he introduced acceleration as the rate of change of velocity. Falling was explained as resulting an increasing impetus.

In Europe, the progress of mechanics was instigated in the 14th century by a French priest and professor at the University of Paris Jean Buridan (1295 – 1358), who knew about the works of Abu'l-Barakat.

It was John Buridan introduced the notion impetus – motion maintained property which was considered in different ways by previous scholars debating Aristotelian view.

Buridan formulated his explanation of the motion of the thrown-up body and the fact that this body accelerates when falling. In did it in his commentaries on Aristotle's Physics. In his view, air cannot be a driver. Rather, it provides resistance to the movement while the moving body is pushing to split and dissect the air. He points to the strange fact: in the absence of the hand, the body does not experience any push against the wind. Why it all depends on the hand? Also, how to explain the movement of a potter's wheel? It is set in motion by hand and then continues to move for a long time. One cannot explain it by the air. If Aristotle is right, then: 6

. . . you would throw a feather farther than a stone and something less heavy farther than something heavier, assuming equal magnitudes and shapes. Experiences show this to be false

6 Buridan, J. (1509). Questions on the eight books of Physics of Aristotle. In Clagett, M. (1959). The

Science of Mechanics in the Middle Ages. The University of Wisconsin Press, Oxford University Press, London, Document 8.2, p. 534.

Avicenna

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Instead of the explanation by Aristotle, Buridan offers another one: the engine of motion is the throwing hand that transmits an "impetus" (pressure, power, desire, ability to move), which propels the body: 7

Thus we can and ought to say in the stone or other projectile there is impressed something which is the motive force (virtus motiva) of the projectile. And this is evidently better than falling back on the statement that the air continues to move that projectile. For the air appears rather to resist. Therefore, it seems to me that it ought to be said that the motor in moving a moving body impresses in it a certain impetus or a certain motive force (vis motiva) of the moving body, in the direction toward which the mover was moving the moving body, either up or down, or laterally, or circularly. And by amount the motor moves that moving body more swiftly, by the same amount it will impress in it a stronger impetus. It is by that impetus that the stone is moved after projector ceases to move. But that impetus is continually decreased by the resisting air and by the gravity of the stone, which inclines it in a direction contrary to that in which the impetus was naturally predisposed to move it. Thus the movement of the stone continually becomes slower, and finally that impetus is so diminished or corrupted that the gravity of some stone wins out over it and moves the stone down to its natural place.

This method, it appears to me, ought to be supported because the other methods do not appear to be true and also because all the appearances are in harmony with this method

Although long and a bit repetitive, this explanation of motion attracts by its simplicity. The impetus is indeed similar to the momentum in the classical mechanics. Both are in direct proportion to speed and, in a way, to the mass. The latter is approximately true, because Buridan had no notion of mass, and used the quantity of matter. Here is how Buridan justified the latter dependence:8

For if anyone seeks why I project a stone father than a feather, and iron or lead fitted to my hand father than just as much wood, I answer that cause of this is that the reception of all forms and natural dispositions is in matter and by reason of matter. Hence by the amount more there is of matter, by that amount can the body receive more of that impetus and more intensely. Now in a dense and heavy body, other things being equal, there is more of prime matter than in rare and light one. Hence a dense and heavy body receives more of that impetus and more intensely, just as iron can receive more validity than wood or water of the same quantity. Moreover, a feather receives such an impetus so weakly that such an impetus is immediately destroyed by the resisting air. And so also if light wood and heavy iron of the same volume and of the same shape are moved equally fast by a projector, the iron will be moved farther because there is impressed in it a more intense impetus, which is not so quickly corrupted as the lesser impetus would be corrupted. This also is the reason why it is more difficult to bring to rest a large smith’s mill which is moving swiftly than a small one, evidently because in the large one, other things being equal, there is more impetus.

7 Clagett, op.cit. p. 534-535. 8 Ibid., p. 535.

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With regad to the falling bodies Buridan wrote: 9

From this theory also appears the cause of why the natural motion of a heavy body downward is continually accelerated. For from the beginning only the gravity was moving it. Therefore, it moved more slowly, but in moving it impressed in the heavy body an impetus. This impetus now together with its gravity moves it. Therefore, the motion becomes faster; and by the amount it is faster, so the impetus becomes more intense. Therefore, the movement evidently becomes continually faster.

This is a new mechanism: impetus by itself, not only supports but also accelerates the movement. In a way, this inference follows from the Aristotelian law of speed of motion being in a direct proportion with the motion cause.

Buridan addressed the common phenomenon of long jump of a person:10

[The impetus then explains why] one who wishes to jump a long distance drops back a way in order to run faster, so that by running he might acquire an impetus which would carry him a longer distance in the jump. Whence the person so running and jumping does not feel the air moving him, but feels the air in front strongly resisting him.

Impetus and falling Although the major effort to explain motion by impetus was related to violent motion, medieval scholars did leave natural motion without applying to it the same idea of impetus – the charge of motion. Here is the account for

motion of falling by Buridan:11

…it follows that one must imagine that a heavy body not only acquires motion unto itself from its principle mover, i.e. its gravity, but that it also acquires unto itself certain impetus with that motion. This impetus has the power of moving the heavy body in conjunction with the permanent natural gravity. And because that impetus is acquired in common with the motion, hence the swifter motion is, the greater and stronger the impetus is. So, therefore, from the beginning the heavy body is moved by its natural gravity only; hence it is moved slowly. Afterwards it is moved by that same gravity and by the impetus acquired at the same time; consequently, it is moved more swiftly. And because the movement becomes swifter, therefore the impetus also becomes greater and stronger, and thus the heavy body is moved by its natural gravity and by that greater impetus simultaneously and so it will again be moved faster; it will always and continuously be accelerated to the end.

One may see that the context of natural motion – falling, brought the scholar to the new feature of impetus. If in the violent motion the impetus was normally dispensed on maintaining motion, in the context of falling it was initially caused by the natural gravity (or weight) of the falling body and then could cause together with

9 Ibid., p. 535-536. 10 Clagett, M. (1959). Op.cit., p. 536. 11 Clagett, M. (1959). Op. cit., pp. 551-552.

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the natural gravity the production of more impetus which was identified with the accidental gravity (or weight) of the falling object. Clearly this was a merely descriptive account for the fact the apparent accelerated motion of the falling body. The relation to "natural weight" at the beginning, as well as labeling the additional motion by "accidental weight" in the course of motion are seemingly of operational origin. Indeed, both weights reflect the effort to stop the falling: it is obvious that to support a body at rest is easier than to stop it at motion, and much easier than to stop it falling already at high speed.

The birth of pendulum One of the central tools used by scholars of the medieval physics was thought experiment. It was understood as a theoretical application of the considered principles to a description of certain imaginary situation following by its account: analysis and inferences. One such thought experiment became especially famous. It suggested imagining a tunnel across the Earth globe (a totally impossible to realize) and a person who threw a small object into the tunnel (Fig. 10). What would happen then?

In accordance with Aristotle's theory, the body should fall towards the center of the universe (the Earth) and stop there. However, the scholars of the 14 th century thought already differently. Here is what Albert of Saxony (1316-1390), one of the followers of Buridan wrote12:

… it would be said also that if the earth were completely perforated, and through that hole a heavy body were descending quite rapidly toward the center, then when the center of gravity of the descending body was at the center of the world, that body would be moved on still further in the other direction, i.e., toward the heavens, because of the impetus in it not yet corrupted.

This way the scholars of mechanics discovered a new type of motion: oscillation of an object with respect to a certain point. Thus, Albert proceeded:

And, in so ascending, when the impetus would be spent, it would conversely descend. And in such a descent it would again acquire unto itself a certain small impetus by which it would be moved again beyond the center. When this impetus was spent, it would descend again. And so it would be moved, oscillating [titubando] about the center until there no longer would be any such impetus in it, and then it would come to rest.

The new theory of motion implied a new scenario to this imaginary situation. Due to the accumulated impetus, the body should not stop at the center of the Earth but continue to swing around the center. It was common that the scholars of that time did not look for any empirical evidence of support to the theoretical claim. In this case,

12 Ibid., p. 566.

Albert of Saxony

Figure 10. Dropping a body towards the Earth's center.

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however, the comparison with the motion of a body suspended by a thread was made. A new object of physics, which played a great role in the following history of physical theories – pendulum – was invented and oscillatory motion of things in reality – discovered by physics.

Projectiles and impetus The medieval scholars of the 14th century upgraded a descriptive theory of Aristotle by employing the idea of impetus. In the simplest version (Fig. 11) a projectile (a cannon ball) received impetus when launched and followed a straight line until it "lost its impetus", at which point it fell abruptly to the ground.

Albert of Saxony suggested a more sophisticated understanding, better corresponding to the observed reality (Fig. 12).

In his vision, the trajectory of a projectile was comprised from three intervals and in each of them the motion was either rectilinear or circular. The first rectilinear fragment was totally determined by the provided impetus. At the second,

circular interval, the externally provided impetus competed with the gravity until the former was fully dispensed. The second rectilinear interval was totally determined by gravity, falling down, to the ground following a straight line.

The medieval progress in kinematics

Until now the account for motion suggested by the scholars took place as a rather philosophical, that is to say, qualitative, debate in a sense that it used vague and not precisely determined concepts. Thus Aristotle addressed the swiftness of a moving body not in term of speed, but expressed the same addressing the time needed to cover certain distance. This vision id different from the modern one and rather corresponds to our description in terms of average speed:

timeelapseddistancespeed = (5)

This was, however, not sufficient to account for the reality. The distinguished medieval scholar Jordanus Nemorarius13 of the 13th century found a clever demonstration that the falling water becomes more and more swift. Indeed, when we

13 Not much is known about this brilliant scholar beyond his belonging to Dominican order. His

outstanding results were the account for the static equilibrium of two connected weights on inclined planes by means of a special principle he introduced. The latter is equivalent to the principle of virtual displacements – the first study which introduced the product of force times distance (work!).

Figure 11. Simple trajectory of a projectile explained through using impetus theory.

Figure 12. A more realistic trajectory of a projectile explained by using impetus theory.

16

observe the falling jet of water we see that it converges, becomes narrower downwards. Jordanus understood that this indicates growing the speed and in fact introduced acceleration concept. This was also the first empirical evidence for the accelerated motion of the falling bodies.

Gerard of Brussels14, the important Flemish physicist of the same time, explored the important question describing motion.15 Suppose the body moves non-uniformly, swift and slow in variation. Gerard introduced a representative velocity of such motion: the speed of uniform motion that would cover the same distance (s) during the same time interval (Δt). This is equivalent to what we consider an average speed v*:

tvS Δ⋅= * (6)

The idea of average speed was very important. It provided a way to describe the motion of solid bodies with final dimensions. For example, for the first time, Gerard proved that if the point a (Fig. 13) describes a circle around O, then if point b at the middle of the radius were going uniformly forward at its speed the area described by the point a (the sector) at certain time would be equal to the area described by the radius if it were moving uniformly with the speed of point b (the shaded rectangular). This treatment of the 13th century allowed, in fact, our modern understanding of rotational movement which we express usually by the dependence between the linear velocity v, angular velocity ω the radius of rotation:

v = ω·R (7)

The following 14th century brought extremely important development of motion description – kinematics. This progress took place in Oxford, at Merton College, by the group of scholars, known as the Merton school: Thomas Bradwardine, William Heytesbury, Richard Swineshead and John Dumbldon. These scholars came to mechanics from the philosophical debates of scholastic philosophy16 addressing the need to describe qualitative accounts for motion by the more precise quantitative ones.17

14 This scholar of the 13th century is also known mainly for his survived works in kinematics. 15 Gerard's Liber de Motu was the first treatise entirely devoted to kinematics. 16 The discussion of manipulating with different qualities comes back to Aristotle and later to Thomas

Aquinas who questioned the meaning of adding different types of qualities. 17 Clagett, M. (1950). Richard Swineshead and Late Medieval Physics: I. The Intension and Remission

of Qualities. Osiris, 9, 131-161.

v

v/2

a

b

O

R R

Figure 13. Description of circular motion of a disc by Gerard.

17

With regard to physical quantities the problem was to identify intensive versus extensive qualities, determine characteristics of intension and remission in the essential meaning. Such were, for example, the problem to distinguish between the quantity of mass and the intensive quantity (density), the quantity of heat and the intensive quantity- effectiveness of an agent in a heat action (in future – temperature). With regard to motion, this was the distance covered versus the intensive quantity – speed. In fact, this problem was far from trivial and actually led to the problem of infinitesimals: the meaning of specific change and dividing a small by another small, the way one can measure intension and remission of qualities.

With regard to uniform motion Thomas Bradwardine wrote:18

A motion is called uniform when equal distances are traversed in equal times with the same velocity

This is, in fact, a definition of a constant velocity. As for the instantaneous velocity, William Heytesbury, suggested:19

In a non-uniform motion the velocity at a given instant of time may be conceived as the distance which the body would traverse if, in a certain interval of time, it was moved uniformly with the velocity it had at the given instant.

This was a great step in physics progress – the introduction of instantaneous velocity. Although this definition by Heytesbury was defected and suffered by tautology, it introduced into use the concept of central importance and, of course, indicated the enormous difficulty to define instantaneous velocity before the introduction of calculus and the concepts of limits.

Furthermore, the same Heytesbury defined the uniformly accelerated motion:20

Any motion is uniformly accelerated when the velocity is increased with equal amounts in arbitrary, but equal, intervals of time.

And despite a limited correctness of instantaneous quantities, the Merton scholars could achieve several results of far-reaching importance for the development of mechanics. The famous among them was regarding the distance traversed by the body maintaining the uniformly accelerated motion:21

If the velocity of a body is increased uniformly from v0 to v1 during the time interval t the distance traversed in that time will be:

tvvvs ⋅⎥⎦⎤

⎢⎣⎡ −

+=2

010 (8)

This formula for zero initial velocity provides tvs ⋅= 121 22 which together with

the change for velocity tav ⋅=1 leads to the most famous for our school students:

18 Pedersen, O. & Phil, M. (1974). Early Physics and Astronomy. Macdonald & Janes, London, p. 220. 19 Ibid., pp. 220-221. 20 Ibid., p. 221. 21 Ibid., p. 222. 22 This formula is known as "Merton mean speed rule" and will be represented again in the following

section.

18

2

21 tas ⋅= (9)

This result was known to the scholars since after (14c.).

Result (8) matched the previous program of Gerard regarding the average velocity v* (6). It is therefore called Mean Speed Rule. Indeed:

tvtvvtvvvs *22

01010 =⋅

+=⋅⎥⎦

⎤⎢⎣⎡ −

+= (10)

However, the most obvious meaning the formula 10 received in the hands of another bright mind of the medieval science: Nicole Oresme.

Nicole Oresme

The greatest step in the theoretical analysis of motion belongs to the renowned medieval scholar from the University of Paris – Nicole Oresme (1320-1382), a pupil of Johannes Buridan. He was the first to introduce a graphical representation of qualities which vary in time.

This method made visual the progress in kinematics achieved before Oresme and allowed further highly important progress.

It is a commonplace today to represent the changing in time quality, such as velocity v, ascribing its magnitude to the vertical axis (latitude) placed against horizontal axis (longitude) which points represent different instances of time. As a result one obtains a graph representing the variation of velocity in time. Thus, for the case of the uniform motion (v=const) the graph will be a straight line (Fig. 14a). Clearly the area under the graph represents the distance covered by such moving body (s=v⋅t). In case of a non-uniform velocity, we obtain the graphical representation of motion (Fig. 14b). And in the case of the uniformly non-unifom motion (a=const) one receives the trapezium (Fig. 14c) for which the expression for the area reproduces the result (10) obtained by Merton school (Fig. 14d) – Mean Speed Rule.

v

t

s=v⋅t

(a) v

t

tvsi

iΔ=∑

(b)

Oresme in 1377

19

Figure 14. Graphical representation of velocity dependences in (a) uniform, (b) non-uniform, (c) constantly accelerated motions. (d) Merton theorem regarding the

distanced traversed at constantly accelerated motion.

On Figure 15 one may observe the way Oresme presented in his book his graphical methods and applied it to various motions. Figure 15. Graphical representation of velocity dependences at different motions by Oresme. One may see that Oresme thought in terms of vertical lines with small but finit width and actually replaced the motion with varying velocity to the motion with constant and different velocities during small intervals of time. This is actually the idea of definite integral introduce into mathematics only in the 17th century in the new field of calculus.

Two hundred years later Galileo wrote in his book regarding the naturally accelerated body in falling:23

THEOREM I, PROPOSITION I The time in which any space is traversed by a body starting from rest and uniformly accelerated is equal to the time in which that same space would be traversed by the same body moving at a uniform speed whose value is the mean of the highest speed and the speed just before acceleration began.

In this piece we recognize the idea of Gerard of Brussels, Merton school and Oresme. Galileo adopted them all to explain the motion of the naturally falling bodies (compare Fig. 14d with Fig. 16).

23 Galilei, G. (1638/1914). Dialogues concerning Two New Sciences. Dover, New York, p.173.

v0

t

tvvs ⋅+

=2

10

(d) v*

v1

t

(c)

tvvs ⋅+

=2

10v0

v1

Figure 16. Galileo's drawing in 1638.

20

Being equipped with the powerful method of graphical representation, Oresme obtain the important characteristic of the uniformly accelerated motion. Considering the graph of velocity versus time for such motion (Fig. 17) allows obtaining the ratio between the distances traversed by the body moving at a constant acceleration.

Figure 17. Uniformly accelerated motion. Comparison between the distances traversed in subsequent equal intervals.

Comparison of the marked areas leads to:

s1 : s2: s3 : s4 : … = 1 : 3 : 5 : 7 :… (11)

This result indicates that the ratio between the traversed by a body moving at constant acceleration is equal to the ratio of odd numbers. This is an interesting result of Oresme,24 which presents a feature by which the motion at constant acceleration could be recognized and distinguished from any other. Oresme did not give this result any use, but Galileo did. About two hundred years later, he gave to (11) a direct use and proved by it that the body while descending along the inclined plane moved at a constant acceleration (Fig. 18). Galileo did not cite Oresme and contemporary teachers of physics use to refer this result to Galileo only.

24 Drake, S. (1999). Essays on Galileo and the History and Philosophy of Science. University of

Toronto Press, Toronto, Vol. II, p.249.

t

v

1 0

2 3

1

2

3

s1 =1/2 s2 =3/2 s3 =5/2

21

Figure 18. (a) Galileo's experiment with inclined plane. (b) Galileo's drawing supporting his proof that the distances measured traversed by a ball in equal time intervals "bear to one another the ratio of the series of odd numbers 1, 3, 5, 7…".25

More about the medieval progress in dynamics The scholars of Merton College applied mathematical tools to describe also the dynamics of motion, that is, to relate the motion to forces. Thomas Bradwardine in 1328 suggested his own relationship that similarly to Aristotle's (2) and Philoponus' (3) laws related between the velocity of a body on one side and the exerted force F and the resistance force R, on the other.

The form of this dependence was special and we exemplify it with particular numbers. Suppose that forces F and R have the values: F=3 and R=1 and they cause velocity v. Then, in accordance with Bradwardine, the increased force F=9 and the same R=1 would cause double velocity 2v. In other words, double (or triple and so on) velocities were produced by squaring or cubing the ratio F/R, that is, (F/R)2 (or (F/R)3 and so on).

In general terms, Bradwardine's law postulated that the velocity v was such a function ϕ of the ratio F/R that satisfied the demand:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡=×

n

RFvn ϕ (11)

In other words, the rise of v by factor n is caused by the exponential rise of the ratio (F/R). In terms of the later introduced logarithmic function, this law obtains the following form:

( )RFv /log∝ (12)

The advantage of this form was that when F approached in value to R, the velocity v smoothly approached to zero. The formula requires F be greater than R (F > R) and preserved the divergence when R is diminishing to zero:

∞→→ then v0Rwhen (13)

25 Ibid., Corollary I, pp. 175-176.

22

This behavior of (13) is similar to the behavior of Aristotelian law (2) when 0Rwhen → and this similarity to Aristotelian law attracted some scholastic

scientists. Importantly, the scientists of that time (14th century) considered different dependence according to their matching certain theoretical principles, but not any kind of data corresponding real experiment.

III. TRANSITION TO THE CLASSICAL PHYSICS

Galileo Galileo Galiley (1564-1642), at the beginning of his scientific activity adopted the impetus as an engine and the cause of movement. However the critique aroused too. In addressing Galileo's early work on motion – De Motu – Koyre26 he wrote:27

[Galileo thought that] it is unthinkable and absurd not to admit that the cause or force which produces it must necessarily spend and finally exhaust itself in this production [of motion]. It can never remain unchanged for two consecutive moments, and therefore the movement that it produces must necessarily slow down and come to an end. Thus, it is very important lesson from the young Galileo. He teaches us that impetus physics, though compatible with movement in a vacuum, is like that of Aristotle incompatible with the principle of inertia.

Galileo stated that the idea of impetus as a charge for motion contradicts the spontaneous continuation of motion independent on length and time, there's no place for inertia if the charge of motion must be spent.

In the course of his experiments Galileo found empirically that:28

Figure 19a. The ball acquires the same speed when discern from the same height, regardless the way.

The speeds acquired by one and the same body moving down planes of different inclinations equal when the heights of these planes are equal.

This means that the final velocity of fall depends on the vertical height of descent and not on the way actually made by the moving body (Fig. 19a).

26 Alexandre Koyré (1892 –1964) was a distinguished historian of science who provided numerous

interpretations of the history and philosophy of science. 27 Koyre, A. (1943). Galileo and Plato. Journal of the History of Ideas, 4(4), 400-428. 28 Galilei, G. (1638/1914). Op. cit. p. 169

1

2 3

1 2 3

Figure 19b. The ball climbs to the same height, regardless the way it takes.

23

Moreover, the same held regarding the elevation that the same body can climb on different planes (Fig. 16b). If there were a payment for motion itself – the impetus possessed – this would not be the case.

Galileo demonstrated this rule quite clearly by his famous pendulum experiment (Fig. 20)29.

Imagine this page to represent a vertical wall, with a nail driven into it; and from the nail let there be suspended a lead

bullet of one or two ounces by means of a fine vertical thread, AB, say from four to six feet long, on this wall draw a horizontal line DC, at right angles to the vertical thread AB, which hangs about two finger-breadths in front of the wall. Now bring the thread AB with the attached ball into the position AC and set it free; first it will be observed to descend along the arc CBD, to pass the point B, and to travel along the arc BD, till it almost reaches the horizontal CD, a slight shortage being caused by the resistance of the air and the string; from this we may rightly infer that the ball in its descent through the arc CB acquired a momentum [impeto] on reaching B, which was just sufficient to carry it through a similar arc BD to the same height. Having repeated this experiment many times, let us now drive a nail into the wall close to the perpendicular AB, say at E or F, so that it projects out some five or six finger-breadths in order that the thread, again carrying the bullet through the arc CB, may strike upon the nail E when the bullet reaches B, and thus compel it to traverse the arc BG, described about E as center. From this we can see what can be done by the same momentum [impeto] which previously starting at the same point B, carried the same body through the arc BD to the horizontal CD. Now, gentlemen, you will observe with pleasure that the ball swings to the point G in the horizontal, and you would see the same thing happen if the obstacle were placed at some lower point, say at F, about which the ball would describe the arc BI, the rise of the ball always terminating exactly, on the line CD. But when the nail is placed so low that the remainder of the thread below it will not reach to the height CD (which would happen if the nail were placed nearer B than to the intersection of AB with the horizontal CD) then the thread leaps over the nail and twists itself about it.

Galileo arrived to the conclusion: the quality possessed by the body in motion is tested by the height of its elevation or discern (or just falling). Therefore, to maintain the horizontal motion, that is to say, without elevation, one does not need any such "force", provided there is no friction with the air or the surface. This was not only the claim for inertial motion, but at the same time, a refutation of impetus. Instead, the intensity of motion was characterized by Galileo by using the notion of momentum but not impetus. The latter, as well as the similar to it "force of motion", were left by

29 Ibid., pp. 170-171.

Figure 20. The constraint pendulum experiment by Galileo: freed from point C the mob of the pendulum stops on the opposite point D, G, I at the same level DC, regardless the trajectory it is compelled to go it takes.

24

Galileo for the use by Sagredo, who expressed the naïve views, and Simplicio, who represented the old physics30:

Sagredo: From these considerations it appears to me that we may obtain a proper solution of the problem discussed by philosophers, namely, what

causes the acceleration in the natural motion of heavy bodies? Since, as it seems to me, the force impressed by the agent projecting the body upwards diminishes continuously, this force, so long as it was greater than the contrary force of gravitation, impelled the body upwards; when the two are in equilibrium the body ceases to rise and passes through the state of rest in which the impressed impetus is not destroyed, but only its excess over the weight of the body has been consumed - the excess which caused the body to rise. Then as the diminution of the outside impetus continues, and gravitation gains the upper hand, the fall begins,

but slowly at first on account of the opposing impetus, a large portion of which still remains in the body; but as this continues to diminish it also continues to be more and more overcome by gravity, hence the continuous acceleration of motion.

One may recognize in this piece the old view of Hipparchus and many others scholars after him who account for the thrown up body.

In his attack on the concept of impetus Galileo tried to show that the state of rest and motion were no so much different qualitatively. This was a severe blow to the concept of impetus which assumed the essential difference between the two states the rest and the movement: 31

Sagredo. When I think of a heavy body falling from rest, that is, starting with zero speed and gaining speed in proportion to the time from the beginning of the motion; such a motion as would, for instance, in eight beats of the pulse acquire eight degrees of speed; having at die end of the fourth beat acquired four degrees; at the end of the second, two; at the end of the first, one: and since time is divisible without limit, it follows from all these considerations that if the earlier speed of a body is less than its present speed in a constant ratio, then there is no degree of speed however small (or, one may say, no degree of slowness however great) with which we may not find this body traveling after starting from infinite slowness, i. e., from rest.

The state of rest appears here actually continuously connected to the state of motion, as if replaced with motion at an infinitely small speed. The qualitative opposition between the two preserved in mind of scholars for thousands years, collapsed. What is then the rational of impetus? Is it there, in the moving body, at all? This was a radical conceptual change.

Two more steps of Galileo one may mention in the direction of impetus refutation towards the new theory of motion: the new account for falling bodies and the principle of relativity.

30 Ibid., p. 168. 31 Ibid., p. 162.

Galileo Galiley

25

The principle of relativity It seems natural that after comprehending the fact that horizontal motion of a body is continuously related to the state rest and the horizontal motion does not need any support (as the one that was provided by impetus), to pose a more general inquiry: is there any difference at all that takes place in objects when they move horizontally at a constant velocity? Here Galileo made an unprecedented conjecture, he claimed that: nothing changes with the whole reality in a room which moves at constant velocity. Nothing means now all the objects, their behavior and the laws that govern reality. To represent this totality, Galileo depicts the new principle in a very mundane manner: 32

Salviati: Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe

carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on

the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping toward the stern, although while the drops are in the air the ship runs many spans. The fish in their

water will swim toward the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during

32 Galiley, G. (1953). Dialogue Concerning the Two Chief World Systems. University of California

Press, Berkeley, CA, pp. 186 - 187 (Second Day).

William Hogarth (1745). Captain Lord Georg Graham in his cabin

26

long intervals by keeping themselves in the air. And if smoke is made by burning some incense, it will be seen going up in the form of a little cloud, remaining still and moving no more toward one side than the other. The cause of all these correspondences of effects is the fact that the ship's motion is common to all the things contained in it, and to the air also. That is why I said you should be below decks; for if this took place above in the open air, which would not follow the course of the ship, more or less noticeable differences would be seen in some of the effects noted.

This claim of absolute equivalence of the reality regardless the motion excludes any manifestation of the special entity that entered to the body to maintain its motion, as it was imagined by people for so long time. And if there is no manifestation of impetus, why to ascribe it to the moving bodies at all? Bodies just move at any velocity without any engine inside them.

All our experience testified that there is no reason whatsoever to introduce impetus or any other special quality of bodies related to the fact of their motion.

Falling That far for the horizontal motion which since Aristotle was related to the violent, imposed on the body, activity, actualization of some potentiality. The other trend of thought addressed the natural motion, which was eventually reduced to the treatment of falling of things – vertical motion. Although the original claim of Aristotle –objects fall with swiftness proportional to their weight (1) – did not match empirical evidence in many cases, in others – it did, and thus, reports of empirical discrepancy were not sufficient to establish the new understanding of this type of motion.

Galileo made here two decisive contributions. Firstly, by performing serious experiments (such as shown in Fig. 18), he was able to establish the precise and quantitatively proven fact: neglecting the air resistance, bodies fall at a constant acceleration, regardless the amount of matter they contain. Secondly, Galileo brought into for the fact that the claim of Aristotle in intrinsically contradictive. Without saying what the true account is for the falling, the intrinsic contradiction dismissed Aristotelian theory of falling.

In refuting Aristotelian claim that heavier things fall faster, Galileo reproduced the argument suggested by Gianbattista Benedetti (1530–1590) an Italian mathematician from Venice, in 155333 (Fig. 18).

Here is how Galileo presented the internal contradition of the Aristotelian theory:34 Salviati. If then we take two bodies whose natural speeds are different, it is clear that on uniting the two, the more rapid one will be partly retarded by the slower, and the slower will be somewhat hastened by

33 Gliozzi, M. (1965). Storia della Fisica. Vol.II, Storia della Scienze, Torino. 34 Galilei, G. (1638/1914). Dialogues concerning Two New Sciences. Dover, New York, p. 63

Figure 21. Thought experiment by Benedetti: small body A should fall at velocity V1 smaller than V2 of the bigger body B. Being connected, however they had to fall at velocity V3. The latter had to be less than V3 (since A impedes the falling of B) and greater than V3 (since A+B is bigger than B) – a contradiction.

V1 V2

V3

A B A+B

27

the swifter. Do you not agree with me in this opinion?

Simplicio. You are unquestionably right.

Salviati. But if this is true, and if a large stone moves with a speed of, say, eight while a smaller moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than ' the lighter one, I infer that the heavier body moves more slowly.

Galileo understood that he could not pose a different theoretical treatment for the falling bodies – he had no better theory (such was provided later by Newton). So the best he could do was to dismiss the old theory and provide empirical evidence of the fact that bodies fall regardless their materials and sizes (given the resistance of the air can be neglected).

On the motion of projectiles

Following the two afore-presented achievements in understanding motion: inertial horizontal and vertical accelerated (falling), Galileo made another great progress – a complete account for the projectile motion. It was presented in the Fourth Day discussion in his Dialogues concerning Two New Sciences in 1638. For the first time in physics, Galileo stated that the motion of projectile is comprised from two, horizontal at permanent velocity and vertical – falling at permanent acceleration. Comprising the two Galileo obtains the precise form of the trajectory – parabola (Fig. 22).35

Figure 22. The drawing of Galileo representing the construction of parabola as the trajectory corresponding to the horizontal uniform and vertical permanently accelerated motions.

In his quantitatively precise treatment, Galileo obtained the theoretical proof for the angle of inclination of 45° for the maximal distance reached by the projectile. Also, a completely new result was discovered: the projectile thrown at two angles which complement to 90° would traverse the same distance (Fig. 23).

All together, the ancient problem of projectile motion received a full treatment by Galileo. However, although complete, this was only a kinematics treatment. The facts why the particular type of motion took place in vertical and horizontal directions

35 Galilei, G. (1638/1914). Dialogues concerning Two New Sciences. Dover, New York, p.249.

Naturally preserved uniform motion

Permanently accelerated falling motion

28

remained open weighting for the dynamic (force-motion) explanation with the new theory of Newton.

Descartes

Rene Descartes (1596–1650), a renowned French philosopher of nature, was familiar with Galileo's results but he went another way. His approach was to construct a new theory of motion basing on correct initial ideas and through using analytical logic and mathematics, whether or not the obtained result matches the empirical evidence. This way, however, could bring only to a partial success, as far as one intends to understand the Nature.

Descartes formulated his vision in a special treatise The Principles of Philosophy, published in 1644. His core ideas regarding the mechanical arrangement in the Nature,

Descatres formulated in three laws, which included new ideas about motion of material bodies, some correct and some erroneous.

Firstly, Descartes, unlike Aristotle, defined the movement as a state and as such requires a cause for change to stay as it was previously. 36

The first law of nature: that any object, in and of itself, always perseveres in the same state; and thus what is moved once always continues to be moved.

Descartes reasoned this statement by the “immutability of God”, which might not convince contemporary learner: why this and not the other state the nature should keep unchanged. However, addressing motion as a state of a body (not a process) and, in other words, introducing the inertial motion: motion without a mover, was a correct guess and it remain the principle in the new classical mechanics.

At the same time, Cartesian opposing motion to rest kept with the old Aristotelian conception contradicted Galilean principle of relativity and was wrong. The rest-motion opposition matches naïve intuition (often addressed as "commonsense").

Our experience testifies that any two rectilinear and uniform movements with arbitrary velocities are indistinguishable, and the state of rest could be one of them. Motion and rest coexist in one body by mere relation to different objects.

Then Descartes stated:37

36 Descartes, R. (1644/2004). The Principles of Philosophy. Kessinger Publishing, Montana, Part II. 37 Ibid.

Horizontal distance

Height

Rene Descartes

Figure 23. Trajectories of projectiles thrown at different angles to the horizon. Note the same distance of the complementary to 90 degrees angles of the launch.

29

The second law of nature: that every motion of itself is rectilinear; and hence what is moved circularly tends always to recede from the center of the circle it describes.

In his second law of nature, Descartes specified the type of motion that remains unchanged without cause to change it – this is the rectilinear motion. Here came to the end the view of ancients that uniform circular motion presents eternal natural state preserved with out any reason in the motion of planets. Descartes again reasoned by "the immutability and simplicity of the operation by which God conserves motion in matter". Quite interestingly, Descartes reasoned rectilinear motion also by saying:38

For He [God] does not conserve it other than precisely the way it is in the moment of time in which He conserves, with no relation to what perhaps was shortly before. Although no motion occurs instantaneously, it is nevertheless manifest that everything that is moved, in the single instants that can be designated while it is moved, is determined to continue its motion toward some direction along a straight line, and never along any curved line.

One observes here the reasoning by going to the small interval of time when the body freed at A faces two options: to continue along ACB (Fig. 24) or deviate along ABF. Descartes state that the former is natural and presents the same state, and the latter is changing the state requires a special cause (the sling and hand) and that is why we

perceive it:39

For example, stone A, rotated in sling EA around circle ABF, at the instant in which it is at point A is determined to motion in some direction, namely along a straight line toward C, such that the straight line AC is tangent to the circle. But one cannot arrange that it be determined to any curved motion; for, even if it previously came from L to A along a curved line, nevertheless nothing of this curvity can be understood to remain in it when it is at point A. This is also confirmed by experience, because if it then left the sling it would not continue to be moved toward B, but toward C. From which it follows that everybody that is moved circularly, perpetually tends to recede from the center of the circle it describes. We experience this by tactile sense in a stone that we move in a circle with a sling. . .

As in the first law, we still miss here “uniform” for the motion that is also required. However, let’s not forget that Galileo still used to think about rotational inertial motion. Descartes explicitly rejected this idea; he provides the rectilinear motion with a special status: only such motion is preserved as inertial.

Another aspect touches on active forces. Descartes' law addresses the tendency of the body compelled to move on a circular path to proceed in the tangential direction, which is realized that moment when the agent imposing the circular motion (the sling)

38 Ibid. 39 Ibid.

Figure 24. Rene Descartes' drawing illustrating the second law of nature.

30

ceases to influence the stone. This Cartesian approach – considering tendency – was adopted also by Newton who in the same spirit stated in his Principia in 1687: 40

Every body perseveres in its state of resting or moving uniformly straight on, except inasmuch as it is not compelled by impressed forces to change that state.

As we see, Newton does not talk simply about the absence of forces but, similar to Descartes, rather about the tendency to preserve the state of resting or moving uniformly which succeeds inasmuch as the external force diminishes.

Finally, Descartes formulated the law of motion following interaction between bodies, which in his mind could solely by contact, in particular, in collisions:41

Third law: that a body, in colliding with another larger one [Fig. 25a], loses nothing of its motion; but, in colliding with a smaller one [Fig. 25b], loses as much as it transfers to that one.

Figure 25. The schematic representation of the stated law of collisions. The law points to two possible case which are different (a) and (b).

The third law of nature is this: where a body that is moved meets another, if it has less force to continue along a straight line than the other has to resist it, then it is turned aside in another direction [Fig. 25a], retaining its quantity of motion, and changing only the determination of motion. If, however it has greater force [Fig. 25b], then it moves the other body with it and loses as much of its motion as it gives to that other.

Descartes stated here a central law in physics related to his name ever since – the conservation of the quantity of motion, or in our terms, momentum. However the momentum, or quantity of motion, is erroneously conceived by Descartes as a scalar quantity. If we however try to represent this idea we may write:

∑∑ =finin

mvmv ?? (14)

40 Newton, I. (1687/1999). The Principia. Mathematical Principles of Natural Philosophy. Translated

by B. Cohen & A. Whitman. University of California Press, Berkeley, CA. In the translation into Russian by Krilov the first law appeared in a more inclusive form that be translated into English is:

Every body continues to preserve its state of rest or uniform motion in right line until it is and so far as it is not compelled to change that state by forces impressed upon it.

41 Descartes, R. (1644/2004), op.cit.

Before collision:

After collision:

Larger SmallerBefore

collision:

After collision:

Larger Smaller

(b) (a)

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The uncertainty in the indexes of velocities are due to the separation the law of collisions into two cases: a collision with smaller or bigger body, which is obscure and generally erroneous.

Descartes correctly separates between the collisions of hard and soft bodies; each implies a different account. This dichotomy of collisions preserved in physics. The claim regarding soft bodies is apparently wrong: inelastic collision does not imply stopping for the colliding bodies:42

… when they [moving bodies] meet a soft body, to which they can easily transfer all their motion, they immediately come to rest. All the particular causes of the changes which occur in [the motion of] bodies are contained in this third law, or at least those that are physical; for whether, and in what way, human or angelic minds have the force to move bodies, we do not now inquire but reserve for our treatise On Man.

Seemingly, Descartes kept in mind dissipation of motion in motion within the "soft" medium, such as when a hard ball rolls into sand. As appropriate for a scientific text, Descartes deliberately excluded all non-physical cases.

On the motion of projectiles In his Principles, Descartes addressed the traditional topic of movement of projectiles. After the triumph of Galileo in treating this subject Descartes' one seems rather weak and non-convincing:43

Certainly, everyday experience of things that are thrown wholly confirms our rule. For there is no other reason why thrown [bodies] should continue in motion for any time after they have been separated from the thrower than that once moved they continue to be moved, until they are slowed by contrary bodies. And it is manifest that they usually are gradually retarded by the air, or some other fluid bodies in which they are moved, and hence their motion cannot last long. For we can experience air resisting the motions of other bodies by our sense of touch if we strike it with a fan; the flight of birds also confirms the same thing. And there is no other fluid which does not, even more manifestly than air, resist the motions of projectiles.

From his first law Descartes deduces the answer to the question that was difficult for Aristotle to answer: why the stone continues to move after it leaves the hand of the person. Aristotle's special mechanism of air turbulence – antiperistasis – was rejected by the new principle of preserving the state of motion, and the air was considered as a factor that impedes motion. Descartes, although worked after Galileo, did not provide a comprehensive account for the projectile motion as Galileo did (trajectory, velocity change, acceleration), and remained on the very general level of understanding. This approach was not sufficient since Descartes considered motion as a scalar quantity. This prevented him to split the motion of a projectile into two essentially different ones, horizontal inertial and vertical accelerated. Therefore, no precise quantitative description of the motion of projectiles was attained.

42 Ibid. 43 Ibid.

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Final refutation of the old theories of motion – Newton Ultimately, the new theory of motion was established due to the contribution of Newton (1643–1727) – an outstanding English physicist of the 17th century. He founded the discipline we label classical mechanics.

Being a teenager student of Cambridge he thoroughly studied every word in the Descartes Principles of Philosophy, copied them to his notebooks and made notes. Newton drew on the previous results obtained by medieval theories of motion as well as by those who participated in the scientific revolution of the 17th century: Galileo, Kepler, Descartes, Hooke, and Huygens.

Unlike the qualitative approach of Descartes that failed to explain the Kepler's precise mathematical statements regarding the motion of planets, Newton did so, after insensitive debates with Hooke in the Royal Society of London. The new theory was presented in 1687 in the treatise that became fundamental for human culture – The Mathematical Principles of Natural Philosophy – Newton's answer to the Principles of Philosophy of Descartes. Newton started his debate with Descartes already from the title: Mathematical Principles instead of Principles, and Natural Philosophy instead of Philosophy.

Right at the beginning Newton presented his laws of motion, which replaced all those introduced before, from Aristotle's to those by Descartes. Here are they in comparison:

Laws of nature in Descartes' Principles Laws of nature in Newton's Principia44

The first law of nature: that any object, in and of itself, always perseveres in the same state; and thus what is moved once always continues to be moved.

The second law of nature: that every motion of itself is rectilinear; and hence what is moved circularly tends always to recede from the center of the circle it describes.

Third law: that a body, in colliding with another larger one, loses nothing of its motion; but, in colliding with a smaller one, loses as much as it transfers to that one.

∑∑ =outin

mvmv ??

Law I. Every body perseveres in its state of being at rest or of moving uniformly straight forward except insofar as it is compelled to change its state by forces impressed.

Law II. A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.

Fmv ∝Δ )(

Law III. To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always opposite in direction.

44 Newton, I. (1687/1999). Op.cit.

Isaac Newton

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What were the major changes introduced by Newton in the laws? The changes were numerous and essential:

1. The first laws of both scholars, Descartes and Newton, might seem rather similar. However, a closer look reveals essential differences.45 Newton' laws includes relation of the body to the impressed force and describes the tendency of the body to preserve the state of motion or rest. These states are addressed as totally equivalent, unlike the conception of Descartes. This was the paradigmatic shift in physics knowledge which became among the central features of the modern physics.

2. The second and third laws of Descartes were removed Newton. In a way, they became direct implications of the second Newton's law. All "tendencies" were removed in favor of using forces.

Regarding the third law of Descartes that states conservation of (quantity of) motion, it was essentially changed. The principle of conservation of quantity of motion (this time it is the quantity sensitive to the direction of motion – vectorial quantity, in our terms) was stated in Corollary III:46

The quantity of motion, which is obtained by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves

This claim is a direct implication of the second Newton's law for a closed system of bodies.

3. The fundamental fallacy of Descartes who considered the interaction of bodies as asymmetrical process (in terms of "winners" and "losers") was removed by Newton in his third law. Perhaps, it was within the debate with Descartes that Newton stated this law in a separate claim although unlike the previous laws, presented as axioms (that is without demonstration), the third law of Newton was proved by him basing on the first one.47 Interactions between any two bodies were stated to be symmetrical: any acting force is equal to the force of its reaction.

Unlike Descartes' laws, Newton's laws matched the experience quantitatively in the great abundance of physical situations, and therefore these laws were unanimously preferred by scholars to Descartes' ones. The empirical results and observations matched to the Newtonian mechanics with the high level of accuracy.

Newton's theory established the basis of classical mechanics that replaced the old theories of motion although remained in a vivid discourse with them, especially in the context of science education.

*******************************************************************

45 Galili, I. & Tseitlin, M. (2003). Newton's first law: text, translations, interpretations, and physics

education. Science and Education, 12 (1), 45-73. 46 Newton, op.cit., p.420. 47 Newton, op.cit., p. 428.

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Summary We may summarize the mentioned in the excurse points in the development of theory of motion through two thousand years: from Aristotle to Newton. This may be done by a flowchart as following: Note two trends of natural and violant motions (arrows) united only by Newton.

Aristotelian theory of motion

Natural motion: intention to move to accord

with weight/levity

Violent motion: motion by the external mover -IVc

Projectiles: antiperistasis

Violent motion: motion by the

provided impetus

Hellenistic-Medieval theory of motion

Kinematics Velocity, instant velocity, acceleration,

distance traversed in uniform and uniformly accelerated motion, geometrical

representation of motion as a change in time

Vertical motion of constant

acceleration Galileo

Transition to the Classical Mechanics

Horisontal inertial motion of constant

velocity Galileo

Principle of relativity: linear uniform motion=rest

Galileo

Motion (uniform and linear) is a natural state of

bodies Descartes

Classical Mechanics

Principle of relativity: linear uniform motion=rest

Galileo

Laws of motion in terms of force and vectorial

momentum Newton

Conservation of motion Descartes

Natural motion: Self enhanced impetus (natural and accidental

gravity/weight)

Projectiles: Parabolic trajectory

Galileo

-IIIc –

15c

16c –

17c

Projectiles: Special case of dynamics in

two dimensions

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Historical and philosophical background including nature of science Following the historical development of the history of theory of motion reveals three steps of knowledge progress. These theories identified different types of motion and describe them with general theoretical framework. The three theories of motion were: Hellenic (Aristotle's theory), Hellenistic-Medieval (the theory of impetus), and the Classical mechanics (inertial motion and dynamics of forces). The three periods and each saw motion in different ontological and epistemological perspective. We will address them here. Firstly, regarding to the ontology, one identifies three stages of conception.

Motion as actualization

According to Aristotle, the world is organized in some order (cosmos) and split into two areas, celestial and terrestrial. Motion in celestial world is natural, eternal and uniform. Motion in the terrestrial world splits into natural, caused by a violation of the world order the desire to take the proper place and rest there, and violated, caused by a mover. In both cases motion reflects causes and so could be understood as the process of their actualization – the process. Void is impossible. Therefore, motion to the natural place faces resistance from the medium, other bodies, causing them to move.

Motion as charged activity

In general, the Greek cosmos was preserved in the medieval world. Although the scholars departed from Aristotle in many points and criticized him. In the new impetus physics, the cause of violent motion was already local (independent of the world order) and was placed within the body itself. Impetus as a sort of a charge, provided entity, which may increase itself (as in falling) or be dispensed (due to the resistance of the medium). The antiperistasis was abandoned. Void became a logical possibility. Impetus corresponded to the finite world of Aristotle. It was also finite, derived from the mover, which itself was owned by another mover, and so on, similar to the chain of movers in Aristotle's vision.

Motion as a state

In classical physics the uniform, rectilinear motion of a body is a state and does not require cause independent and existents of any other bodies. The "natural" falling is not natural motion but caused by a force of gravity. The inertia (inertial mass) is a property which manifests itself in the resistance to the change of the state of uniform, rectilinear motion. Any other than uniform and rectilinear motion is considered as changing of different states, expressed by acceleration. The change in the states of motion is caused by other bodies interacting by forces. 48

According to Koyre,49 two important features characterized the scientific revolution of the 17-th century. First, the destruction of the Greek cosmos, i.e. replacement of the finite and hierarchically ordered world of Aristotle and the Middle Ages by the infinite universe, comprised of void and point masses and governed by uniform laws (paradigm of Newton). Second, the geometrization of space, i.e.

48 The equivalence of opposites was stated as a principle by another medieval philosopher – Nicholas

of Cusa (1401-1464). In fact this was the development of the dialectical method starting from Socrates.

49 Koyre, A. (1968). Metaphysics and Measurement: Essays in Scientific Revolution. Harvard University Press, Cambridge, Mass.

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substitution of the Aristotelian concrete space (aggregate of "seats" in plenum) by the abstract space of Euclidean geometry – void regarded as real entity.

The most important for the new theory of motion was that the world of all bodies was considered to be placed into the infinite, empty, absolute space (and absolute time.) The motion in this space established absolute motion. In practice, however, one cannot but consider body’s positions and motions relatively to other bodies. Newton provided the evidence of the absolute space in his thought experiment of a rotating bucket with water.50 The invention of absolute space (as well as absolute time and absolute movement) was of essential ontological importance.

In light of the need to interpret inertial motion, it was important for Newton's mechanics to separate between internal and external forces. Inertia was related to the internal forces, and the interaction with other bodies – to the external. To distinguish between the two one need to imagine a single body in the universe. Then, facing the question how one can determine motion of a single body in absence of anything else in the whole universe one gives the answer – in relation to the absolute space.51

This explanation was invalidated only three hundred yeas later, within the modern science. Then, in his famous thought experiment of elevator, Einstein demonstrated the equivalence of inertia and gravitation. Since inertia presented an intrinsic property of bodies, and gravitation – their external interaction, Einstein's discovery questioned the very idea of separation between internal and external and consequently dismissed the idea of Newtonian absolute space. In the new world described by modern physics, that one of Einstein, the concepts of absolute space cannot be justified: all motions are solely relative and the space given to measurement.

The metaphysical turn of the medieval science Why Aristotle himself, or his many followers, didn't come to the impetus idea, which now seems for us so obvious, and teachers are evident that students spontaneously think about dynamics in similar way. Why in the late Middle Ages (without any new facts) the scholars took this idea so easily? The reason can be attributed to the change in the type of representation of reality as adopted in different periods of science. In the Middle Ages it changed mainly due to theological reasons. Scholars started to consider qualities as independent of things. If in the Hellenic scholars considered changes in terms of changing of things, the medieval view considered changes in qualities. The concept of motion led scholars to consider bodies in motion in relation to the changing of specific quality. Therefore, motion ceased to be a process of actualization of certain causes in the moving body as Aristotle postulated. Serious changes took place in dynamics too.

For Aristotle, the violent motion required an engine (the mover) in contact with the moved body. Therefore, violent motion required other pushing bodies and thus excluded void. When the engine/agent was replaced by the changing quality, motion became somewhat like heat and could be transferred from body to body. Similarly, the conception of force as independent of its agent was introduced. (Note that without such ideas Newtonian mechanics would have been impossible.) This theoretical

50 Newton, I. (1687/1999). Op. cit., pp. 412-413. The invalidity of this argument of Newton was shown

in physics by E. Mach in his The Science of Mechanics (1889). Mach pointed on the unjustified neglecting of the presence of other objects around, including the distant stars.

51 Euler, L. (1765/2009). Theory of the motion of solid or rigid bodies. Online: http://www.17centurymaths.com/contents/mechanica3.html

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framework of isolated forces preserved, however, also in the classical mechanics unlike the opposition of motion and rest, reined in the medieval science.

The epistemological change: the Nature of Science This excurse presented historical development of the theory of motion and the change of the conceptual epistemological perspective on motion. As known, the nature of science is historically dependent. In the course of history, three great theories of motion corresponded to the three different epistemologies.

The ancient Greek Science At that period, scholars mainly sought for the ideas standing behind the things. Such was the first science. It totally separated itself from the world of practice which was considered as voluntary and subjective. Hellenic science looked beyond simple monitoring of things, but tried to see the unchangeable behind the observed changes. Scientists looked for the permanent ideal essences. In the process of contemplation of the changing reality, the scholars wanted to guess about the nature of the cosmos, the ordered, regulated, perfect space in which everything has its goal, place and destination. This was not an easy task in the world full of changes. Plato refused constructing a theory of motion for the complexity of change description in mathematical terms. Aristotle, who tried to distinguish and separate between physics and mathematics, chose a different way. Aristotelian science presented the motion as transitions between certain points in which the body normally exists in the state of rest. The common aspiration of the Hellenic scholars was to understand the world through the immutable ideas (principles) and contemplation was the way to reveal and analyze them. The process of contemplation had to be careful and cautious, dismissing non essential and misleading factors.

The medieval science During this period of science, all existed objects and in all possible aspects were understood in their relation to God. The medieval science tested Aristotelian epistemology by the Christian theology. The considered phenomena were checked in their meaning with relation to central idea and principle – God. Thus, for example, in considering the existence of void, the arguments of Aristotle were replaced by the theological arguments like: would the divine mind create nothing, is it pointless to create nothing? So, the experts in science, scientists, were at the same time theologians, and science was all interwoven with theological concepts and ideas. The soul, as the residence of God, was separated from the body. Since soul was considered a form, form obtained independent existence from the substrate. Although the authority of God was stronger than that of Aristotle, the latter could say much more about the nature and the principles in according to which this nature was arranged ("designed" in the view of scholastic philosophers). Therefore, the texts of Aristotle were considered to be the major resource in science of the middle ages, more important than the reality itself.

The important epistemological progress in understanding motion was performed in the middle ages. In particular, there was a breakthrough in kinematics – the theory of describing motion. Due to the contribution of Gerard of Brussels, Oresme of Paris and the Merton school of Oxford motion was described terms of new concepts which allow a quantitative description of motion. Among the central results were: posing the question of instant velocity, concepts of average velocity, acceleration, uniform and uniformly deformed movements, the functional dependence for the distance

38

traversed in the uniformly deformed motion (mean speed theorem). The extremely important epistemological progress of Oresme was the graphical representation of motion. All these results, although not applied to the reality (this was the agenda of the coming revolution), established a conceptual basis and were antecedent to the similar and often identical results attained by Galileo, Descartes and Newton, after the scientific revolution of the 17th century.

The modern science In the scientific revolution if the 17th century and thereafter in classical science, the framework of treating motion changed. One can represent this change by terming it as a balanced synthesis of epistemology. If the Aristotelian theory was very holistic, all embracing treatment of reality, based on general principles of empirical origin and avoided precise mathematical accounts; if the medieval theory of motion was, to the much extant, formal conceptual, mathematical, rational, and excluded experimental verification; the modern science, established after the revolution, combined both approaches in a well balanced, complimentary synthesis of rational and empirical.

Theories preserved the features of rationally arranged, hierarchical and coherent knowledge systems which were based on the firm empirical basis and referred to controlled experiments. There was no more a single, all embracing theory as that by Aristotle, but theories in different areas of physics knowledge (mechanics, electricity, magnetism, thermodynamics) started to consolidate. The first of them was in mechanics. Classical mechanics addressed moving bodies and the forces of interaction between them.

Previously, science was not interested in the artificially created things, and directly observed the nature, avoiding intervention. In contrast, the modern science scholars designed and investigated the reality they create, although the observer remained distant from the observed phenomenon and claimed that it exists regardless the observer. By using apparatus scholars produced conditions not simply available in nature. For example, evidence for the law of inertia could be illustrated in laboratory easier than in everyday environment prevailed by friction. Francis Bacon, in the early 17th century, called the process of investigation of nature – elicitation of its secrets through "interrogation". Such epistemology would seem absurd to Aristotle and medieval scholars. Modern science often deals with artificial (technical) "reality" and applies to it the established the laws and theories. Experimentation, as monitoring objects in laboratory became the central tool of science.

Cumulative nature of science

Objectivity of scientific knowledge is the major feature of science. At the same time, each scientist is an individual whose knowledge includes objective as well as subjective knowledge. It is quite a challenge to see how the scholars, each influenced by a range of factors causing subjectivity to his or her knowledge, succeed, despite of that, to create the common objective knowledge of science. Several factors stipulate this success. One of them is the cumulative and discursive nature of scientific knowledge. This feature was pronounced from the beginning of science. At the middle age period it was already the paradigmatic slogan of scholars. The well known metaphor introduced by Bernard of Chartres who was a twelfth-century French Neo-Platonist philosopher and scholar presents the rational of the cumulative nature of science. It is known from John of Salisbury who in 1159, wrote:52

52 Crombie, A. C. (1959). Medieval and Early Modern Science. Doubleday Anchor Books, New York,

p. 27.

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Bernard of Chartres used to say that we are like dwarfs on the shoulders of giants, so that we can see more than they, and things at a greater distance, not by virtue of any sharpness of sight on our part, or any physical distinction, but because we are carried high and raised up by their giant stature.

Figure 26 presents the stain glass window in Chartres which represents the same idea.

Isaac Newton used the same metaphor in the letter to his rival Robert Hooke dated

February 5, 1676:

If I have seen a little further it is by standing on the shoulders of Giants.

The development of theory of motion, started by Aristotle, demonstrates the construction process in which many scholars contributed. Those who lived later essentially used the results obtained in the previous research. This idea was addressed, for example, by the historian Edward Grant who wrote about using the mean speed theorem by Galileo:53

Oresme's geometric proof and numerous arithmetic proofs of the mean speed theorem were widely disseminated in Europe during the fourteenth and fifteenth centuries and were especially popular in Italy. Through printed editions of the late fifteenth and early sixteenth centuries, it is quite likely that Galileo became reasonably familiar with them. He made the mean speed theorem the first proposition of the Third Day of his Discourses on Two New Sciences where it served as a foundation of the new science of motion. Not only is Galileo's proof strikingly similar to Oresme's, but the accompanying geometric figure is virtually identical, despite a 90° reorientation that had already been made by some medieval authors.

In fact, it could not be different: the scientific knowledge is too complex and extended and the life span of an individual as well as one's abilities are too limited to match such a knowledge and promote its development. Scientific knowledge is essentially a collective knowledge and resides in numerous minds.

At the same time, the development of the theory of motion demonstrates that the accumulation is not equal to simple addition: in the course of progress people refine

53 Grant, E. (1977). Physical Science in the Middle Ages. Cambridge University Press, Cambridge,

Mass, p. 58.

Figure 26. Illustration of the idea of cumulative nature of knowledge on the stainlglass window in Chartres Cathedral. Four evangelists are sitting on the shoulders of four prophets: Isaiah, Jeremiah, Ezekiel and Daniel.

40

the knowledge and change the epistemological as well as ontological paradigms and frameworks. This is done within the ongoing synchronic and diachronic discourse of the participants.

*******************************************************************

Target group, curricular relevance and didactical benefit Theory of motion presents the major part of mechanics curricula regardless the level of instruction. Therefore, the target group of learning the materials presented in this excurse, mainly qualitatively, is, first of all, the teachers of physics, pre- and in-service. In addition the materials may be recommended for learning by the students who show interest in causal justification of the theory of motion of classical mechanics: why the fundamental statements of the theory (such as the equivalence of the uniform rectilinear motion and the state of rest) are like they are presented in class and not as they seem to naïve learner prior instruction.

In presenting the excurse, one need to justify the curricular relevance of the materials included in the excurse. This is because despite of the revealed relevance and importance of these contents, the old theories of motion, which comprise this historical excurse, are normally out of the contents learned in regular physics classes either at schools or in colleges. However, the new vision of physics curriculum, promoted by this project54, implies the relevance and necessity of the exposure of the genesis of the contemporary adopted knowledge, and therefore considering previously adopted theories as a condition for genuine learning of motion within the classical physics.

In particular, the arguments making relevant learning the old theories of motion could be summarized as following:

1. The presented materials create an authentic image of physics as a domain of science in which the knowledge was constructed in a diachronic discourse converging to the presently adopted vision. The excurse started with the description of the Aristotelian theory and principles of explanation adopted by natural philosophy. The excurse presents the important theory of impetus, which was introduced, and ultimately dominated, in the very long period of physics, after Aristotle and until Galileo and Newton.

2. The alternative accounts of motion historically preceding the current understanding of this subject matter create a contrast for the learned theory within the classical mechanics and thus emphasize the crucial points of the subject matter.

3. The alternative accounts of the historically preceding views on motion create a space of learning in which the considered subject matter can be effectively learned through the construction of the goal concepts by variation of their theoretical contents and major points. The important feature of the excurse is its promoting learning of theory instead of the blind adopting its anti-intuitive claims and focus only on the applications in the problem solving.

4. The old theories of motion present the cultural heritage and establish cultural knowledge of physics. Indeed, the discourse on the account for motion spread over the long history and so included different cultural periods in science. Qualitative knowledge of the major accomplishments of those times enriches the appreciation of

54 See the theoretical introduction to the HIPST project.

41

the achievements of the human culture, which stipulates the humanistic education, often lacking in the youth who specialize on the disciplinary training. General education, humanistic cultural values, and principles (such as non-pragmatic aspects of scientific inquiry, and its essentially international character) should not abandon science classes. These aspects are demonstrated in the way the theory of motion consolidated historically in different cultural environments and reflecting their infuences.

5. Learning physics should not ignore demonstration of the epistemology of science (the nature of science). The old theories of motion spread on three major periods of the history of physics and practice different epistemologies. The complexity of the subject is that the contemporary epistemology fuses in variety of ways the features of the previous periods also in this aspect. For example out modern scientific method synthesizes rationalism with empiricism in a well-balanced proportion. The previous periods, although incorporated both aspects, often showed prevalence of one them, theoretical (rational) or empirical (experimental). Learning these experiences, their successes and failures, is pedagogically effective for the same reasons as the variation in the ontological domain (subject matter aspects) addressed above.

6. Learning physics very often interacts with is issues of religious nature. The history of development the theory of motion may be very useful and informative in this aspect. It showed different approaches of scientists to faith and irrational aspects of their scientific activities. Aristotle, Hipparchus, Avicenna, medieval scholars, Galileo, Descartes and Newton were all very different in their religious views and all significantly contributed to the construction of the theory of motion as we use today. Much educational benefits may be reached in exposure of them in the mediated by the teacher discussion. These will lead to the more mature general cultural knowledge. *******************************************************************

Activities, methods and media for learning The purpose of this historic-philosophical excursus into the theory of motion is not simply providing the previously unknown knowledge, but the acquaintance with alternative conceptions and approaches to the scientific account for motion. Such a goal presumes a paradigmatic change in students' and teachers' perception of science in general and physics, in particular. To reach it one need more than lecturing, but rather a discussion on behalf of the students. The following questions might illustrate possible topics of such discussions.

Topics for discussion

• Aristotelian motion theory explains the motion of projectile by a special mechanism of antiperistatis. Is there any conservation law of classical mechanics that contradicts to this mechanism?

• Early history of impetus explains the motion of projectile by providing them with a charge of motion that changes their quality. Is a purely internal cause of movement possible from the classical mechanics point of view?

• Discuss an experiment, which would disprove the theory of impetus. A possibility of such an experiment one may find in the following thought experiment.

42

Thought experiment

Imagine a tossed up a coin. We know how it moves: rises up, stops for an instant, and falls down. By throwing the coin one violates the world order in the Aristotelian world. Hipparchus suggested the way to compensate the tendency of heavy bodies to seek the center of the world – impetus that the thrower put into the coin. We may ask now, what would happen if we compensated the gravity of the body when it reached the top of its fly? Obviously, the uncompensated impetus remains in it… Thus, at the very moment the gravity is switched off, the body will immediately fly up with the impetus it still keeps. The story will

be very different in the classical mechanics. If one switches off the gravity at the moment the body is at the top of its fly, it would remain at rest and would stay there as long as there is no gravitational force. The moment the gravitation returns, the body starts to fall.

Although we cannot not switch off the gravitation, it is possible to compensate it by using electrostatic field. If we perform such an experiment this would be a devastating blow to the impetus physics: the coin would immediately fall down. Note, however, that just this possibility to disprove the impetus theory make it scientific, as claimed the renowned philosopher of science – Sir Karl Popper.

Questions for reflection and discussion

• Suppose that the impetus theory is correct. What would be motion of the body if at the highest point of its fly one were support the pebble by hand put below the coin?

• Suppose that the classical mechanics is correct. What would happen with the body in the situation described in the previous question?

• Suppose that the impetus theory is correct and the body is tossed up. It moves up and then falls. Is there a moment when the impetus, provided to the body, nullifies? Where will it happen (if at all)?

• Suppose that the classical mechanics is true. In the situation described above, is there during the motion a point in which the impulse nullifies? If so, where?

One may say that impetus took the explanatory role by force, momentum, and energy. It, however, is not always successful, even at a qualitative level. The following questions may show such limitation of the impetus theory, even for a qualitative use (and hence, the power of the classical mechanics).

Questions for reflection

• How does the impetus theory explain (if at all) that the body thrown up begins to fall down? In other words, why does not the body remain at the top?

• How does the classical mechanics answer the same question? What are the concepts of classical physics that play the major role in this answer?

• How would the impetus theory explain the ball bouncing from a wall? How does classical mechanics do this?

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• Can the impetus theory explain the collision (for example, of two identical) balls? Why don’t the balls stick together after collision?

• How does the classical mechanics explain the same situation? What physical quantity requires the balls to recede, momentum and/or kinetic energy?

******************************************************************* Obstacles to teaching and learning

Content related obstacles It is a well-documented fact that students face many difficulties in learning classical theory of motion, which implies teaching difficulties of the educators. Misunderstanding of the physics account of motion is widespread and investigated by education research.55 It is easy to perform testing. Simply present the idea of impetus to anybody. If one cannot see whether it is wrong or right, this would indicate lacking conceptual understanding of the Newtonian dynamics. The idea of impetus is, indeed, very close to the naive conceptions of students.56

The state of the motion The first difficulty of understanding of inertia law is adoption of the idea that the concept of "state" includes rest and rectilinear uniform motion on equal bases. Indeed, the identity of motion and rest contradicts the common sense and experience in everyday situations.

The further difficulty comes from motion characteristics of instant velocity. Indeed, velocity always corresponds to a certain period of time and distance traversed. How can one talk, then, about the velocity (or rest!) "at the moment"? This difficulty was addressed by the medieval scholars who introduced understanding of instant velocity as a velocity of another, imaginary motion which was uniformed and traversed the equal space in the same time interval.

The term “state” conveys some idea of permanency, how can we, then, apply it to the motion, which is a sort of change? In this regard, one should not confuse the state of motion with the state of the moving body. The latter is described by the position and instantaneous velocity of the body at each moment. The physics teacher should promote separation of the two in students understanding.

Finaly, it is a challenge to understand that Newtonian inertia is not about the desire to preserve motion (the obsolete understanding from Aristotle and the impetus theory), but rather resisting the change of motion states.

Teaching dynamics

In classical mechanics, motion may be described purely in kinematic terms (position, velocity, trajectory, acceleration, distance). Such description is not sufficiently explanatory regarding motion and to be accomplished in term of dynamics – causal factors of the change the state of motion. Being in a state of motion does not require intrusion (force), whereas the transition between states requires the "engine" to cause such a transfer of the moving body.

55 Galili, I. & Bar, V. (1992), Motion implies force. Where to expect vestiges of the misconception?

International Journal of Science Education, 14 (1), 63-81. 56 McCloskey, M. (1983). Intuitive Physics, Scientific American, 248(4), 122-130.

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The difficulty of teaching dynamics is in the intuition of students, which is based on their life experience. Intuition tells them that any motion requires force. The motion in our regular environment occurs when gravitation and all kinds of friction are present. This may impede understanding of general laws that govern motion. The situation without friction is more suggestive with respect to the idea that motion (uniform and rectilinear), as a state, does need any force of support. The inertial law addresses preservation of the same state of motion.

The creators of the new mechanics, Galileo and Newton, arrived to the new understanding by thinking about ideal cases. Galileo analyzed real situations with gradually decreasing friction of the medium. Newton treated the solar system – planets moving in vacuum – and inferred regarding the laws of their motion.

Teaching the nature of science Science is a special manner of humans to account for reality. Other areas of human culture – such as art, literature, philosophy etc. – do the same by using different means. It is beneficial to contrast the nature of science with the methodology in other domains of human knowledge.

The difficulties of teaching and learning the nature of science could be in several dimensions. Some of them we address in the following.

Language difficulties

This difficulty stems from the two levels of the language used in physics teaching. The first language is used to make sense of the natural phenomena and artifacts, to described and explain them. While doing so physics drafts tools from the philosophy of science which provides conceptual meaning to the science contents and thus defines what is used to be named as the nature of science.

Thus at the first level, physics describes the motion, phenomena and regularities of the motion of celestial bodies. Physics makes by using cognitive tools such as principles, laws, definitions, theories, experiments, models, representations, etc. These terms comprise a more abstract level of language, often called met-language.

In science teaching the educator often uses these notions (Ohm's law, Ampere's law, Newton's laws), but practically never define them (that is never defines the concept of law itself). This feature often causes confusion in laws application, such as when the area of validity and kind of reliability of the laws are neglected and unknown. Further confusion takes place between different elements: laws are confused with principles, theories with models, etc.

The remedy of this difficulty is in explicit addressing the used tools by explanation of their meaning, providing explicit definitions (operational and nominal). Although not easy, the need of addressing meta-language, which reveals the nature of science, is evident and the confusion of students and teachers who neglect these aspects is documented.57

Objective versus subjective

Evidently, the scientific knowledge is an intellectual object. Such objects are produced through contribution of numerous individuals. It is easy to agree that

57 Galili, I. & Lehavi, Y. (2006). Definitions of physical concepts: a study of physics teachers’

knowledge and views. International Journal of Science Education, 28 (5), 521-541.

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scientific knowledge resides in many personal frames of understanding. Karl Popper introduced the concept of the third world:58

By world 3 I mean the world of the products of the human mind, such as languages; tales and stories and religious myths; scientific conjectures or theories, and mathematical constructions; songs and symphonies; paintings and sculptures. But also aeroplanes and airports and other feats of engineering.

However unlike other elements of the third world and despite possible subjective nature of individual knowledge, scientific intends and purports to be objective. It is a great challenge of the physics teacher to show, demonstrate and explain by convincing examples and concept refinements how elements of knowledge, which comprise an amalgam of subjective and objective nature in each particular mind, succeed to create objective physics knowledge in World 3 – collective knowledge.

The scientific knowledge is objective knowledge by its definition, that is, the knowledge which is not voluntary and independent of supernatural and mystic elements. The scientific knowledge states laws, rules and principles which may be tested each time one reproduce the conditions of their validity. This was always the standard of philosophers of nature and physicists, whether or not they were religious in their worldviews.

These features of objectivity one may find in all old theories of motion. Objectiveness, however, should not be confused with correctness. Despite their intention to match the reality of motion, the attempts of people to account for it was of limited success and often failed. This excurse displays several such attempts which preceded classical mechanics. Today we know that classical physics is also limited in its validity to the macroscopic objects of our close environment. However, the limited area of validity does not make this theory subjective. We use the objectiveness of classical theory of motion by innumerous implications we make to this knowledge, from successful monitoring of cars, planes and ships to sending people to the Moon and automatic laboratories to all other planets of the solar system.

Historical thinking

When teachers address the knowledge from different cultural periods, they should relate the old and new meanings of different terms and concepts. All scientific revolutions caused radical changes in the nature of science and in the vocabulary of science (language and meta-language).

For example, when the teacher tells about the observation in Aristotelian science, he should carefully distinguish between observation (contemplation) and experiment. Teacher should be aware that theory for Aristotle meant something different from the contemporary meaning of the same notion. Also, the notion model is completely pointless within Hellenic science. Greek philosophers of nature believed that their knowledge reflected the reality as it was. These conceptual modifications require appropriate comments.

Reversing Holderlin's aphorism59, one may say that where salvation appears the danger appears as well. Historical research materials can, as many educators believe,

58 Popper, K. (1978). Three Worlds. The Tanner Lecture on Human Values. The University of

Michigan. Online: http://www.tannerlectures.utah.edu/lectures/documents/popper80.pdf 59 A known quote of German poet Friederich Johann Hölderlin reads: "Where there is danger some

Salvation grows there too", Patmos. F. Holderlin, Poems and Fragments. Ann Arbor, The University of Michigan Press 1966, pp. 462-463.

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bring salvation for teaching difficult physics issues. However, the same step may cause undesirable difficulties in understanding and interpretation. Naturally, the old texts often include obsolete ideas surpassed in the course of history and resolved by other scholars in subsequent studies. Much of careful and critical attitude to the materials is required on behalf of the teachers to identify the conceptually problematic points, to understand the limited validity of the claims and not blindly adopt them, regardless the perceived authority of the author.

Historical materials may include comments and ideas, interpretations, which contradict each other. This is what normally happens when the Aristotelian theory of motion and the impetus physics are compared. Superficial reading seeking for the straight and simple answer may lead to a confusion regarding the explanations. The superficially reasonable nature of impetus may convince naïve perception. Despite of that, the concept of impetus leads to false explanations and should be replaced, as it took place in the history, by the concepts of momentum and inertia, which are different and each took something from the role played by impetus in the old theory.

Also the idea of inertia changed its meaning in the course of history from a feature of laziness and resistance to motion (Kepler) to the inertial mass of Newton.60

******************************************************************* Research evidence Research results agree that students often develop and hold views on motion similar to those of the impetus theory. This does not exclude the ideas close the Aristotelian theory too. Particular context determines which of the alternative ideas is preferred by the student. For example, the context of dragging and pushing bodies is the one that invites Aristotelian force-motion understanding,61 whereas in the context of projectiles, the idea of impetus prevails.62 The latter often draws on the analogy with a car, which needs fuel to go.

******************************************************************* Further development Continued development of knowledge regarding the subject of using the history and philosophy of science in physics teaching may start with using materials of this excurse. However, to be effective and allow mature application to other topics, it requires reading introductory texts on the philosophy of science such as those by Kuhn,63 as well as enriching texts on the history of science, that could be found in the writings of Koyré64 and Collingwood.65

60 See our excurse to the history of inertia. 61 Whitaker, R. J. (1983). Aristotle is not dead: Student Understanding of Trajectory Motion. American

Journal of Physics, 51(4), 352-357 62 McCloskey, M. (1983). Op.cit., McCloskey, M. (1983). Naive Theories of Motion, in Gentner and

Stevens, eds. Mental Models, Lawrence Erlbaum Associates, Hillsdale, New Jersey, pp. 299-324. 63 Kuhn, T. The Structure of Scientific Revolutions. 64 Koyré, A. (1957). From the Closed World to the Infinite Umverse. Baltimore. 65 Collingwood, J. (1994). The Idea of History, Oxford University Press.