On the Historical Role of Aristotelian Mechanics1

O.S. Bakai

National Scientific Center "Kharkiv Institute of Physics and Technology" (61108 Kharkiv 108, Akademichna st., 1, Ukraine)

he Ancient Greek philosophers created the concept of the nature of motion and determined the general principles of motion. Descriptions of the laws of motion in the Aristotelian physics are so Tfull and constructive that they allow formalization. This paper contains the equations of motion in a general form, obtained in strict accordance with the laws of motion of Hellenic scientists. Aristotelian mechanics is reviewed, and compared with Newtonian mechanics. The connection between Aristotelian mechanics and Newtonian mechanics is investigated.

INTRODUCTION

There appears to be a wide gap in the history of civilization on the European continent, because of the long period of the Early Middle Ages, which separates the Hellenic epoch from the Renaissance and modern civilization. It was a period of substantial change in the public and social order, accompanied by the formation and expansion of Christianity. As to the development of culture and science, that period was not too fortunate. It was rather a period that lacked noticeable progress. And furthermore, the destruction of libraries, the islands of culture, and the overall destruction of the extremely thin and brittle layer of creators and scientists all over Europe, broke the organic connection between the Hellenic epoch and the Renaissance for nearly two thousand years. The achievements of the Hellenes in the sphere of science were undoubtedly the basis of the revival and bloom during the Renaissance. At the same time, because of the lack of vital connection between these two epochs, starting from the XVII century, the grandeur of heritage of the great predecessors was somewhat obscured. In the long run it has led to an under-appreciation of the earlier work based on not fully correct estimations of the significance of this heritage. Let us appeal to the history of physics. A large number of conceptual works about the achievements of Ancient Greeks in the sphere of physics have been written, but the conclusions are reduced to the idea that they (the Ancient Greeks) were the authors of brilliant conjectures and hypotheses (for example, the atomistic

1 A preliminary version of this study was published in Ukrainian in the journal UKR.FIZ.ZH.43/3, 378, 1998.

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hypothesis), which later, many centuries after, were proved, and acquired their modern constructive forms. The emphasis is placed on the ascertainment by the great Hellenes of the laws of action of static forces, which were used in simple instruments (levers, inclined planes, tackle blocks, etc.), Archimedes's law, and the geometric laws of optics. As for their contribution to the discovery of such fundamental physical laws as the laws of mechanics, it is commonly thought that there was more confusion than advancement. This is what the famous physicist and popularizer of physics G. Gamov writes about the contribution of Aristotle to the development of sciences and, among others, physics, in his book Biography of Physics l\l: "Most of the works of Aristotle preserved until our time are the "treatises" which probably represent the texts of lectures which he delivered in the Lyceum on various branches of science. There are the treatises on logic and psychology of which he was the inventor, the treatises on political science, and on various biological problems, especially on classification of plants and animals. But, whereas in all these fields Aristotle made tremendous contributions which influenced human thought for two millennia after his death, probably his most important contribution in the field of physics was the invention of the name of that science, which was derived from the Greek word φυσιζ which means nature. The shortcomings of Aristotelian philosophy in the study of physical phenomena should be ascribed to the fact that Aristotle's great mind was not mathematically inclined as were the minds of many other ancient Greek philosophers. His ideas concerning the motion of terrestrial objects and celestial bodies did probably more harm than service to the progress of science. At the rebirth of scientific thinking during the Renaissance, people like Galileo had to struggle hard to throw off the yoke of Aristotelian philosophy, which was generally considered at that time as 'the last word in knowledge', making further inquiries into the nature of things quite unnecessary".

The above quote nearly exhausts the contents of the section dealing with Aristotle in III. It should be noticed that Gamov was a widely and diversely gifted man who made considerable contributions to the solution of actual problems in physics, astrophysics and even in biology. The elucidation of the bright pages in the history of physics that he gives is deep and exciting. As to his assessment of the significance and role of Aristotle (along with his predecessors) to the history of physics, he made no original statement and simply repeated the thought that was widespread even before his time. Even though there were serious reasons for making such a judgement, there are still more serious reasons for reconsideration of this judgement and those are the actual works of Aristotle (see 121 and works quoted in this book and comments on it) and his predecessors. Aristotle used and discussed the main ideas of his predecessors in the sphere of physics and, in particular, in mechanics. He developed these ideas, put them in order, and at the same time made an effort to put them in agreement with the main conceptions of his philosophy. For a long time his "Physics" was considered as the peak of scientific and philosophic thought. The formation and strengthening of the position of Christianity demanded if not scientific validation then, at least, agreement between the canons of religion and the leading scientific achievements. It turned out that the picture of the Universe drawn by Aristotle, showing it as the sequence of concentric spheres with the Earth situated in the center, fits this purpose in the best way. Aristotle describes the last sphere as indivisible and

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moving by itself, and being the motive force for all the lower spheres. The upholders of religion had only to connect this last sphere with the name of the Holy, and so that was what they did. In such a way the mechanics of Aristotle and the planetary system of Ptolemeus were canonized. This was exactly the reason why, while creating a new mechanics, Galileo had to endure a hard struggle with the church reaction, which widely looked like a struggle with Aristotelian mechanics (but not with some incorrect pages of it). The mechanics of Newton was based not only on known principles, but on a constructive mathematical basis as well - created by him (and simultaneously by Leibnitz) the apparatus of differential and integral calculus. The striking achievements of Newton, particularly in celestial mechanics, marked a huge, revolutionary step in the development of mechanics. Thanks to the above-mentioned struggle with the reactionary mind, the mechanics of Newton was interpreted (and, as one can see from the quotation given above, is interpreted at present) not as the development of Aristotelian mechanics, but as the refutation of it. Actually, both Galileo and Newton were the lucky heirs who eventually were able to make use of the invaluable Hellenic treasures and to enlarge them, rather than poor prisoners of confusion, that had to release themselves from heavy chains, prepared with the help of Aristotle and his predecessors.

The aim of this work is to determine the connections and the differences between Aristotelian mechanics and Newtonian mechanics. To achieve this aim, I decided to formalize Aristotelian mechanics by applying mathematical apparatus to it, but in strict accord with the main principles of this mechanics, as it is done in Newtonian mechanics.

1. MAIN IDEAS AND PRINCIPLES OF ARISTOTELIAN MECHANICS

Let us use the term "Aristotelian mechanics" for the contribution of Ancient Greeks into this sphere without separating and considering their contributions individually. Aristotle had discussed the most important ideas on the nature of matter and motion lows thoroughly. (It is worth noting that his formulations, objections and suggestions were never presented as a final and only right word. As a form in which many ideas were presented very often dialogues were used including different suggestions, positions and points of view). In the list of the main ideas and principles of Aristotelian mechanics let us concentrate on those that played a significant role in the process of creating and developing mechanics and that have preserved their significance up to now.

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Divisibility and interaction of systems.

Here any object being described or investigated is called a system. It was assumed in Aristotelian mechanics that systems can consist of parts, subsystems, which interact with each other. And motion, i.e. the change of state of the subsystem, may occur (and occurs) because of the influence of another subsystem. At the same time Aristotle assumed the existence of such systems that are indivisible, move by themselves and can influence the motion of another systems.

Motion as changing of observable values with time and as a form of existence.

The idea of motion is defined as the changes with time of any observable value that characterizes a system. It was supposed that no system exists without motion. Thus, every observable value is changing with time or can be changed. There were also introduced the idea and definition of a continuous variable value as such that every segment of it can be divided up to infinity. In particular, the time and the coordinate of a body were considered by Aristotle as continuous variables. This is seen from his analysis of the well-known Zenon's aporias.

Causality principle

According to this principle all events that occur at the current moment are connected by causal relationships with the events that occurred at a previous moment. However not all events of the past are connected by causal relationships with all events that occurred later, i.e. "after this" does not automatically mean "as a result of this". So, among the sequence of events that occurred at previous moments of time and those that occur at the current moment one can distinguish such a series of events that have causal relationships. Concerning mechanical motion it means that the state of a system at the current moment depends on its state at the previous moment, but not vice versa.

Principle of determinism

According to this principle, the state of a system at the next moment is unambiguously determined by the state of a system at the current moment. This principle was clearly formulated by Democrites while considering the motion of atoms.

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Remark 1. One of the opponents of this principle was Epicurius. His objection was based on the thesis that if the causality principle was realized strictly then it would be impossible to bring any changes into the course of events. In particular, there would be no place for the manifestation of will. According to Epicurius, moving atoms occasionally make undetermined changes of motion. And thanks to this, an infinite number of different forms and realizations of motion do exist. Epicurius also used the idea of the smallest possible undetermined change of motion.

In Aristotelian time these were the main ideas about the nature of motion and the principles it obeys. Let us notice the fact that here we do not find any consideration of counteractive forces in the process of interaction between systems, not to mention the principle of the equality of active and counteractive forces, which is one of the three principles of Newtonian mechanics.

2. THE FORM OF EQUATIONS OF MOTION IN ARISTOTELIAN MECHANICS

The above principles of motion allow for the statement of the equations of motion in general form and, in particular, equations of motion of bodies in Aristotelian mechanics. Let us show how this can be done, based only on what was stated in the previous section. Every system is characterized by some set of observable values, and we denote them as = q\,...,q„ .

Each of these values can and does change with time,

Ii = 9i(0' '=1,2,..., η, (1) time t being a continuous variable. Let us also suppose that values q, are continuous (this restriction can be left out through the formal complications). According to the causality principle the state of a system at some next moment {^(ί + Δ/)} is connected through a causal relationship with the state of the system at the current moment (<7(f)} , and according to the

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principle of determinism this connection is unambiguous. Let us consider the increase of the value of variable q, on the interval of time [i, t + Δί]:

= + -qt(t). (2)

Owing to the continuity of t and qt the value below makes sense:

v,· (r) = lim (Δ?, /At) = dqt /dt, (3) Δ/->0

which is the instantaneous rate of change of the value qt, and if Δ/ -> 0

qi(t + At) = qi(t) + vi(t)At. (4)

From this we can see that if {^(ί + Δί)} is unambiguously determined by the state of system at the moment t, the same also can be said about the set of instantaneous velocities {v(i)} = V| (ί),..., v„ (t).

So, one can write

ν,· (0 = ίί 0(')}>'] (')- ->9n (')>']> (5) or

dq i/dt = Ft [q] (f), ...,qn (*),*], i=l, ...,n , (6)

where Ft (i), ...,qn (ί),ί] is some function being determined by the nature of the system and the character of forces that affect it. This is just a general form of the equations of motion that follows from the principles of Aristotelian mechanics. In the case when a system can be divided into two interacting subsystems such that one of them is affected by another, but the second one is moving independently from the first, the system of equations (6) should be somewhat modified. Suppose that a subsystem being affected by another one is described by the set of parameters Ξ Ξ <7ι,···,and the other one is described by the set {q)2 qm+\, qn • Then the set {q}2 is satisfied by equations in form (6), and for the set {9}] one obviously has

dqi/dt=Fi[qi,...,qm;qm+u...,qn,l] = Fi[(h,...,qm,t], i-1, ...,m. (7)

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Remark 2. In the dynamics of Epicurius one has additionally to take into account the effect of an undetermined change of values q{. This can be done by introducing a generalized undetermined (random) force into the equations of motion:

dqi/dt = Fi[{q(t)},t] + Fi[{q{t)},t], (8)

where F, [J

t0

F ( i [fa (')}"] = Σ ft [fa (')} · •] (t -zk ). (9) k=\

Here /,· (r^.) is a force that in the general case depends on dynamic variables {17}. If one assumes, as Epicurius did, that there is a smallest possible uncontrolled change of q,, then f, should be chosen as a constant in absolute value but random in sign (at different moments r*).

As we can see, the equations of motion in Aristotelian mechanics are differential in their form. To use them for determining the state of a system at any future moment t' > /q , as it is seen from (4), (6), one has to know only what was the state of a system at some initial moment IQ :

fa(<)}|,=,0=fao}· (10)

Thus, equations in form (6) or (7) together with initial conditions (10) must unambiguously determine the state of a system at any future moment. It is not difficult to make sure that the above mentioned equations satisfy this requirement. Indeed, according to the well-known existence theorem about solutions of systems of differential equations of the form (6) or (7), such solutions exist with quite general properties of the right-hand parts. And the theorem of uniqueness establishes that for the given initial conditions this solution is unique. So, there exists, and exists uniquely, the solution of system (6), and it is determined by the initial conditions. As we can see, being derived with the help of the principles of causality and determinism, the form of the equations of motion in Aristotelian mechanics does not lead to contradiction with the principles on which it is based.

3. SENSE OF EQUATIONS OF MOTION IN ARISTOTELIAN MECHANICS

As one can see, the main principles of Aristotelian mechanics lead to the form of the equations of motion that are similar to the ones we use today. And as for the sense of these equations, it is just the place where, as

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Aristotle's compatriots would say, the Achilles heel of his mechanics is situated. From the laws of motion, expressed in the form of equations (6), (10), one can see that after these equations have been derived the main difficulty of the next step in the progression is transferred onto the problem of making sense of the nature of forces Fi. Hellenic scientists mostly dealt with manifestations of the action of different forces, but they did not achieve a great deal of new insight in studying their nature. Their conceptions of the fundamental forces that act between atoms were quite naive. Even a question about the nature of forces, about interactions between systems by themselves, as far as we know, wasn't formulated at that time, although a force as a cause of motion was introduced. The statement that all observable forms of motion in the sublunary sphere belong to the category of forced, i.e. occur as a result of action of forces is probably the only one that was added by Aristotle to the principles mentioned in the first section. But, as we shall see, it is already enough to lead to an important result of Aristotelian mechanics. Consider the dynamics of a moving body within the framework of Aristotelian mechanics. For the description of such motion Aristotle confined himself to consideration of the location (coordinate) of a body only, and asserted that the velocity of a body is proportional to the magnitude of the force that acts on it. To give this assertion a form of equation of motion, let us denote the coordinate of a body as q and confine ourselves to consideration of one-dimensional motion in a determined direction. Then q is a one-component scalar value and equations of motion (6) acquire a simple form:

dq/dt = cf(q,t), (11) where the coefficient c takes account of those features of a body that determine its response to the action of a force. The differences in consequences of the action of the same force on different systems are described by exactly this coefficient. The nature of the force, as well as the nature of this coefficient, is left out of our (as well as Aristotelian in his time) scope. Equation (11) should be supplemented with the initial conditions:

(12) «ML -*>•

Let us rewrite equation (11) in more convenient form:

(13)

It is this equation that makes the main sense of Aristotelian mechanics. As for the coordinate q, its dependence on time can be found by integrating equation (4). In particular, if a force does not depend on q and is a known function of time, one has

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t t

<7 = <70+ H'') dt' = q0+c jf(t') dt\ (14) Ό Ό

In the general case we deal with solving a more complicated differential equation.

Remark 3 Taking into account remarks 1 and 2, we can write down the equation of motion of an atom in Epicurean approach:

dq/dt = cf(q,t) + c f(q,t), (15)

where / is a random force. If a random force has the form of (9) and does not depend on q, then integration (14) leads to the random leaps of an atom at the moments τ*.

4. A BRIEF DISCUSSION ABOUT THE FORM AND SENSE OF NEWTONIAN MECHANICS

To clarify the main differences between Aristotelian and Newtonian mechanics, let us consider the motion of a body under the influence of external forces.

Difference 1.

In Newtonian dynamics the state of a moving system is determined not only by a coordinate, but also by a velocity. Thus, a full set of dynamic variables consists of a coordinate q and a velocity ν (for the simplification of notation here, as in (11), we confine ourselves to the case of one-dimensional motion in a specified direction).

Difference 2.

A force, as a cause of a change of state of motion, affects only a velocity, causing an acceleration. As to the form of the equations of motion in Newtonian dynamics, it is the same as the one obtained for Aristotelian dynamics. The principles of determinism and causality are put by Newton into the equations of motion without a preliminary discussion, a priori. As shown in the preceding section, equations having the form (6), (10) "automatically" satisfy these principles. Thus Newton, in spite of deriving equations of motion in a general form, had proposed concrete dynamical equations. Later, the equations of motion of a more

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general form for other types of motion (in the Aristotelian sense) were obtained, but they, in the majority of cases, may be reduced to Newtonian equations. It is because of this fact that there appeared a new trend of philosophy - mechanicalism - in which all natural phenomena are described by the equations of Newtonian dynamics. With the appearance of equations of electrodynamics and then of general relativistic theory, not to mention the equations of quantum fields theory, it became clear that Newtonian equations, although they have retained their honorary place, can't be a base of description of all natural phenomena. But the causality and determinism principles, discovered in Ancient Greece, have retained their importance in all parts of quantum and relativistic physics. Let us write the equations of motion of a body taking into account the differences stated above:

dq/dt = f0(q,v,t) = v(t), (16)

dv/dt=bf(q, v,t) with initial conditions:

9(')L0=<70, v(r)Uo=v0. (17)

Coefficient b is introduced into the second equation of (16) for the same reason as coefficient c in equation (11). In Newtonian dynamics it is connected with the measure of inertia of the body, the mass m, and is equal to 1 Im. As to the force, j{q, v, t), in Newton dynamics, it has the same meaning as in Aristotelian dynamics, because, in the long run, in both cases a force causes a body that was at rest to move. In spite of the essential differences between dynamics (13), (14) and (16), (17), it turns out that there are some cases when they both give very similar results. Such cases are those in which a body moves with negligibly small accelerations, i.e. when the value a (?) =dv / dt may be neglected compared with the terms that stand in the right-hand part of the second equation of (16). To avoid consideration of the complicated (by the form, not in its meaning) general case, let us restrict ourselves to the more simple, but very widespread case whenyfg, v, t) consists of the force of friction,/^, v, t):

fd{q,v,t) = -Kv, (19) where at is a friction coefficient, and the external motive force, fe(q, v, t), that does not depend on velocity. Then the second equation in (16) acquires the following form:

dv/dt = -(K/m)v + (l/m)fe(q,t). (20)

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To make the acceleration negligibly small, the forces in the right-hand part in (20) must almost completely compensate one another, thus, with high accuracy, the next equation should be presented as:

fd+fe= 0 (21)

v(t) = (l/K)fe{q,t). (22)

Comparing (13) and (22), one can see that these expressions coincide. So, the dynamics of Aristotle coincide with Newtonian dynamics when the motion caused by the external force occurs without noticeable acceleration thanks to dissipative processes. Let us list some examples of laws of motion which we use nowadays, and which are presented in Aristotelian form. First of all, let us consider Ohm's law:

/ = σΕ , (23)

according to which the electric current strength I is proportional to the electric field strength E, the coefficient of proportionality being conductance σ. Remembering that

I=Nev, (24)

where Ν is a number of charge carriers with charge e moving with (average) velocity v, one can see that each charge carrier moves with velocity

ν = (σ/Νβ)Ε. (25)

Having compared (25) and (13), we see that Ohm's law has the form of an Aristotelian law of motion. The same concerns the dynamic of bodies that move in viscous media due to quasi-stationary forces, when the velocity turns out to be proportional to the acting force F:

ν = nF . (26)

Coefficient η is called a mobility coefficient. As we see, the mistake of Aristotle was not that the law formulated by him is never true. Aristotle erroneously considered equation (13) to be a general law of motion. Newton proved that Aristotelian

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equations are not general, but, as we see, the complete disproof of Aristotelian mechanics is a mistake. The results of qualitative observations caused Aristotle to formulate the law of motion (13), and the desire to comprehend all the phenomena of motion at once led to erroneous generalization. Let us remind ourselves that the laws of Newtonian dynamics also have a limited area of applicability. They rely not only on principles of mechanics formulated by him, but also (implicitly) on the conceptions of absolute time and space. These conceptions, as we know, turned out to be erroneous. Formulation of the relativistic principle in fact led to the disproof of Newton's equations of motion and Galileo's principle of addition of velocities, although the principles of Newtonian mechanics can be preserved if one considers the mass m to be time dependent. As to Newtonian dynamics, it appeared to be approximate and correct for the cases of small (much less than the velocity of light) velocities of bodies. Fortunately, this area covers practically all forms of motion of macroscopic bodies. Thanks to this Newtonian mechanics has retained its great importance. The main principles of Aristotelian mechanics appeared to be much more powerful than his law of motion. These principles have not lost their importance up to now, though the law of motion proposed by Aristotle has a quite limited range of application. Almost the same can be said about Newtonian mechanics too, with the difference that its range of application is much wider and that Aristotelian dynamics is a particular case of Newtonian dynamics.

Remark 4. Now let us turn to Epicurean dynamics. It was seldom considered and discussed because it was never put on a constructive basis. Although the equation of motion (15) is obtained in strict accordance with Epicurean ideas about the nature of motion and the main principles of Aristotelian mechanics. However, Epicurus himself, together with his contemporaries, was far from mathematical formalization of his thoughts. The main Epicurean thesis about non-determination, the unpredictability of all events, about the importance of randomness in the phenomena of motion is hard to oppose, as one has not only personal evidence but also the results of numerous scientific experiments plus the whole part of mechanics - statistical mechanics - that is practically based on this thesis. On the other hand, the proof of this thesis turns out to be not so simple because, as we know, it conflicts with Newtonian mechanics, which is based on the principle of determinism. It turned out to be a central problem of statistical mechanics that one needs to prove kinetic equations, which have to be consistent with the second law of thermodynamics (non-decreasing of entropy of an isolated system with time) and to describe irreversible relaxation processes. Kinetic equations are obtained through the process of "roughening" of the exact dynamic equations, which are equations of Newtonian dynamics and thus based on the causality principle and, as it is known, describe processes reversible in time. So, the reversibility in time of Newtonian dynamical equations disproves the irreversibility of statistical mechanics equations. L. Boltzman was the scientist who derived the first kinetic equation of statistical physics and, as it seemed to him, proved the famous //-theorem that grounds the second law of

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thermodynamics. He was also the first who had to endure powerful criticism because of the incompatibility of the Boltzman equation with the equations of dynamics. Let us not stop on the course of long-lasting, extremely keen, exciting and dramatic discussions on this problem. Let us be reminded that P. Ehrenfest was the first who showed that the arbitrarily small uncertainty in the state of a system (if Epicurus was right!) is enough to ground the irreversibility of thermodynamic systems. Much later a more thorough study of dynamical systems described by Newtonian equations led to the discovery of the fact that even comparatively simple dynamic systems possess stochastic states of motion. Those states of motion are called stochastic in which the correlation between the initial state of a system and its state in the current moment disappears with time. More exactly, autocorrelation functions of dynamic variables in the case of stochastic motion decrease with time and remain near to zero. This means that it is practically impossible to determine what the exact state of a system will be in quite a long period of time, even if its initial state is known exactly and its dynamics are governed by deterministic Newtonian equations. Dealing with such a system, we cannot restrict ourselves to the consideration of any one solution of equations of motion with one or another initial conditions, because a system quickly forgets about them, and the arbitrarily small perturbation of initial conditions quickly leads to the arbitrarily large difference of states of a system. So, we must take into account such peculiarities of dynamics and resort to the statistical description of it, neglecting the exact knowledge of the initial conditions, because in reality this knowledge does not give an opportunity to predict the behavior of a system with acceptable accuracy. With the discovery of stochastic solutions of deterministic dynamic equations, the kinetic equations of statistical mechanics obtained their foundation. At the same time the Epicurean hypothesis about the non- determination of atomic motion found its proof (or disproof?). Having no possibility for more detailed consideration of the stochastic dynamics of deterministic systems in the frame of this work, we propose that the reader address the conceptual and popular scientific publications devoted to this problem (see /3, 4/).

Isaak Newton

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Isaak Newton (1643-1727) published Philosophia Naturalis Principia Mathematica (The Mathematical Principles of Philosophy or The Principia) in 1687. Newton begins The Principia by defining the concepts of mass, momentum, and three types of forces: inertial, impressed and centripetal. He also gives definitions of absolute time, space, and motion, offering evidence for the existence of absolute space and motion in his famous "bucket experiment". The three Laws of Motion are proposed, with consequences derived from them. On this solid foundation the three books rest. They are formulated in rigorously logical Euclidean fashion in the form of propositions, lemmas, corollaries and scholia. Book One, Of The Motion of Bodies, applies the laws of motion to the behaviour of bodies in various orbits. Book Two continues with the motion of resisted bodies in fluids, and with the behaviour of fluids themselves. In the Third Book, The System of the World, Newton applies the Law of Universal Gravitation to the motion of planets, moons and comets within the Solar System. He explains a diversity of phenomena from this unifying concept. The Principia brought Newton fame, publicity, and financial security. Preface Newton begins with a reference on the mechanics of Ancient Greeks: "Since the ancients (as we are told by Pappas), made great account of the science of mechanics in the investigation of natural things; and the moderns, lying aside substantial forms and occult qualities, have endeavoured to subject the phenomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it regards philosophy. The ancients considered mechanics in a twofold respect; as rational, which proceeds accurately by demonstration; and practical. To practical mechanics all the manual arts belong, from which mechanics took its name." And further, "...rational mechanics will be the science of motions resulting from any forces whatsoever, and of the forces required to produce any motions, accurately proposed and demonstrated. This part of mechanics was cultivated by the ancients in the five powers which relate to manual arts, who considered gravity (it not being a manual power) no otherwise than as it moved weights by those powers." In the final part of the Third Book, General Scholium, Newton tries to establish a connection of his mechanics and the System of the World with God. The following citations elucidate his attitude. "This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being." "This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God παντοκρατωρ , or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants." "...by way of allegory, God is said to see, to speak, to laugh, to love, to hate, to desire, to give, to receive, to rejoice, to be angry, to fight, to frame, to work, to build; for all our notions of God are taken from the ways of mankind by a certain similitude, which, though not perfect, has some likeness, however. And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to Natural Philosophy."

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5. A TROUBLESOME HERITAGE

As can be seen, the main doctrine and principles of Aristotelian mechanics are implicitly included in Newtonian mechanics. No doubt, Newton was more aware of the Ancient Greek heritage than representatives of the following generations, trained on his Principia. But this heritage was not a proper treasure to be used explicitly because of the secular dogmas of the Church, which were based on Aristotelian scholastics. Therefore, waiving the scholastics and dogmas of the Church, Newton was nearly forced to repudiate officially the Aristotelian heritage, despite the fact that he could not reject it completely. Moreover, Newton not only kept silent about the importance of the material inherited from the Greeks, but he went much further (see Box, above). To wit, he put forth considerable effort to set the new mechanics in the place which at that time was occupied by Aristotelian philosophy. And he succeeded in this excellently. As Bernal has written 151, Newton's triumph was so huge that a danger appears that he would became a new Aristotle (in a way) creating insurmountable obstacles for further progress. This danger was not a mythical one. It was only after a century had passed that scientists in England became sufficiently free of his authority that they could start their own investigations in fields where he worked.

6. EPILOGUE

Our aim did not consist in a trial of rehabilitation or reconsideration and apology of Aristotlean mechanics. We think it doesn't need this. Our aim was a deeper understanding of what was done for the creation of mechanics by the representatives of different epochs. We tried to show how much the principles and ideas yield into the building of a constructive basis of physics and how the principles that were formulated based on observations and general arguments two thousand years ago have retained their importance up to now.

Having passed along the way from the principles to equations of motion first in Aristotlean mechanics and then in Newtonian mechanics, one could assert that progress in physics occurred through the searching of fundamental principles. But, in fact, the history of physics knows examples when the equations of motion that were proposed for the description of empirical data and, thus, seemed to have quite a limited range of applicability, have not only retained their importance, but also led to the discovery of new fundamental principles. In connection with this let us recall Maxwell's equations. The author (J. C. Maxwell) had considered them as mathematical relations that were empirically established by M. Faraday. What is more, for a long time he tried to find the grounds for these equations in the frame of mechanistic models. It was a big success for Maxwell when he had constructed the Lagrangean of the electromagnetic field. In this way it was shown that his equations could be presented in the framework of formalism developed on the basis of Newtonian

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mechanics. At long last it turned out that his equations not only don't need mechanistic grounds, but they substantially can't have them, and enforced formulation of the new fundamental principles of physics - the principle of relativity and the quantum hypothesis. Maxwell's equations are probably the most unexpected discovery in physics and the most productive in their consequences. Worth noting is that in form they are the same as the equations of Aristotelian mechanics should be, despite the fact that Maxwell's equations are written for field variables. As to their substance, they contain a whole new area of physics, with the richest set of physical phenomena. If we have succeeded in clarifying to some extent the deep connection between Aristotelian mechanics and Newtonian mechanics and the later achievements of physics, we shall consider the goal of this work to have been achieved.

ACKNOWLEDGMENT

I am very grateful to Loren Jacobson for taking up the challenging task of transforming my amateur English into a more standard form.

REFERENCES

1. Gamov G. Biography of Physics, New York: Harper & Row, 1981. 2. Aristotle. Works in 4 v., - M.: Mysl', 1981. (In Russian). 3. Bakai A.S., Sigov Yu.S. Multifaceted turbulence. Fluid Mech. Res. 22, 7-74 (1995) 4. Zaslavskiy G.M. Stochasticity of dynamic systems. M.: Nauka, 1987. (In Russian). 5. Russell B. A History of Western Philosophy, New York: Symon & Shuster, 1972

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