An Essay on the History of the Establishment of the Fundamental Principles of Mechanics

(An introductory lecture to the course in theoretical mechanics delivered at the Naval Academy).

A. N. Krylov.

§ 1. Of all the phenomena of the world around us, the motion of bodies is the most ordinary and the most widespread; and one may say that not a single human action is performed without the motion of something.

From the remotest antiquity there have remained enormous structures showing that already then there existed the ability to employ auxiliary means and machines for lifting and moving loads.

Likewise, from time immemorial—prehistoric times—there has existed the use of vessels for sailing on seas, rivers, and lakes.

It would seem that all this should have attracted the attentive notice of inquiring minds and become the subject of study and scientific investigation.

Meanwhile, reality shows that then, as now, practice far outstripped theory, for only in Aristotle’s Physics, he who lived from 384 to 322 B.C., can one find an exposition of the doctrine of motion and forces and the foundations of the doctrine of machines. Then, in two works of Archimedes (287–212 B.C.), there is found an exposition, quite different from Aristotle’s, of the doctrine of the equilibrium of the simplest of machines—the lever; the doctrine of centers of gravity; and, finally, the doctrine of the equilibrium of bodies floating in a liquid. These last two doctrines were created entirely by Archimedes.

The great philosopher Aristotle, although he spent twenty years in Plato’s school, whose motto was “let no one ignorant of geometry enter here,” was not a true geometer; in his doctrine of motion he proceeded from a basic proposition accepted a priori, which seemed to agree with the everyday experience of everyone and was so simple that the most elementary reasoning made it self-evident. But here it turned out that, in interpreting ordinary everyday phenomena, a very important circumstance had been overlooked or falsely understood, so that the accepted fundamental proposition did not correspond to reality. Nevertheless, twenty centuries passed before this was discovered by Galileo.

Aristotle’s doctrine of the equilibrium of the lever and of other machines reducible to it leads to correct conclusions, but in its presentation it did not possess the geometric rigor characteristic of the classical works of the ancient mathematicians and, apparently, did not satisfy Archimedes, who, in giving his own exposition, does not mention Aristotle. However, it must be borne in mind that Aristotle’s authentic manuscripts lay hidden for 120 years after his death, walled up in a cellar, and therefore could not have been known to Archimedes.

§ 2. The works of the great Syracusan mathematician represent, to this day, an unsurpassed model of geometric rigor and consistency of reasoning.

Archimedes begins his treatise “On the Equilibrium of Planes and on Centers of Gravity” with the following words:

“Postulates: 1) Equal weights suspended at equal lengths are in equilibrium.

2) Equal weights suspended at unequal lengths are not in equilibrium: that suspended at the greater length descends.

3) If weights suspended at any lengths whatever are in equilibrium and something is added to one of them, then they will no longer be in equilibrium—the weight to which something has been added will descend.

4) Similarly, if something is taken away from one of these weights, then they will no longer be in equilibrium—the one from which nothing has been taken away will descend.”

From these four postulates, which may be regarded either as self-evident or as established by immediately accessible experience, Archimedes, with the full rigor of ancient geometry, derives first of all the laws of equilibrium of the lever.

To these four postulates he adds, at the very beginning, four others relating to centers of gravity; using them and the proven properties of the lever, he finds the positions of the centers of gravity of various kinds of rectilinear figures, and then of a parabolic segment.

His second work, “On Floating Bodies,” likewise presents an unattainable model of geometric investigation, based on two simple and self-evident assumptions, from which everything is then derived by exact, rigorous arguments and proofs. One should not think that Archimedes confined himself only to the most elementary derivations of the laws of equilibrium of floating bodies—no, he applies these laws to determining the equilibrium positions of a segment of a paraboloid of revolution for different ratios of its density to the density of the liquid. It turns out that such a paraboloid can be in a stable position of equilibrium

not always in the position in which its axis is vertical and its vertex downward, but sometimes, and even more often, precisely the opposite, i.e. with its vertex upward and with the axis inclined.

Taking into account that the very doctrine of the center of gravity, that the determination of areas and volumes bounded by curved lines and surfaces, belongs to Archimedes, and that the entire mathematical apparatus of that time was reduced to the doctrine of proportions, we can only to a small degree form an idea of his genius.

Thus, one may say that Archimedes established the principles of a new science, “Mechanics,” including it in the domain of the exact speculative sciences, whose rigor was not inferior to geometry; for in his work “On the Quadrature of the Parabola” he uses the results obtained in the doctrine of the equilibrium of the lever to prove purely geometrical truths.

Centuries and millennia pass; to the doctrines of Archimedes and Aristotle not only is nothing added, but much of them is even lost. Only eighteen centuries after Archimedes does the Dutch engineer and scholar Stevin make a substantial addition to the doctrine of equilibrium: he finds the conditions of equilibrium of a heavy body on an inclined plane; from these conditions it was already not difficult for him to pass to the condition of equilibrium of three forces lying in one plane and applied at the same point, i.e. to the so-called triangle or parallelogram of forces, which later became the fundamental theorem in the doctrine of equilibrium in general.

Stevin lays at the basis of his reasoning the proposition: “perpetual motion is impossible.” This truth is far from being as obvious as the propositions of Archimedes, but Stevin, as an engineer, boldly applies it, for his practical experience admits of no doubt.

Almost at the same time Galileo gave new, precise foundations to the doctrine of motion. But let us return somewhat and add a few details concerning the works of the founder of ancient natural science, Aristotle.

§ 3. Aristotle touches upon mechanics in three of his works: “Mechanics,” “Physics,” and “On the Heavens and the World.”

The first of these works is essentially a kind of collection of questions or problems with briefly stated answers or explanations. Among these problems let us note Problem IV, stated as follows: “How is it that by means of a lever small forces set great loads in motion?” In the explanation, among other things, it is said: “With respect to the lever one must consider three things: the point of support, representing the center, and two loads: the moving and the moved. The moved load is to the moving one in the inverse ratio of the lengths, and the farther the load is from the point of support, the more easily it is moved. The reason for this is the same as that already given in the explanation of weights: the point which is farther from the center describes a larger circle; therefore, under the action of the same force, the moving load will describe the larger circle the farther it is from the point of support. Let \(AB\) be a lever, whose support is \(E\), the moved

the load \(C\), moving \(D\), then before the end of the motion this latter will arrive at \(G\), while the load being raised will arrive at \(K''\).

With this the explanation ends; moreover, according to the custom of ancient authors, the reader must draw up the diagram himself; but, comparing this explanation with what was said in Question 1, that “everything relating to weights is reduced to the consideration of the properties of the circle, the lever to weights, and all the other peculiarities of mechanical motions to the lever,” one can discern the general principle to which Aristotle reduces the finding of the conditions of equilibrium of all machines, namely: in equilibrium the velocities of displacement of loads in the vertical direction are inversely proportional to the magnitudes of these loads. But this general principle is obviously not formulated and is not expressed clearly and firmly.

Be that as it may, in his mechanics the conditions of equilibrium of the lever and of other machines reducible to it are stated correctly, although they are obtained by means of long, not entirely distinct and not entirely clear arguments.

Let us also turn our attention to Problem XXX: “Why, when two men carry on a pole, or on some other object, some load, do they experience unequal pressure, if only the load is not in the middle, and is that pressure greater on the one who is nearer to the load?—Because under such conditions the pole is a lever, whose point of support is the load; of the two bearers, the one nearer to the load represents, as it were, the thing moved (the resistance), while the farther one represents the mover; and the farther he is from the load, the easier it is for him to move it and the stronger is the pressure downward on the other bearer, for the load represents the same sort of fulcrum as if it were the point of support. But when the load is in the middle, neither one bears a greater burden, and neither is the mover, but each carries an equal weight.” In these words everyone will find the correct method of resolving a given force into two forces parallel to it, applied at the given points.

§ 4. As has already been said, Aristotle did not confine himself only to statics, or the doctrine of equilibrium; his Physics is wholly devoted to the doctrine of motion.

Aristotle’s Physics is considered one of the most remarkable works of this all-embracing thinker and serves as the foundation for those of his other writings in which he sets forth the whole totality of teachings about nature, i.e. all the natural science of his time.

In present terminology this work belongs to the domain of pure philosophy, and not to that group of knowledge which we now call physics, although a considerable part of the work is devoted to the doctrine of motion, but from a different point of view than that from which this phenomenon is considered in present-day physics and mechanics.

Present-day physics and mechanics, based largely on experiment and observation, and therefore also on the testimony of the senses and on measurements with the inevitable errors inherent in them, would likewise have done little to satisfy the inclination of the minds of the ancient Greeks toward exact, abstract reasoning,

since these arguments, which seem to us in many respects not to pertain to natural science and its applications, do not satisfy us much.

Aristotle is not concerned with establishing exact quantitative relations among the various quantities considered in the study of the motion of bodies; he seeks, as it were, to penetrate into the very essence of this phenomenon and to establish whether it exists in itself or only in our representation. He wants to comprehend and explain what space and time are, what infinity is, what a void is, whether it can exist, and other questions of the same kind. These questions give him the opportunity to display the full force of his logic and all the art of his dialectic and subtlety of reasoning. As an example let us cite several excerpts from what he says about time, quoting from the French translation of Barthélemy de St. Hilaire.

“Here are several considerations,” says Aristotle, “which may be adduced to prove that time does not exist at all, or, if it does exist, then only in a manner scarcely perceptible and very obscure.”

“Thus, one part of time has been and is no more; another must be and is not yet. Yet it is only of these elements that both infinite time and the time which we reckon in unceasing succession are composed. But that which is composed of non-existent elements appears itself not to possess true existence. To this it must be added that, for every divisible object, it is necessary, by the very property of its divisibility, when it exists, that some of its parts, or even all its parts, should also exist. But for time, although it is divisible, some parts have been, others will be, while not one exists in the present. The present—the moment or instant—is not a part of time, for the part of any thing serves as the measure of that thing; on the other hand, the whole must be made up of the union of parts, whereas time does not consist of successive moments of the present. Moreover, the moment itself, the very present, which separates the past from the future—is it one or not; does it always remain identical and immutable, or does it constantly change and constantly differ?

“All these are likewise questions that are not easy to resolve. Indeed, if the moment is always and constantly different, then a moment which is no longer now, but which was before this, must have disappeared at some instant; and then consecutive moments cannot exist together with one another, for the preceding one has always, of necessity, already vanished. But it is impossible that a moment should have vanished in itself, for then it did not yet exist; and it is impossible that the preceding moment should have vanished in some other moment; consequently it is necessary to admit that it is impossible for moments to follow one another continuously, just as it is impossible for points on a line to be arranged continuously one after another”...

“We imagine, however, that time cannot be apprehended without change; for we ourselves, if we experience in our thoughts no change, or if the change that occurs in them,

escapes us, then we reckon that no time at all has elapsed, just as it does not exist for those fabulous people who sleep near the heroes in Sardis and who, upon awakening, have no sensation of time, for they join the preceding moment with the subsequent one and make of them one, destroying all the intervening moments which they did not notice. Thus, just as there would be no time if one moment did not become another, but were one and the same single moment, so also in the case when people do not notice that this is another moment, it seems that there has been no time throughout the whole interval. But if time is removed when we notice no change, and our spirit appears to us to be abiding in a single and indivisible instant, and, conversely, when we feel and distinguish change, then we assert that there is elapsed time, it is therefore clear that for us time exists only on condition of motion. Thus, there is no doubt that without motion time is impossible, and that time is not motion.”

Having then examined the relation between moments or instants and time, the similarity and difference between a point and a moment, Aristotle continues: “Thus, time is the number of motion, but this number does not belong to one and the same point, which would be at the same time both beginning and end, as happens on a line, whose boundary between parts it constitutes, though it is not itself part of the line. We saw the reason for this in the fact that a point taken on a line, for example at its middle, has a double significance, for it is at once both beginning and end, and this, in the motion of a body, would necessarily include a certain time of stoppage or rest. But nothing of the kind can be the case with respect to time, which flows incessantly without the slightest interruptions. But it is clear that the moment—the present—does not constitute a part of time, just as a subdivision of motion is not motion, just as points are not parts of a line, whereas lines, when two of them are distinguished in one, constitute parts of this one line and are not its points. Thus, the present moment, considered as the boundary between the past and the future, is not time; it is only a mark of time, which it delimits and determines. But, since it serves for the counting of motion and time, it is a number, with this difference, however: boundaries necessarily belong to the object which they bound, whereas an abstract number may serve for counting anything whatever; and the number ten, for example, after it has been applied to these ten horses before our eyes, may in exactly the same way be applied to a multitude of other objects whose number is also ten.”

These excerpts give some idea of the spirit or character in which Aristotle’s Physics is set forth. One must remember that this composition was written 2200 years ago, and if now, in the study of motion, we “take time as an independent variable,” then not

is tantamount to Aristotle’s wishing to express in words: “time is the number of motion.”

As may be seen, the Physics is not a mathematical work, but a philosophical-critical one; therefore, out of its total extent of 400 pages, in the edition that I have before me, only two pages are devoted to establishing quantitative relations between the elements of motion, forming the concluding chapter of Book VII. This is the passage: “After it has been shown in what way motions may be compared with one another, it remains to show those relations which may exist between them. I shall first return to certain principles which I have already indicated. Every mover (force) always moves something movable in something and in some measure. It acts upon this movable thing in something, i.e. in time; it moves it in some measure, i.e. carries it over some distance; for the mover moves continuously throughout the time of its action. Motion is always some quantity and always advances through some magnitude. Let us denote the mover by A, the moved thing by B, and by C the magnitude through which the moved thing has been advanced during the time D in which the motion took place. In the same time the power denoted by A will make half of the movable body B accomplish a motion twice as great as C; the same distance C it will make it pass in half the time D, for in this case proportionality is preserved. Thus, when the force remains the same, the movable thing, reduced by half, traverses the same path in half the time.”

From these words, twice repeated, it follows without doubt that Aristotle supposed that a constant force, acting upon a given body, causes it to move uniformly throughout the whole time of its action, and moreover that the velocity of this motion is proportional to the force, as is evident from his subsequent words: “Consequently, one may establish two other rules which are consequences of the preceding one: if the force and the movable thing remain the same, then the motion in half the time will be twice as small; but if the force is also reduced, then it will make the same movable thing accomplish half the motion in the same time. Let, for example, the force E be half the force A, and let F be a new movable thing amounting to half of B; under these assumptions the relations will remain the same, and the forces will be proportional to the movable loads—such two forces will produce equal motion in equal time.”

“However, one should not think,” Aristotle continues, “that if E, equal to half of A, can make F, equal to half of B, traverse the path C in the time D, then E can also make the doubled F traverse, in the same time, a path equal to half of C; it may turn out that a force capable of moving half of a movable body is incapable of moving the whole; thus, if A advances B by the magnitude C in the time D, then it may be that the force denoted by E will not be able to move B during the time D. This half-force may be,”

will not even be able to make B traverse any part of C, not merely such a part as constitutes the same share of C as A does of E, for it may turn out that there will be no motion at all. Thus, for example, if the full force is needed to move some load, then half of it will not be able to produce any motion in any interval of time; for otherwise one sailor would suffice to set a ship in motion, if only the force of all the shipmen could be subdivided both with respect to number and with respect to the distance which they, acting jointly, make the ship traverse.”

These words show how far the ancient world was from the conception of the “law of inertia,” first divined by Galileo and finally formulated by Newton.

But let us cite one more famous passage from Aristotle’s work On the Heavens, where he also speaks of motion. This work is devoted to an exposition of the doctrine of the structure of the world as it was then represented by other philosophers, whom Aristotle in places refutes and in places supplements in his own teaching.

At that time it was believed that everything consists of four elements—earth, water, air, and fire. Earth possesses only heaviness, fire—only lightness, while air and water possess both the one and the other. Everything heavy strives downward, everything light—upward; therefore: “heavy is everything that is capable of being borne toward the middle, light everything that is borne away from the middle or center of the world”... “The heaviest is that which is situated beneath everything that strives downward; the lightest is that which is situated above everything that goes upward.” Accordingly, motion too is considered as according to nature or contrary to nature. It is self-evident that in this doctrine there is no question of establishing any quantitative relations, and in Aristotle’s exposition everything is reduced to dialectical reasonings like those cited above.

In Chapter VI of Book I he considers the following question: can there be an infinite body, and will its weight be finite or infinite. It is clear that, in the absence of a definition of what, in such a case, should be understood by the word weight, this question is so indeterminate that the reasonings relating to it cannot possess sufficient clarity and precision. In this very reasoning, Aristotle says, among other things: “From what has been said it follows that the weight of an infinite body cannot be finite; therefore it is infinite. Consequently, if this cannot be, then an infinite body cannot exist in nature either. That weight cannot be infinite is evident from the following: if some body traverses some space in a certain time, then the greater body is in motion for a shorter time; but times are in the inverse ratio of weights, i.e., if half the weight traverses so much in a certain time, then double the weight will traverse the same amount in half the time. Moreover, every finite weight traverses a finite space in a finite time. Since all

occurs, as has been said, then it turns out that if any weight were infinite, it would at the same time both move and not move. It must move, for it is of the same kind as a finite one, but it also follows of necessity that it does not move, for in its motion the ratio of the time of traversing a given distance to the time of traversing this distance by a finite body must be less than any inverse ratio of the greater to the lesser (less than any number).” And, having shown that this leads to a contradiction, Aristotle concludes: “It is impossible that there should be an infinite weight, and likewise lightness.”

In this excerpt it is necessary to pay attention to Aristotle’s words that the times during which heavy bodies traverse one and the same space are inversely proportional to the weights; this was in his time an undisputed, generally accepted doctrine, which he uses in his speculative argument.

But from his writings it is not clear what other conclusions were drawn from this proposition and from the others cited above, and whether the motion of some load under the action of a force different from its weight was conceived exactly as one might suppose from what is said in the Mechanics, or not. Apparently not; and in general the ancients did not know a definition of the motion of a body under the action of a given force.

Aristotle’s Physics consists, as it were, of two parts, of which the first is entitled simply Physics, and the second Lectures on Physics.

In ch. XI of book IV of this latter work there is the following passage, which leads one to think that the motion of bodies was explained by the action upon them of the surrounding medium. Here is the passage: “Obviously there are two possible reasons why a given load or a given body receives a more rapid motion. This happens either because the medium through which it passes, being different according as the body moves in water, in earth, or in air, or because the moving bodies themselves are in themselves different and, other conditions being equal, possess greater heaviness or lightness. The medium through which the body passes is a cause impeding the motion of the body to the greatest degree when this medium has a motion directed oppositely, than when it is at rest. This resistance is the greater the more difficult it is for the medium to be divided, and it resists the more strongly the denser it is. Suppose, for example, that A traverses path B in one medium in time C, and in a rarer medium path D in time E. If the length B is equal to the length D, then the motion will be proportional to the resistance of the medium. Let us suppose that B is water and D, for example, air; the lighter and less corporeal the air is in comparison with water, the sooner A will traverse path D than B. It is obvious that the first speed will stand to the second in the same ratio as air to water, and if we assume that air is twice as light as water, then the body will pass through B in twice as much time as through D, and the time C will be twice the time E.”

“Consequently, the motion of a body will be the faster, the less corporeal, the less resisting, and the more easily divisible the medium through which it passes. But there is no relation for comparing a body with the void (i.e., a relation of the density of a body to the void) which would show how many times the body surpasses the void, just as there is no relation of a number to zero... Consequently there is none such for motion in the void either; and if a body traverses some distance in the lightest medium in a given time, then its speed in the void will exceed every possible ratio”...

The long excerpts cited above have been given in order to show that Aristotle’s doctrine was taken in large part from what the simplest and most immediate observation provides—for example, the sight of loads being moved by draught animals, when, owing to friction, a constant force must be applied to the body in order to maintain uniform motion. The phenomenon seemed so simple, and the explanation given appeared so much in accord with what everyone constantly saw and experienced for himself, that the essence of the matter escaped Aristotle’s critical analysis, and the mighty genius of Galileo was needed in order to reveal the delusion in what for twenty centuries had seemed to arouse no doubts whatever.

§ 5. Galileo, a most skillful observer and experimenter, an outstanding mathematician, a practical mechanic, a thinker and poet, was able, with his all-embracing genius, to renounce Aristotle’s authority. We shall return, in its proper place, to a more detailed acquaintance with his establishment of the true foundations of mechanics on the basis of precise experiments generalized into strictly mathematical reasoning.

In contradiction to Aristotle’s doctrine, he showed that a body under the action of a constant force moves uniformly accelerated, and not uniformly, as was assumed in Aristotle’s doctrine; that bodies fall from the same height in one and the same time regardless of their weight; that a body thrown obliquely to the horizon describes a parabola, and Galileo showed how to calculate its parameter for a given speed and angle of projection; he studied in detail the motion of a heavy body down an inclined plane and, having very ingeniously and precisely arranged the experiment, used it to confirm his laws of motion for falling bodies. Passing to another limiting case, i.e., when the plane is horizontal, he concluded that a body not subject to the force of gravity will move uniformly and in a straight line, thus establishing the law of inertia and the independence of the action of a force from the state of the body.

All these discoveries, fundamental for mechanics, were set forth by him in the work entitled: Discorsi e dimostrazioni mathematiche intaro a dua scienze alla mecanica e i movimenti locali, published in Leiden in 1638.

In this work, divided into six days or conversations, the exposition proceeds, as it were, by conversations among three persons: Salviati, Sagredo, and Simplicio, the first of whom sets forth his new views to the other two interlocutors, who put various questions to him. Sagredo, for the most—

part of him supports it, while Simplicio stubbornly defends the views of Aristotelian philosophy, but, overcome by the arguments of his opponents, is forced to surrender.

We have seen that, in Aristotle’s opinion, bodies fall with velocities proportional to their weight, i.e., they traverse the same path in the less time the greater the weight, so that according to this doctrine: “a hundred-pound weight falls from a height of 100 feet in the same time as a one-pound weight from a height of 1 foot,”—Salviati gives this example. In exactly the same way, according to Aristotle, when a body falls in some medium—water or air, or in general in media of different density—it moves “toward its place” with velocities inversely proportional to the density of the medium; and the natural free motion of bodies, i.e., motion without the action of any external cause, is uniform circular motion.

Galileo, in refuting these views, first of all changes the very formulation of the question: Aristotle always asks why this or that phenomenon occurs, and then gives his metaphysical explanation, taking it as established that the phenomenon occurs precisely as indicated in the question. Galileo first of all asks how the phenomenon actually occurs, and only after establishing this does he seek out the cause, resorting to exact experiments, very cleverly arranged, although with the simplest means.

Thus, concerning the fall of bodies, he says, among other things: “If there were no resistance of the air, all bodies would fall equally, i.e., with the same velocity from equal heights of fall”… “while moving in this way with uniform acceleration, so that in equal intervals of time the velocity increases by equal amounts”… “In motion in a resisting medium, as the velocity increases, the falling body encounters a constantly increasing resistance, as a result of which there occurs a constant decrease of acceleration and, finally, the resistance reaches such a magnitude that there will be no acceleration and the body will continue to move thereafter uniformly”… “When a body falls in some medium, it must be borne in mind that the body is acted upon not by its full weight, but only by the excess of this weight over the weight of the liquid or medium displaced by the body. Thus, for example, if one accepts that lead is 10,000 times heavier than air, and ebony 1,000 times, then from the magnitude of the velocity which in a vacuum would be the same (meaning at the end of equal intervals of time counted from the beginning of the fall), the air will subtract for lead one part out of 10,000, and from the velocity of ebony one out of 1,000, i.e., 10 parts out of 10,000. Therefore, if a piece of lead and a piece of ebony fall from a certain height which in a vacuum they would traverse in the same time, then in air, out of 10,000 units of the distance traversed, the lead will lose one unit, while the wood will lose 10; hence, after the stated time has elapsed, the piece of lead will outstrip the wood by 9 units”… “In a similar manner one may calculate the velocities which the same body will have when falling in various media; moreover, one must proceed not from a consideration of the different resistance exerted—”

medium, but consider the excess of a body’s weight over the weight of the medium displaced by it”... “Thus, if one takes a body which is only slightly heavier than water, for example, mountain oak, then, if a piece of it weighs 1000 drachmas, the corresponding quantity of water 950 drachmas, and of air 2 drachmas, from the full speed of 1000 units in air there remains 998, whereas in water only 50; and, consequently, this body will fall in air approximately 20 times faster than in water”.... “It must also be remembered that motion downward in water occurs only when the specific weight of the body is greater than that of water, and such bodies are several hundred times heavier than air; therefore, in order to find the speeds in water and in air, one may neglect the loss of speed in air and assume that this speed is the same as in a void; then one may say that the speeds in water and in air are related to one another as the weight of the body is to the excess of this weight over the weight of the water displaced”.... “We would see that all this agrees far better with experiments than Aristotle’s arguments.”

Galileo confirms all his assertions by experiments carried out with the simplest means, but exact thanks to their remarkably ingenious arrangement. Thus, for example, in order to prove that bodies do indeed fall with uniformly accelerated motion, he first proves mathematically that in such motion the distances traversed from the beginning of the motion are proportional to the squares of the times, and he then verifies this property by experiment.

He arranged this experiment as follows: having first shown by strict reasoning, subsequently also confirmed by experiment, that in motion along an inclined plane a body acquires, for a given height of fall, always one and the same speed, independently of the inclination of the plane, he concluded that motion along an inclined plane is also uniformly accelerated, the acceleration constituting of the full acceleration in free fall the same fraction as the height of the plane is of its length. Taking a board 18 feet long, 9 inches wide, and 3 inches thick, and making along its length a groove a little more than an inch wide, he lined it with smooth parchment. Along this groove he let a perfectly smooth, polished copper ball roll, giving the board various inclinations. To measure the time, he was no longer content, as in other cases, with counting the beats of his pulse, but took a bucket of water and inserted into its bottom a thin tube, which he opened when releasing the ball and closed with his finger when it passed the marked divisions; the water that flowed out was collected in a vessel placed underneath and weighed; the quantity of water was proportional to the corresponding intervals of time, while the distances traversed from the beginning of the motion proved proportional to their squares, i.e., twice as great a quantity of outflowing water corresponded to a fourfold distance, and this was so whatever the inclination of the plane.

To show that all bodies fall with the same acceleration if air resistance is neglected, Galileo resorted to experiments

with pendulums, suspending from thin threads of equal length balls made of various materials; all the pendulums swung in the same way, i.e. making the same number of oscillations in equal intervals of time, although the amplitude of the oscillations of the light bodies decreased more rapidly than that of the heavy ones.

Having established and verified by experiment the laws of falling bodies, i.e. of their rectilinear motion under the action of gravity, Galileo proceeds to consider the motion of bodies thrown either horizontally or obliquely to the horizon. Beginning his exposition of this doctrine, he says: “If a body meets no resistance to motion along a horizontal plane, then from what was explained in considering motion along an inclined plane it follows that this motion is uniform and on an infinite plane would continue without end; but if this plane is bounded, and the body is heavy, then, after it reaches the edge of the plane, with the further continuation of its motion there is added to its indestructible horizontal motion a motion produced by gravity, and a composite motion is formed, which I call the motion of a thrown body and which is composed of uniform horizontal motion and uniformly accelerated vertical motion.” Having expressed this proposition, Galileo examines this motion in detail, shows that the trajectory is a parabola, and studies its properties.

From these excerpts one can see that the true laws of falling bodies, i.e. of their free motion under the action of gravity, discovered and established by Galileo, are in all respects completely contrary to Aristotle’s ideas. At the same time one can see the principles by means of which the motion of a body under the action of any constant force can be determined. Motion here is assumed to be translational, and the body mentally reduced, as it were, to a single point.

§ 6. The direct continuer of Galileo’s work in the further course of the development of mechanics was Huygens, but it is necessary to mention Galileo’s contemporary, the great astronomer Kepler, who, although he himself left no works on mechanics, by his astronomical discoveries contributed to the establishment in it of the greatest of the laws of nature. Kepler, having set himself the goal of finding the true laws of planetary motion, applied to their discovery an empirical method: first, on the basis of processing observations, he established the form of the orbits independently of any assumptions; then he set up some hypothesis, sometimes a most fantastic one, and subjected it to verification by means of calculations, comparing the positions of the planet computed on the basis of the hypothesis made, whose motion was being studied, with the observed ones, drawing the latter chiefly from the many years of observations inherited by him from Tycho Brahe, the most accurate of that time. The laws of planetary motion discovered by Kepler then gave Newton, fifty years later, the possibility of establishing the law of universal gravitation.

§ 7. Huygens, a distinguished Dutchman by birth, who spent a considerable part of his life in Paris, in the diversity of his talents was not inferior to Galileo, and perhaps even surpassed him in mathematical gifts. Huygens’s principal work on mechanics is the one published in 1673: “Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae.” This work comprises five parts, in the first of which a description is given of the construction of a clock with a pendulum, set in motion by a weight or a spring. Huygens’s chief invention consisted in the device of the “escapement,” by virtue of which the motion of the clock is regulated by the oscillations of the pendulum, and at the same time, at each swing, the pendulum receives a certain impulse sustaining its motion, which otherwise would quickly die out from friction and the resistance of the air. In the second part he develops Galileo’s doctrine of the fall of bodies, of the motion of bodies along an inclined plane, generalizes the concept of acceleration and applies it to the consideration of the motion of a body thrown vertically upward, and finally examines the remarkable properties of the motion of a body along a cycloid whose plane is vertical. The third part is purely geometrical; here the doctrine of the curvature of curves, their evolutes and involutes, is established. The fourth chapter is devoted to the doctrine of the center of oscillation, in other words to the theory of the “physical” pendulum, i.e. a solid body of finite dimensions oscillating about a fixed horizontal axis, and to finding the length of such a “mathematical pendulum” whose time of swing would be equal to the time of swing of the given physical one.

The work ends with a series of theorems on “centrifugal force,” stated without proof. In this same work Huygens gives, in the form of a special theorem, the dependence between the time of one swing of a pendulum, its length, and the time of fall from a height equal to the length of the pendulum. This theorem makes it possible for him, from the time of one swing of the pendulum, to find the height from which a body falls in the first second of time—in other words, to find the “acceleration of the force of gravity.”

§ 8. In 1686 there appeared “the greatest of all, in Lagrange’s words, productions of the human mind,” namely Newton’s work: “Philosophiae naturalis principia mathematica.”

In this work Newton definitively establishes the foundations of dynamics, i.e. the doctrine of the motion of bodies under the action of any given forces whatever, and brings this doctrine to a high degree of perfection. Having discovered the world law of universal gravitation, Newton applies it to the investigation of the motions of the heavenly bodies and lays those unshakable foundations of celestial mechanics, the agreement of whose most complex conclusions with the most exact observations provides irrefutable confirmation of the principles set forth by Newton.

In the preface to this work Newton expresses his view of the significance of mechanics, its place among the natural sciences, and its fundamental task. We shall cite several excerpts from this preface.

“The ancients attached great importance to the study of mechanics in the investigation of natural phenomena... They considered mechanics in two ways: theoretical, developed by exact proofs and reasoning, and applied; to this latter belong all those manufactures and crafts which are called mechanical and from which the science itself received its name. But since craftsmen in their work are satisfied with only a small degree of precision, the opinion arose that mechanics differs from geometry precisely in that everything completely exact belongs to geometry, while the less exact is assigned to mechanics. But the errors lie not in the craft or art itself, but belong to the executor of the work—the one who works with less precision is the worse mechanic, and if anyone could perform a work with the most perfect precision, he would be the best of all mechanics.

“However, the very drawing of straight lines and circles, which serves as the foundation of geometry, in essence belongs to mechanics. Geometry does not teach how to draw these lines, but presupposes the feasibility of such constructions. It is also presupposed that one who begins the study of geometry has already learned previously to draw circles and straight lines accurately; in geometry it is shown only in what manner, by means of drawing these lines, various questions and problems are solved. The drawing itself of a straight line and of a circle also constitutes a problem, but not a geometrical one. The solution of this problem is borrowed from mechanics—geometry teaches only the use of these solutions. Geometry is therefore glorified because, having borrowed from without so few fundamental propositions, it accomplishes so much.

“Thus, geometry is founded upon mechanical practice and is nothing other than that part of general mechanics in which the art of exact measurement is set forth and proved. But since in crafts and manufactures one has for the most part to deal with motion, usually everything that concerns only magnitudes is assigned to geometry, and everything that concerns motion—to mechanics.

“In this sense, rational mechanics is the doctrine of motions produced by whatever forces there may be, and of the forces required for the production of whatever motions there may be, exactly set forth and proved.”

Newton, who is the true creator of modern theoretical or rational mechanics, gave the above most exact and most complete definition of this science.

In order to show that such a science is not an abstract speculation, Newton sets as the principal problem of mechanics: “From the consideration of the phenomena of motions taking place, to find the forces of nature, and by the forces found to prove and explain the other phenomena.”

The problem of mechanics, thus posed, makes this science the foundation of modern natural science and modern technology; it is only necessary, not-

it ceased, following Newton, to study the phenomena occurring in nature and the forces that produce them.

In such a study of phenomena one must single out what is chief and essential in them. This chief and essential element is to be generalized and clothed in an abstract, ideal form, just as this has been done in geometry with those images that constitute the subject of its study; and concerning this generalized and ideal element one must state precise propositions and derive the consequences that follow from them, and then compare them with observation and experiment. For this path of investigating nature Newton stated the following four rules of reasoning.

1) One must not admit more causes of natural phenomena than those which are true and sufficient for explaining what occurs.

“Nature is simple and does not resort to the luxury of an excessive number of causes for its phenomena,” says Newton, explaining this rule.

2) To identical phenomena one should, as far as possible, assign identical causes.

“Thus, for example, to the breathing of men and animals, to the fall of a stone in Europe and America, to the shining of the sun and of a kitchen hearth, to the reflection of light on the earth and on the planets.”

3) Such properties of bodies as can neither be intensified nor weakened, and which belong to all bodies accessible to experiment, must be regarded as the general properties of all bodies.

For example, the extension of bodies is apprehended by our senses, and we ascribe it to all bodies; that all bodies are impenetrable is likewise inferred on the basis of experience and not of speculation—everything with which we have to do is impenetrable; this property, too, is therefore regarded as general.

4) In experimental natural science one must regard propositions obtained by induction from the consideration of phenomena as entirely exact, or very close to complete exactness, regardless of the contrary hypotheses that might be devised, until such phenomena are discovered by which these propositions are further made more precise or prove to admit exceptions.

“One must proceed in this way so that assumptions do not prevail over conclusions.”

§ 9. Newton begins the exposition of his Principia with the definition of those fundamental concepts with which one has to deal. These definitions are as follows:

1) The quantity of matter (mass) is the measure of such, proportional to its density and volume.

2) The quantity of motion is the measure of such, proportional to velocity and mass.

3) The inherent force of matter is its innate capacity for resistance, by which every individual body, left to itself, maintains its state of rest or of uniform rectilinear motion.

“This force is always proportional to the mass and, if it differs from the inertia of the mass, then only in the way in which we conceive it.”

4) An impressed force is an action exerted upon a body in order to change its state of rest or of uniform rectilinear motion.

“Force manifests itself solely in action, and when the action ceases it does not remain in the body. The body thereafter continues to maintain its new state by inertia alone.”

5) Centripetal force is that by which bodies are drawn, driven, or in some way tend toward a certain point as to a center.

“Such is the force of gravity, under whose action bodies tend toward the center of the earth; the magnetic force by which iron is attracted to a magnet; and that force, whatever it may be, by which the planets are continually deflected from rectilinear motion and compelled to move along curved lines”...

6) The absolute magnitude of a centripetal force is the measure of the greater or lesser power of the source itself of its propagation from the center into the surrounding space.

7) The accelerative magnitude of a centripetal force is the measure proportional to the velocity which it produces during a given time.

8) The motive magnitude of a centripetal force is its measure, proportional to the quantity of motion which is produced by it during a given time.

Having then explained the meaning of these definitions and of the basic concepts of time and space, and having explained what is meant by relative motion, Newton states the following:

Axioms or laws of motion.

1) Every body continues to maintain its state of rest or of uniform and rectilinear motion, until and insofar as it is compelled by impressed forces to change that state.

2) The change in the quantity of motion is proportional to the impressed motive force and takes place in the direction of the straight line along which that force acts.

3) To every action there is always an equal and opposite reaction; that is, the mutual actions of two bodies upon one another are equal and directed in opposite directions.

As is evident, these laws, generalizing the propositions of Galileo and Huygens, fundamentally contradict the Aristotelian teaching that every body possesses a property of heaviness or lightness which makes it go “to its own place.” Matter possesses its basic property, “inertia,” by virtue of which every body, without the action of an external cause, either remains at rest or moves uniformly and rectilinearly; a body cannot

excite motion in itself, but it can act upon other bodies, causing them to move; moreover, these actions between bodies are always mutual and are subject to Newton’s third law.

It is from these propositions, adopted as axioms—though axioms that are speculative and that constitute a generalization of observed phenomena—that all the remaining propositions of mechanics and the properties of motion are derived by a purely mathematical route.

The derivation of the fundamental propositions of mechanics in accordance with the rules set forth above, from the consideration of the simplest and most obvious phenomena; the strictly mathematical development from these few propositions of all its further and more complex conclusions; and their complete agreement with observed phenomena—all this gives this science that degree of certainty such that every conclusion of it, if correctly and mathematically rigorously founded, is regarded as an equally immutable truth as the truths of geometry.

§ 10. Problems of mechanics require, for their solution, the use of the calculus of infinitesimals, which was likewise discovered in the 1670s by Newton and Leibniz. Although Newton possessed this calculus, in his Principia he gives only brief indications of it and for the most part uses geometrical methods, analogous to the manner employed by the ancient authors.

The first half of the eighteenth century was a period of rapid development of the newly discovered differential and integral calculi and of their applications. Naturally, mechanics, too, was not left aside. Thus, in 1736 our Academy of Sciences published, in two volumes, Euler’s Mechanics, the first complete treatise in which this subject was expounded in a purely analytical manner.

There followed the works of d’Alembert, of the same Euler on the motion of a rigid body, separate articles and works by the Bernoulli brothers, and so forth. Alongside theoretical mechanics there also developed its applications to questions of astronomy, i.e. to determining the motion of the heavenly bodies and those “perturbations” which their mutual attraction introduces into their motion, which depends chiefly on the attractive force of the sun. The theoretical study of the moon’s motion presented particular difficulties: the displacement of its perigee seemed not to agree with the law of attraction, until Clairaut found the cause of the discrepancy in the fact that, in the series obtained, it had not been permissible to discard those terms which, before detailed investigation, appeared small, but afterward proved to have substantial significance.

In the last quarter of the eighteenth century two names acquired predominant importance—Lagrange and Laplace: the first in questions of pure mathematics and in the development of general methods for solving problems of theoretical mechanics; the second in applications to questions concerning the motion of the heavenly bodies.

Finally, in 1788 Lagrange’s work Mécanique Analytique appeared, in which this science is set forth in a coherent system,

By a purely analytical path, without a single drawing, proceeding from one general principle. In our course we shall make use of many of Lagrange’s derivations, thereby becoming acquainted with the simplest and most general results and with the methods by which they are obtained.

In 1799 the first of the five volumes of Laplace’s Celestial Mechanics appeared. This work is incomparably more difficult to study than Lagrange’s work, and only a very few sections of it find a place in a general course of mechanics.

In the nineteenth century, on the one hand, the analytical techniques for solving problems of mechanics were perfected; here the foremost name is that of the famous German mathematician Jacobi, whose methods for integrating the differential equations of mechanics form the subject of special courses. On the other hand, the applications of mechanics to the study of the motion of fluids and elastic bodies were developed.

Standing quite apart is the name of the French geometer Poinsot, who succeeded in giving—first, in 1808, to statics, and then, in 1851, to the theory of the rotational motion of a rigid body—a geometrical form of extraordinary simplicity and elegance.