[the Rise of the Mechanical View

The great mystery story ... The first clew ... Vectors

...

(Albert Einstein & Leopold Infeld)

In imagination there exists the perfect mystery story. Such a story presents all the essential clews, and compels us to form our own theory of the case. If we follow the plot carefully we arrive at the complete solution for ourselves just before the author's disclosure at the end of the book. The solution itself, contrary to those of inferior mysteries, does not disappoint us; moreover, it appears at the very moment we expect it.

Can we liken the reader of such a book to the scientists, who throughout successive generations continue to seek solutions of the mysteries in the book of nature? The comparison is false and will have to be abandoned later, but it has a modicum of justification which may be extended and modified to make it more appropriate to the endeavor of science to solve the mystery of the universe.

This great mystery story is still unsolved. We cannot even be sure that it has a final solution. The reading has already given us much; it has taught us the [4] The Evolution of Physics

rudiments of the language of nature; it has enabled us to understand many of the clews, and has been a source of joy and excitement in the oftentimes painful advance of science. But we realize that in spite of all the volumes read and understood we are still far from a complete solution, if, indeed, such a thing exists at all. At every stage we try to find an explanation consistent with the clews already discovered. Tentatively accepted theories have explained many of the facts, but no general solution compatible with all known clews has yet been evolved. Very often a seemingly perfect theory has proved inadequate in the light of further reading. New facts appear, contradicting the theory or unexplained by it. The more we read, the more fully do we appreciate the perfect construction of the book, even though a complete solution seems to recede as we advance.

The scientist reading the book of nature, if we may

(Albert Einstein & Leopold Infeld) The Rise of the Mechanical View (5)

be allowed to repeat the trite phrase, must find the solution for himself, for he cannot, as impatient readers of other stories often do, turn to the end of the book. In our case the reader is also the investigator, seeking to explain, at least in part, the relation of events to their rich context. To obtain even a partial solution the scientist must collect the unordered facts available and make them coherent and understandable by creative thought.

It is our aim, in the following pages, to describe in broad outline that work of physicists which corresponds to the pure thinking of the investigator. We shall be chiefly concerned with the role of thoughts and ideas in the adventurous search for knowledge of the physical world.

A most fundamental problem, for thousands of years wholly obscured by its complications, is that of motion. All those motions we observe- in nature, that of a stone thrown into the air, a ship sailing the sea, a cart pushed along the street, are in reality very intricate.

(Albert Einstein & Leopold Infeld)

(~6~) The Evolution of Physics

To understand these phenomena it is wise to begin with the simplest possible cases, and proceed gradually to the more complicated ones. Consider a body at rest, where there is no motion at all. To change the position of such a body it is necessary to exert some influence upon it, to push it or lift it, or let other bodies, such as horses or steam engines, act upon it. Our intuitive idea is that motion is connected with the acts of pushing, lifting or pulling. Repeated experience would make us risk the further statement that we must push harder if we wish to move the body faster. It seems natural to conclude that the stronger the action exerted on a body, the greater will be its speed. A four-horse carriage goes faster than a carriage drawn by only two horses. Intuition thus tells us that speed is essentially connected with action.

It is 2 familiar fact to readers of detective fiction that a false clew muddles the story and postpones the solution. The method of reasoning dictated by intuition was wrong and led to false ideas of motion which were held for centuries. Aristotle's great authority throughout Europe was perhaps the chief reason for the long belief in this intuitive idea. We read in the Mechanics, for two thousand years attributed to him:

The moving body comes to a standstill when the force which pushes it along can no longer so act as to push it.

The discovery and use of scientific reasoning by Galileo was one of the most important achievements in the history of human thought, and marks the real beginning of physics. This discovery taught us that intuitive conclusions based on immediate observation

(Albert Einstein & Leopold Infeld)

{([The Rise of the Mechanical View 7])}

-are not always to be trusted, for they sometimes lead to the wrong clews.

But where does intuition go wrong? Can it possibly be wrong to say that a carriage drawn by four horses must travel faster than one drawn by only two?

Let us examine the fundamental facts of motion more closely, starting with simple everyday experiences familiar to mankind since the beginning of civilization and gained in the hard struggle for existence.

Suppose that someone going along a level road with

1-1

a pushcart suddenly stops pushing. The cart will go on, Mn movin' for a short distance before coming to rest. 9

We ask: how is it possible to increase this distance? There are various ways, such as oiling the wheels, and making the road very smooth. The more easily the wheels turn, and the smoother the road, the longer the cart will go on moving. And just what has been done by the oiling and smoothing? Only this: the external influences have been made smaller. The effect of what is called friction has been diminished, both in the wheels and between the wheels and the road. This is already a theoretical interpretation of the observable evidence, an interpretation which is, in fact, arbitrary. One significant step further and we shall have the right clew. Imagine a road perfectly smooth, and wheels with no friction at all. Then there would be nothing to stop the cart, so that it would run forever. This conclusion is reached only by thinking of an idealized experiment, which can never be actually performed, since it is impossible to eliminate all external influences. The idealized experiment shows the clew which really formed the foundation of the mechanics of motion.

(Albert Einstein & Leopold Infeld)

8 The Evolution of Physics

Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

We have seen that this law of inertia cannot be derived directly from experiment, but only by speculative thinking consistent with observation. The idealized experiment can never be actually performed, although it leads to a profound understanding of real experiments.

From the variety of complex motions in the world around us we choose as our first example uniform motion. This is the simplest, because there are no external forces acting. Uniform motion can, however, never be realized; a stone thrown from a tower, a cart pushed along a road can never move absolutely uniformly because we cannot eliminate the influence of external forces.

In a good mystery story the most obvious clews often lead to the wrong suspects. In our attempts to understand the laws of nature we find, similarly, that the most obvious intuitive explanation is often the wrong one.

But a further question concerning motion arises immediately. If the velocity is no indication of the external forces acting on a body, what is? The answer to this fundamental question was found by Galileo and still more concisely by Newton, and forms a further clue in our investigation.

To find the correct answer we must think a little more deeply about the cart on a perfectly smooth road. In our idealized experiment the uniformity of the motion was due to the absence of all external forces. Let us now imagine that the uniformly moving cart is given a push in the direction of the motion. What happens now? Obviously its speed is increased. Just as obviously, a push in the direction opposite to that of the motion would decrease the speed. In the first case the cart is accelerated by the push, in the second case decelerated, or slowed down. A conclusion follows at once: the action of an external force changes the velocity. Thus not the velocity itself but its change is a consequence of pushing or pulling. Such a force either increases or decreases the velocity according to whether it acts in the direction of motion or in the opposite direction. Galileo saw this clearly and wrote in his(einstein & Infeld)

(Albert Einstein & Leopold Infeld){[10] The Evolution of Physics
... any velocity once imparted to a moving body will be rigidly maintained as long as the external causes of acceleration or retardation are removed, a condition which is found only on horizontal planes; for in the case of planes which slope downwards there is already present a cause of acceleration; while on planes sloping upward there is retardation; from this it follows that motion along a horizontal plane is perpetual; for, if the velocity be uniform, it cannot be diminished or slackened, much less destroyed.
By following the right clew we achieve a deeper understanding of the problem of motion. The connection between force and the change of velocity and not, as we should think according to our intuition, the connection between force and the velocity itself is the basis of classical mechanics as formulated by Newton.
We have been making use of two concepts which play principal roles in classical mechanics: force and change of velocity. In the further development of science both of these concepts are extended and generalized. They must, therefore, be examined more closely.
What is force? Intuitively, we feel what is meant by this term. The concept arose from the effort of pushing, throwing or pulling; from the muscular sensation accompanying each of these acts. But its generalization goes far beyond these simple examples. We can think of force even without picturing a horse pulling a carriage! We speak of the force of attraction between the sun and the earth, the earth and the moon, and of those forces which cause the tides. We speak of the force by which the earth compels ourselves and all the objects about us to remain within its sphere of influence, and of the force with which the wind makes
(Albert Einstein & Leopold Infeld) The Rise of the Mechanical View [11]

waves on the sea, or moves the leaves of trees. W e and where we observe a change in velocity, an external force, in the general sense, must be held responsible. Newton wrote in his Principia:

An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.

' This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertias only. Impressed forces are of different origins; as from percussion, from pressure, from centripetal force(Albert Einstein & Leopold Infeld)|. [r ~e].

If a stone is dropped from the top of a tower its motion is by no means uniform; the velocity increases as the stone falls. We conclude: an external force is acting in the direction of the motion. Or, in other words: the earth attracts the stone. Let us take another example. What happens when a stone is thrown straight upward? The velocity decreases until the stone reaches its highest point and begins to fall. This decrease in velocity is caused by the same force as the acceleration of a falling body[^]. In one case the force acts in the direction of the motion, in the other'Case in the opposite direction. The force is the same, but it causes acceleration or deceleration according to whether the stone is dropped or thrown upward.

VECTORS

All motions we have been considering are rectilinear, that is, along a straight line. Now we must go one step further. We gain an understanding of the laws of nature by analyzing the simplest cases and by leaving out of our first attempts all intricate complications.

(Albert Einstein & Leopold Infeld)[12] The Evolution of Physics

A straight line is simpler than a curve. It is, however, impossible to be satisfied with an understanding of rectilinear motion alone. The motions of the moon, the earth and the planets, just those to which the principles of mechanics have been applied with such brilliant success, are motions along curved paths. Passing from rectilinear motion to motion along a curved path brings new difficulties. We must have the courage to overcome them if we wish to understand the principles of classical mechanics which gave us the first clews and so formed the starting point for the development of science.

Let us consider another idealized experiment, in which a perfect sphere rolls uniformly on a smooth table. We know that if the sphere is given a push, that is, if an external force is applied, the velocity will be changed. Now suppose that the direction of the blow is not, as in the example of the cart, in the line of motion, but in a quite different direction, say, perpendicular to that line. What happens to the sphere? Three stages of the motion can be distinguished: the initial motion, the action of the force, and the final motion after the force has ceased to act. According to the law of inertia the velocities before and after the action of the force are both perfectly uniform. But there is a difference between the uniform motion before and after the action of the force: the direction is changed. The initial path of the sphere and the direction of the force are perpendicular to each other. The final motion will be along neither of these two lines, but somewhere between them, nearer the direction of the force if the blow is a hard one and the initial velocity small, nearer the original line of motion if the blow

is gentle and the initial velocity great.

Our new conclusion, based on the law of inertia, is: in general the action of an external force changes not only the speed but also the direction of the motion. An understanding of this fact prepares us for the generalization introduced into physics by the concept of vectors.

We can continue to use our straightforward method of reasoning. The starting point is again Galileo's law of inertia. We are still far from exhausting the consequences of this valuable clew to the puzzle of motion.

Let us consider two spheres moving in different directions on a smooth table. So as to have a [dc-finite] picture we may assume the two directions perpendicular to each other. Since there are no external forces acting, the motions are perfectly uniform. Suppose, further, that the speeds are equal, that is, both cover the same distance in the same interval of time. But is it correct to say that the two spheres have the same velocity? The answer can be yes or no! If the speedometers of two cars both show forty miles per hour it is usual to say that they have the same speed or velocity, no matter in which direction they are traveling. But science must create its own language, its own concepts, for its own use. Scientific concepts often begin with those used in ordinary language for the affairs of everyday life, but they develop quite differently. They are transformed and lose the ambiguity associated with them in ordinary language, gaining in rigorousness so that they may be applied to scientific thought.

From the physicist's point of view it is advantageous to say that the velocities of the two spheres moving in different directions are different. Although purely a (Albert Einstein & Leopold Infeld).

[14] The Evolution of Physics

(Albert Einstein & Leopold Infeld) matter of convention, it is more convenient to say that four cars traveling away from the same traffic circle on different roads do not have the same velocity even though the speeds, as registered on the speedometers, are all forty miles per hour. This differentiation between speed and velocity illustrates how physics, starting with a concept used in everyday life, changes it in a way which proves fruitful in the further development of science.

If a length is measured, the result is expressed as a number of units. The length of a stick may be 3 ft. 7 in.; the weight of some object 2 lb. 3 oz.; a measured time interval so many minutes or seconds. In each of these~ cases the result of the measurement is expressed by a number. A number alone is, however, insufficient for describing some physical concepts. The recognition of this fact marked a distinct advance in scientific investigation. A direction as well as a number is essential for the characterization of a velocity, for example. Such a quantity, possessing both magnitude

[image]

(Albert Einstein & Leopold Infeld)The Rise Of the Mechanical View [15]

If four cars diverge with equal speed from a traffic circle, their velocities can be represented by four vectors of the same length, as seen from our last drawing. In the scale used one inch stands for 40 m.p.h. In this way any velocity may be denoted by a vector, and conversely, if the scale is known, one may ascertain the velocity from such a vector diagram.

if two cars pass each other on the highway and their speedometers both show 40 m.p.h. we characterize their velocities by two different vectors with arrows pointing in opposite directions. So also the arrows indicating "uptown " and "downtown" subway trains

must point in opposite directions. But all trains moving uptown at different stations or on different avenues with the same speed have the same velocity, which may be represented by a single vector. There is nothing about a vector to indicate which stations the train passes or on which of the many parallel tracks it is running. In other words, according to the accepted convention, all such vectors, as drawn below, may be regarded as equal; they lie along the same or parallel lines, are of equal length, and finally, have arrows

[16]

(Albert Einstein & Leopold Infeld)The Evolution Of Physics

pointing in the same direction. The next figure shows vectors all different, because they differ either in length

[image]
or direction, or both. The same four vectors may be drawn in another way, in which they all diverge from

[image]

(Albert Einstein & Leoplod Infeld)

The Rise of the Mechanical View [17]

a common point. Since the starting point does not matter, these vectors can represent the velocities of four cars moving away from the same traffic circle, or the velocities of four cars in different parts of the country traveling with the indicated speeds in the indicated directions.

This vector representation may now be used to describe the facts previously discussed concerning rectilinear motion. We talked of a cart, moving uniformly in a straight line and receiving a push in the direction of its motion which increases its velocity. Graphically this may be represented by two vectors, a shorter one denoting the velocity before the push and a longer one in the same direction denoting the velocity after the

[image]

push. The meaning of the doi:ted (einstein & Infeld) vector is clear; it represents the change in velocity for which, as we know, the push is responsible. For the case where the force is directed against the motion, where the motion is slowed down, the diagram is somewhat different.

----------------
Again the dotted vector corresponds to a change in velocity, but in this case its direction is different. It is clear~ that not only velocities themselves but also their changes are vectors. But every change in velocity is due to the action of an external force; thus the force must also be represented by a vector. In order to characterize a force it is not sufficient to state how hard we (Albert Einstein & Leopold Infeld) <18 >The Evolution of Physics

push the cart; we must also say in which direction we push. The force, like the velocity or its change, must be represented by a vector and not by a number alone. Therefore: the external force is also a vector, and must have the same direction as the change in velocity. In the last two drawings the dotted vectors show the direction of the force as truly as they indicate the change in velocity.

Here the skeptic may remark that he sees no advantage in the introduction of vectors. All that has been accomplished is the translation of previously recognized facts into an unfamiliar and complicated language. At this stage it would indeed be difficult to convince him that he is wrong. For the moment he is, in fact, right. But we shall see that just this strange language leads to an important generalization in which vectors appear to be essential.

THE (einstein & Infeld) RIDDLE OF MOTION

So long as we deal only with motion along a straight line we are far from understanding the motions observed in nature. We must consider motions along curved paths, and our next step is to determine the laws governing such motions. This is no easy task. In the caw of rectilinear motion our concepts of velocity, change of velocity, and force proved most useful. But we do not immediately see how we can apply, them to motion along a curved path. It is indeed possible to imagine that the old concepts are unsuited to the description of general motion, and that new ones must be created. Should we try to follow our old path, or seek a new one?(Albert Einstein & Leopold Infeld)

(Albert Einstein & Leopold Infeld) The generalization of a concept is a process very
often used in science. A method of generalization is
not uniquely determined, for there are usually numer-
ous ways of carrying it out. One requirement, how-
ever, must be rigorously satisfied: any generalized con-cept
must reduce to the original one when the original
conditions are fulfilled.(einstein & Infeld)


[Albert Einstein, ESSAYS IN SCIENCE, © 1934

The whole evolution of our ideas about the processes of nature, with which we have been concerned so far, might be regarded as an organic development of Newton�s ideas. But while the process of perfecting the field theory was still in full swing, the facts of heat, radiation, the spectra, radioactivity, etc., revealed a limit to the serviceableness of the whole intellectual system which today still seems to us absolutely insuperable, in spite of immense success at certain points. Many physicists maintain - and there are weighty arguments in their favor that in the face of these facts not merely the differential law but the law of causation itself hitherto the fundamental postulate of all natural science, has collapsed. Even the possibility of a spatio-temporal construction, which can be unambiguously coordinated with physical events, is denied.�]

(Albert Einstein & Leopold Infeld) We can best explain this by the example with which we are now dealing. We can try to generalize the old concepts of velocity, change of velocity and force for the case of motion along a curved path. Technically, when speaking of curves, we include straight lines. The straight line is a special and trivial example of a curve. If, therefore, velocity, change in velocity and force are introduced for motion along a curved line, then they are automatically introduced for motion along a straight line. But this result must not contradict those results previously obtained. If the curve becomes a straight line, all the generalized concepts must reduce to the familiar ones describing rectilinear motion. But this restriction is not sufficient to determine the generalization uniquely. It leaves open many possibilities. The history of science shows that the simplest generalizations sometimes prove, successful and sometimes not. We must first make a guess. In our case it is a simple matter to guess the right method of generalization. The new concepts prove very successful and help us to understand the motion of a thrown stone as well as that of the planets.

And now just what do the words velocity, change in velocity, and force mean in the general case of motion along a curved line? Let us begin with velocity. Along the curve a very small body is moving from left to

(Albert Einstein & Leopold Infeld)[20] The Evolution of Physics

right.

Such a small body is often called a particle. The dot on the curve in our drawing shows the position of the particle at some instant of time. What is the velocity corresponding to this time and position? Again Galileo's clew hints at a way of introducing the velocity. We must, once more, use our imagination and think about an idealized experiment. The particle moves along the curve, from left to right, under the influence of external forces. Imagine that at a given time and at the point indicated by the dot, all these forces suddenly cease to act. Then, the motion must, according to the law of inertia, be uniform. In practice we can, of course, never completely free a body from all external influences. We can only surmise "what would happen if . . . ?" and judge the pertinence of our guess by the conclusions which can be drawn from it and by their agreement with experiment.

The vector in the next drawing indicates the guessed direction of the uniform motion if all external forces

~

were to vanish. It is the direction of the so-called tangent. Looking at a moving particle through a microscope one sees a very small part of the curve, which

(Albert Einstein & Leopold Infeld)

The Rise of the mechanical View [21]

appears as a small segment. The tangent is its prolongation. Thus the vector drawn represents the velocity at a given instant. The velocity vector lies on the tangent. Its length represents the magnitude of the velocity, or the speed as indicated, for instance, by the speedometer of a car.

Our idealized experiment about destroying the motion in order to find the velocity vector must not be taken too seriously. It only helps us to understand what we should call the velocity vector and enables us to determine it for a. given instant at a given point.

In the next drawing, the velocity vectors for three different positions of a particle moving along a curve

[image]~

are shown. In this case not only the direction but the magnitude of the velocity, as indicated by the length of the vector, varies during the motion.

Does this new concept of velocity satisfy the requirement formulated for all generalizations? That is: does it reduce to the familiar concept if the curve becomes a straight line? Obviously it does. The tangent to a straight line is the line itself. The velocity vector lies in the line of the motion, just as in the case of the moving cart or the rolling spheres.

The next step is the introduction of the change in velocity of a particle moving along a curve. This also may be done in various ways, from which we choose the simplest and most convenient. The last drawing

[image]`~

(Albert Einstein & Leopold Infeld)[22] The Evolution of Physics

showed several velocity vectors representing the motion at various points along the path. The first two of these may be drawn again so that they have a common

[image]

starting point, as we have seen is possible with vectors. The dotted vector, we call the change in velocity. Its starting point is the end of the first vector and its end point the end of the second vector. This definition of the change in velocity may~ at first sight, seem artificial and meaningless. It becomes much clearer in the special case in which vectors (i) and (2) have the same direction. This, of course, means going over to the case of straight-line motion. If both vectors have the same initial point the dotted vector again connects their end points. The drawing is now identical with that on page17

[image]

, and the previous concept is regained as a special case of the new one. We may remark that we had to separate the two lines in our drawing since otherwise they would coincide and be indistinguishable.

We now have to take the last step in our process of generalization. It is the most important of all the guesses we have had to make so far. The connection between force and change in velocity has to be established so that we can formulate the clew which will...(Albert Einstein & Leopold Infeld)

(Albert Einstein & Leopold Infeld) The Rise of the Mechanical View [23]

enable us to understand the general problem of motion. The clew to an explanation of motion along a straight line was simple: external force is responsible for change in velocity: the force vector has the same

""direction as the change. And now what is to be regarded as the clew to curvilinear motion? Exactly the same! The only difference is that change of velocity has now a broader meaning than before. A glance at the dotted vectors of the last two drawings shows this point clearly. If the velocity is known for all points along the curve, the direction of the force at any point can be deduced at once. One must draw the velocity vectors for two instants separated by a very short time-interval and therefore corresponding to positions very near each other. The vector from the end point of the first to that of the second indicates the direction of the acting force. But it is essential that the two velocity vectors should be separated only by a (Albert Einstein & Leopold Infeld)

(Albert Einstein & Leopold Infeld)

very short" time-interval. The rigorous analysis of such words as "very near," "very short" is far from simple. Indeed it was this analysis which led Newton and Leibnitz to the discovery of differential calculus.

It is a tedious and elaborate path which leads to the generalization of Galileo's clew. We cannot show here how abundant and fruitful the consequences of this generalization have proved. Its application leads to simple and convincing explanations of many facts previously incoherent and misunderstood.

From the extremely rich variety of motions we shall take only the simplest and apply to their explanation the law just formulated.

A bullet shot from a gun, a stone thrown at an angle,(Albert Einstein & Leopold Infeld)

{[found inside text scanned in encrypted or error?-] I 11~ ~ i I I i !;;]}

(Albert Einstein & Leopold Infeld)

[24] The Evolution of Physics

a stream of water emerging from a hose, all describe familiar paths of the same type, the parabola. Imagine a speedometer attached to a stone, for example, so that its velocity vector may be drawn for any instant.

and(einstein & Infeld) directed downward. It is exactly the same as when a stone is allowed to fall from the top of a tower. The paths are quite different, as also are the velocities, but the change in velocity has the same direction, that is, toward the center of the earth.

A stone attached to the end of a string and swung around in a horizontal plane moves in a circular path.

All the vectors in the diagram representing this motion have the same length if the speed is uniform. Nevertheless, the velocity is not uniform, for the path is not a straight line. Only in uniform, rectilinear motion are there no forces involved. Here, however, there... (Albert Einstein & Leopold Infeld)

(Albert Einstein & Leopold Infeld) The Rise of the Mechanical View [25]

[IMAGE]

are, and the velocity changes not in magnitude but in direction. According to the law of motion there must be some force responsible for this change, a force in this case between the stone and the hand holding the string. A further question arises immediately: in what direction does the force act? Again a vector diagram shows the answer. The velocity vectors for two very near points are drawn, and the change of velocity

[IMAGE]

found. This last vector is seen to be directed along the string toward the center of the circle, and is always

perpendicular to the velocity vector, or tangent. In other words the hand exerts a force on the stone by means of the string.

[26] The Evolution of Physics

Very similar is the more important example of the revolution of the moon around the earth. This may be represented approximately as uniform circular motion. The force is directed toward the earth for the same reason that it was directed toward the hand in our former example. There is no string connecting the earth and the moon, but we can imagine a line between the centers of the two bodies; the force lies along this line and is directed toward the center of the earth, just as the force on a stone thrown in the air or dropped from a tower.

All that we have said concerning motion can be summed up in a single sentence. Force and change of velocity are vectors having the same direction. This is the initial clew to the problem of motion, but it certainly does not suffice for a thorough explanation of all motions observed. The transition from Aristotle's line of thought to that of Galileo formed a most important cornerstone in the foundation of science. Once this break was made, the line of further development was clear. Our interest here lies in the first stages of development, in following initial clews, in showing how new physical concepts are born in the painful struggle with old ideas. We are concerned only with pioneer work in science, which consists of finding new and unexpected paths of development; with the adventures in scientific thought which create an ever-changing [Enlargening] picture of the [Expansion of the] universe. The initial and fundamental steps are always of a revolutionary character. Scientific imagination finds old concepts too confining, and replaces them by new ones. The continued development along any line already initiated is more in the nature of evolution, until the next turning point is reached when a ?

The Rise of the Mechanical View[27]

still newer field must be conquered. In order to understand, however, what reasons and what difficulties force a change in important concepts, we must know not only the initial clews, but also the conclusions which can be drawn.

One of the most important characteristics of modem physics is that the conclusions drawn from initial clews are not only qualitative but also quantitative. Let us again consider a stone dropped from a tower. We have seen that its velocity increases as it falls, but we should like to know much more. Just how great is this change? And what is the position and the velocity of the stone at any time after it begins to fall? We wish to be able to predict events and to determine by experiment whether observation confirms these predictions and thus the initial assumptions.

To draw quantitative conclusions we must use the language of mathematics. Most of the fundamental ideas of science are essentially simple, and may, as a rule, be expressed in a language comprehensible to everyone. To follow up these ideas demands the knowledge of a highly refined technique of investigation. Mathematics as a tool of reasoning is necessary if we wish to draw conclusions which may be compared with experiment. So long as we are concerned only with fundamental physical ideas we may avoid the language of mathematics. Since in these pages we do this consistently, we must occasionally restrict ourselves to quoting, without proof, some of the results necessary for an understanding of important clews arising in the further development. The price which must be paid for abandoning the language of mathematics is a loss in precision, and the necessity of some ,

The Evolution of Physics[28]

times quoting results without showing how they were reached.

A very important example'of motion is that of the earth around the sun. It is known that the path is a closed curve, called the ellipse. The construction of a vector diagram of the change in velocity shows that the force on the earth is directed toward the sun. But ---- ---------------

sun[image]

this~ after all, is scant information. We should like to be able to predict the position of the earth and the other planets for any arbitrary instant of time, we should like to predict the date and duration of the next solar eclipse and many other astronomical events. It is possible to do these things, but not on the basis of our initial clew alone, for it is now necessary to know not only the direction of the force but also its absolute value, its magnitude. It was Newton who made the inspired guess on this point. According to his low of gravitation the force of attraction between two bodies depends in a simple way on their distance from each other. It becomes smaller when the distance increases. To be specific it becomes, 2 X 2 = 4 times smaller if the distance is doubled, 3 X 3 = 9 times smaller if the distance is made three times as great... (einstein & Infeld)

Mark 4
³¹ "It is like a mustard seed, which, when sown upon the soil, though it is smaller than all the seeds that are upon the soil,

The Rise of the Mechanical View [29]

(einstein & Infeld) Thus we see that in the case of gravitational force we have succeeded in expressing, in a simple way; the dependence of the force on the distance between the moving bodies. We proceed similarly in all other cases where forces of different kinds, for instance, electric, magnetic, and the like, are acting. We try to use a simple expression for the force. Such an expression is justified only when the conclusions drawn from it are confirmed by experiment.

But this knowledge of the gravitational force alone is not sufficient for a description of the motion of the planets. We have seen that vectors representing force and change in velocity for any short interval of time have the same direction, but we must follow Newton one step further and assume a simple relation between their lengths. Given all other conditions the same, that is, the same moving body and changes considered over

equal time intervals, then, according to Newton, the -m

change of velocity is proportional to the force.

Thus just two complementary guesses are needed for quantitative conclusions concerning the motion of the planets. One is of a general character, stating the connection between force and change in velocity. The other is special, and states the exact dependence of the particular kind of force involved on the distance between the bodies., The first is Newton's general law of motion, the second his law of gravitation. Together they determine the motion. This can be made clear by the following somewhat clumsy-sounding reasoning, Suppose that at a given time the position and velocity of a planet can be determined, and that the force is known. Then, according to style='mso-bidi-font-weight:normal'>Newton's laws we know the change in velocity during a short time interval.

(einstein & Infeld) [30] The Evolution of Physics

Knowing the initial velocity and its change, we can find the velocity and position of the planet at the end of the time interval. By a continued repetition of this process the whole path of the motion may be traced without further recourse to observational data. This is, in principle, the way mechanics predicts the course of a body in motion, but the method used here is hardly practical. In practice such a step-by-step procedure would be extremely tedious as well as inaccurate. Fortunately, it is quite unnecessary; mathematics furnishes a short cut, and makes possible precise description of the motion in much less ink than we use for a single sentence. The conclusions reached in this way can be proved or disproved by observation.

The same kind of external force is recognized in the motion of a stone falling through the air and in the revolution of the moon in its orbit, namely, that of the earth's attraction for material bodies. Newton recognized that the motions of falling stones, of the moon, and of planets are only very special manifestations of a universal gravitational force acting between any two bodies. In simple cases the motion may be described and predicted by the aid of mathematics. In remote and extremely complicated cases, involving the action of many bodies on each other, a mathematical description is not so simple, but the fundamental principles are the same.

We find the conclusions, at which we arrived by following our initial clews, realized in the motion of a thrown stone, in the motion of the moon, the earth, and the planets.

It is really our whole system of guesses which is to be either proved or disproved by experiment. No one............

(einstein & Infeld) The Rise of the Mechanical View [31]

of the assumptions can be isolated for separate testing. In the case of the planets moving around the sun it is found that the system of mechanics works splendidly. Nevertheless we can well imagine that another system, based on different assumptions, might work just as well.

Physical concepts are free creations of the human mind, and are not, however it may seem, uniquely determined by the external world. In our endeavor to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. If he is ingenious he may form some picture of a mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility or the meaning of such a comparison. But he certainly believes that, as his knowledge increases, his picture of reality will become simpler and simpler and will explain a wider and wider range of his sensuous impressions. He may also believe in the existence of the ideal limit of knowledge and that it is approached by the human mind. He may call this ideal limit the objective truth.

ONE CLEW REMAINS

When first studying mechanics one has the impression that everything in this branch of science is simple, fundamental and settled for all time. One would hardly suspect the existence of an important clew which no

....(Albert Einstein & Leopold Infeld)[32] The Evolution of Physics

one noticed for three hundred years. The neglected clew is connected with one of the fundamental concepts of mechanics, that of mass.

Again we return to the simple idealized experiment of the cart on a perfectly smooth road. If the cart is initially at rest and then given a push, it afterwards moves uniformly with a certain velocity. Suppose that the action of the force can be repeated as many times as desired, the mechanism of pushing acting in the same way and exerting the same force on the same cart. However many times the experiment is repeated the final velocity is always the same. But what happens if the experiment is changed, if previously the cart was empty and now it is loaded? The loaded cart will have a smaller final velocity than the empty one. The conclusion is: if the same force acts on two different bodies, both initially at rest, the resulting velocities will not be the same. We say that the velocity depends on the mass of the body, being smaller if the mass is greater.

We know, therefore, at least in theory, how to dc-termine the mass of a body or, more exactly, how many times greater one mass is than another. We have identical forces acting on two resting masses; finding that the velocity of the first mass is three times greater than that of the second. We conclude that the first mass is three times smaller than the second. This is certainly not a very practical way of determining the ratio of two masses. We can, nevertheless, well imagine having done it in this, or in some similar way, based upon the application of the law of inertia.

How do we really determine mass in practice? Not, of course, in the way just described. Everyone knows....(einstein & Infeld)