 CHAPTER 11 THE CALCULOUS OF PROBABILITIES No doubt the reader will be astonished to find reflections on the calculus of probabilities in such a volume as this. What is that calculus to do with physical science? The questions I shall raise, without, however, giving them a solution, are naturally raised by the philosopher who is examining the problems of physics. So far is this the case that in the two preceding chapters I have several times used the words probability and chance. Good facts, as I said above, can only be probable. However solidly founded a prediction may appear to be, we are never absolutely certain that experiment will not prove it false, but the probability is often so great that practically it may be accepted. And little farther on I added, to see what a part the belief in simplicity plays in our generalizations, we have verified a simple law in a large number of particular cases, and we refuse to admit that this so often repeated coincidence is a mere effect of chance. Thus in a multitude of circumstances the physicist is often in the same position as the gambler who reckons up his chances. Every time that he reasons by induction he more or less consciously requires the calculus of probabilities, and that is why I am obliged to open this chapter parenthetically and to interrupt our discussion of method in the physical sciences in order to examine a little closer what this calculus is worth and what dependence we may place upon it. The very name of the calculus of probabilities is a paradox. Probability as opposed to certainty is what one does not know, and how can we calculate the unknown? Yet many eminent scientists have devoted themselves to this calculus, and it cannot be denied that science has drawn therefrom no small advantage. How can we explain this apparent contradiction? Has probability been defined? Can it even be defined? And if it cannot, how can we venture to reason upon it? The definition, it will be said, is very simple. The probability of an event is the ratio of the number of cases favorable to the event to the total number of possible cases. A simple example will show how incomplete this definition is. I throw two dice. What is the probability that one of the two, at least, turns up a six? Each can turn up in six different ways. The number of possible cases is six times six equals thirty-six. The number of favorable cases is eleven. The probability is eleven thirty-sixths. That is the correct solution. But why cannot we just as well proceed as follows? The points which turn up on the two dice forms six times seven divided by two equal twenty-one different combinations. Among these combinations six are favorable. The probability is six twenty-firsts. Now why is the first method of calculating the number of possible cases more legitimate than the second? In any case, it is not the definition that tells us. We are therefore bound to complete the definition by saying, to the total number of possible cases provided the cases are equally probable. So we are compelled to define the probable by the probable. How can we know the two possible cases are equally probable? Will it be by a convention? If we insert at the beginning of every problem an explicit convention, well and good. We then have nothing to do but to apply the rules of arithmetic and algebra, and we complete our calculation, when our result cannot be called in question. But if we wish to make the slightest application of this result, we must prove that our convention is legitimate, and even then shall find ourselves in the presence of the very difficulty we thought we had avoided. It may be said that common sense is enough to show us the convention that should be adopted, alas! Mr. Bertrand has amused himself by discussing the following simple problem. What is the probability that a chord of a circle may be greater than the side of the inscribed equilateral triangle? The illustrious geometer successively adopted two conventions, which seemed to be equally imperative in the eyes of common sense, and with one convention he finds one half and with the other one third. The conclusion which seems to follow from this is that the calculus of probabilities is a useless science, that the obscure instinct which we call common sense, and to which we appeal for the legitimization of our conventions must be distrusted. But to this conclusion we can no longer subscribe. We cannot do without that obscure instinct. Without it science would be impossible, and without it we could neither discover nor apply a law. Have we any right, for instance, to enunciate Newton's law? No doubt numerous observations are in agreement with it, but is not that a simple fact of chance? And how do we know besides that this law which has been true for so many generations will not be untrue in the next? To this objection the only answer you can give is, it is very improbable. But grant the law. By means of it I can calculate the position of Jupiter in a year from now. Yet have I any right to say this? Who can tell if a gigantic mass of enormous velocity is not going to pass near the solar system and produce unforeseen perturbations? Here again the only answer is, it is very improbable. From this point of view all the sciences would only be unconscious applications of the calculus of probabilities. And if this calculus be condemned, then the whole of the sciences must also be condemned. I shall not dwell at length on scientific problems in which the intervention of the calculus of probabilities is more evident. In the forefront of these is the problem of interpolation, in which, knowing a certain number of values of a function, we try to discover the intermediary values. I may also mention the celebrated theory of errors of observation, to which I shall return later. The kinetic theory of gases, a well-known hypothesis wherein each gaseous molecule is supposed to describe an extremely complicated path, but in which, through the effect of great numbers, the mean phenomena which are all even observed obey the simple laws of Marriott and Guy Lussac. All these theories are based upon the laws of great numbers, and the calculus of probabilities would evidently involve them in its ruin. It is true that they have only a particular interest, and that, save as far as interpolation is concerned, they are sacrifices to which we might readily be resigned. But I have said above, it would not be these partial sacrifices that would be in question. It would be the legitimacy of the whole of science that would be challenged. I quite see that it might be said, we do not know, and yet we must act. As for action, we have not time to devote ourselves to an inquiry that will suffice to dispel our ignorance. Besides, such an inquiry would demand unlimited time. We must therefore make up our minds without knowing. This must be often done whatever may happen, and we must follow the rules although we may have but little confidence in them. What I know is, not that such a thing is true, but that the best course for me is to act as if it were true. The calculus of probabilities, and therefore science itself, would be no longer of any practical value. Unfortunately, the difficulty does not thus disappear. A gambler wants to try a coup, and he asks my advice. If I give it him, I use the calculus of probabilities, but I shall not guarantee success. That is what I shall call subjective probability. In this case we might be content with the explanation of which I have just given a sketch. But assume that an observer is present at the play, that he knows of the coup, and that play goes on for a long time, and that he makes a summary of his notes. He will find that events have taken place in conformity with the laws of the calculus of probabilities. That is what I shall call objective probability, and it is this phenomenon which has to be explained. There are numerous insurance societies which apply the rules of the calculus of probabilities, and they distribute to their shareholders dividends, the objective reality of which cannot be contested. In order to explain them we must do more than invoke our ignorance and the necessity of action. Thus absolute skepticism is not admissible. We may distrust, but we cannot condemn on block. Action is necessary. 1. CLASSIFICATION OF THE PROBLEMS OF PROBABILITY In order to classify the problems which are presented to us with reference to probabilities, we must look at them from different points of view, and first of all from that of generality. I set above that probability is the ratio of the number of favorable to the number of possible cases. What for want of a better term I call generality will increase with the number of possible cases. This number may be finite as, for instance, if we take a throw of the dice in which the number of possible cases is thirty-six, that is the first degree of generality. But if we ask, for instance, what is the probability that a point within a circle is within the inscribed square, there are as many possible cases as there are points in the circle, that is to say, an infinite number. This is the second degree of generality. Generality can be pushed farther still. We may ask the probability that a function will satisfy a given condition. There are then as many possible cases as one can imagine different functions. This is the third degree of generality, which we reach, for instance, when we try to find the most probable law after a finite number of observations. Yet we may place ourselves at a quite different point of view. If we were not ignorant there would be no probability. There could only be certainty. But our ignorance cannot be absolute, for then there would be no longer any probability at all. Thus the problems of probability may be classed according to the greater or lesser depth of this ignorance. In mathematics we may set ourselves problems in probability. What is the probability that the fifth decimal of a logarithm taken at random from a table is a nine? There is no hesitation in answering that this probability is one-tenth. Here we possess all the data of the problem. We can calculate our logarithm without having recourse to the table, but we need not give ourselves the trouble. This is the first degree of ignorance. In the physical sciences our ignorance is already greater. The state of a system at a given moment depends on two things, its initial state and the law according to which that state varies. If we know both this law and this initial state we have a simple mathematical problem to solve and we fall back upon our first degree of ignorance. And it often happens that we know the law and do not know the initial state. It may be asked, for instance, what is the present distribution of the minor planets. We know that from all time they have obeyed the laws of Kepler, but we do not know what was their initial distribution. In the kinetic theory of gases we assume that the gaseous molecules follow rectilinear paths and obey the laws of impact and elastic bodies. Yet as we know nothing of their initial velocities we know nothing of their present velocities. The calculation of probabilities alone enables us to predict the mean phenomena which will result from a combination of these velocities. This is the second degree of ignorance. Finally it is possible that not only the initial conditions but the laws themselves are unknown. We then reach the third degree of ignorance and in general we can no longer affirm anything at all as to the probability of a phenomenon. It often happens that instead of trying to discover an event by means of a more or less imperfect knowledge of the law, the events may be known and we want to find the law, or that instead of deducing effects from causes we wish to deduce the causes from the effects. Now these problems are classed as probability of causes and are the most interesting of all from their scientific applications. I play at a carte with a gentleman whom I know to be perfectly honest. What is the chance that he turns up the king? It is one-eighth. This is a problem of the probability of effects. I play with a gentleman whom I do not know. He has dealt ten times and he has turned the king up six times. What is the chance that he is a sharper? This is a problem in the probability of causes. It may be said that it is the essential problem of the experimental method. I have observed n values of x and the corresponding values of y. I have found that the ratio of the latter to the former is practically constant. There is the event. What is the cause? Is it probable that there is a general law according to which y would be proportional to x and that small divergences are due to errors of observation? This is the type of question that we are ever asking and which we are unconsciously solved whenever we aren't engaged in scientific work. I am now going to pass and review these different categories of problems by discussing in succession what I have called subjective and objective probability. 2. Probability in Mathematics The impossibility of squaring the circle was shown in 1885, but before that date all geometers considered this impossibility, as so probable that the Académie des Sciences rejected without examination, the alas, two numerous memoirs on this subject that a few unhappy madmen sent in every year. Was the academy wrong? Evidently not, and it knew perfectly well that by acting in this manner it did not run the least risk of stifling a discovery of moment. The academy could not have proved that it was right, but it knew quite well that its instinct did not deceive it. If you had asked the academicians they would have answered. We have compared the probability that an unknown scientist should have found out what has been vainly sought for so long, with the probability that there is one madman the more on the earth, and the latter has appeared to us the greater. These are very good reasons, but there is nothing mathematical about them. They are purely psychological. If you had pressed them further they would have added. Why do you expect a particular value of a transcendental function to be an algebraical number? If R be the root of an algebraical equation, why do you expect this root to be a period of the function sine 2x? And why is it not the same with the other roots of the same equation? To sum up, they would have invoked the principle of sufficient reason in its vaguest form. Yet what information could they draw from it? At most a rule of conduct for the employment of their time, which could be more usefully spent at their ordinary work than in reading a lucubration that expired in them a legitimate distrust. But what I called above objective probability has nothing in common with this first problem. It is otherwise with a second. Let us consider the first ten thousand logarithms that we find in a table. Among these ten thousand logarithms I take one at random. What is the probability that its third decimal is an even number? You will say without any hesitation that the probability is one-half, and in fact if you pick out in a table the third decimals in these ten thousand numbers you will find nearly as many even digits as odd. Or if you prefer it, let us write ten thousand numbers corresponding to our ten thousand logarithms writing down for each of these numbers plus one if the third decimal of the corresponding logarithm is even and minus one if odd. And then let us take the mean of these ten thousand numbers. I do not hesitate to say that the mean of these ten thousand units is probably zero, and if I were to calculate it practically I would verify that it is extremely small. But this verification is needless. I might have rigorously proved that this mean is smaller than .003. To prove this result I should have had to make a rather long calculation for which there is no room here, and for which I may refer the reader to an article that I published in the Revue Generale des Sions, April 15th, 1899. The only point to which I wish to draw attention is the following. In this calculation I had occasion to rest my case on only two facts, namely that the first and second derivatives of the logarithm remain in the interval considered between certain limits. Hence our first conclusion is that the property is not only true of the logarithm, but of any continuous function whatever, since the derivatives of every continuous function are limited. If I was certain beforehand of the result it is because I have often observed analogous facts for other continuous functions, and next it is because I went through in my mind in a more or less unconscious and imperfect manner the reasoning which led me to the preceding inequalities, just as a skilled calculator, before finishing his multiplication, takes into account what it ought to come to approximately. And besides, since what I call my intuition was only an incomplete summary of a piece of true reasoning, it is clear that observation has confirmed my predictions, and that the objective and subjective probabilities are in agreement. As a third example I shall choose the following. The number u is taken at random, and n is given very large integer. What is the mean value of sine and u? This problem has no meaning by itself. To give it one a convention is required, namely we agree that the probability for the number u to lie between a and a plus dA is function of a dA, that it is therefore proportional to the infinitely small interval dA, and is equal to this multiplied by a function, function of a, only depending on a. As for this function I choose it arbitrarily, but I must assume it to be continuous. The value of sine and u, remaining the same when u increases by 2p, I may without loss of generality assume that u lies between 0 and 2, and I shall thus be led to suppose that a is a periodic function whose period is 2, and I shall thus be led to suppose that a is a periodic function whose period is 2p. The mean value that we seek is readily expressed by a simple integral, and it is easy to show that this integral is smaller than 2m sub k divided by n to the kth power, m sub k being the maximum value of the kth derivative of u. We see then that if the kth derivative is finite, our mean value will tend toward zero when it increases indefinitely, and that more rapidly than 1 divided by n to the k minus 1 power. The mean value of sine and u, when n is very large, is therefore zero. To define this value I required a convention, but the result remains the same whatever that convention may be. I have imposed upon myself but slight restrictions when I assumed that the function a is continuous and periodic, and these hypotheses are so natural that we may ask ourselves how they can be escaped. Examination of the three preceding examples, so different in all respects, has already given us a glimpse on the one hand of the role of what philosophers call the principle of sufficient reason, and on the other hand of the importance of the fact that certain properties are common to all continuous functions. The study of probability in the physical sciences will lead us to the same result. 3. Probabilities in the physical sciences We now come to the problems which are connected with what I have called the second degree of ignorance, namely those in which we know the law but do not know the initial state of the system. I could multiply examples, but I shall take only one. What is the probable present distribution of the minor planets on the zodiac? We know they obey the laws of Kepler. We may even, without changing the nature of the problem, suppose that their orbits are circular and situated in the same plane, a plane which we are given. On the other hand, we know absolutely nothing about their initial distribution. However, we do not hesitate to affirm that this distribution is now nearly uniform. Why? Let B be the longitude of a minor planet in the initial epoch, that is to say the epoch zero. Let A be its mean motion. Its longitude at the present time, i.e. at the time t, will be A t plus B. To say that the present distribution is uniform is to say that the mean value of the signs and cosines of multiples of A t plus B is zero. Why do we assert this? Let us represent our minor planet by a point in a plane, namely the point whose coordinates are A and B. All these representative points will be contained in a certain region of the plane, but as they are very numerous this region will appear dotted with points. We know nothing else about the distribution of the points. Now what do we do when we apply the calculus of probabilities to such a question as this? What is the probability that one or more representative points may be found in a certain portion of the plane? In our ignorance we are compelled to make an arbitrary hypothesis. To explain the nature of this hypothesis I may be allowed to use, instead of a mathematical formula, a crude but concrete image. Let us suppose that over the surface of our plane has been spread imaginary matter, the density of which is variable, but varies continuously. We shall then agree to say that the probable number of representative points to be found on a certain portion of the plane is proportional to the quantity of this imaginary matter which is found there. If there are then two regions of the plane of the same extent, the probabilities that a representative point of one of our minor planets is in one or other of these regions will be as the mean densities of the imaginary matter in one or other of the regions. Here then are two distributions, one real, in which the representative points are very numerous, very close together, but discrete, like the molecules of matter in the atomic hypothesis. The other, remote from reality, in which our representative points are replaced by imaginary continuous matter. We know that the latter cannot be real, but we are forced to adopt it through our ignorance. If again we had some idea of the real distribution of the representative points, we could arrange it so that in a region of some extent the density of this imaginary continuous matter may be nearly proportional to the number of representative points, or, if it is preferred, to the number of atoms which are contained in that region. Even that is impossible, and our ignorance is so great that we are forced to choose arbitrarily the function which defines the density of our imaginary matter. We shall be compelled to adopt a hypothesis from which we can hardly get away. We shall suppose that this function is continuous. That is sufficient, as we shall see, to enable us to reach our conclusion. What is at the instant t, the probable distribution of the minor planets, or rather, what is the mean value of the sine of the longitude at the moment t, i.e., of sine at plus b? We made at the outset an arbitrary convention, but if we adopt it, this probable value is entirely defined. Let us decompose the plane into elements of surface. Consider the value of sine of at plus b at the center of each of these elements. Multiply this value by the surface of the element and by the corresponding density of the imaginary matter. Let us then take the sum of all the elements of the plane. This sum, by definition, will be the probable mean value we seek, which will thus be expressed by a double integral. It may be thought at first that this mean value depends on the choice of the function, phi, which defines the density of the imaginary matter, and as this function, phi, is arbitrary, we can, according to the arbitrary choice which we make, obtain a certain mean value. But this is not the case. A simple calculation shows us that our double integral decreases very rapidly as f increases. Thus I cannot tell what hypothesis to make as to the probability of this or that initial distribution, but when once the hypothesis is made, the result will be the same, and this gets me out of my difficulty. Whatever the function may be, the mean value tends toward zero as t increases, and as the minor planets have certainly accomplished a very large number of revolutions, I may assert that this mean value is very small. I may give to any value I choose with one restriction. This function must be continuous, and in fact, from the point of view of subjective probability, the choice of a discontinuous function would have been unreasonable. What reason could I have, for instance, for supposing that the initial longitude might be exactly zero degrees, but that it could not lie between zero degrees and one degree? The difficulty reappears if we look at it from the point of view of objective probability. If we pass from our imaginary distribution in which the supposititious matter was assumed to be continuous, to the real distribution in which our representative points are formed as discrete atoms, the mean value of sine of At plus B will be represented quite simply by 1 divided by n times sine of At plus B, it being the number of minor planets. Instead of a double integral referring to a continuous function, we shall have a sum of discrete terms. However, no one will seriously doubt that this mean value is practically very small. Our representative points being very close together, our discrete sum will in general differ very little from an integral. An integral is the limit towards which a sum of terms tends when the number of these terms is indefinitely increased. If the terms are very numerous, the sum will differ very little from its limit. That is to say, from the integral, and what I said of the latter will still be true of the sum itself. But there are exceptions. If, for example, for all the minor planets, B equals slash 2 minus At, the longitude of all the planets at the time t would be slash 2, and the mean value in question would be evidently unity. For this to be the case of the time zero, the minor planets must have all been lying on a kind of spiral of peculiar form, with its spires very close together. All will admit that such an initial distribution is extremely improbable, and even if it were realized, the distribution would not be uniform at the present time. For example, on the first January 1900, but it would become so a few years later. Why then do we think this initial distribution improbable? This must be explained for if we are wrong in rejecting as improbable this absurd hypothesis, our inquiry breaks down, and we can no longer affirm anything on the subject of the probability of this or that present distribution. Once more we shall invoke the principle of a sufficient reason to which we must always recur. We might admit that at the beginning the planets were distributed almost in a straight line. We might admit that they were irregularly distributed, but it seems to us that there is no sufficient reason for the unknown cause that gave them birth to have acted along a curve so regular and yet so complicated, which would appear to have been expressly chosen so that the distribution at the present day would not be uniform. The questions raised by games of chance, such as roulette, are fundamentally quite analogous to those we have just treated. For example, a wheel is divided into thirty-seven equal compartments, alternately red and black. A ball is spun round the wheel, and after having moved round a number of times it stops in front of one of these subdivisions. The probability that the division is red is obviously one-half. The needle describes an angle, including several complete revolutions. I do not know what is the probability that the ball is spun with such a force that this angle should lie between zero and plus d, but I can make a convention. I can suppose that this probability is parenthesis d. As for the function parenthesis, I can choose it in an entirely arbitrary manner. I have nothing to guide me in my choice, but I am naturally induced to suppose the function to be continuous. Let a length measured on the circumference of the circle of radius unity of each red and black compartment. We have to calculate the integral of parenthesis d, extending it on the one hand to all the red, and on the other hand to all the black compartments, and to compare the results. Consider an interval two, comprising two consecutive red and black compartments. Let capital M and M be the maximum and minimum values of the function parenthesis in this interval. The integral extended to the red compartments will be smaller than capital M. Extended to the black it will be greater than M. The difference will therefore be smaller than capital M minus M. But if the function is supposed to be continuous, and if on the other hand the interval is very small, with respect to the total angle described by the needle, the difference capital M minus M will be very small. The difference of the two integrals will be therefore very small, and the probability will be very nearly one-half. We see that without knowing anything of the function we must act as if the probability were one-half. And on the other hand it explains why, from the objective point of view, if I watch a certain number of coins observation will give me almost as many black coups as red. All the players know this objective law, but it leads them into a remarkable error, which has often been exposed, but into which they are always falling. When the red is one, for example, six times running, they bet on black, thinking that they are playing an absolutely safe game, because they say it is a very rare thing for the red to win seven times running. In reality their probability of winning is still one-half. Observation shows, it is true, that the series of seven consecutive reds is very rare, but series of six reds followed by a black are also very rare. They have noticed the rarity of the series of seven reds, if they have not remarked the rarity of six reds and a black it is only because such series strike the attention less. 5. The Probability of Causes We now come to the problems of the probability of causes the most important from the point of view of scientific applications. Two stars, for instance, are very close together on the celestial sphere. Is this apparent contiguity a mere effect of chance? Are these stars, although almost on the same visual ray, situated at very different distances from the earth, and therefore very far indeed from one another? Or does the apparent correspond to a real contiguity? This is a problem on the probability of causes. First of all, I recall that at the outset of all problems of probability of effects that have occupied our attention up to now, we have had to use a convention which was more or less justified, and if in most cases the result was to a certain extent independent of this convention, it was only the condition of certain hypotheses which enabled us, a priori, to reject discontinuous functions. For example, it was only the condition of certain hypotheses which enabled us, a priori, to reject discontinuous functions, for example, or certain absurd conventions. We shall again find something analogous to this when we deal with the probability of causes. An effect may be produced by the cause A, or by the cause B. The effect has just been observed. We asked the probability that it is due to the cause A. This is an a posteriori probability of cause, but I could not calculate it if a convention more or less justified did not tell me in advance what is the a priori probability for the cause A to come into play. I mean the probability of this event to someone who had not observed the effect. To make my meaning clearer, I go back to the game of Eccarte mentioned before. My adversary deals for the first time and turns up a king. What is the probability that he is a sharper? The formulae, ordinarily taught, gives eight nights, a result which is obviously rather surprising. If we look at it closer, we see that the conclusion is arrived at, as if, before sitting down at the table, I had considered that there was one chance and two that my adversary was not honest. An absurd hypothesis, because in that case I should certainly not have played with him, and this explains the absurdity of the conclusion. The function on the a priori probability was unjustified, and that is why the conclusion of the a posteriori probability led me into an inadmissible result. The importance of this preliminary convention is obvious. I shall even add that if none were made, the problem of the a posteriori probability would have no meaning. It must be always made either explicitly or tacitly. Let us pass on to an example of a more scientific character. I require to determine an experimental law. This law, when discovered, can be represented by a curve. I make a certain number of isolated observations, each of which may be represented by a point. When I have obtained these different points, I draw a curve between them as carefully as possible, giving my curve a regular form, avoiding sharp angles, accentuated inflections, and any sudden variation of the radius of curvature. This curve will represent to me the probable law, and not only will it give me the values of the functions intermediary to those which have been observed, but it also gives me the observed values more accurately than direct observation does. That is why I make the curve pass near the points and not through the points themselves. Here then is a problem in the probability of causes. The effects are the measurements I have recorded. They depend on the combination of two causes, the true law of the phenomenon, and errors of observation. Knowing the effects, we have to find the probability that the phenomenon shall obey this law or that, and that the observations have been accompanied by this or that error. The most probable law, therefore, corresponds to the curve we have traced, and the most probable error is represented by the distance of the corresponding point from that curve. But the problem has no meaning if before the observations I had an a priori idea of the probability of this law or that, or of the chances of error to which I am exposed. If my instruments are good, and I knew whether this is so or not before beginning the observations, I shall not draw the curve far from the points which represent the rough measurements. If they are inferior, I may draw it a little farther from the points so that I may get a less sinuous curve, much will be sacrificed to regularity. Why then do I draw a curve without sinuosities? Because I consider a priori law represented by a continuous function, or function the derivatives of which to a high order are small, as more probable than a law not satisfying those conditions. But for this conviction the problem would have no meaning. Interpolation would be impossible. No law could be deduced from a finite number of observations. Science would cease to exist. Fifty years ago physicists considered, other things being equal, a simple law as more probable than a complicated law. This principle was even invoked in favor of Marriott's law as opposed to that of Regnault. But this belief is now repudiated, and yet how many times are we compelled to act as though we still held it? However that may be. What remains of this tendency is the belief in continuity, and as we have just seen, if the belief in continuity were to disappear, experimental science would become impossible. 6. THE THEORY OF ERRORS We are thus brought to consider the theory of errors which is directly connected with the problem of the probability of causes. Here again we find effects. To wit, a certain number of irreconcilable observations, and we try to find the causes which are, on the one hand, the true value of the quantity to be measured, and, on the other, the error made in each isolated observation. We must calculate the probable a posteriori value of each error, and therefore the probable value of the quantity to be measured. But, as I have just explained, we cannot undertake this calculation unless we admit a priori, i.e., before any observations are made, that there is a law of the probability of errors. Is there a law of errors? The law to which all calculators are sent is Gauss's law, that is represented by a certain transcendental curve known as the bell. But it is first of all necessary to recall the classic distinction between systematic and accidental errors. If the meter with which we measure a length is too long, the number we get will be too small, and it will be no use to measure several times, that is a systematic error. If we measure with an accurate meter, we may make a mistake and find the length sometimes too large and sometimes too small, and when we take the mean of a large number of measurements, the error will tend to grow small. These are accidental errors. It is clear that systematic errors do not satisfy Gauss's law, but do accidental errors satisfy it? Numerous proofs have been attempted, almost all of them crude paralogisms. But starting from the following hypotheses, we may prove Gauss's law. The error is the result of a very large number of partial and independent errors. Each partial error is very small and obeys any law of probability whatever, provided the probability of a positive error is the same as that of an equal negative error. It is clear that these conditions will be often but not always fulfilled, and we may reserve the name of accidental for errors which satisfy them. We see that the method of least squares is not legitimate in every case. In general, physicists are more distressful of it than astronomers. This is no doubt because the latter, apart from the systematic errors to which they and the physicists are subject alike, have to contend with an extremely important source of error which is entirely accidental. I mean atmospheric undulations. So it is very curious to hear a discussion between a physicist and an astronomer about a method of observation. The physicist persuaded that one good measurement is worth more than many bad ones is preeminently concerned with the elimination by means of every precaution of the final systematic errors. The astronomer retorts, but you can only observe a small number of stars and accidental errors will not disappear. What conclusion must we draw? Must we continue to use the method of least squares? We must distinguish. We have eliminated all the systematic errors of which we have any suspicion. We are quite certain that there are others still, but we cannot detect them. And yet we must make up our minds and adopt a definitive value which will be regarded as the probable value, and for that purpose it is clear that the best thing we can do is to apply Gauss's law. We have only applied a practical rule referring to subjective probability, and there is no more to be said. Yet we want to go farther and say that not only the probable value is so much, but that the probable error in the result is so much. This is absolutely invalid. It would be true only if we were sure that all the systematic errors were eliminated, and of that we know absolutely nothing. We have two series of observations. By applying the law of least squares we find that the probable error in the first series is twice as small as in the second. The second series may, however, be more accurate than the first, because the first is perhaps affected by a large systematic error. All that we can say is that the first series is probably better than the second, because its accidental error is smaller, and that we have no reason for affirming that the systematic error is greater for one of the series than for the other, our ignorance on this point being absolute. 7 Conclusions In the preceding lines I have set several problems and have given no solution. I do not regret this, for perhaps they will invite the reader to reflect on these delicate questions. However that may be, there are certain points which seem to be well established. To undertake the calculation of any probability, and even for that calculation to have any meaning at all, we must admit, as a point of departure, an hypothesis or convention which is always something arbitrary about it. In the choice of this convention we can be guided only by the principle of sufficient reason. Unfortunately, this principle is very vague and very elastic, and in the cursory examination we have just made we have seen it assume different forms. The form under which we meet it most often is the belief in continuity, a belief which it will be difficult to justify by apodietic reasoning, but without which all science would be impossible. Finally, the problems to which the calculus of probabilities may be applied with profit are those in which the result is independent of the hypothesis made at the outset, provided only that this hypothesis satisfies the condition of continuity. CHAPTER XII. Fresnel's theory. The best example that can be chosen is the theory of light and its relations to the theory of electricity. It is owing to Fresnel that the science of optics is more advanced than any other branch of physics. The theory called the theory of undulations forms a complete whore which is satisfying to the mind, but we must not ask from it what it cannot give us. The object of mathematical theories is not to reveal to us the real nature of things that would be an unreasonable claim. Their only object is to coordinate the physical laws with which physical experiments make us acquainted. The annunciation of which, without the aid of mathematics, we should be unable to effect. Whether the ether exists or not matters little. Let us leave that to the metaphysicians. What is essential for us is that everything happens as if it existed, and that this hypothesis is found to be suitable for the explanation of phenomena. After all, have we any other reason for believing in the existence of material objects? That, too, is only a convenient hypothesis. Only it will never cease to be so, while some day no doubt the ether will be thrown aside as useless. But at the present moment the laws of optics and the equations which translate them into the language of analysis hold good, at least as a first approximation. It will therefore be always useful to study a theory which brings these equations into connection. The undulatory theory is based on a molecular hypothesis. This is an advantage to those who think they can discover the cause under the law. But others find in it a reason for distrust. And this distrust seems to me as unfounded as the illusions of the former. These hypotheses play but a secondary role. They may be sacrificed, and the sole reason why this is not generally done is that it would involve a certain loss of lucidity in the explanation. In fact if we look at it a little closer we shall see that we borrow from molecular hypotheses but two things, the principle of the conservation of energy and the linear form of the equations which is the general law of small movements as of all small variations. This explains why most of the conclusions of Fresnel remain unchanged when we adopt the electromagnetic theory of light. Maxwell's theory. We all know that it was Maxwell who connected by a slender tie two branches of physics, optics and electricity, until then unsuspected of having anything in common. Thus blended in a larger aggregate in a higher harmony Fresnel's theory of optics did not perish. Parts of it are yet alive, and their mutual relations are still the same. Only the language which we use to express them has changed. And on the other hand Maxwell has revealed to us other relations hitherto unsuspected between the different branches of optics and the domain of electricity. The first time a French reader opens Maxwell's book his admiration is tempered with a feeling of uneasiness and often of distrust. It is only after prolonged study and at the cost of much effort that this feeling disappears. Some minds of high calibre never lose this feeling. Why is it so difficult for the ideas of this English scientist to be acclimatized among us? No doubt the education received by most enlightened Frenchmen predisposes them to appreciate precision and logic more than any other qualities. In this respect the old theories of mathematical physics gave us complete satisfaction. All our masters from Laplace to Katci proceeded along the same lines. Starting with clearly enunciated hypotheses they deduced from them all their consequences with mathematical rigor and then compared them with experiment. It seemed to be their aim to give to each of the branches of physics the same precision as to celestial mechanics. A mind accustomed to admire such models is not easily satisfied with a theory. Not only will it not tolerate the least appearance of contradiction but it will expect the different parts to be logically connected with one another and will require the number of hypotheses to be reduced to a minimum. This is not all. There will be other demands which appear to me to be less reasonable. Behind the matter of which our senses are aware and which is made known to us by experiment such a thinker will expect to see another kind of matter, the only true matter in its opinion which will no longer have anything but purely geometrical qualities and the atoms of which will be mathematical points subject to the laws of dynamics alone and yet he will try to represent to himself by an unconscious contradiction these invisible and colourless atoms and therefore to bring them as close as possible to ordinary matter. Then only will he be thoroughly satisfied and he will then imagine that he has penetrated the secret of the universe. Even if the satisfaction is fallacious it is nonetheless difficult to give it up. Thus on opening the pages of Maxwell a Frenchman expects to find a theoretical whole as logical and as precise as the physical optics that is founded on the hypotheses of the ether. He is thus preparing for himself a disappointment which I should like the reader to avoid. So I will warn him at once of what he will find and what he will not find in Maxwell. Maxwell does not give a mechanical explanation of electricity and magnetism. He can find himself to showing that such an explanation is possible. He shows that the phenomenon of optics are only a particular case of electromagnetic phenomena. From the whole theory of electricity a theory of light can be immediately deduced. Unfortunately the converse is not true. It is not always easy to find a complete explanation of electrical phenomena. In particular it is not easy if we take as our standing point Fresnel's theory to do so no doubt would be impossible. But nonetheless we must ask ourselves if we are compelled to surrender admirable results which we thought we had definitely acquired. That seems a step backwards and many sound intellects will not willingly allow of this. Should the reader consent to set some bounds to his hopes he will still come across other difficulties. The English scientist does not try to erect a unique definitive and well arranged building. He seems to raise rather a large number of provisional and independent constructions between which communication is difficult and sometimes impossible. Take for instance the chapter in which electrostatic attractions are explained by the pressures and tensions of the dialectic medium. This chapter might be suppressed without the rest of the book being thereby less clear or less complete and yet it contains a theory which is self-sufficient and which can be understood without reading a word of what precedes or follows. But it is not only independent of the rest of the book it is difficult to reconcile it with the fundamental ideas of the volume. Maxwell does not even attempt to reconcile it. He merely says, I have not been able to make the next step, namely to account by mechanical considerations for these stresses in the dialectic. This example will be sufficient to show what I mean. I could quote many others. Thus who would suspect on reading the pages devoted to magnetic rotary polarization that there is an identity between optical and magnetic phenomena? We must not flatter ourselves that we have avoided every contradiction but we ought to make up our minds. Two contradictory theories provided that they are kept from overlapping and that we do not look to find in them the explanation of things may in fact be very useful instruments of research and perhaps the reading of Maxwell would be less suggestive if he had not opened up to us so many new and divergent ways. But the fundamental ideas masked as it were. So far is this the case that in most works that are popularized this idea is the only point which is left completely untouched. To show the importance of this I think I ought to explain in what this fundamental idea consists but for that purpose a short digression is necessary. The Mechanical Explanation of Physical Phenomena In every physical phenomenon there is a certain number of parameters which are reached directly by experiment and which can be measured. I shall call them the parameters Q. Observation next teaches us the laws of the variations of these parameters and these laws can be generally stated in the form of differential equations which connected together the parameters Q and time. What can be done to give a mechanical interpretation to such a phenomenon? We may endeavor to explain it either by the movements of ordinary matter or by those of one or more hypothetical fluids. These fluids will be considered as formed of a very large number of isolated molecules M. When may we say that we have a complete mechanical explanation of the phenomenon? It will be on the one hand when we know the differential equations which are satisfied by the coordinates of these hypothetical molecules in equations which must in addition conform to the laws of dynamics and on the other hand when we know the relations which define the coordinates of the molecules in as functions of the parameters Q attainable by experiment. These equations as I have said should conform to the principles of dynamics and in particular to the principle of the conservation of energy and to that of least action. First of these two principles teaches us that the total energy is constant and may be divided into two parts. One kinetic energy or V's Viva which depends on the masses of the hypothetical molecules M and on their velocities. This I shall call T. Two the potential energy which depends only on the coordinates of these molecules and this I shall call U. It is the sum of the energies T and U that is constant. Now what are we taught by the principle of least action? It teaches us that to pass from the initial position occupied at the instant T to the final position occupied at the instant T1. The system must describe such a path that in the interval of time between the instant T0 and T1 the mean value of the action i.e. the difference between the two energies T and U must be as small as possible. The first of these two principles is moreover a consequence of the second. If we know the functions T and U this second principle is sufficient to determine the equations of motion. Among the paths which enable us to pass from one position to another there is clearly one for which the mean value of the action is smaller than for all the others. In addition there is only such path and it follows from this that the principle of least action is sufficient to determine the path followed and therefore the equations of motion. We thus obtain what are called the equations of language. In these equations the independent variables are the coordinates of the hypothetical molecules M. But I now assume that we take for the variables the parameters Q which are directly accessible to the experiment. The two parts of the energy should then be expressed as a function of the parameters Q and the derivatives. It is clear that it is under this form that they will appear to the experimenter. The latter will naturally endeavour to define kinetic and potential energy by the aid of quantities he can directly observe. If this be granted the system will always proceed from one position to another by such a path that the mean value of the action is a minimum. It matters little that T and U are now expressed by the aid of the parameters Q and their derivatives. It matters little that it is also by the aid of these parameters that we define the initial and final positions. The principle of least action will always remain true. Now here again of the whole of the parts which lead from one position to another there is one and only one for which the mean action is a minimum. The principle of least action is therefore sufficient for the determination of the differential equations which define the variations of the parameter Q. The equations thus obtained are another form of Lagrange's equations. To form these equations we need not know the relations which connect the parameters Q with the coordinates of the hypothetical molecules nor the masses of the molecules nor the expression of U as a function of the coordinates of these molecules. All we need know is the expression of U as a function of the parameters Q and that of T as a function of the parameters Q and the derivatives that is the expressions of the kinetic and potential energy in terms of experimental data. One of two things must now happen. Either for a convenient choice of T and U the Lagrangian equations constructed as we have indicated will be identical with the differential equations deduced from the experiment or there will be no functions T and U for which this identity takes place. In a latter case it is clear that no mechanical explanation is possible. The necessary condition for a mechanical explanation to be possible is therefore this, that we must choose the functions T and U so as to satisfy the principle of least action and of the conservation of energy. Besides this condition is sufficient suppose in fact that we have found a function U of the parameters Q which represents one of the parts of energy and that the part of the energy which we represent by T is a function of the parameters Q and the derivatives that it is a polynomial of the second degree with respect to its derivatives and finally to the Lagrangian equations formed by the aid of these two functions T and U are in conformity with the data of the experiment. How can we deduce from this a mechanical explanation? U must be regarded as the potential energy of a system of which T is the kinetic energy. There is no difficulty as far as U is concerned but can T be regarded as the V's Viva of a material system? It is easily shown that this is always possible and in an unlimited number of ways. I will be content with referring the reader to the pages of the preface of my electricité et optique for further details. Thus if the principle of least action cannot be satisfied no mechanical explanation is possible. If it can be satisfied once it follows that since there is one there must be an unlimited number. One more remark. Among the quantities that may be reached by experiment directly we shall consider some as the coordinates of our hypothetical molecules. Some will be our parameters Q and the rest will be regarded as dependent not only on the coordinates but on the velocities. Or what comes to the same thing? We look on them as derivatives of the parameters Q or as a combination of these parameters and their derivatives. Here then a question occurs. Among all these quantities measured experimentally which shall we choose to represent the parameters Q and which shall we prefer to regard as the derivatives of these parameters? This choice remains arbitrary to a large extent but a mechanical explanation will be possible if it is done so as to satisfy the principle of least action. Next Maxwell asks, Can this choice and that of the two energies T and U be made so that electric phenomena will satisfy this principle? Experiment shows us that the energy of an electromagnetic field decomposes into electrostatic and electrodynamic energy. Maxwell recognized that if we regard the former as the potential energy U and the latter as the kinetic energy T and that if on the other hand we take the electrostatic charges of the conductors as the parameters Q and the intensity of the currents as derivatives of other parameters Q under these conditions Maxwell has recognized that electric phenomena satisfies the principle of least action. He was then certain of a mechanical explanation. If he had expounded this theory at the beginning of his first volume instead of relegating it to a corner of the second it would not have escaped the attention of most readers. If therefore a phenomenon allows of a complete mechanical explanation it allows of an unlimited number of others which will equally take into account all the particulars revealed by experiment. And this is confirmed by the history of every branch of physics. In optics for instance Fresno believed vibration to be perpendicular to the plane of polarization. Newman holds that it is parallel to that plane. For a long time in experimentum Crucius was sought for, which would enable us to decide between these two theories, but in vain. In the same way without going out of the domain of electricity we find that the theory of two fluids and a single fluid theory equally account in a satisfactory manner for all the laws of electrostatics. All these facts are easily explained thanks to the properties of the Langrange equations. It is easy now to understand Maxwell's fundamental idea. To demonstrate the possibility of a mechanical explanation of electricity we need not trouble to find the explanation itself. We need only know the expression of the two functions T and U, which are the two parts of energy, and to form with these two functions Lagrange's equations, and then to compare these equations with the experimental laws. How shall we choose from all the possible explanations, one in which the help of experiment will be wanting? The day will perhaps come when physicists will no longer concern themselves with questions which are inaccessible to positive methods, and will leave them to the metaphysicians. That day has not yet come, man does not so easily resign himself to remaining forever ignorant of the causes of things. Our choice cannot be, therefore, any longer guided by considerations in which personal appreciation plays too large a part. There are, however, solutions which all will reject because of their fantastic nature, and others which all will prefer because of their simplicity. As far as magnetism and electricity are concerned, Maxwell abstained from making any choice. It is not that he has a systemic contempt for all that positive methods cannot reach. As may be seen from time to time, he is devoted to the kinetic theory of gases. I may add that if in his magnum opus he develops no complete explanation, he is attempted one in an article in the Philosophical Magazine. The strangeness and the complexity of the hypotheses he found himself compelled to make led him afterwards to withdraw it. The same spirit is found throughout his whole work. He throws into relief the essential, i.e., what is common to all theories. Everything that suits only a particular theory is passed over, almost in silence. The reader, therefore, finds himself in the presence of a form nearly devoid of matter, which at first he is tempted to take as a fugitive and unassailable phantom. But the efforts he is thus compelled to make force him to think, and eventually he sees that there is often something rather artificial in the theoretical aggregates which he once admired. CHAPTER XIII. OF SCIENCE AND HYPOTHOSIS This is our LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, please visit LibriVox.org. Recording by Ashwin Jain. SCIENCE AND HYPOTHOSIS by Henry Poincaré. CHAPTER XIII. ELECTRODYNAMICS The history of electrodynamics is very instructive from our point of view. The title of MPU's immortal work is, THEORY OF THOSE PHENOMENOUS ELECTRODYNAMICS, ONICAMENTA FONDE SU EXPERIENCE. He therefore imagined that he had made no hypothesis, but as we shall not be long in recognizing, he was mistaken. Only of these hypothesis he was quite unaware. On the other hand, his successors see them clearly enough, because their attention is attracted by the weak points in compared solution. They made fresh hypothesis, but this time deliberately. How many times they had to change them before they reached the classic system, which is perhaps even now not quite definitive, we shall see. 1. AMPU'S THEORY In AMPU's experimental study of the mutual action of currents, he has operated, and he could operate only with closed currents. This was not because he denied the existence or possibility of open currents. If two conductors are positively and negatively charged and brought into communication by a wire, his current is set up which passes from one to another until the two potentials are equal. According to the ideas of AMPU's time, this was considered to be an open current. Their current was known to pass from the first conductor to the second, but they did not know it returned from the second to the first. All currents of this kind were therefore considered by AMPU to be open currents. For instance, the currents of discharge of a condenser, he was unable to experiment on them, their duration being too short. Another kind of open current may be imagined. Suppose we have two conductors, A and B, connected by a wire, AMB. Small conducting masses in motion are first of all placed in contact with the conductor, B. Receive an electric charge and leaving B as set in motion along a path B and A, carrying their charge with them. On coming into contact with A, they lose their charge, which then returns to B along the wire AMB. Now here we have, in a sense, a closed circuit. Since the electricity describes the closed circuit B and AMB, but the two parts of the current are quite different. In the wire AMB, the electricity is displaced through a fixed conductor like a voltaic current. Overcoming an ohmic resistance and developing heat. We say that it is displaced by conduction. In the part B and A, the electricity is carried by a moving conductor and is said to be displaced by convection. If, therefore, the convection current is considered to be perfectly analogous to the conduction current, the circuit B and AMB is closed. If, on the contrary, the convection current is not a true current and, for instance, does not act on the magnet, there is only the conduction current AMB, which is open. For example, if you connect by a wire the poles of a Holtz machine, the charge rotating disc transfers the electricity by convection from one pole to the other and it returns to the first pole by conduction to the wire. But currents of this kind are very difficult to produce with appreciable intensity. In fact, with the means of Ampere's disposal, we almost say it was impossible. To sum up, Ampere could conceive of the existence of two kinds of open currents, but he could experiment on neither because they were not strong enough or because their duration was too short. Experiment, therefore, could only show him the action of a closed current on a closed current or, more accurately, the action of a closed current on a portion of current because a current can be made to describe a closed circuit of which part may be in motion and the other part fixed. The displacements of the moving part may be studied under the action of another closed current. On the other hand, Ampere had no means of studying the action of an open current either on a closed or on another open current. Two, the case of closed currents. In the case of mutual action of two closed currents, experiment revealed to Ampere remarkably simple laws. The following will be useful to us in the sequel. One, is the intensity of the currents is kept constant and if the two circuits, after having undergone displacements and deformations, whatever, return finally to the initial positions, the total work done by the electrodynamical actions is zero. In other words, there is an electrodynamical potential of the two circuits proportional to the product of their intensities and depending on the form and relative positions of the circuits. The work done by the electrodynamical actions is equal to the change of this potential. The action of a closed solenoid is zero. Three, the action of a circuit C on another multi-circuit C dash depends only on the magnetic field developed by the circuit C. At each point in space, we can in fact define in magnitude and direction a certain force called magnetic force which enjoys the following properties. A, the force exercised by C on a magnetic pole is applied to that pole and is equal to the magnetic force multiplied by the magnetic mass of the pole. B, a very short magnetic needle tends to take the direction of the magnetic force and the couple to which it tends to reduce is proportional to the product of the magnetic force, the magnetic moment of the needle and the sign of the dip of the needle. If the circuit C dash is displaced, the amount of work done by the electrodynamic action of C on C dash will be equal to the increment of flow of magnetic force which passes through the circuit. Two, action of a closed current on a portion of current. Ampere, being unable to produce the open current, properly so-called, had only one way of setting the action of a closed current on a portion of current. This was by operating on a circuit C composed of two parts, one movable and the other fixed. The movable part was, for instance, a movable wire AB, the ends A and B of which were slide along a fixed wire. In one of the positions of the movable wire, the end A rested on the point A and the end B on the point B of the fixed wire. The current ran from A to B that is from A to B along the movable wire and then from B to A along the fixed wire. This current was therefore closed. In a second position, the movable wire having slipped the points A and B were respectively at A dash and B dash on the fixed wire. The current ran from A to B that is from A dash to B dash on the movable wire and returned from B dash to B and then from B to A and then from A to A dash all on the fixed wire. This current was also closed. If a similar circuit be exposed to the action of a closed current C, the movable part will be displaced just as if it were acted on by a force. Ampere admits that the force apparently acting on the movable part AB representing the action of C on the portion beta of the current remains the same whether an open current runs through beta stopping at A and beta or whether a closed current runs falls to beta and then returns to through the fixed portion of the circuit. This hypothesis seemed natural enough and Ampere innocently assumed it. Nevertheless, the hypothesis is not necessary for we should presently see that Helmholtz rejected it. However, that may be it enabled Ampere although he had never produced an open current to lay down the laws of the action of a closed current on an open current or even on an element of current. They are simple. One, the force acting on an element of current is applied to that element. It is normal to the element and to the magnetic force in proportion to that component of the magnetic force which is normal to the element. Two, the action of a closed solenoid on an element of current is zero where the electrodynamic potential has disappeared. That is, when a closed and an open current of constant intensities return to the initial positions, the total work done is not zero. Three, continuous rotations. The most remarkable electrodynamical experiments are those in which continuous rotations are produced and which are unipolar induction experiments. A magnet may turn about its axis. A current passes force through a fixed wire and then enters the magnet by the pole end. For instance, passes through half the magnet and emerges by a sliding contact and re-enters the fixed wire. The magnet then begins to rotate continuously. This is Verdi's experiment. How is it possible? If it were a question of two circuits of invariable form, C fixed and C dash movable about an axis, the latter would never take up a position of continuous rotation. In fact, there is an electrodynamical potential. There must therefore be a position of equilibrium when the potential is a maximum. Continuous rotations are therefore possible only when the circuit C dash is composed of two parts. One fixed and the other movable about an axis as in the case of Verdi's experiment. Here again it is convenient to draw a distinction. The passage from the fixed to the movable part or vice versa may take place either by simple contact. The same point of movable part remaining constantly in contact with the same point of the fixed part or by sliding contact the same point of the movable part coming successfully into contact with the different points of the fixed part. It is only in the second case that there can be continuous rotation. This is what then happens. The system tends to take up a position of equilibrium but when at the point of reaching that position the sliding contact puts the moving part in contact the fresh point in the fixed part. It changes the connections and therefore the conditions of equilibrium so that as the position of equilibrium is ever eluding so to speak the system which is trying to reach it rotation may take place indefinitely. Ampere admits that the action of the circuit on the movable part of C dash is the same as if the fixed part of C dash did not exist. Therefore if the current passing through the movable part were an open current it concluded that the action of a closed on an open current or vice versa that of an open current on a fixed current may give rise to continuous rotation. But this conclusion depends on the hypothesis which I have enunciated and to which as I said above helmets decline to subscribe for mutual action of two open circuits. As far as the mutual action of two open currents and in particular that of two elements of current is concerned all experiment break down. Ampere falls back on hypothesis. He assumes one that the mutual action of two elements reduced to a force acting along the joint. Two that the action of two closed currents is the resultant of the mutual actions of the different elements which are the same as if these elements were isolated. The remarkable thing is that here again Ampere makes two hypothesis without being aware of it. However that may be these two hypothesis together with the experiments on closed currents suffice to determine completely the law of mutual action of the two elements. But then most of the simple laws we have met in the case of closed currents are no longer true. In the first place there is no electrodynamical potential nor was there any as we have seen in the second case of a closed current acting on an open current. Next there is properly speaking no magnetic force and we have above defined this force in three different ways. One by the action of a magnetic pole. Two by the director couple which orientates the magnetic needle. Three by the action on element of current. In the case with which we are immediately concerned not only are these three definitions not in harmony but each has lost its meaning. One a magnetic pole is no longer acted on by a unique force applied to that pole. We have seen in fact the action of an element of current on a pole is not applied to the pole but to the element it may moreover be replaced by a force applied to the pole and by a couple. Two the couple which acts on the magnetic needle is no longer a simple director couple. For its moment with respect to the axis of the needle is not zero. It decomposes into a director couple properly so-called and a supplementary couple which tends to produce the continuous rotation of which we have spoken above. Three finally the force acting on an element of a current is not normal to that element. In other words the unity of the magnetic force has disappeared. Let us see in what this unity consists. Two systems which exercise the same action on a magnetic pole will also exercise the same action on an indefinitely small magnetic needle or on an element of current placed at the point in space at which the pole is. Well this is true. The two systems only contain close currents and according to Ampere it would not be true if the systems contain open currents. It is sufficient to remark for instance that if a magnetic pole is placed at A and an element at B the direction of the element being in A produced this element which will exercise no action on the pole will exercise an action either on a magnetic needle placed at A or on an element of current at A. Five induction. We know that the discovery of electrodynamical induction followed not long after the immortal work of Ampere. As long as it is only a question of close currents there is no difficulty and Helmholtz has even remarked that the principle of conservation of energy is sufficient for us to reduce the laws of induction from the electrodynamical laws of Ampere but on the condition as we are trying has shown that we make a certain number of hypothesis. The same principle again enables this deduction to be made in a case of open currents although the result cannot be tested by experiment since such currents cannot be produced. If we wish to compare this method of analysis with Ampere's theorem on open currents we get results which are calculated to surprises. In the first place induction cannot be reduced from the variation of the magnetic field by the well-known formula of scientists and practical men. In fact as if as I have said properly speaking there is no magnetic field but further if a circuit C is subjected to the induction of a variable voltage system S and if this system S is displaced and deformed in any way whatever so that the intensity of the currents of the system varies according to any law whatever then so long as after these variations the system eventually returns to its initial position it seems natural to suppose that the mean electromotive force induced in the currency is zero this is true if the circuit C is closed and if the system S only contains closed currents it is no longer true if we accept the theory of Ampere since there would be open currents so that not only will induction no longer be the variation of the flow of magnetic force in any of the usual senses of the world but it cannot be represented by the variation of that force whatever it may be but to Helmholtz theory I have dwelt upon the consequences of Ampere's theory and on his method of explaining the action of open currents it is difficult disregard the paradoxical and artificial character of the propositions to which we are thus led we feel bound to think it cannot be so we may imagine that the Helmholtz has been led to look for something else he rejects the fundamental hypothesis ampere namely that the mutual action of two elements of current reduces to a force along the iron he admits that an element of current is not acted upon by a single force but by a force and a couple and this is what gave rise to a celebrated polemic between Bertrand and Helmholtz Helmholtz replaces Ampere's hypothesis by the following two elements of current always admit of a electrodynamic potential depending solely upon their position and orientation and the work of the forces that they exercise one on the other is equal to the variation of this potential thus Helmholtz can no more do without hypothesis that ampere but at least he does not do so without explicitly announcing it in the case of closed currents which alone are accessible to experiment the two theories agree in all other cases they differ in the first place contrary to what ampere supposed the force which seems to act on the movable portion of a closed current is not the same as that acting on the movable portion if it were isolated and if it constituted an open current let us return to the circuit c dash of which we spoke a verb and which was formed by a movable wire sliding on a fished wire in the only experiment that can be made the movable portion B is not isolated but is part of a closed circuit when it passes from A B to A dash B dash the total electrodynamic potential varies for two reasons first it has a slight increment because the potential of A dash B dash with respect to the current C is not the same as that of A B secondly it has a second increment because it must be increased by the potential of the elements A A dash and B dash B with respect to C it is this double increment which represents the work of the force acting upon the portion A B if on the contrary be isolated the potential would only have the first increment and this first increment alone would measure the work of the force acting on a B in the second place there could be no continuous rotation without sliding contact and in fact that as we have seen in the case of closed currents is an immediate consequence of the existence of an electrodynamic potential in Faraday's experiment if the magnet is fixed and if the part of current external to the magnet runs along a movable wire that movable wire may undergo continuous rotation but it does not mean that if the contacts of the wire with the magnet were suppressed and an open current were to run along the wire the wire would still have a movement of continuous rotation I have just said in fact that an isolated element is not acted on in the same way as a movable element making part of a closed circuit but there is another difference the action of a solenoid on a closed current is zero according to experiment and according to the two theories its action on an open current would be zero according to ampere and it would not be zero according to Helmholtz from this follows an important consequence we are given above three definitions of the magnetic force the third has no meaning here since an element of current is no longer acted upon by a single force nor has the first any meaning what in fact is a magnetic pole it is the extremity of an indefinite linear magnet this magnet may be replaced by an indefinite solenoid for the definition of magnetic force to have any meaning the action exercised by an open current on an indefinite solenoid would only depend on the position of the extremity of that solenoid that is the action of a closed solenoid is zero now we have just seen this is not the case on the other hand there is nothing to prevent us from adopting the second definition which is founded on the measurement of the direct couple which tends to orientate the magnetic needle but if it is adopted neither the effect of induction nor electronic effects will depend solely on the distribution of the lines of force in this magnetic field part three difficulty is raised by these theories Helmholtz theory is an advance on that of ampere it is necessary however that every difficulty should be removed in both the name magnetic field has no meaning or if you give it one by a more or less artificial convention the ordinary laws so familiar to the electricians no longer apply and it is thus that the electromotive force induced in a wire is no longer measured by the number of lines of force met by that wire and our objections do not proceed only from the fact that it is difficult to give up deeply rooted habits of language and thought there is something more if you do not believe in actions at a distance an electrodynamic phenomena must be explained by a modification of the medium and this medium is precisely what we call magnetic field and then the electromagnetic effects should only depend on that field all these difficulties arise from the hypothesis of hoaking currents part four maxville's theory such were the difficulties raised by the current theories when maxville with the stroke of the pen caused them to vanish to his mind in fact all currents are close currents maxville admits that if in a dielectric the electric field happens to vary this dielectric becomes the seat of a particular phenomenon acting on the galvanometer like current and called the current of displacement if then two conductors bearing positive and negative charges are placed in connection by means of a wire during the discharge there is an open current of conduction that wire but they are produced at the same time in the surrounding dielectric currents displacement which closed this current of conduction we know that maxville's theory leads to the explanation of optical phenomena which would be due to extremely rapid electrical oscillations at that period such a conception was only a daring hypothesis which could be supported by no experiment but after 20 years maxville's idea received the confirmation of experiment herds succeeded in producing systems of electric oscillations which he produced all the properties of light and only differ the length of their wave that is to say as violet differs from red in some measure he made a synthesis of light it may be said it hurts had not directly proved maxville's fundamental idea of the action of the current of displacement in the galvanometer this is true in a sense what he has shown directly is that electromagnetic induction is not instantaneously propagated as was supposed but its speed is the speed of light yet to suppose that there is no current of displacement and that induction is with the speed of light or rather to suppose the currents of displacement produce inductive effects and that the induction takes place instantaneously comes to the same thing this cannot be seen at the first glance but it is proved by an analysis of which i must not even think of giving even a summary here part five roland's experiment but as i said above there are two kinds of open conduction currents there are first the currents of discharge of condenser or of any conductor whatever there are also cases in which the electric charges describe a closed contour being displaced by conduction in one part of the circuit and by convection in the other part the question might be regarded as solved for open currents of the first kind they were closed by currents of displacement for open currents of the second kind the solution appeared still more simple it seemed that if the current were closed it could only be the it seemed that if the current were closed it could only be by the current of convection itself for that purpose it was sufficient to admit that a convection current that is a charged conductor in motion could act on the galvanometer but experimental confirmation was lacking it appeared difficult in fact to obtain a sufficient intensity even by increasing as much as possible the charge and velocity of the conductors roland an extremely skillful experimentalist with a force to triumph or to seem to triumph over these difficulties a disk received a strong electrostatic charge and a very high speed of rotation an astatic magnetic system placed beside the disk and event deviations the experiment was made twice by roland once in berlin and once in baltimore it was afterwards repeated by himsted these physicists even believed that they could announce that they had succeeded in making quantitative measurements 20 years roland's law was admitted without objection by all physicists and indeed everything seemed to conform it the spark certainly does produce a magnetic effect and does it not seem extremely likely that the spark discharged is due to particles taken from one of the electrodes and transferred to the other electrode with the charge is not the very spectrum of the spark in which we recognize the lines of metal on the electrode a proof of it this spark would then be a real current of induction on the other hand it is also admitted that in an electrolyte the electricity is carried by the ions in motion the current in electrolyte would therefore also be a current of convection but it acts on the magnetic need and in the same way for cathodic rays crooks attributed these rays to very subtle matter charged with negative electricity and moving with very high velocity he looked upon them in other words as currents of convection now these cathodic rays are deviated by the magnet in virtue of the principle of action and reaction they should in turn deviate the magnetic needle it is true the thirds believed he had proved that the cathodic rays do not carry negative electricity and that they do not act on the magnetic needle it hurts was wrong first of all parents succeeded in collecting the electricity carried by these rays electricity of which hurts denied the existence the german scientist appears to have been deceived by the effects due to the action of x-rays which were not yet discovered afterwards in quite recently the action of the cathodic rays on the magnetic needle has been brought to light those all these phenomena looked upon as currents of convection electric sparks electrolytic currents cathodic rays act in the same manner on the galvanometer and in conformity to rollins law part six lauren's theory we need not go much further according to lauren's theory currents of conduction would themselves be true convection currents electricity would remain indissolubly connected with certain material particles called electrons the circulation of these electrons through bodies would produce voltage currents and what would distinguish conductors from insulators would be that the one could be traversed by these electrons while the others would check the movement of the electron lauren's theory is very attractive it gives a very simple explanation of certain phenomena which the earlier theories even maxwell's in its primitive form could only deal within an unsatisfactory manner for example the aberration of light the partial impulse of luminous waves magnetic polarization and zeeman's experiment a few objections still remain the phenomena of an electric system seem to depend on the absolute velocity of translation of the center of gravity of the system which in contrary to the idea that we have of the relativity of space supported by mr m creamy m lip man has presented this objection in a very striking form imagine two charge conductors with the same velocity of translation they're relatively at rest however each of them being equivalent to a current of convection they ought to attract one another and by measuring this attraction you could measure their absolute velocity no reply the partitions of morange what we could measure in that way is not their absolute velocity but their relative velocity with respect to the either so that the principle of relativity is safe whatever there may be in these objections the edifice of electronics seem at any rate in its broad lines definitively constructed everything was presented under the most satisfactory aspect the theories of ampere and helmholtz which are made for the open currents that no longer existed seem to have no more than purely historic interest and the inextricable complications to which these theories led have been almost forgotten this guisance has been recently disturbed by the experiments of mr creamy which have contradicted or at least have seemed to contradict the results formally obtained by roland numerous investigators have endeavored to solve the question and fresh experiments have been undertaken what result will they give i shall take care not to risk a prophecy which might be falsified between the day this book is ready for the press and the day on which it is placed before the public end of chapter 13 recording by ashwin jen end of book science and