What is the present distribution of the minor planets?

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 we know nothing of their initial velocities. So we know nothing of their present velocities.

The calculus 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, instead of deducing effects from causes, we wish to deduce the causes from the effects.

These problems are classified as probability of causes.

I play at écarté with an honest gentleman.

What is the chance that he turns up the king? It is 18.

This is a problem of the probability of effects.

I play with a gentleman whom I do not know.

He has dealt 10 times. He has turned the king up 6 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 divergencies are due to errors of observation?

This is the type of question that we are ever asking, and which we unconsciously solve whenever we are engaged in scientific work. I am now going to pass in review these different categories of problems by discussing in succession what I have called subjective and objective probability.

II. Probability in Mathematics

The impossibility of squaring the circle was shown in 1885. Before then, all geometers considered this impossibility as so “probable” that the Académie des Sciences rejected without examination the, alas! too numerous memoirs on this subject that a few unhappy madmen sent in every year.

Was the Académie 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 Académie 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 π be the root of an algebraical equation, why do you expect this root to be a period of the function sin 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 would be more usefully spent at their ordinary work than in reading a lucubration that inspired in them a legitimate distrust.

But what I called above objective probability has nothing in common with this first problem. It is otherwise with the second. Let us consider the first 10, 000 logarithms that we find in a table.

Among these 10,000 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 12 , and in fact if you pick out in a table the third decimals in these 10,000 numbers you will find nearly as many even digits as odd.

Or, if you prefer it, let us write 10, 000 numbers corresponding to our 10,000 logarithms, writing down for each of these numbers +1 if the third decimal of the corresponding logarithm is even, and −1 if odd; and then let us take the mean of these 10,000 numbers.

I do not hesitate to say that the mean of these 10, 000 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 0.003. 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 générale des Sciences, 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;

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.

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 a given very large integer. What is the mean value of sin nu? 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 + da is φ(a) da; that it is therefore proportional to the infinitely small interval da, and is equal to this multiplied by a function φ(a), only depending on a.

As for this function I choose it arbitrarily, but I must assume it to be continuous. The value of sin nu remaining the same when u increases by 2π, I may without loss of generality assume that u lies between 0 and 2π, and I shall thus be led to suppose that φ(a) is a periodic function whose period is 2π.

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

2πM K , n K M K being the maximum value of the Kth derivative of φ(u). We see then that if the Kth derivative is finite, our mean value will tend towards zero when n increases 1 indefinitely, and that more rapidly than K+1. n

The mean value of sin nu 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 φ(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 rôle 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.