sciences, he uses with all the candour which characterises him these expressions, page 362 — “This application, notwithstanding the happy efforts of some geometricians, is still, if I may so say, but in its first elements, and it must open to following generations a source of intelligence truly inexhaustible.” Yet he had then made not only the learned essay of which we are speaking, but also a work greatly superior, the Elements of the Calculation of Probabilities and of its Application to games of chance, to lotteries and to the judgments of men, which were not published till the year 1805.
I believe, then, that I have advanced nothing rash in observing that in subjects difficult by the number, subtility, complexity and intimate connexion of the circumstances to be considered, without the omission of any of them, the great talent of well combining those, not sufficiently numerous, which have been perceived, has not been sufficient to preserve the most skilful calculators from important errors and great misreckonings. We perceive that that was to be expected. But now I must go further, and all this leads me to a last reflection, which flows from the nature of things, like those which have just been read, which confirms several important principles established in the preceding volumes, which far from annihilating the great hopes of Condorcet tends to assure and realise them, by restraining them within certain limits; but which appear to me to show manifestly, how far the calculation of probabilities is from being the same thing with the theory of probability. Observe in what this observation consists.
The principal object of the theory of probability and its great utility, is in setting out from the reunion of a certain number of given causes, to determine the degree of the probability of the effects which ought to follow; and setting out from the reunion of a certain number of known effects, to determine the degree of the probability of the causes, which have been able to produce them. We may even say that all the results of this theory are but different branches of this general result, and may be traced to be nothing more than parts of it.
Now we have previously seen, and on different occasions, that for beings of any kind, to be successfully submitted to the action of calculation, it is necessary they should be susceptible of adaptation to the clear, precise and invariable divisions of the ideas of quantity, and to the series of the names of numbers and of cyphers, which express them. This is a condition necessary to the validity of every calculation from which that which has probability for its object, cannot be any more exempt, than that which conducts to absolute certainty.
Hence it rigorously follows, that there is a multitude of subjects of which it would be absolutely impossible to calculate the data, if even (which is not always the case) it should be possible to collect them all without over looking any.
Assuredly the degrees of the capacity, of the probity of men, those of the energy and the power of their passions, prejudices and habits, cannot possibly be estimated in numbers. It is the same as to the degrees of influence of certain institutions, or of certain functions, of the degrees of importance of certain establishments, of the degrees of difficulty of certain discoveries, of the degrees of utility of certain inventions, or of certain processes. I know that of these quantities, truly inappreciable and innumerable in all the rigour of the word, we seek and even attain to a certain point, in determining the limits, by means of number, of the frequency and extent of their effects; but I also know that in these effects which we are obliged to sum and number together as things perfectly similar, in order to deduce results, it is almost always and I may say always impossible to unravel the alterations and variations of concurrent causes, of influencing circumstances, and of a thousand essential considerations, so that we are necessitated to arrange together as similar a multitude of things very different, to arrive only at those preparatory results which are afterwards to lead to others which cannot fail to become entirely fantastical.
Is an example desired, very striking, drawn from a subject which surely does not present as many difficulties of this kind as moral ideas? Here is one. Certainly none of those who have undertaken to estimate the effort of the muscles of the heart, have erred against the rules of calculation, nor, what is more, against the laws of animated mechanics, the certainty of which should still preserve them from many errors. Yet some have been led to estimate this effort at several thousands of pounds, and others only at some ounces; and nobody knows with certainty which are nearest to truth. What succour then can we derive from calculation, when even availing ourselves properly of it we are subject to such aberrations and to such prodigious incertitudes?
It is then true, and I repeat it, that there is a multitude of things to which the calculation of probabilities like every other calculation is completely inapplicable. These things are much more numerous than is generally believed, and even by many very skilful men, and the first step to be taken in the science of probability is to know how to distinguish them. It is for the science of the formation of our ideas, for that of the operations of our intelligence, in a word for sound ideology, to teach us the number of these things, to enable us to know their nature, and to show us the reasons why they are so refractory. And it is a great service which it will render to the human mind, by preventing it in future from making a pernicious use of one of its most excellent instruments. It already shows us that the science of probability is a thing very distinct from the calculation of probability with which it has been confounded, since it extends to many objects to which the other cannot attain. This is what I principally proposed to elucidate.
Finally, as I have before announced, this observation does not destroy the great hopes which the piercing genius of Condorcet had made him conceive from the employment of calculation in general, and from that of probability in particular, in the advancement of the moral sciences; for if the different shades of our moral ideas cannot be expressed in numbers, and if there are many other things relative to social science, which are equally incapable of being estimated and calculated directly, these things depend on others which often render them reducible to calculable quantities, if we may use the expression. Thus for example, the degrees of the value of all things useful and agreeable, that is to say, the degrees of interest we attach to their possession cannot be noted directly by figures, but all those which can be represented by quantities of weight or extension of a particular thing, become calculable and even comparable the one with the other; in like manner the energy and durability of the secret springs which cause and preserve the action of the organs constituting our life are not susceptible of direct appreciation, but we judge of them by their effects. Time and different kinds of resistance and waste are susceptible of very exact divisions. This is sufficient for us, and we derive thence a great multitude of results and of valuable combinations; now there is an infinity of things in the moral sciences which offer us similar resources; but there are also many which offer none, and once more it is of great importance to discriminate perfectly between them: For first, in respect to these latter, every employment of calculation is abusive; and moreover there are often species of quantities presented which appear calculable, but which are inextricably complicated by mixture with those other species of quantities which I permit myself to call refractory, and then if calculation be applied thereto, the most skilful mathematicians are inevitably led into enormous errors; against this in my opinion they have not always been sufficiently on their guard. As to these two latter cases we may say of calculation what has been said of the syllogistic art as to all our reasonings whatsoever; that is, that it conducts our mind much less correctly than the simple light of good sense aided by sufficient attention.
This is all I had to observe on the science and calculation of probability, and I draw from it the following consequences: The theory of probability is neither a part of nor a supplement to logic. This theory moreover is not a science separate and distinct from all others. All sciences have a positive and a conjectural part. In all of them the positive part consists in distinguishing the effects which always and necessarily follow certain causes, and the causes which always and necessarily produce certain effects. In all of them also the conjectural part consists in proceeding from the reunion of a certain number of given causes to determine the degrees of probability of the effects which ought to follow from them, and in proceeding from the reunion of a certain number of known effects to determine the degree of probability of the causes which have been able to produce them. In these two parts, when the ideas compared
