The Logic of Chance, 3rd edition / An Essay on the Foundations and Province of the Theory of Probability, With Especial Reference to Its Logical Bearings and Its Application to Moral and Social Science and to Statistics

by John Venn

[3] We might express it thus:—a few instances are not sufficient to display a law at all; a considerable number will suffice to display it; but it takes a very great number to establish that a change is taking place in the law.

[4] The mathematician may illustrate the nature of this substitution by the analogies of the ‘circle of curvature’ in geometry, and the ‘instantaneous ellipse’ in astronomy. In the cases in which these conceptions are made use of we have a phenomenon which is continuously varying and also changing its rate of variation. We take it at some given moment, suppose its rate at that moment to be fixed, and then complete its career on that supposition.

[5] So called from its first mathematical treatment appearing in the Commentarii of the Petersburg Academy; a variety of notices upon it will be found in Mr Todhunter's History of the Theory of Probability.

CHAPTER II.

FURTHER DISCUSSION UPON THE NATURE OF THE SERIES MENTIONED IN THE LAST CHAPTER.

§ 1. In the course of the last chapter the nature of a particular kind of series, that namely, which must be considered to constitute the basis of the science of Probability, has received a sufficiently general explanation for the preliminary purpose of introduction. One might indeed say more than this; for the characteristics which were there pointed out are really sufficient in themselves to give a fair general idea of the nature of Probability, and of the sort of problems with which it deals. But in the concluding paragraphs an indication was given that the series of this kind, as they actually occur in nature or as the results of more or less artificial production, are seldom or never found to occur in such a simple form as might possibly be expected from what had previously been said; but that they are almost always seen to be associated together in groups after a somewhat complicated fashion. A fuller discussion of this topic must now be undertaken.

We will take for examination an instance of a kind with which the investigations of Quetelet will have served to familiarize some readers. Suppose that we measure the heights of a great many adult men in any town or country. These heights will of course lie between certain extremes in each direction, and if we continue to accumulate our measures it will be found that they tend to lie continuously between these extremes; that is to say, that under those circumstances no intermediate height will be found to be permanently unrepresented in such a collection of measurements. Now suppose these heights to be marshalled in the order of their magnitude. What we always find is something of the following kind;—about the middle point between the extremes, a large number of the results will be found crowded together: a little on each side of this point there will still be an excess, but not to so great an extent; and so on, in some diminishing scale of proportion, until as we get towards the extreme results the numbers thin off and become relatively exceedingly small.

The point to which attention is here directed is not the mere fact that the numbers thus tend to diminish from the middle in each direction, but, as will be more fully explained directly, the law according to which this progressive diminution takes place. The word ‘law’ is here used in its mathematical sense, to express the formula connecting together the two elements in question, namely, the height itself, and the relative number that are found of that height. We shall have to enquire whether one of these elements is a function of the other, and, if so, what function.

§ 2. After what was said in the last chapter, it need hardly be insisted upon that the interest and significance of such investigations as these are almost entirely dependent upon the statistics being very extensive. In one or other of Quetelet's works on Social Physics[1] will be found a selection of measurements of almost every element which the physical frame of man can furnish:—his height, his weight, the muscular power of various limbs, the dimensions of almost every part and organ, and so on. Some of the most extensive of these express the heights of 25,000 Federal soldiers from the Army of the Potomac, and the circumferences of the chests of 5738 Scotch militia men taken many years ago. Those who wish to consult a large repertory of such statistics cannot be referred to any better sources than to these and other works by the same author.[2]

Interesting and valuable, however, as are Quetelet's statistical investigations (and much of the importance now deservedly attached to such enquiries is, perhaps, owing more to his efforts than to those of any other person), I cannot but feel convinced that there is much in what he has written upon the subject which is erroneous and confusing as regards the foundations of the science of Probability, and the philosophical questions which it involves. These errors are not by any means confined to him, but for various reasons they will be better discussed in the form of a criticism of his explicit or implicit expression of them, than in any more independent way.