606 THE POPULAR SCIENCE MONTHLY. 



tion of errors which Quetelet, Galton, and others, have applied with 

 so much success to the study of biological and social matters. This 

 application of continuity to cases where it does not really exist illus- 

 trates, also, another point which will hereafter demand a separate 

 study, namely, the great utility which fictions sometimes have in 

 science. 



II. 



The theory of probabilities is simply the science of logic quantita- 

 tively treated. There are two conceivable certainties with reference 

 to any hypothesis, the certainty of its truth and the certainty of its 

 falsity. The numbers one and zero are appropriated, in this calculus, 

 to marking these extremes of knowledge ; while fractions having 

 values intermediate between them indicate, as we may vaguely say, 

 the degrees in which the evidence leans toward one or the other. 

 The general problem of probabilities is, from a given state of facts, 

 to determine the numerical probability of a possible fact. This is the 

 same as to inquire how much the given facts are worth, considered as 

 evidence to prove the possible fact. Thus the problem of probabilities 

 is simply the general problem of logic. 



Probability is a continuous quantity, so that great advantages may 

 be expected from this mode of studying logic. Some writers have gone 

 so far as to maintain that, by means of the calculus of chances, every 

 solid inference may be represented by legitimate arithmetical opera- 

 tions upon the numbers given in the premises. If this be, indeed, 

 true, the great problem of logic, how it is that the observation of one 

 fact can give us knowledge of another independent fact, is reduced to 

 a mere question of arithmetic. It seems proper to examine this pre- 

 tension before undertaking any more recondite solution of the paradox. 



But, unfortunately, writers on probabilities are not agreed in re- 

 gard to this result. This branch of mathematics is the only one, I 

 believe, in which good writers frequently get results entirely errone- 

 ous. In elementary geometry the reasoning is frequently fallacious, 

 but erroneous conclusions are avoided; but it may be doubted if 

 there is a single extensive treatise on probabilities in existence which 

 does not contain solutions absolutely indefensible. This is partly 

 owing to the want of any regular method of procedure ; for the sub- 

 ject involves too many subtilties to make it easy to put its problems 

 into equations without such an aid. But, beyond this, the fundamental 

 principles of its calculus are more or less in dispute. In regard to 

 that class of questions to which it is chiefly applied for practical pur- 

 poses, there is comparatively little doubt; but in regard to others 

 to which it has been sought to extend it, opinion is somewhat un- 

 settled. 



the appearance of exactitude where none exists. Certain newspapers which affect a 

 learned tone talk of " the average man," when they simply mean most men, and have no 

 idea of striking an average. 



