ERRORS OF OBSERVATION. 411 



with a new A and the other with a new B: it is now an even chance for A, But have we 

 not simply restored the original state of things ? The addition of one more A or B does not 

 alter the ratio of ^s to ^s in an infinite number. What was the original collection of 

 arrangements except A followed by every possible arrangement and B followed by every 

 possible arrangement ? How then are there more ways of beginning with A than of begin- 

 ning with B ? No beginner can answer this sophism : no proficient can make sure of having 

 avoided the like, if he should take an assumption about the long run, or derived from 

 the long run, until he has obtained verification from fundamental principles. 



The science is essentially enumerative of equally probable cases, and draws all its con- 

 clusions from distribution of these cases under heads, and subsequent enumeration of the 

 numbers under the several heads. The cases may be infinitely many, and it may require all 

 the power of algebraic development or of the integi'al calculus to present the results of the 

 enumerations : but this does not affect the truth of my assertion, though it places an array 

 of symbolic reasonings between the beginner and clear perception of the fundamental method. 

 In the subject of this paper there has always been a leaning towards the assumption of some 

 complex results upon native evidence ; especially on points connected with the average : and 

 the probable whole has not infrequently been assumed to be a congeries of the most probable 

 parts. This turns out, on proper examination, to be true in some very marked cases : and 

 the conclusion is made welcome for its own sake, as well as for the letters of introduction 

 which sound demonstration furnishes. But in the theory of probabilities, no less than in 

 the conduct of life, if we open our houses to strangers upon the strength of pleasant looks 

 and plausible stories, we shall certainly be swindled at last. That a probable whole must be 

 composed entirely of probable parts is a fallacy of almost universal sway : it resembles the 

 mistake made by Frankenstein, who constructed every limb and feature of his man upon the 

 most approved model of separate beauty, and produced the ugliest monster that ever was 

 seen. Stories which are throughout of the highest probability may be true ; there are such 

 truths: but those who note actual occurrences see that very complex wholes without impro- 

 bable parts are extremely rare. The common mind weighs the probable against the particular 

 improbable which the evidence seems to favour; it always forgets that in a priori reasoning, 

 it is the probable against one or other of all the improbables. 



••• I now proceed to the statement of my own views : 



If a? be a quantity which may take various values, x^, w^, .ra,... X in number; we have 

 \"'2a?, \~'2a?^ &sc. for the average value, average square, &c. If y, z,... be other quantities, 

 having /x, »/,... values severally, it is clear that average product and product of averages are 

 convertible terms, if combination of values may take place in any manner. Thus in "^.x^yz^ , 

 we have Xfxv terms, which are the terms of the product "S^v' .^y .^z* : and division by Xuv 

 shews that 2:«^y«*= 2:a7* . S:y . S:«*, where Sra?^ means ^«- : X, the average x'. Let us 

 now examine the average of all the values of {x + y -^ % +•••)*' which are X/jli/... in number. 

 Take a product of the type t^lylxl, in which a + fi + 'y=k: and let Xfxv... = N. For 

 given values of a, b, c, this term (with a?, y, z) occurs in every value of {x + ...)" in which 

 J?^ + ^j + z^ is seen : that is, in N:\[xv of the terms: and the same of every term of exponents 



