CHAPTER V.

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THE

LOGIC OF CHANCE

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, Sc.D., F.R.S.,

FELLOW AND LECTURER IN THE MORAL SCIENCES, GONVILLE AND CAIUS COLLEGE,

CAMBRIDGE.

LATE EXAMINER IN LOGIC AND MORAL PHILOSOPHY IN THE

UNIVERSITY OF LONDON.

"So careful of the type she seems

So careless of the single life."

_THIRD EDITION, RE-WRITTEN AND ENLARGED._

London:

MACMILLAN AND CO.

AND NEW YORK

1888

[_All Rights reserved._]

_First Edition printed_ 1866.

_Second Edition_ 1876.

_Third Edition_ 1888.

PREFACE TO FIRST EDITION

Any work on Probability by a Cambridge man will be so

likely to have its scope and its general treatment of the

subject prejudged, that it may be well to state at the outset

that the following Essay is in no sense mathematical. Not

only, to quote a common but often delusive assurance, will

'no knowledge of mathematics beyond the simple rules of

Arithmetic' be required to understand these pages, but it is

not intended that any such knowledge should be acquired by

the process of reading them. Of the two or three occasions

on which algebraical formulæ occur they will not be found to

form any essential part of the text.

The science of Probability occupies at present a somewhat

anomalous position. It is impossible, I think, not to

observe in it some of the marks and consequent disadvantages

of a _sectional_ study. By a small body of ardent students it

has been cultivated with great assiduity, and the results they

have obtained will always be reckoned among the most extraordinary

products of mathematical genius. But by the

general body of thinking men its principles seem to be

regarded with indifference or suspicion. Such persons may

admire the ingenuity displayed, and be struck with the profundity

of many of the calculations, but there seems to

them, if I may so express it, an _unreality_ about the whole

treatment of the subject. To many persons the mention of

Probability suggests little else than the notion of a set of

rules, very ingenious and profound rules no doubt, with which

mathematicians amuse themselves by setting and solving

puzzles.

It must be admitted that some ground has been given

for such an opinion. The examples commonly selected by

writers on the subject, though very well adapted to illustrate

its rules, are for the most part of a special and peculiar

character, such as those relating to dice and cards. When

they have searched for illustrations drawn from the practical

business of life, they have very generally, but unfortunately,

hit upon just the sort of instances which, as I shall endeavour

to show hereafter, are among the very worst that could

be chosen for the purpose. It is scarcely possible for any

unprejudiced person to read what has been written about the

credibility of witnesses by eminent writers, without his experiencing

an invincible distrust of the principles which they

adopt. To say that the rules of evidence sometimes given

by such writers are broken in practice, would scarcely be

correct; for the rules are of such a kind as generally to defy

any attempt to appeal to them in practice.

This supposed want of harmony between Probability and

other branches of Philosophy is perfectly erroneous. It

arises from the belief that Probability is a branch of mathematics

trying to intrude itself on to ground which does not

altogether belong to it. I shall endeavour to show that this

belief is unfounded. To answer correctly the sort of questions

to which the science introduces us does generally demand

some knowledge of mathematics, often a great knowledge,

but the discussion of the fundamental principles on which

the rules are based does not necessarily require any such

qualification. Questions might arise in other sciences, in

Geology, for example, which could only be answered by the

aid of arithmetical calculations. In such a case any one

would admit that the arithmetic was extraneous and accidental.

However many questions of this kind there might

be here, those persons who do not care to work out special

results for themselves might still have an accurate knowledge

of the principles of the science, and even considerable

acquaintance with the details of it. The same holds true in

Probability; its connection with mathematics, though certainly

far closer than that of most other sciences, is still of

much the same kind. It is principally when we wish to

work out results for ourselves that mathematical knowledge

is required; without such knowledge the student may still

have a firm grasp of the principles and even see his way to

many of the derivative results.

The opinion that Probability, instead of being a branch of

the general science of evidence which happens to make much

use of mathematics, _is_ a portion of mathematics, erroneous as

it is, has yet been very disadvantageous to the science in

several ways. Students of Philosophy in general have thence

conceived a prejudice against Probability, which has for the

most part deterred them from examining it. As soon as a

subject comes to be considered 'mathematical' its claims

seem generally, by the mass of readers, to be either on the

one hand scouted or at least courteously rejected, or on the

other to be blindly accepted with all their assumed consequences.

Of impartial and liberal criticism it obtains little

or nothing.

The consequences of this state of things have been, I

think, disastrous to the students themselves of Probability.

No science can safely be abandoned entirely to its own devotees.

Its details of course can only be studied by those who

make it their special occupation, but its general principles

are sure to be cramped if it is not exposed occasionally to

the free criticism of those whose main culture has been of

a more general character. Probability has been very much

abandoned to mathematicians, who as mathematicians have

generally been unwilling to treat it thoroughly. They have

worked out its results, it is true, with wonderful acuteness,

and the greatest ingenuity has been shown in solving various

problems that arose, and deducing subordinate rules. And

this was all that they could in fairness be expected to do.

Any subject which has been discussed by such men as

Laplace and Poisson, and on which they have exhausted all

their powers of analysis, could not fail to be profoundly

treated, so far as it fell within their province. But from this

province the real principles of the science have generally

been excluded, or so meagrely discussed that they had better

have been omitted altogether. Treating the subject as mathematicians

such writers have naturally taken it up at the

point where their mathematics would best come into play,

and that of course has not been at the foundations. In the

works of most writers upon the subject we should search in

vain for anything like a critical discussion of the fundamental

principles upon which its rules rest, the class of

enquiries to which it is most properly applicable, or the

relation it bears to Logic and the general rules of inductive

evidence.

This want of precision as to ultimate principles is perfectly

compatible here, as it is in the departments of Morals

and Politics, with a general agreement on processes and

results. But it is, to say the least, unphilosophical, and

denotes a state of things in which positive error is always

liable to arise whenever the process of controversy forces us

to appeal to the foundations of the science.

With regard to the remarks in the last few paragraphs,

prominent exceptions must be made in the case of two recent

works at least.[1] The first of these is Professor de Morgan's

_Formal Logic_. He has there given an investigation into the

foundations of Probability as conceived by him, and nothing

can be more complete and precise than his statement of

principles, and his deductions from them. If I could at all

agree with these principles there would have been no necessity

for the following essay, as I could not hope to add

anything to their foundation, and should be far indeed from

rivalling his lucid statement of them. But in his scheme

Probability is regarded very much from the Conceptualist

point of view; as stated in the preface, he considers that

Probability is concerned with formal inferences in which the

premises are entertained with a conviction short of absolute

certainty. With this view I cannot agree. As I have entered

into criticism of some points of his scheme in one of the

following chapters, and shall have occasion frequently to refer

to his work, I need say no more about it here. The other

work to which I refer is the profound _Laws of Thought_ of

the late Professor Boole, to which somewhat similar remarks

may in part be applied. Owing however to his peculiar

treatment of the subject, I have scarcely anywhere come

into contact with any of his expressed opinions.

The view of the province of Probability adopted in this

Essay differs so radically from that of most other writers on

the subject, and especially from that of those just referred to,

that I have thought it better, as regards details, to avoid all

criticism of the opinions of others, except where conflict was

unavoidable. With regard to that radical difference itself

Bacon's remark applies, behind which I must shelter myself

from any change of presumption.--"Quod ad universalem

istam reprehensionem attinet, certissimum vere est rem reputanti,

eam et magis probabilem esse et magis modestam,

quam si facta fuisset ex parte."

Almost the only writer who seems to me to have expressed

a just view of the nature and foundation of the rules

of Probability is Mr Mill, in his _System of Logic_.[2] His

treatment of the subject is however very brief, and a considerable

portion of the space which he has devoted to it is

occupied by the discussion of one or two special examples.

There are moreover some errors, as it seems to me, in what

he has written, which will be referred to in some of the

following chapters.

The reference to the work just mentioned will serve to

convey a general idea of the view of Probability adopted in

this Essay. With what may be called the Material view of

Logic as opposed to the Formal or Conceptualist,--with that

which regards it as taking cognisance of laws of things and

not of the laws of our own minds in thinking about things,--I

am in entire accordance. Of the province of Logic, regarded

from this point of view, and under its widest aspect, Probability

may, in my opinion, be considered to be a portion. The

principal objects of this Essay are to ascertain how great a

portion it comprises, where we are to draw the boundary between

it and the contiguous branches of the general science

of evidence, what are the ultimate foundations upon which its

rules rest, what the nature of the evidence they are capable

of affording, and to what class of subjects they may most fitly

be applied. That the science of Probability, on this view of

it, contains something more important than the results of a

system of mathematical assumptions, is obvious. I am convinced

moreover that it can and ought to be rendered both

interesting and intelligible to ordinary readers who have any

taste for philosophy. In other words, if the large and growing

body of readers who can find pleasure in the study of

books like Mill's _Logic_ and Whewell's _Inductive Sciences_,

turn with aversion from a work on Probability, the cause in

the latter case must lie either in the view of the subject or

in the manner and style of the book.

I take this opportunity of thanking several friends,

amongst whom I must especially mention Mr Todhunter, of

St John's College, and Mr H. Sidgwick, of Trinity College,

for the trouble they have kindly taken in looking over the

proof-sheets, whilst this work was passing through the Press.

To the former in particular my thanks are due for thus

adding to the obligations which I, as an old pupil, already

owed him, by taking an amount of trouble, in making suggestions

and corrections for the benefit of another, which few

would care to take for anything but a work of their own.

His extensive knowledge of the subject, and his extremely

accurate judgment, render the service he has thus afforded

me of the greatest possible value.

GONVILLE AND CAIUS COLLEGE

_September_ 1866.

1. I am here speaking, of course, of

those only who have expressly treated

of the foundations of the science. Mr

Todhunter's admirable work on the

_History of the Theory of Probability_

being, as the name denotes, mainly

historical, such enquiries have not

directly fallen within his province.

2. This remark, and that at the

commencement of the last paragraph,

having been misunderstood, I ought

to say that the only sense in which

originality is claimed for this Essay

is in the thorough working out of the

Material view of Logic as applied to

Probability. I have given a pretty

full discussion of the general principles

of this view in the tenth

chapter, and have there pointed out

some of the peculiarities to which it

leads.

PREFACE TO SECOND EDITION

The principal reason for designating this volume a second

edition consists in the fact that the greater portion of what

may be termed the first edition is incorporated into it. Besides

various omissions (principally where the former treatment

has since seemed to me needlessly prolix), I have added

new matter, not much inferior in amount to the whole of

the original work. In addition, moreover, to these alterations

in the matter, the general arrangement of the subject

as regards the successive chapters has been completely

changed; the former arrangement having been (as it now

seems to me) justly objected to as deficient and awkward in

method.

After saying this, it ought to be explained whether any

change of general view or results will be found in the present

treatment.

The general view of Probability adopted is quite unchanged,

further reading and reflection having only confirmed

me in the conviction that this is the soundest and most

fruitful way of regarding the subject. It is the more necessary

to say this, as to a cursory reader it might seem

otherwise; owing to my having endeavoured to avoid the

needlessly polemical tone which, as is often the case with

those who are making their first essay in writing upon any

subject, was doubtless too prominent in the former edition.

I have not thought it necessary, of course, except in one or

two cases, to indicate points of detail which it has seemed

necessary to correct.

A number of new discussions have been introduced upon

topics which were but little or not at all treated before. The

principal of these refer to the nature and physical origin of

Laws of Error (Ch. II.); the general view of Logic, and consequently

of Probability, termed the Material view, adopted

here (Ch. X.); a brief history and criticism of the various

opinions held on the subject of Modality (Ch. XII.); the

logical principles underlying the method of Least Squares

(Ch. XIII.); and the practices of Insurance and Gambling,

so far as the principles involved in them are concerned

(Ch. XV.). The Chapter on the Credibility of Extraordinary

Stories is also mainly new; this was the portion of the former

work which has since seemed to me the least satisfactory,

but owing to the extreme intricacy of the subject I am far

from feeling thoroughly satisfied with it even now.

I have again to thank several friends for the assistance

they have so kindly afforded. Amongst these I must prominently

mention Mr C. J. Monro, late fellow of Trinity. It is

only the truth to say that I have derived more assistance from

his suggestions and criticisms than has been consciously obtained

from all other external sources together. Much of this

criticism has been given privately in letters, and notes on

the proof-sheets; but one of the most elaborate of his discussions

of the subject was communicated to the Cambridge

Philosophical Society some years ago; as it was not published,

however, I am unfortunately unable to refer the reader to it.

I ought to add that he is not in any way committed to any

of my opinions upon the subject, from some of which in fact

he more or less dissents. I am also much indebted to

Mr J. W. L. Glaisher, also of Trinity College, for many hints

and references to various publications upon the subject of

Least Squares, and for careful criticism (given in the midst

of much other labour) of the chapter in which that subject is

treated.

I need not add that, like every one else who has had to

discuss the subject of Probability during the last ten years,

I have made constant use of Mr Todhunter's History.

I may take this opportunity of adding that a considerable

portion of the tenth chapter has recently appeared in the

January number of _Mind_, and that the substance of several

chapters, especially in the more logical parts, has formed part

of my ordinary lectures in Cambridge; the foundation and

logical treatment of Probability being now expressly included

in the Schedule of Subjects for the Moral Sciences Tripos.

_March_ 1876.

PREFACE TO THIRD EDITION

The present edition has been revised throughout, and in fact

rewritten. Three chapters are new, viz. the fifth (On the

conception of Randomness) and the eighteenth and nineteenth

(On the nature, and on the employment, of Averages).

The eighth, tenth, eleventh, and fifteenth chapters have been

recast, and much new matter added, and numerous alterations

made in the remaining portions.[1] On the other hand

three chapters of the last edition have been nearly or

entirely omitted.

These alterations do not imply any appreciable change of

view on my part as to the foundations and province of

Probability. Some of them are of course due to the necessary

changes involved in the attempt to write up to date upon a

subject which has not been stationary during the last eleven

years. For instance the greatly increased interest now

taken in what may be called the Theory of Statistics has

rendered it desirable to go much more fully into the Nature

and treatment of Laws of Error. The omissions are mainly

due to a wish to avoid increasing the bulk of this volume

more than is actually necessary, and to a feeling that the

portions treating specially of Inductive Logic (which occupied

some space in the last edition) would be more

suitable to a regular work on that subject. I am at present

engaged on such a work.

The publications which I have had occasion to notice

have mostly appeared in various scientific journals. The

principal authors of these have been Mr F. Galton and

Mr F. Y. Edgeworth: to the latter of whom I am also

personally much obliged for many discussions, oral and

written, and for his kindness in looking through the proof-sheets.

His published articles are too numerous for separate

mention here, but I may say generally, in addition to the

obligations specially noticed, that I have been considerably

indebted to them in writing the last two chapters. Two

authors of works of a somewhat more substantial character,

viz. Prof. Lexis and Von Kries, only came under my notice

unfortunately after this work was already in the printer's

hands. With the latter of these authors I find myself in

closer agreement than with most others, in respect of his

general conception and treatment of Probability.

_December_ 1887.

1. I have indicated the new chapters and sections by printing them in

italics in the Table of Contents.

TABLE OF CONTENTS.[1]

1. Chapters and sections which are nearly or entirely new are printed

in _italics_.

PART I.

PHYSICAL FOUNDATIONS OF THE SCIENCE OF PROBABILITY. Chh. I-V.

CHAPTER I.

THE SERIES OF PROBABILITY.

§§ 1, 2. Distinction between the proportional propositions of Probability,

and the propositions of Logic.

3, 4. The former are best regarded as presenting a series of individuals,

5. Which may occur in any order of time,

6, 7. And which present themselves in groups.

8. Comparison of the above with the ordinary phraseology.

9, 10. These series ultimately fluctuate,

11. Especially in the case of moral and social phenomena,

12. Though in the case of games of chance the fluctuation is practically

inappreciable.

13, 14. In this latter case only can rigorous inferences be drawn.

15, 16. The Petersburg Problem.

CHAPTER II.

ARRANGEMENT AND FORMATION OF THE SERIES. LAWS OF ERROR.

§§ 1, 2. Indication of the nature of a Law of Error or Divergence.

3. Is there necessarily but one such law,

4. Applicable to widely distinct classes of things?

5, 6. This cannot be proved directly by statistics;

7, 8. _Which in certain cases show actual asymmetry._

9, 10. Nor deductively;

11. Nor by the Method of Least Squares.

12. Distinction between Laws of Error and the Method of Least

Squares.

13. Supposed existence of types.

14-16. Homogeneous and heterogeneous classes.

17, 18. _The type in the case of human stature, &c._

19, 20. The type in mental characteristics.

21, 22. Applications of the foregoing principles and results.

CHAPTER III.

ORIGIN OR PROCESS OF CAUSATION OF THE SERIES.

§ 1. The causes consist of (1) 'objects,'

2, 3. Which may or may not be distinguishable into natural kinds,

4-6. And (2) 'agencies.'

7. Requisites demanded in the above:

8, 9. Consequences of their absence.

10. Where are the required causes found?

11, 12. Not in the direct results of human will.

13-15. Examination of apparent exceptions.

16-18. _Further analysis of some natural causes._

CHAPTER IV.

HOW TO DISCOVER AND PROVE THE SERIES.

§ 1. The data of Probability are established by experience;

2. Though in practice most problems are solved deductively.

3-7. Mechanical instance to show the inadequacy of any à priori proof.

8. The Principle of Sufficient Reason inapplicable.

9. Evidence of actual experience.

10, 11. Further examination of the causes.

12, 13. Distinction between the succession of physical events and the

Doctrine of Combinations.

14, 15. Remarks of Laplace on this subject.

16. Bernoulli's Theorem;

17, 18. Its inapplicability to social phenomena.

19. Summation of preceding results.

CHAPTER V.

THE CONCEPTION OF RANDOMNESS.

§ 1. _General Indication._

2-5. _The postulate of ultimate uniform distribution at one stage or

another._

6. _This area of distribution must be finite:_

7, 8. _Geometrical illustrations in support:_

9. _Can we conceive any exception here?_

10, 11. _Experimental determination of the random character when the

events are many:_

12. _Corresponding determination when they are few._

13, 14. _Illustration from the constant π._

15, 16. _Conception of a line drawn at random._

17. _Graphical illustration._

PART II.

LOGICAL SUPERSTRUCTURE ON THE ABOVE PHYSICAL FOUNDATIONS.

Chh. VI-XIV.

CHAPTER VI.

MEASUREMENT OF BELIEF.

§§ 1, 2. Preliminary remarks.

3, 4. Are we accurately conscious of gradations of belief?

5. Probability only concerned with part of this enquiry.

6. Difficulty of measuring our belief;

7. Owing to intrusion of emotions,

8. And complexity of the evidence.

9. And when measured, is it always correct?

10, 11. Distinction between logical and psychological views.

12-16. Analogy of Formal Logic fails to show that we can thus detach

and measure our belief.

17. Apparent evidence of popular language to the contrary.

18. How is full belief justified in inductive enquiry?

19-23. Attempt to show how partial belief may be similarly justified.

24-28. Extension of this explanation to cases which cannot be repeated

in experience.

29. Can other emotions besides belief be thus measured?

30. Errors thus arising in connection with the Petersburg Problem.

31, 32. The emotion of surprise is a partial exception.

33, 34. Objective and subjective phraseology.

35. The definition of probability,

36. Introduces the notion of a 'limit',

37. And implies, vaguely, some degree of belief.

CHAPTER VII.

THE RULES OF INFERENCE IN PROBABILITY.

§ 1. Nature of these inferences.

2. Inferences by addition and subtraction.

3. Inferences by multiplication and division.

4-6. Rule for independent events.

7. Other rules sometimes introduced.

8. All the above rules may be interpreted subjectively, i.e. in terms

of belief.

9-11. Rules of so-called Inverse Probability.

12, 13. Nature of the assumption involved in them:

14-16. Arbitrary character of this assumption.

17, 18. _Physical illustrations._

CHAPTER VIII.

THE RULE OF SUCCESSION.

§ 1. Reasons for desiring some such rule:

2. Though it could scarcely belong to Probability.

3. Distinction between Probability and Induction.

4, 5. Impossibility of reducing the various rules of the latter under

one head.

6. Statement of the Rule of Succession;

7. _Proof offered for it._

8. _Is it a strict rule of inference?_

9. _Or is it a psychological principle?_

CHAPTER IX.

INDUCTION.

§§ 1-5. Statement of the Inductive problem, and origin of the Inductive

inference.

6. Relation of Probability to Induction.

7-9. The two are sometimes merged into one.

10. Extent to which causation is needed in Probability.

11-13. Difficulty of referring an individual to a class:

14. This difficulty but slight in Logic,

15, 16. But leads to perplexity in Probability:

17-21. Mild form of this perplexity;

22, 23. Serious form.

24-27. Illustration from Life Insurance.

28, 29. Meaning of 'the value of a life'.

30, 31. Successive specialization of the classes to which objects are

referred.

32. Summary of results.

CHAPTER X.

CHANGE, CAUSATION AND DESIGN.

§ 1. Old Theological objection to Chance.

2-4. Scientific version of the same.

5. Statistics in reference to Free-will.

6-8. Inconclusiveness of the common arguments here.

9, 10. _Chance as opposed to Physical Causation._

11. _Chance as opposed to Design in the case of numerical constants._

12-14. _Theoretic solution between Chance and Design._

15. _Illustration from the dimensions of the Pyramid._

16, 17. _Discussion of certain difficulties here._

18, 19. _Illustration from Psychical Phenomena._

20. Arbuthnott's Problem of the proportion of the sexes.

21-23. Random or designed distribution of the stars.

(_Note on the proportion of the sexes_.)

CHAPTER XI.

MATERIAL AND FORMAL LOGIC.

§§ 1, 2. _Broad distinction between these views;_

2, 3. _Difficulty of adhering consistently to the objective view;_

4. _Especially in the case of Hypotheses._

5. The doubtful stage of our facts is only occasional in Inductive

Logic.

6-9. But normal and permanent in Probability.

10, 11. Consequent difficulty of avoiding Conceptualist phraseology.

CHAPTER XII.

CONSEQUENCES OF THE DISTINCTIONS OF THE PREVIOUS CHAPTER.

§§ 1, 2. Probability has no relation to time.

3, 4. Butler and Mill on Probability before and after the event.

5. Other attempts at explaining the difficulty.

6-8. What is really meant by the distinction.

9. Origin of the common mistake.

10-12. Examples in illustration of this view,

13. Is Probability relative?

14. What is really meant by this expression.

15. Objections to terming Probability relative.

16, 17. In suitable examples the difficulty scarcely presents itself.

CHAPTER XIII.

ON MODALITY.

§ 1. Various senses of Modality;

2. Having mostly some relation to Probability.

3. Modality must be recognized.

4. Sometimes relegated to the predicate,

5, 6. Sometimes incorrectly rejected altogether.

7, 8. Common practical recognition of it.

9-11. Modal propositions in Logic and in Probability.

12. Aristotelian view of the Modals;

13, 14. Founded on extinct philosophical views;

15. But long and widely maintained.

16. Kant's general view.

17-19. The number of modal divisions admitted by various logicians.

20. Influence of the theory of Probability.

21, 22. Modal syllogisms.

23. Popular modal phraseology.

24-26. Probable and Dialectic syllogisms.

27, 28. Modal difficulties occur in Jurisprudence.

29, 30. Proposed standards of legal certainty.

31. Rejected formally in English Law, but possibly recognized practically.

32. How, if so, it might be determined.

CHAPTER XIV.

FALLACIES.

§§ 1-3. (I.) Errors in judging of events after they have happened.

4-7. Very various judgments may be thus involved.

8, 9. (II.) _Confusion between random and picked selections._

10, 11. (III.) Undue limitation of the notion of Probability.

12-16. (IV.) Double or Quits: the Martingale.

17, 18. Physical illustration.

19, 20. (V.) Inadequate realization of large numbers.

21-24. Production of works of art by chance.

25. Illustration from doctrine of heredity.

26-30. (VI.) Confusion between Probability and Induction.

31-33. (VII.) Undue neglect of small chances.

34, 35. (VIII.) _Judging by the event in Probability and in Induction._

PART III.

VARIOUS APPLICATIONS OF THE THEORY OF PROBABILITY. Chh. XV-XIX.

CHAPTER XV.

INSURANCE AND GAMBLING.

§§ 1, 2. The certainties and uncertainties of life.

3-5. Insurance a means of diminishing the uncertainties.

6, 7. Gambling a means of increasing them.

8, 9. Various forms of gambling.

10, 11. _Comparison between these practices._

12-14. Proofs of the disadvantage of gambling:--

(1) on arithmetical grounds:

15, 16. _Illustration from family names._

17. (2) from the 'moral expectation'.

18, 19. _Inconclusiveness of these proofs._

20-22. _Broader questions raised by these attempts._

CHAPTER XVI.

APPLICATION OF PROBABILITY TO TESTIMONY.

§§ 1, 2. Doubtful applicability of Probability to testimony.

3. Conditions of such applicability.

4. Reasons for the above conditions.

5, 6. Are these conditions fulfilled in the case of testimony?

7. The appeal here is not directly to statistics.

8, 9. Illustrations of the above.

10, 11. Is any application of Probability to testimony valid?

CHAPTER XVII.

CREDIBILITY OF EXTRAORDINARY STORIES.

§ 1. Improbability before and after the event.

2, 3. Does the rejection of this lead to the conclusion that the

credibility of a story is independent of its nature?

4. General and special credibility of a witness.

5-8. Distinction between alternative and open questions, and the

usual rules for application of testimony to each of these.

9. _Discussion of an objection._

10, 11. Testimony of worthless witnesses.

12-14. Common practical ways of regarding such problems.

15. Extraordinary stories not necessarily less probable.

16-18. Meaning of the term extraordinary, and its distinction from

miraculous.

19, 20. Combination of testimony.

21, 22. Scientific meaning of a miracle.

23, 24. Two distinct prepossessions in regard to miracles, and the

logical consequences of these.

25. Difficulty of discussing by our rules cases in which arbitrary

interference can be postulated.

26, 27. Consequent inappropriateness of many arguments.

CHAPTER XVIII.

ON THE NATURE AND USE OF AN AVERAGE, AND ON THE DIFFERENT KINDS OF AVERAGE.

§ 1. _Preliminary rude notion of an average,_

2. _More precise quantitative notion, yielding_

(1) _the Arithmetical Average,_

3. (2) _the Geometrical._

4. _In asymmetrical curves of error the arithmetic average must be

distinguished from,_

5. (3) _the Maximum Ordinate average,_

6. (4) _and the Median._

7. _Diagram in illustration._

8-10. _Average departure from the average, considered under the above

heads, and under that of_

11. (5) _The (average of) Mean Square of Error,_

12-14. _The objects of taking averages._

15. _Mr Galton's practical method of determining the average._

16, 17. _No distinction between the average and the mean._

18-20. _Distinction between what is necessary and what is experimental

here._

21, 22. _Theoretical defects in the determination of the 'errors'._

23. _Practical escape from these._

(_Note about the units in the exponential equation and integral._)

CHAPTER XIX.

THE THEORY OF THE AVERAGE AS A MEANS OF APPROXIMATION TO THE TRUTH.

§§ 1-4. _General indication of the problem: i.e. an inverse one

requiring the previous consideration of a direct one._

[I. _The direct problem:--given the central value and law of

dispersion of the single errors, to determine those of the

averages._ §§ 6-20.]

6. (i) _The law of dispersion may be determinable _à priori_,_

7. (ii) _or experimentally, by statistics._

8, 9. _Thence to determine the modulus of the error curve._

10-14. _Numerical example to illustrate the nature and amount of

the contraction of the modulus of the average-error curve._

15. _This curve is of the same general kind as that of the single errors;_

16. _Equally symmetrical,_

17, 18. _And more heaped up towards the centre._

19, 20. _Algebraic generalization of the foregoing results._

[II. _The inverse problem:--given but a few of the errors to

determine their centre and law, and thence to draw the above

deductions._ §§ 21-25.]

22, 23. _The actual calculations are the same as before,_

24. _With the extra demand that we must determine how probable are the

results._

25. _Summary._

[III. _Consideration of the same questions as applied to certain

peculiar laws of error._ §§ 26-37.]

26. (i) _All errors equally probable._

27, 28. (ii) _Certain peculiar laws of error._

29, 30. _Further analysis of the reasons for taking averages._

31-35. _Illustrative examples._

36, 37. _Curves with double centre and absence of symmetry._

38, 39. _Conclusion._

THE LOGIC OF CHANCE.

CHAPTER I.

_ON CERTAIN KINDS OF GROUPS OR SERIES AS THE FOUNDATION OF PROBABILITY._

1. It is sometimes not easy to give a clear definition of a

science at the outset, so as to set its scope and province before

the reader in a few words. In the case of those sciences

which are more immediately and directly concerned with

what are termed objects, rather than with what are termed

processes, this difficulty is not indeed so serious. If the

reader is already familiar with the objects, a simple reference

to them will give him a tolerably accurate idea of the

direction and nature of his studies. Even if he be not

familiar with them, they will still be often to some extent

connected and associated in his mind by a name, and

the mere utterance of the name may thus convey a fair

amount of preliminary information. This is more or less

the case with many of the natural sciences; we can often

tell the reader beforehand exactly what he is going to study.

But when a science is concerned, not so much with objects

directly, as with processes and laws, or when it takes for the

subject of its enquiry some comparatively obscure feature

drawn from phenomena which have little or nothing else in

common, the difficulty of giving preliminary information

becomes greater. Recognized classes of objects have then

to be disregarded and even broken up, and an entirely novel

arrangement of the objects to be made. In such cases it is

the study of the science that first gives the science its unity,

for till it is studied the objects with which it is concerned

were probably never thought of together. Here a definition

cannot be given at the outset, and the process of obtaining it

may become by comparison somewhat laborious.

The science of Probability, at least on the view taken of

it in the following pages, is of this latter description. The

reader who is at present unacquainted with the science

cannot be at once informed of its scope by a reference to

objects with which he is already familiar. He will have

to be taken in hand, as it were, and some little time

and trouble will have to be expended in directing his

attention to our subject-matter before he can be expected to

know it. To do this will be our first task.

2. In studying Nature, in any form, we are continually

coming into possession of information which we sum up in

general propositions. Now in very many cases these general

propositions are neither more nor less certain and accurate

than the details which they embrace and of which they are

composed. We are assuming at present that the truth of

these generalizations is not disputed; as a matter of fact

they may rest on weak evidence, or they may be uncertain

from their being widely extended by induction; what is

meant is, that when we resolve them into their component

parts we have precisely the same assurance of the truth of

the details as we have of that of the whole. When I know,

for instance, that all cows ruminate, I feel just as certain

that any particular cow or cows ruminate as that the whole

class does. I may be right or wrong in my original statement,

and I may have obtained it by any conceivable mode

in which truths can be obtained; but whatever the value of

the general proposition may be, that of the particulars is

neither greater nor less. The process of inferring the particular

from the general is not accompanied by the slightest

diminution of certainty. If one of these 'immediate inferences'

is justified at all, it will be equally right in every

case.

But it is by no means necessary that this characteristic

should exist in all cases. There is a class of immediate inferences,

almost unrecognized indeed in logic, but constantly

drawn in practice, of which the characteristic is, that as they

increase in particularity they diminish in certainty. Let me

assume that I am told that _some_ cows ruminate; I cannot

infer logically from this that any particular cow does so,

though I should feel some way removed from absolute disbelief,

or even indifference to assent, upon the subject; but

if I saw a herd of cows I should feel more sure that some of

them were ruminant than I did of the single cow, and my

assurance would increase with the numbers of the herd about

which I had to form an opinion. Here then we have a class

of things as to the individuals of which we feel quite in

uncertainty, whilst as we embrace larger numbers in our

assertions we attach greater weight to our inferences. It is

with such classes of things and such inferences that the

science of Probability is concerned.

3. In the foregoing remarks, which are intended to

be purely preliminary, we have not been able altogether to

avoid some reference to a subjective element, viz. the degree

of our certainty or belief about the things which we are

supposed to contemplate. The reader may be aware that

by some writers this element is regarded as the subject-matter

of the science. Hence it will have to be discussed

in a future chapter. As however I do not agree with the

opinion of the writers just mentioned, at least as regards

treating this element as one of primary importance, no

further allusion will be made to it here, but we will pass on

at once to a more minute investigation of that distinctive

characteristic of certain classes of things which was introduced

to notice in the last section.

In these classes of things, which are those with which

Probability is concerned, the fundamental conception which

the reader has to fix in his mind as clearly as possible, is, I

take it, that of a series. But it is a series of a peculiar kind,

one of which no better compendious description can be given

than that which is contained in the statement that it combines

individual irregularity with aggregate regularity. This

is a statement which will probably need some explanation.

Let us recur to an example of the kind already alluded to,

selecting one which shall be in accordance with experience.

Some children will not live to thirty. Now if this proposition

is to be regarded as a purely indefinite or, as it would

be termed in logic, 'particular' proposition, no doubt the

notion of a series does not obviously present itself in connection

with it. It contains a statement about a certain

unknown proportion of the whole, and that is all. But it is

not with these purely indefinite propositions that we shall

be concerned. Let us suppose the statement, on the contrary,

to be of a numerical character, and to refer to a given

proportion of the whole, and we shall then find it difficult to

exclude the notion of a series. We shall find it, I think,

impossible to do so as soon as we set before us the aim of

obtaining accurate, or even moderately correct inferences.

What, for instance, is the meaning of the statement that

two new-born children in three fail to attain the age of

sixty-three? It certainly does not declare that in any given

batch of, say, thirty, we shall find just twenty that fail:

whatever might be the strict meaning of the words, this

is not the import of the statement. It rather contemplates

our examination of a large number, of a long succession

of instances, and states that in such a succession we shall

find a numerical proportion, not indeed fixed and accurate at

first, but which tends in the long run to become so. In

every kind of example with which we shall be concerned we

shall find this reference to a large number or succession

of objects, or, as we shall term it, _series_ of them.

A few additional examples may serve to make this plain.

Let us suppose that we toss up a penny a great many

times; the results of the successive throws may be conceived

to form a series. The separate throws of this series seem to

occur in utter disorder; it is this disorder which causes our

uncertainty about them. Sometimes head comes, sometimes

tail comes; sometimes there is a repetition of the same face,

sometimes not. So long as we confine our observation to a

few throws at a time, the series seems to be simply chaotic.

But when we consider the result of a long succession we find

a marked distinction; a kind of order begins gradually to

emerge, and at last assumes a distinct and striking aspect.

We find in this case that the heads and tails occur in about

equal numbers, that similar repetitions of different faces do

so also, and so on. In a word, notwithstanding the individual

disorder, an aggregate order begins to prevail. So again if

we are examining the length of human life, the different lives

which fall under our notice compose a series presenting the

same features. The length of a single life is familiarly uncertain,

but the average duration of a batch of lives is becoming

in an almost equal degree familiarly certain. The

larger the number we take out of any mixed crowd, the

clearer become the symptoms of order, the more nearly will

the average length of each selected class be the same.

These few cases will serve as simple examples of a property

of things which can be traced almost everywhere, to a

greater or less extent, throughout the whole field of our experience.

Fires, shipwrecks, yields of harvest, births, marriages,

suicides; it scarcely seems to matter what feature we

single out for observation.[1] The irregularity of the single

instances diminishes when we take a large number, and at

last seems for all practical purposes to disappear.

In speaking of the effect of the average in thus diminishing

the irregularities which present themselves in the details,

the attention of the student must be prominently directed to

the point, that it is not the _absolute_ but the _relative_ irregularities

which thus tend to diminish without limit. This

idea will be familiar enough to the mathematician, but to

others it may require some reflection in order to grasp it

clearly. The absolute divergences and irregularities, so far

from diminishing, show a disposition to increase, and this (it

may be) without limit, though their relative importance shows

a corresponding disposition to diminish without limit. Thus

in the case of tossing a penny, if we take a few throws, say

ten, it is decidedly unlikely that there should be a difference

of six between the numbers of heads and tails; that is, that

there should be as many as eight heads and therefore as few

as two tails, or _vice versâ_. But take a thousand throws, and

it becomes in turn exceedingly likely that there should be

as much as, or more than, a difference of six between the

respective numbers. On the other hand the _proportion_ of

heads to tails in the case of the thousand throws will be very

much nearer to unity, in most cases, than when we only took

ten. In other words, the longer a game of chance continues

the larger are the spells and runs of luck in themselves, but

the less their relative proportions to the whole amounts

involved.

4. In speaking as above of events or things as to the

details of which we know little or nothing, it is not of course

implied that our ignorance about them is complete and universal,

or, what comes to the same thing, that irregularity

may be observed in all their qualities. All that is meant is

that there are _some_ qualities or marks in them, the existence

of which we are not able to predicate with certainty in the

individuals. With regard to all their other qualities there

may be the utmost uniformity, and consequently the most

complete certainty. The irregularity in the length of human

life is notorious, but no one doubts the existence of such

organs as a heart and brains in any person whom he happens

to meet. And even in the qualities in which the irregularity

is observed, there are often, indeed generally, positive limits

within which it will be found to be confined. No person,

for instance, can calculate what may be the length of any

particular life, but we feel perfectly certain that it will not

stretch out to 150 years. The irregularity of the individual

instances is only shown in certain respects, as e.g. the length

of the life, and even in these respects it has its limits. The

same remark will apply to most of the other examples with

which we shall be concerned. The disorder in fact is not

universal and unlimited, it only prevails in certain directions

and up to certain points.

5. In speaking as above of a series, it will hardly be

necessary to point out that we do not imply that the objects

themselves which compose the series must occur successively

in time; the series may be formed simply by their coming

in succession under our notice, which as a matter of fact

they may do in any order whatever. A register of mortality,

for instance, may be made up of deaths which took place

simultaneously or successively; or, we might if we pleased

arrange the deaths in an order quite distinct from either of

these. This is entirely a matter of indifference; in all these

cases the series, for any purposes which we need take into

account, may be regarded as being of precisely the same description.

The objects, be it remembered, are given to us in

nature; the order under which we view them is our own private

arrangement. This is mentioned here simply by way of

caution, the meaning of this assertion will become more plain

in the sequel.

I am aware that the word 'series' in the application with

which it is used here is liable to some misconstruction, but I

cannot find any better word, or indeed any as suitable in all

respects. As remarked above, the events need not necessarily

have occurred in a regular sequence of time, though

they often will have done so. In many cases (for instance,

the throws of a penny or a die) they really do occur in succession;

in other cases (for instance, the heights of men, or

the duration of their lives), whatever may have been the

order of their actual occurrence, they are commonly brought

under our notice in succession by being arranged in statistical

tables. In all cases alike our processes of inference involve

the necessity of examining one after another of the members

which compose the group, or at least of being prepared to do

this, if we are to be in a position to justify our inferences.

The force of these considerations will come out in the course

of the investigation in Chapter VI.

The late Leslie Ellis[2] has expressed what seems to me

a substantially similar view in terms of genus and species,

instead of speaking of a series. He says, "When individual

cases are considered, we have no conviction that the ratios of

frequency of occurrence depend on the circumstances common

to all the trials. On the contrary, we recognize in the determining

circumstances of their occurrence an extraneous

element, an element, that is, extraneous to the idea of the

genus and species. Contingency and limitation come in (so

to speak) together; and both alike disappear when we consider

the genus in its entirety, or (which is the same thing)

in what may be called an ideal and practically impossible

realization of all which it potentially contains. If this be

granted, it seems to follow that the fundamental principle

of the Theory of Probabilities may be regarded as included

in the following statement,--The conception of a genus

implies that of numerical relations among the species subordinated

to it." As remarked above, this appears a substantially

similar doctrine to that explained in this chapter,

but I do not think that the terms genus and species are by

any means so well fitted to bring out the conception of a

tendency or limit as when we speak of a series, and I therefore

much prefer the latter expression.

6. The reader will now have in his mind the conception

of a series or group of things or events, about the individuals

of which we know but little, at least in certain respects,

whilst we find a continually increasing uniformity as we

take larger numbers under our notice. This is definite

enough to point out tolerably clearly the kind of things

with which we have to deal, but it is not sufficiently definite

for purposes of accurate thought. We must therefore attempt

a somewhat closer analysis.

There are certain phrases so commonly adopted as to

have become part of the technical vocabulary of the subject,

such as an 'event' and the 'way in which it can

happen.' Thus the act of throwing a penny would be called

an event, and the fact of its giving head or tail would be

called the way in which the event happened. If we were

discussing tables of mortality, the former term would denote

the mere fact of death, the latter the age at which

it occurred, or the way in which it was brought about,

or whatever else in it might be the particular circumstance

under discussion. This phraseology is very convenient, and

will often be made use of in this work, but without explanation

it may lead to confusion. For in many cases the

way in which the event happens is of such great relative

importance, that according as it happens in one way or

another the event would have a different name; in other

words, it would not in the two cases be nominally the same

event. The phrase therefore will have to be considerably

stretched before it will conveniently cover all the cases to

which we may have to apply it. If for instance we were

contemplating a series of human beings, male and female,

it would sound odd to call their humanity an event, and

their sex the way in which the event happened.

If we recur however to any of the classes of objects

already referred to, we may see our path towards obtaining

a more accurate conception of what we want. It will easily

be seen that in every one of them there is a mixture of

similarity and dissimilarity; there is a series of events

which have a certain number of features or attributes in

common,--without this they would not be classed together.

But there is also a distinction existing amongst them; a

certain number of other attributes are to be found in some

and are not to be found in others. In other words, the

individuals which form the series are compound, each being

made up of a collection of things or attributes; some of

these things exist in all the members of the series, others

are found in some only. So far there is nothing peculiar

to the science of Probability; that in which the distinctive

characteristic consists is this;--that the occasional attributes,

as distinguished from the permanent, are found on

an extended examination to tend to exist _in a certain definite

proportion of the whole number of cases_. We cannot tell in

any given instance whether they will be found or not, but

as we go on examining more cases we find a growing uniformity.

We find that the proportion of instances in which

they are found to instances in which they are wanting, is

gradually subject to less and less comparative variation, and

approaches continually towards some apparently fixed value.

The above is the most comprehensive form of description;

as a matter of fact the groups will in many cases take a

far simpler form; they may appear, e.g. simply as a succession

of things of the same kind, say human beings, with

or without an occasional attribute, say that of being left-handed.

We are using the word attribute, of course, in its

widest sense, intending it to include every distinctive feature

that can be observed in a thing, from essential qualities

down to the merest accidents of time and place.

7. On examining our series, therefore, we shall find

that it may best be conceived, not necessarily as a succession

of events happening in different ways, but as a succession

of groups of things. These groups, on being analysed, are

found in every case to be resolvable into collections of substances

and attributes. That which gives its unity to the

succession of groups is the fact of some of these substances or

attributes being common to the whole succession; that which

gives their distinction to the groups in the succession is the

fact of some of them containing only a portion of these substances

and attributes, the other portion or portions being

occasionally absent. So understood, our phraseology may

be made to embrace every class of things of which Probability

can take account.

8. It will be easily seen that the ordinary expression

(viz. the 'event,' and the 'way in which it happens') may be

included in the above. When the occasional attributes are

unimportant the permanent ones are sufficient to fix and

appropriate the name, the presence or absence of the others

being simply denoted by some modification of the name or

the addition of some predicate. We may therefore in all such

cases speak of the collection of attributes as 'the event,'--the

same event essentially, that is--only saying that _it_ (so as

to preserve its nominal identity) happens in different ways

in the different cases. When the occasional attributes however

are important, or compose the majority, this way of

speaking becomes less appropriate; language is somewhat

strained by our implying that two extremely different assemblages

are in reality the same event, with a difference only

in its mode of happening. The phrase is however a very

convenient one, and with this caution against its being misunderstood,

it will frequently be made use of here.

9. A series of the above-mentioned kind is, I apprehend,

the ultimate basis upon which all the rules of

Probability must be based. It is essential to a clear comprehension

of the subject to have carried our analysis up

to this point, but any attempt at further analysis into the

intimate nature of the events composing the series, is not

required. It is altogether unnecessary, for instance, to form

any opinion upon the questions discussed in metaphysics as

to the independent existence of substances. We have discovered,

on examination, a series composed of groups of

substances and attributes, or of attributes alone. At such

a series we stop, and thence investigate our rules of inference;

into what these substances or attributes would themselves

be ultimately analysed, if taken in hand by the

psychologist or metaphysician, it is no business of ours to

enquire here.

10. The stage then which we have now reached is

that of having discovered a quantity of things (they prove

on analysis to be groups of things) which are capable of

being classified together, and are best regarded as constituting

a series. The distinctive peculiarity of this series is

our finding in it an order, gradually emerging out of disorder,

and showing in time a marked and unmistakeable uniformity.

The impression which may possibly be derived from the

description of such a series, and which the reader will probably

already entertain if he have studied Probability before,

is that the gradual evolution of this order is indefinite, and

its approach therefore to perfection unlimited. And many of

the examples commonly selected certainly tend to confirm

such an impression. But in reference to the theory of the

subject it is, I am convinced, an error, and one liable to lead

to much confusion.

The lines which have been prefixed as a motto to this

work, "So careful of the type she seems, so careless of the

single life," are soon after corrected by the assertion that

the type itself, if we regard it for a long time, changes,

and then vanishes and is succeeded by others. So in Probability;

that uniformity which is found in the long run,

and which presents so great a contrast to the individual

disorder, though durable is not everlasting. Keep on watching

it long enough, and it will be found almost invariably to

fluctuate, and in time may prove as utterly irreducible to

rule, and therefore as incapable of prediction, as the individual

cases themselves. The full bearing of this fact upon

the theory of the subject, and upon certain common modes

of calculation connected with it, will appear more fully in

some of the following chapters; at present we will confine

ourselves to very briefly establishing and illustrating it.

Let us take, for example, the average duration of life.

This, provided our data are sufficiently extensive, is known to

be tolerably regular and uniform. This fact has been already

indicated in the preceding sections, and is a truth indeed

of which the popular mind has a tolerably clear grasp at the

present day. But a very little consideration will show that

there may be a superior as well as an inferior limit to

the extent within which this uniformity can be observed;

in other words whilst we may fall into error by taking too

few instances we may also fail in our aim, though in a very

different way and from quite different reasons, by taking too

many. At the present time the average duration of life in

England may be, say, forty years; but a century ago it was

decidedly less; several centuries ago it was presumably very

much less; whilst if we possessed statistics referring to a still

earlier population of the country we should probably find that

there has been since that time a still more marked improvement.

What may be the future tendency no man can say for

certain. It may be, and we hope that it will be the case,

that owing to sanitary and other improvements, the duration

of life will go on increasing steadily; it is at least conceivable,

though doubtless incredible, that it should do so without

limit. On the other hand, and with much more likelihood,

this duration might gradually tend towards some fixed

length. Or, again, it is perfectly possible that future generations

might prefer a short and a merry life, and therefore

reduce their average longevity. The duration of life cannot

but depend to some extent upon the general tastes, habits

and employments of the people, that is upon the ideal which

they consciously or unconsciously set before them, and he would

be a rash man who should undertake to predict what this ideal

will be some centuries hence. All that it is here necessary

however to indicate is, that this particular uniformity (as we

have hitherto called it, in order to mark its relative character)

has varied, and, under the influence of future eddies

in opinion and practice, may vary still; and this to any

extent, and with any degree of irregularity. To borrow a

term from Astronomy, we find our uniformity subject to what

might be called an irregular _secular_ variation.

11. The above is a fair typical instance. If we had

taken a less simple feature than the length of life, or one

less closely connected with what may be called by comparison

the great permanent uniformities of nature, we should

have found the peculiarity under notice exhibited in a far

more striking degree. The deaths from small-pox, for example,

or the instances of duelling or accusations of witchcraft,

if examined during a few successive decades, might

have shown a very tolerable degree of uniformity. But these

uniformities have risen possibly from zero; after various and

very great fluctuations seem tending towards zero again, at

least in this century; and may, for anything we know, undergo

still more rapid fluctuations in future. Now these

examples must be regarded as being only extreme ones, and

not such very extreme ones, of what is the almost universal

rule in nature. I shall endeavour to show that even the few

apparent exceptions, such as the proportions between male

and female births, &c., may not be, and probably in reality

are not, strictly speaking, exceptions. A type, that is, which

shall be in the fullest sense of the words, persistent and

invariable is scarcely to be found in nature. The full import

of this conclusion will be seen in future chapters. Attention

is only directed here to the important inference that, although

statistics are notoriously of no value unless they are in sufficient

numbers, yet it does not follow but that in certain cases

we may have too many of them. If they are made too extensive,

they may again fall short, at least for any particular

time or place, of their greatest attainable accuracy.

12. These natural uniformities then are found at

length to be subject to fluctuation. Now contrast with them

any of the uniformities afforded by games of chance; these

latter seem to show no trace of secular fluctuation, however

long we may continue our examination of them. Criticisms

will be offered, in the course of the following chapters, upon

some of the common attempts to prove _à priori_ that there

must be this fixity in the uniformity in question, but of its

existence there can scarcely be much doubt. Pence give

heads and tails about equally often now, as they did when

they were first tossed, and as we believe they will continue

to do, so long as the present order of things continues. The

fixity of these uniformities may not be as absolute as is

commonly supposed, but no amount of experience which we

need take into account is likely in any appreciable degree to

interfere with them. Hence the obvious contrast, that,

whereas natural uniformities at length fluctuate, those afforded

by games of chance seem fixed for ever.

13. Here then are series apparently of two different

kinds. They are alike in their initial irregularity, alike in

their subsequent regularity; it is in what we may term

their ultimate form that they begin to diverge from each

other. The one tends without any irregular variation

towards a fixed numerical proportion in its uniformity; in

the other the uniformity is found at last to fluctuate, and to

fluctuate, it may be, in a manner utterly irreducible to rule.

As this chapter is intended to be little more than explanatory

and illustrative of the foundations of the science,

the remark may be made here (for which subsequent justification

will be offered) that it is in the case of series of the

former kind only that we are able to make anything which

can be interpreted into strict scientific inferences. We shall

be able however in a general way to see the kind and extent

of error that would be committed if, in any example, we were

to substitute an imaginary series of the former kind for any

actual series of the latter kind which experience may present

to us. The two series are of course to be as alike as possible

in all respects, except that the variable uniformity has been

replaced by a fixed one. The difference then between them

would not appear in the initial stage, for in that stage the

distinctive characteristics of the series of Probability are not

apparent; all is there irregularity, and it would be as impossible

to show that they were alike as that they were

different; we can only say generally that each shows the

same kind of irregularity. Nor would it appear in the next

subsequent stage, for the real variability of the uniformity

has not for some time scope to make itself perceived. It

would only be in what we have called the ultimate stage,

when we suppose the series to extend for a very long time,

that the difference would begin to make itself felt.[3] The

proportion of persons, for example, who die each year at the

age of six months is, when the numbers examined are on a

small scale, utterly irregular; it becomes however regular

when the numbers examined are on a larger scale; but if we

continued our observation for a very great length of time, or

over a very great extent of country, we should find this

regularity itself changing in an irregular way. The substitution

just mentioned is really equivalent to saying, Let

us assume that the regularity is fixed and permanent. It is

making a hypothesis which may not be altogether consistent

with fact, but which is forced upon us for the purpose of

securing precision of statement and definition.

14. The full meaning and bearing of such a substitution

will only become apparent in some of the subsequent

chapters, but it may be pointed out at once that it is in this

way only that we can with perfect strictness introduce the

notion of a 'limit' into our account of the matter, at any

rate in reference to many of the applications of the subject

to purely statistical enquiries. We say that a certain proportion

begins to prevail among the events in the long run;

but then on looking closer at the facts we find that we have

to express ourselves hypothetically, and to say that if present

circumstances remain as they are, the long run will show its

characteristics without disturbance. When, as is often the

case, we know nothing accurately of the circumstances by

which the succession of events is brought about, but have

strong reasons to suspect that these circumstances are likely

to undergo some change, there is really nothing else to be

done. We can only introduce the conception of a limit, towards

which the numbers are tending, by assuming that

these circumstances do not change; in other words, by substituting

a series with a fixed uniformity for the actual one

with the varying uniformity.[4]

15. If the reader will study the following example,

one well known to mathematicians under the name of the

Petersburg[5] problem, he will find that it serves to illustrate

several of the considerations mentioned in this chapter. It

serves especially to bring out the facts that the series with

which we are concerned must be regarded as indefinitely

extensive in point of number or duration; and that when so

regarded certain series, but certain series only (the one in

question being a case in point), take advantage of the indefinite

range to keep on producing individuals in it whose

deviation from the previous average has no finite limit

whatever. When rightly viewed it is a very simple problem,

but it has given rise, at one time or another, to a good

deal of confusion and perplexity.

The problem may be stated thus:--a penny is tossed up;

if it gives head I receive one pound; if heads twice running

two pounds; if heads three times running four pounds, and

so on; the amount to be received doubling every time that

a fresh head succeeds. That is, I am to go on as long as it

continues to give a succession of heads, to regard this succession

as a 'turn' or set, and then take another turn, and so

on; and for each such turn I am to receive a payment; the

occurrence of tail being understood to yield nothing, in fact

being omitted from our consideration. However many times

head may be given in succession, the number of pounds I

may claim is found by raising two to a power one less

than that number of times. Here then is a series formed by

a succession of throws. We will assume,--what many persons

will consider to admit of demonstration, and what

certainly experience confirms within considerable limits,--that

the rarity of these 'runs' of the same face is in direct

proportion to the amount I receive for them when they do

occur. In other words, if we regard only the occasions on

which I receive payments, we shall find that every other

time I get one pound, once in four times I get two pounds,

once in eight times four pounds, and so on without any end.

The question is then asked, what ought I to pay for this

privilege? At the risk of a slight anticipation of the results

of a subsequent chapter, we may assume that this is equivalent

to asking, what amount paid each time would on the

average leave me neither winner nor loser? In other words,

what is the average amount I should receive on the above

terms? Theory pronounces that I ought to give an _infinite_

sum: that is, no finite sum, however great, would be an

adequate equivalent. And this is really quite intelligible.

There is a series of indefinite length before me, and the

longer I continue to work it the richer are my returns, and

this without any limit whatever. It is true that the very

rich hauls are extremely rare, but still they do come, and

when they come they make it up by their greater richness.

On every occasion on which people have devoted themselves

to the pursuit in question, they made acquaintance, of course,

with but a limited portion of this series; but the series on

which we base our calculation is unlimited; and the inferences

usually drawn as to the sum which ought in the long

run to be paid for the privilege in question are in perfect

accordance with this supposition.

The common form of objection is given in the reply, that

so far from paying an infinite sum, no sensible man would

give anything approaching to £50 for such a chance. Probably

not, because no man would see enough of the series to

make it worth his while. What most persons form their

practical opinion upon, is such small portions of the series

as they have actually seen or can reasonably expect. Now

in any such portion, say one which embraces 100 turns, the

longest succession of heads would not amount on the average

to more than seven or eight. This is observed, but it is forgotten

that the formula which produced these, would, if it

had greater scope, keep on producing better and better ones

without any limit. Hence it arises that some persons are

perplexed, because the conduct they would adopt, in reference

to the curtailed portion of the series which they are practically

likely to meet with, does not find its justification in inferences

which are necessarily based upon the series in the completeness

of its infinitude.

16. This will be more clearly seen by considering the

various possibilities, and the scope required in order to exhaust

them, when we confine ourselves to a _limited_ number of

throws. Begin with three. This yields eight equally likely

possibilities. In four of these cases the thrower starts with

tail and therefore loses: in two he gains a single point

(i.e. £1); in one he gains two points, and in one he gains four

points. Hence his total gain being eight pounds achieved in

four different contingencies, his average gain would be two

pounds.

Now suppose he be allowed to go as far as n throws, so

that we have to contemplate 2^{n} possibilities. All of these

have to be taken into account if we wish to consider what

happens on the average. It will readily be seen that, when

all the possible cases have been reckoned once, his total gain

will be (reckoned in pounds),

2^{n-2} + 2^{n-3}·2 + 2^{n-4}·2^{2} + ... + 2·2^{n-3} + 2^{n-2} + 2^{n-1},

viz.

(n + 1) 2^{n-2}.

This being spread over 2^{n-1} different occasions of gain his

average gain will be 1/2 (n + 1).

Now when we are referring to averages it must be remembered

that the minimum number of different occurrences

necessary in order to justify the average is that which enables

each of them to present itself once. A man proposes to stop

short at a succession of ten heads. Well and good. We tell

him that his average gain will be £5. 10_s_. 0_d_.: but we also

impress upon him that in order to justify this statement he

must commence to toss at least 1024 times, for in no less

number can all the contingencies of gain and loss be exhibited

and balanced. If he proposes to reach an average gain of £20,

he will require to be prepared to go up to 39 throws,

To _justify_ this payment he must commence to throw 2^{39} times,

i.e. about a million million times. Not before he has

accomplished this will he be in a position to prove to any

sceptic that this is the true average value of a 'turn' extending

to 39 successive tosses.

Of course if he elects to toss to all eternity we must

adopt the line of explanation which alone is possible where

questions of infinity in respect of number and magnitude are

involved. We cannot tell him to pay down 'an infinite sum,'

for this has no strict meaning. But we tell him that, however

much he may consent to pay each time runs of heads occur,

he will attain at last a stage in which he will have won

back his total payments by his total receipts. However

large n may be, if he perseveres in trying 2^{n} times he _may_

have a true average receipt of 1/2 (n + 1) pounds, and if he

continues long enough onwards he _will_ have it.

The problem will recur for consideration in a future

chapter.

1. The following statistics will give a fair idea of the wide range of

experience over which such regularity is found to exist: "As

illustrations of equal amounts of fluctuation from totally

dissimilar causes, take the deaths in the West district of London in

seven years (fluctuation 13.66), and offences against the person

(fluctuation 13.61); or deaths from apoplexy (fluctuation 5.54), and

offences against property, without violence (fluctuation 5.48); or

students registered at the College of Surgeons (fluctuation 1.85),

and the number of pounds of manufactured tobacco taken for home

consumption (fluctuation 1.89); or out-door paupers (fluctuation

3.45) and tonnage of British vessels entered in ballast (fluctuation

3.43), &c." [Extracted from a paper in the Journal of the

Statistical Society, by Mr Guy, March, 1858; the 'fluctuation' here

given is a measure of the amount of irregularity, that is of

departure from the average, estimated in a way which will be

described hereafter.]

2. Transactions of the Cambridge Philosophical Society, Vol. IX. p. 605.

Reprinted in the collected edition of his writings, p. 50.

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.

3. In the first place then, he always, or almost always,

assumes that there can be but one and the same law of arrangement

for the results of our observations, measurements,

and so on, in these statistical enquiries. That is, he assumes

that whenever we get a group of such magnitudes

clustering about a mean, and growing less frequent as

they depart from that mean, we shall find that this diminution

of frequency takes place according to one invariable

law, whatever may be the nature of these magnitudes,

and whatever the process by which they may have

been obtained.

That such a uniformity as this should prevail amongst

many and various classes of phenomena would probably seem

surprising in any case. But the full significance of such a

fact as this (if indeed it were a fact) only becomes apparent

when attention is directed to the profound distinctions in the

nature and origin of the phenomena which are thus supposed

to be harmonized by being brought under one comprehensive

principle. This will be better appreciated if we take a brief

glance at some of the principal classes into which the things

with which Probability is chiefly concerned may be divided.

These are of a three-fold kind.

4. In the first place there are the various combinations,

and runs of luck, afforded by games of chance. Suppose

a handful, consisting of ten coins, were tossed up a

great many times in succession, and the results were tabulated.

What we should obtain would be something of the

following kind. In a certain proportion of cases, and these

the most numerous of all, we should find that we got five

heads and five tails; in a somewhat less proportion of cases

we should have, as equally frequent results, four heads six

tails, and four tails six heads; and so on in a continually

diminishing proportion until at length we came down, in a

very small relative number of cases, to nine heads one tail,

and nine tails one head; whilst the least frequent results

possible would be those which gave all heads or all tails.[3]

Here the statistical elements under consideration are, as

regards their origin at any rate, optional or brought about

by human choice. They would, therefore, be commonly

described as being mainly artificial, but their results ultimately

altogether a matter of chance.

Again, in the second place, we might take the accurate

measurements--i.e. the actual magnitudes themselves,--of

a great many natural objects, belonging to the same genus

or class; such as the cases, already referred to, of the heights,

or other characteristics of the inhabitants of any district.

Here human volition or intervention of any kind seem to

have little or nothing to do with the matter. It is optional

with us to _collect_ the measures, but the things measured are

quite outside our control. They would therefore be commonly

described as being altogether the production of nature,

and it would not be supposed that in strictness chance had

anything whatever to do with the matter.

In the third place, the result at which we are aiming

may be some fixed magnitude, one and the same in each

of our successive attempts, so that if our measurements

were rigidly accurate we should merely obtain the same

result repeated over and over again. But since all our

methods of attaining our aims are practically subject to

innumerable imperfections, the results actually obtained

will depart more or less, in almost every case, from the

real and fixed value which we are trying to secure. They

will be sometimes more wide of the mark, sometimes less

so, the worse attempts being of course the less frequent.

If a man aims at a target he will seldom or never hit it

precisely in the centre, but his good shots will be more[4]

numerous than his bad ones. Here again, then, we have

a series of magnitudes (i.e. the deflections of the shots from

the point aimed at) clustering about a mean, but produced

in a very different way from those of the last two cases.

In this instance the elements would be commonly regarded

as only partially the results of human volition, and chance

therefore as being only a co-agent in the effects produced.

With these must be classed what may be called _estimates_,

as distinguished from measurements. By the latter are

generally understood the results of a certain amount of

mechanism or manipulation; by the former we may understand

those cases in which the magnitude in question is

determined by direct observation or introspection. The

interest and importance of this class, so far as scientific

principles are concerned, dates mainly from the investigations

of Fechner. Its chief field is naturally to be found amongst

psychological data.

Other classes of things, besides those alluded to above,

might readily be given. These however are the classes about

which the most extensive statistics are obtainable, or to

which the most practical importance and interest are attached.

The profound distinctions which separate their

origin and character are obvious. If they all really did

display precisely the same law of variation it would be a

most remarkable fact, pointing doubtless to some deep-seated

identity underlying the various ways, apparently

so widely distinct, in which they had been brought about.

The questions now to be discussed are: Is it the case,

with any considerable degree of rigour, that only one law

of distribution does really prevail? and, in so far as this

is so, how does it come to pass?

5. In support of an affirmative answer to the former

of these two questions, several different kinds of proof are,

or might be, offered.

(I.) For one plan we may make a direct appeal to

experience, by collecting sets of statistics and observing

what is their law of distribution. As remarked above, this

has been done in a great variety of cases, and in some

instances to a very considerable extent, by Quetelet and

others. His researches have made it abundantly convincing

that many classes of things and processes, differing widely

in their nature and origin, do nevertheless appear to conform

with a considerable degree of accuracy to one and the

same[5] law. At least this is made plain for the more

central values, for those that is which are situated most

nearly about the mean. With regard to the extreme values

there is, on the other hand, some difficulty. For instance

in the arrangements of the heights of a number of men,

these extremes are rather a stumbling-block; indeed it has

been proposed to reject them from both ends of the scale

on the plea that they are monstrosities, the fact being that

their relative numbers do not seem to be by any means

those which theory would assign.[6] Such a plan of rejection

is however quite unauthorized, for these dwarfs and giants

are born into the world like their more normally sized

brethren, and have precisely as much right as any others

to be included in the formulæ we draw up.

Besides the instance of the heights of men, other classes

of observations of a somewhat similar character have been

already referred to as collected and arranged by Quetelet.

From the nature of the case, however, there are not many

appropriate ones at hand; for when our object is, not to

illustrate a law which can be otherwise proved, but to

obtain actual direct proof of it, the collection of observations

and measurements ought to be made upon such a large

scale as to deter any but the most persevering computers

from undergoing the requisite labour. Some of the remarks

made in the course of the note on the opposite page will

serve to illustrate the difficulties which would lie in the way

of such a mode of proof.

We are speaking here, it must be understood, only of

_symmetrical_ curves: if there is asymmetry, i.e. if the Law of

Error is different on different sides of the mean,--a comparatively

very small number of observations would suffice

to detect the fact. But, granted symmetry and rapid

decrease of frequency on each side of the mean, we could

generally select some one species of the exponential curve

which should pretty closely represent our statistics in the

neighbourhood of the mean. That is, where the statistics are

numerous we could secure agreement; and where we could

not secure agreement the statistics would be comparatively

so scarce that we should have to continue the observations

for a very long time in order to prove the disagreement.

6. Allowing the various statistics such credit as they

deserve, for their extent, appropriateness, accuracy and so

on, the general conclusion which will on the whole be drawn

by almost every one who takes the trouble to consult them,

is that they do, in large part, conform approximately to one

type or law, at any rate for all except the extreme values.

So much as this must be fully admitted. But that they do

not, indeed we may say that they cannot, always do so in

the case of the extreme values, will become obvious on

a little consideration. In some of the classes of things to

which the law is supposed to apply, for example, the successions

of heads and tails in the throws of a penny, there is

no limit to the magnitude of the fluctuations which may and

will occur. Postulate as long a succession of heads or of tails

as we please, and if we could only live and toss long enough

for it we should succeed in getting it at length. In other

cases, including many of the applications of Probability

to natural phenomena, there can hardly fail to be such

limits. Deviations exceeding a certain range may not be

merely improbable, that is of very rare occurrence, but they

may often from the nature of the case be actually impossible.

And even when they are not actually impossible it

may frequently appear on examination that they are only

rendered possible by the occasional introduction of agencies

which are not supposed to be available in the production

of the more ordinary or intermediate values. When, for

instance, we are making observations with any kind of

instrument, the nature of its construction may put an

absolute limit upon the possible amount of error. And even

if there be not an absolute limit under all kinds of usage

it may nevertheless be the case that there is one under

fair and proper usage; it being the case that only when

the instrument is designedly or carelessly tampered with will

any new causes of divergence be introduced which were not

confined within the old limits.

Suppose, for instance, that a man is firing at a mark.

His worst shots must be supposed to be brought about by

a combination of such causes as were acting, or prepared

to act, in every other case; the extreme instance of what

we may thus term 'fair usage' being when a number of

distinct causes have happened to conspire together so as

to tend in the same direction, instead of, as in the other

cases, more or less neutralizing one another's work. But

the aggregate effect of such causes may well be supposed

to be limited. The man will not discharge his shot nearly

at right angles to the true line of fire unless some entirely

new cause comes in, as by some unusual circumstance

having distracted his attention, or by his having had some

spasmodic seizure. But influences of this kind were not

supposed to have been available before; and even if they

were we are taking a bold step in assuming that these

occasional great disturbances are subject to the same kind

of laws as are the aggregates of innumerable little ones.

We cannot indeed lay much stress upon an example

of this last kind, as compared with those in which we

can see for certain that there is a fixed limit to the range

of error. It is therefore offered rather for illustration than

for proof. The enormous, in fact inconceivable magnitude

of the numbers expressive of the chance of very rare combinations,

such as those in question, has such a bewildering

effect upon the mind that one may be sometimes apt to confound

the impossible with the higher degrees of the merely

mathematically improbable.

7. At the time the first edition of this essay was composed

writers on Statistics were, I think, still for the most

part under the influence of Quetelet, and inclined to overvalue

his authority on this particular subject: of late however

attention has been repeatedly drawn to the necessity of

taking account of other laws of arrangement than the binomial

or exponential.

Mr Galton, for instance,--to whom every branch of the

theory of statistics owes so much,--has insisted[7] that the

"assumption which lies at the basis of the well-known law of

'Frequency of Error'... is incorrect in many groups of vital

and social phenomena.... For example, suppose we endeavour

to match a tint; Fechner's law, in its approximative and

simplest form of sensation = log stimulus, tells us that a

series of tints, in which the quantities of white scattered on a

black ground are as 1, 2, 4, 8, 16, 32, &c., will appear to the

eye to be separated by equal intervals of tint. Therefore, in

matching a grey that contains 8 portions of white, we are

just as likely to err by selecting one that has 16 portions as

one that has 4 portions. In the first case there would be an

error in excess, of 8; in the second there would be an error,

in deficiency, of 4. Therefore, an error of the same magnitude

in excess or in deficiency is not equally probable." The consequences

of this assumption are worked out in a remarkable

paper by Dr D. McAlister, to which allusion will have to be

made again hereafter. All that concerns us here to point out

is that when the results of statistics of this character are

arranged graphically we do _not_ get a curve which is symmetrical

on both sides of a central axis.

8. More recently, Mr F. Y. Edgeworth (in a report of

a Committee of the British Association appointed to enquire

into the variation of the monetary standard) has urged the

same considerations in respect of prices of commodities. He

gives a number of statistics "drawn from the prices of twelve

commodities during the two periods 1782-1820, 1820-1865.

The maximum and minimum entry for each series having

been noted, it is found that the number of entries above the

'middle point,' half-way between the maximum and minimum,[8]

is in every instance less than half the total number of entries

in the series. In the twenty-four trials there is not a single

exception to the rule, and in very few cases even an approach

to an exception. We may presume then that the curves are

of the lop-sided character indicated by the accompanying

diagram." The same facts are also ascertained in respect to

_place_ variations as distinguished from time variations. To

these may be added some statistics of my own, referring to

the heights of the barometer taken at the same hour on more

than 4000 successive days (v. _Nature_, Sept. 2, 1887). So far

as these go they show a marked asymmetry of arrangement.

In fact it appears to me that this want of symmetry

ought to be looked for in all cases in which the phenomena

under measurement are of a 'one-sided' character; in the

sense that they are measured on one side only of a certain

fixed point from which their possibility is supposed to start.

For not only is it impossible for them to fall below this point:

long before they reach it the influence of its proximity is felt

in enhancing the difficulty and importance of the same

amount of absolute difference.

Look at a table of statures, for instance, with a mean

value of 69 inches. A diminution of three feet (were this

possible) is much more influential,--counts for much more,

in every sense of the term,--than an addition of the same

amount; for the former does not double the mean, while the

latter more than halves it. Revert to an illustration. If

a vast number of petty influencing circumstances of the kind

already described were to act upon a swinging _pendulum_ we

should expect the deflections in each direction to display

symmetry; but if they were to act upon a _spring_ we should

not expect such a result. Any phenomena of which the

latter is the more appropriate illustration can hardly be

expected to range themselves with symmetry about a mean.[9]

9. (II.) The last remarks will suggest another kind of

proof which might be offered to establish the invariable nature

of the law of error. It is of a direct deductive kind, not

appealing immediately to statistics, but involving an enquiry

into the actual or assumed nature of the causes by which the

events are brought about. Imagine that the event under

consideration is brought to pass, in the first place, by some

fixed cause, or group of fixed causes. If this comprised all

the influencing circumstances the event would invariably

happen in precisely the same way: there would be no errors

or deflections whatever to be taken account of. But now

suppose that there were also an enormous number of very

small causes which tended to produce deflections; that these

causes acted in entire independence of one another; and that

each of the lot told as often, in the long run, in one direction

as in the opposite. It is easy[10] to see, in a general way, what

would follow from these assumptions. In a very few cases

nearly all the causes would tell in the same direction; in

other words, in a very few cases the deflection would be

extreme. In a greater number of cases, however, it would

only be the most part of them that would tell in one direction,

whilst a few did what they could to counteract the rest;

the result being a comparatively larger number of somewhat

smaller deflections. So on, in increasing numbers, till we

approach the middle point. Here we shall have a very large

number of very small deflections: the cases in which the

opposed influences just succeed in balancing one another,

so that no error whatever is produced, being, though actually

infrequent, relatively the most frequent of all.

Now if all deflections from a mean were brought about in

the way just indicated (an indication which must suffice for

the present) we should always have one and the same law of

arrangement of frequency for these deflections or errors, viz.

the exponential[11] law mentioned in §5.

10. It may be readily admitted from what we know

about the production of events that something resembling

these assumptions, and therefore something resembling the

consequences which follow from them, is really secured in a

very great number of cases. But although this may prevail

approximately, it is in the highest degree improbable that it

could ever be secured, even artificially, with anything approaching

to rigid accuracy. For one thing, the causes of

deflection will seldom or never be really independent of one

another. Some of them will generally be of a kind such that

the supposition that several are swaying in one direction,

may affect the capacity of each to produce that full effect

which it would have been capable of if it had been left to do

its work alone. In the common example, for instance, of

firing at a mark, so long as we consider the case of the tolerably

good shots the effect of the wind (one of the causes of

error) will be approximately the same whatever may be the

precise direction of the bullet. But when a shot is considerably

wide of the mark the wind can no longer be regarded as

acting at right angles to the line of flight, and its effect in

consequence will not be precisely the same as before. In

other words, the causes here are not strictly independent, as

they were assumed to be; and consequently the results to be

attributed to each are not absolutely uninfluenced by those

of the others. Doubtless the effect is trifling here, but I

apprehend that if we were carefully to scrutinize the modes

in which the several elements of the total cause conspire

together, we should find that the assumption of absolute

independence was hazardous, not to say unwarrantable, in a

very great number of cases. These brief remarks upon the

process by which the deflections are brought about must

suffice for the present purpose, as the subject will receive a

fuller investigation in the course of the next chapter.

According, therefore, to the best consideration which

can at the present stage be afforded to this subject, we may

draw a similar conclusion from this deductive line of argument

as from the direct appeal to statistics. The same

general result seems to be established; namely, that approximately,

with sufficient accuracy for all practical purposes, we

may say that an examination of the causes by which the

deflections are generally brought about shows that they are

mostly of such a character as would result in giving us the

commonly accepted 'Law of Error,' as it is termed.[12] The

two lines of enquiry, therefore, within the limits assigned,

afford each other a decided mutual confirmation.

11. (III.) There still remains a third, indirect and

mathematical line of proof, which might be offered to establish

the conclusion that the Law of Error is always one and

the same. It may be maintained that the recognized and

universal employment of one and the same method, that

known to mathematicians and astronomers as the Method of

Least Squares, in all manner of different cases with very

satisfactory results, is compatible only with the supposition

that the errors to which that method is applied must be

grouped according to one invariable law. If all 'laws of

error' were not of one and the same type, that is, if the

relative frequency of large and small divergences (such as we

have been speaking of) were not arranged according to one

pattern, how could one method or rule equally suit them all?

In order to preserve a continuity of treatment, some

notice must be taken of this enquiry here, though, as in the

case of the last argument, any thorough discussion of the

subject is impossible at the present stage. For one thing, it

would involve too much employment of mathematics, or at

any rate of mathematical conceptions, to be suitable for the

general plan of this treatise: I have accordingly devoted a

special chapter to the consideration of it.

The main reason, however, against discussing this argument

here, is, that to do so would involve the anticipation of

a totally different side of the science of Probability from that

hitherto treated of. This must be especially insisted upon, as

the neglect of it involves much confusion and some error.

During these earlier chapters we have been entirely occupied

with laying what may be called the physical foundations of

Probability. We have done nothing else than establish, in one

way or another, the existence of certain groups or arrangements

of things which are found to present themselves in

nature; we have endeavoured to explain how they come to

pass, and we have illustrated their principal characteristics.

But these are merely the foundations of Inference, we have

not yet said a word upon the logical processes which are to

be erected upon these foundations. We have not therefore

entered yet upon the _logic_ of chance.

12. Now the way in which the Method of Least Squares

is sometimes spoken of tends to conceal the magnitude of

this distinction. Writers have regarded it as synonymous

with the Law of Error, whereas the fact is that the two are

not only totally distinct things but that they have scarcely

even any necessary connection with each other. The Law of

Error is the statement of a physical fact; it simply assigns,

with more or less of accuracy, the relative frequency with

which errors or deviations of any kind are found in practice

to present themselves. It belongs therefore to what may be

termed the physical foundations of the science. The Method

of Least Squares, on the other hand, is not a law at all in the

scientific sense of the term. It is simply a rule or direction

informing us how we may best proceed to treat any group of

these errors which may be set before us, so as to extract the

true result at which they have been aiming. Clearly therefore

it belongs to the inferential or logical part of the subject.

It cannot indeed be denied that the methods we employ

must have some connection with the arrangement of the facts

to which they are applied; but the two things are none the

less distinct in their nature, and in this case the connection

does not seem at all a necessary one, but at most one of propriety

and convenience. The Method of Least Squares is

usually applied, no doubt, to the most familiar and common

form of the Law of Error, namely the exponential form with

which we have been recently occupied. But other forms of

laws of error may exist, and, if they did, the method in

question might equally well be applied to them. I am not

asserting that it would necessarily be the best method in

every case, but it would be a possible one; indeed we may

go further and say, as will be shown in a future chapter,

that it would be a good method in almost every case. But

its particular merits or demerits do not interfere with its

possible employment in every case in which we may choose

to resort to it. It will be seen therefore, even from the few

remarks that can be made upon the subject here, that the

fact that one and the same method is very commonly employed

with satisfactory results affords little or no proof that

the errors to which it is applied must be arranged according

to one fixed law.

13. So much then for the attempt to prove the prevalence,

in all cases, of this particular law of divergence. The

next point in Quetelet's treatment of the subject which deserves

attention as erroneous or confusing, is the doctrine maintained

by him and others as to the existence of what he terms a _type_

in the groups of things in question. This is a not unnatural

consequence from some of the data and conclusions of the

last few paragraphs. Refer back to two of the three classes

of things already mentioned in §4. If it really were the case

that in arranging in order a series of incorrect observations

or attempts of our own, and a collection of natural objects

belonging to some one and the same species or class, we found

that the law of their divergence was in each case identical in

the long run, we should be naturally disposed to apply the

same expression 'Law of Error' to both instances alike,

though in strictness it could only be appropriate to the

former. When we perform an operation ourselves with a

clear consciousness of what we are aiming at, we may quite

correctly speak of every deviation from this as being an

error; but when Nature presents us with a group of objects

of any kind, it is using a rather bold metaphor to speak in

this case also of a law of error, as if she had been aiming at

something all the time, and had like the rest of us missed

her mark more or less in almost every instance.[13]

Suppose we make a long succession of attempts to measure

accurately the precise height of a man, we should from one

cause or another seldom or never succeed in doing so with

absolute accuracy. But we have no right to assume that these

imperfect measurements of ours would be found so to deviate

according to one particular law of error as to present the

precise counterpart of a series of actual heights of _different_

men, supposing that these latter were assigned with absolute

precision. What might be the actual law of error in a series

of direct measurements of any given magnitude could hardly

be asserted beforehand, and probably the attempt to determine

it by experience has not been made sufficiently often to

enable us to ascertain it; but upon general grounds it seems

by no means certain that it would follow the so-called exponential

law. Be this however as it may, it is rather a

licence of language to talk as if nature had been at work in

the same way as one of us; aiming (ineffectually for the most

part) at a given result, that is at producing a man endowed

with a certain stature, proportions, and so on, who might

therefore be regarded as the typical man.

14. Stated as above, namely, that there is a fixed

invariable human type to which all individual specimens of

humanity may be regarded as having been meant to attain,

but from which they have deviated in one direction or

another; according to a law of deviation capable of _à priori_

determination, the doctrine is little else than absurd. But

if we look somewhat closer at the facts of the case, and the

probable explanation of these facts, we may see our way to

an important truth. The facts, on the authority of Quetelet's

statistics (the great interest and value of which must be

frankly admitted), are very briefly as follows: if we take any

element of our physical frame which admits of accurate

measurement, say the height, and determine this measure in

a great number of different individuals belonging to any

tolerably homogeneous class of people, we shall find that

these heights do admit of an orderly arrangement about a

mean, after the fashion which has been already repeatedly

mentioned. What is meant by a homogeneous class? is a

pertinent and significant enquiry, but applying this condition

to any simple cases its meaning is readily stated. It implies

that the mean in question will be different according to the

nationality of the persons under measurement. According to

Quetelet,[14] in the case of Englishmen the mean is about

5 ft. 9 in.; for Belgians about 5 ft. 7 in.; for the French about

5 ft. 4 in. It need hardly be added that these measures are

those of adult males.

15. It may fairly be asked here what would have

been the consequence, had we, instead of keeping the English

and the French apart, mixed the results of our measurements

of them all together? The question is an important one, as it

will oblige us to understand more clearly what we mean by

homogeneous classes. The answer that would usually be

given to it, though substantially correct, is somewhat too decisive

and summary. It would be said that we are here

mixing distinctly heterogeneous elements, and that in consequence

the resultant law of error will be by no means of

the simple character previously exhibited. So far as such an

answer is to be admitted its grounds are easy to appreciate.

In accordance with the usual law of error the divergences

from the mean grow continuously less numerous as they

increase in amount. Now, if we mix up the French and

English heights, what will follow? Beginning from the

English mean of 5 feet 9 inches, the heights will at first follow

almost entirely the law determined by these English

conditions, for at this point the English data are very numerous,

and the French by comparison very few. But, as we

begin to approach the French mean, the numbers will cease

to show that continual diminution which they should show,

according to the English scale of arrangement, for here the

French data are in turn very numerous, and the English by

comparison few. The result of such a combination of heterogeneous

elements is illustrated by the figure annexed, of

course in a very exaggerated form.

[Figure: Superposition of two Gaussians.]

16. In the above case the nature of the heterogeneity,

and the reasons why the statistics should be so collected and

arranged as to avoid it, seemed tolerably obvious. It will be

seen still more plainly if we take a parallel case drawn from

artificial proceedings. Suppose that after a man had fired a

few thousand shots at a certain spot, say a wafer fixed somewhere

on a wall, the position of the spot at which he aims

were shifted, and he fired a few thousand more shots at the

wafer in its new position. Now let us collect and arrange all

the shots of both series in the order of their departure from

either of the centres, say the new one. Here we should

really be mingling together two discordant sets of elements,

either of which, if kept apart from the other, would have

been of a simple and homogeneous character. We should

find, in consequence, that the resultant law of error betrayed

its composite or heterogeneous origin by a glaring departure

from the customary form, somewhat after the fashion indicated

in the above diagram.

The instance of the English and French heights resembles

the one just given, but falls far short of it in the stringency

with which the requisite conditions are secured. The

fact is we have not here got the most suitable requirements,

viz. a group consisting of a few fixed causes supplemented by

innumerable little disturbing influences. What we call a

nation is really a highly artificial body, the members of

which are subject to a considerable number of local or occasional

disturbing causes. Amongst Frenchmen were included,

presumably, Bretons, Provençals, Alsatians, and so on,

thus commingling distinctions which, though less than those

between French and English, regarded as wholes, are very

far from being insignificant. And to these differences of

race must be added other disturbances, also highly important,

dependent upon varying climate, food and occupation.

It is plain, therefore, that whatever objections exist against

confusing together French and English statistics, exist also,

though of course in a less degree, against confusing together

those of the various provincial and other components which

make up the French people.

17. Out of the great variety of important causes

which influence the height of men, it is probable that those

which most nearly fulfil the main conditions required by the

'Law of Error' are those about which we know the least.

Upon the effects of food and employment, observation has

something to say, but upon the purely physiological causes

by which the height of the parents influences the height of

the offspring, we have probably nothing which deserves to

be called knowledge. Perhaps the best supposition we can

make is one which, in accordance with the saying that 'like

breeds like', would assume that the purely physiological

causes represent the constant element; that is, given a homogeneous

race of people to begin with, who freely intermarry,

and are subject to like circumstances of climate, food,

and occupation, the standard would remain on the whole

constant.[15]

In such a case the man who possessed the mean height,

mean weight, mean strength, and so on, might then be

called, in a sort of way, a 'type'. The deviations from this

type would then be produced by innumerable small influences,

partly physiological, partly physical and social, acting

for the most part independently of one another, and resulting

in a Law of Error of the usual description. Under such

restrictions and explanations as these, there seems to be no

reasonable objection to speaking of a French or English type

or mean. But it must always be remembered that under

the present circumstances of every political nation, these

somewhat heterogeneous bodies might be subdivided into

various smaller groups, each of which would frequently exhibit

the characteristics of such a type in an even more

marked degree.

18. On this point the reports of the Anthropometrical

Committee, already referred to, are most instructive. They

illustrate the extent to which this subdivision could be

carried out, and prove,--if any proof were necessary,--that

the discovery of Quetelet's _homme moyen_ would lead us a

long chase. So far as their results go the mean 'English'

stature (in inches) is 67.66. But this is composed of Scotch,

Irish, English and Welsh constituents, the separate means of

these being, respectively; 68.71, 67.90, 67.36, and 66.66.

But these again may be subdivided; for careful observation

shows that the mean English stature is distinctly greater in

certain districts (e.g. the North-Eastern counties) than in

others. Then again the mean of the professional classes is

considerably greater than that of the labourers; and that of

the honest and intelligent is very much greater than that of

the criminal and lunatic constituents of the population.

And, so far as the observations are extensive enough for the

purpose, it appears that every characteristic in respect of the

grouping about a mean which can be detected in the more

extensive of these classes can be detected also in the narrower.

Nor is there any reason to suppose that the same

process of subdivision could not be carried out as much

farther as we chose to prolong it.

19. It need hardly be added to the above remarks

that no one who gives the slightest adhesion to the Doctrine

of Evolution could regard the type, in the above qualified

sense of the term, as possessing any real permanence and

fixity. If the constant causes, whatever they may be, remain

unchanged, and if the variable ones continue in the

long run to balance one another, the results will continue to

cluster about the same mean. But if the constant ones

undergo a gradual change, or if the variable ones, instead of

balancing each other suffer one or more of their number to

begin to acquire a preponderating influence, so as to put a

sort of bias upon their aggregate effect, the mean will at

once begin, so to say, to shift its ground. And having once

begun to shift, it may continue to do so, to whatever extent

we recognize that Species are variable and Development is a

fact. It is as if the point on the target at which we aim, instead

of being fixed, were slowly changing its position as we

continue to fire at it; changing almost certainly to some extent

and temporarily, and not improbably to a considerable

extent and permanently.

20. Our examples throughout this chapter have been

almost exclusively drawn from physical characteristics,

whether of man or of inanimate things; but it need not be

supposed that we are necessarily confined to such instances.

Mr Galton, for instance, has proposed to extend the same

principles of calculation to mental phenomena, with a view

to their more accurate determination. The objects to be

gained by so doing belong rather to the inferential part of

our subject, and will be better indicated further on; but

they do not involve any distinct principle. Like other attempts

to apply the methods of science in the region of the

mind, this proposal has met with some opposition; with very

slight reason, as it seems to me. That our mental qualities,

if they could be submitted to accurate measurement, would

be found to follow the usual Law of Error, may be assumed

without much hesitation. The known extent of the correlation

of mental and bodily characteristics gives high probability

to the supposition that what is proved to prevail, at

any rate approximately, amongst most bodily elements which

have been submitted to measurement, will prevail also

amongst the mental elements.

To what extent such measurements could be carried

out practically, is another matter. It does not seem to

me that it could be done with much success; partly because

our mental qualities are so closely connected with,

indeed so run into one another, that it is impossible to

isolate them for purposes of comparison.[16] This is to some

extent indeed a difficulty in bodily measurements, but it is

far more so in those of the mind, where we can hardly get

beyond what can be called a good guess. The doctrine,

therefore, that mental qualities follow the now familiar law

of arrangement can scarcely be grounded upon anything

more than a strong analogy. Still this analogy is quite

strong enough to justify us in accepting the doctrine and

all the conclusions which follow from it, in so far as our

estimates and measurements can be regarded as trustworthy.

There seems therefore nothing unreasonable in the attempt

to establish a system of natural classification of mankind

by arranging them into a certain number of groups above

and below the average, each group being intended to correspond

to certain limits of excellency or deficiency.[17] All

that is necessary for such a purpose is that the rate of

departure from the mean should be tolerably constant under

widely different circumstances: in this case throughout all

the races of man. Of course if the law of divergence is

the same as that which prevails in inanimate nature we

have a still wider and more natural system of classification

at hand, and one which ought to be familiar, more or less, to

every one who has thus to estimate qualities.

21. Perhaps one of the best illustrations of the legitimate

application of such principles is to be found in Mr

Galton's work on _Hereditary Genius_. Indeed the full force

and purport of some of his reasonings there can hardly be

appreciated except by those who are familiar with the conceptions

which we have been discussing in this chapter. We

can only afford space to notice one or two points, but the

student will find in the perusal, of at any rate the more

argumentive parts, of that volume[18] an interesting illustration

of the doctrines now under discussion. For one thing it

may be safely asserted, that no one unfamiliar with the Law

of Error would ever in the least appreciate the excessive

rapidity with which the superior degrees of excellence tend

to become scarce. Every one, of course, can see at once, in

a numerical way at least, what is involved in being 'one of a

million'; but they would not at all understand, how very

little extra superiority is to be looked for in the man who is

'one of two million'. They would confound the mere numerical

distinction, which seems in some way to imply

double excellence, with the intrinsic superiority, which

would mostly be represented by a very small fractional advantage.

To be 'one of ten million' sounds very grand, but

if the qualities under consideration could be estimated in

themselves without the knowledge of the vastly wider area

from which the selection had been made, and in freedom

therefore from any consequent numerical bias, people would

be surprised to find what a very slight comparative superiority

was, as a rule, thus obtained.

22. The point just mentioned is an important one in

arguments from statistics. If, for instance, we find a small

group of persons, connected together by blood-relationship,

and all possessing some mental characteristic in marked

superiority, much depends upon the comparative rarity of

such excellence when we are endeavouring to decide whether

or not the common possession of these qualities was accidental.

Such a decision can never be more than a rough

one, but if it is to be made at all this consideration must

enter as a factor. Again, when we are comparing one nation

with another,[19] say the Athenian with any modern European

people, does the popular mind at all appreciate what sort of

evidence of general superiority is implied by the production,

out of one nation, of such a group as can be composed of

Socrates, Plato, and a few of their contemporaries? In this

latter case we are also, it should be remarked, employing the

'Law of Error' in a second way; for we are assuming that

where the extremes are great so will also the means be, in

other words we are assuming that every amount of departure

from the mean occurs with a (roughly) calculable degree

of relative frequency. However generally this truth may

be accepted in a vague way, its evidence can only be appreciated

by those who know the reasons which can be given

in its favour.

But the same principles will also supply a caution in

the case of the last example. They remind us that, for

the mere purpose of comparison, the _average_ man of any

group or class is a much better object for selection than

the eminent one. There may be greater difficulties in the

way of detecting him, but when we have done so we have

got possession of a securer and more stable basis of comparison.

He is selected, by the nature of the case, from

the most numerous stratum of his society; the eminent

man from a thinly occupied stratum. In accordance therefore

with the now familiar laws of averages and of large

numbers the fluctuations amongst the former will generally

be very few and small in comparison with those amongst the

latter.

1. _Essai de Physique Sociale_, 1869. _Anthropométrie_, 1870.

2. As regards later statistics on the same subject the reader can

refer to the Reports of the Anthropometrical Committee of the

British Association (1879, 1880, 1881, 1883;--especially this

last). These reports seem to me to represent a great advance on the

results obtained by Quetelet, and fully to justify the claim of the

Secretary (Mr C. Roberts) that their statistics are "unique in range

and numbers". They embrace not merely military recruits--like most

of the previous tables--but almost every class and age, and both

sexes. Moreover they refer not only to stature but to a number of

other physical characteristics.

3. As every mathematician knows, the relative numbers of each of these

possible throws are given by the successive terms of the expansion

of (1 + 1)^{10}, viz. 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1.

4. That is they will be more densely aggregated. If a space _the size

of the bull's-eye_ be examined in each successive circle, the number

of shot marks which it contains will be successively less. The

_actual_ number of shots which strike the bull's-eye will not be the

greatest, since it covers so much less surface than any of the other

circles.

5. Commonly called the exponential law; its equation being of the form

y = Ae^{-hx^{2}}. The curve corresponding to it cuts the axis of y

at right angles (expressing the fact that near the mean there are a

large number of values approximately equal); after a time it begins

to slope away rapidly towards the axis of x (expressing the fact

that the results soon begin to grow less common as we recede from

the mean); and the axis of x is an asymptote in both directions

(expressing the fact that no magnitude, however remote from the

mean, is strictly impossible; that is, every deviation, however

excessive, will have to be encountered at length within the range of

a sufficiently long experience). The curve is obviously symmetrical,

expressing the fact that equal deviations from the mean, in excess

and in defect, tend to occur equally often in the long run.

[Figure: Gaussian normal distribution.]

A rough graphic representation of the curve is given above. For the

benefit of those unfamiliar with mathematics one or two brief

remarks may be here appended concerning some of its properties.

(1) It must not be supposed that all specimens of the curve are

similar to one another. The dotted lines are equally specimens of

it. In fact, by varying the essentially arbitrary units in which x

and y are respectively estimated, we may make the portion towards

the vertex of the curve as obtuse or as acute as we please. This

consideration is of importance; for it reminds us that, by varying

one of these arbitrary units, we could get an 'exponential curve'

which should tolerably closely resemble any symmetrical curve of

error, provided that this latter recognized and was founded upon the

assumption that extreme divergences were excessively rare. Hence it

would be difficult, by mere observation, to prove that the law of

error in any given case was not exponential; unless the statistics

were very extensive, or the actual results departed considerably

from the exponential form. (2) It is quite impossible by any graphic

representation to give an adequate idea of the excessive rapidity

with which the curve after a time approaches the axis of x. At the

point R, on our scale, the curve would approach within the

fifteen-thousandth part of an inch from the axis of x, a distance

which only a very good microscope could detect. Whereas in the

hyperbola, e.g. the rate of approach of the curve to its asymptote

is continually decreasing, it is here just the reverse; this rate is

continually increasing. Hence the two, viz. the curve and the axis

of x, appear to the eye, after a very short time, to merge into one

another.

6. As by Quetelet: noted, amongst others, by Herschel, _Essays_, page

409.

7. _Proc. R. Soc._ Oct. 21, 1879.

8. We are here considering, remember, the case of a _finite_ amount of

statistics; so that there are actual limits at each end.

9. It must be admitted that experience has not yet (I believe) shown

this asymmetry in respect of heights.

10. The above reasoning will probably be accepted as valid at this

stage of enquiry. But in strictness, assumptions are made here,

which however justifiable they may be in themselves, involve

somewhat of an anticipation. They demand, and in a future chapter

will receive, closer scrutiny and criticism.

11. A definite numerical example of this kind of concentration of

frequency about the mean was given in the note to §4. It was of a

binomial form, consisting of the successive terms of the expansion

of (1 + 1)^{m}. Now it may be shown (Quetelet, _Letters_, p. 263;

Liagre, _Calcul des Probabilités_, §34) that the expansion of such

a binomial, as m becomes indefinitely great, approaches as its limit

the exponential form; that is, if we take a number of equidistant

ordinates proportional respectively to 1, m, m(m - 1)/(1·2) &c., and

connect their vertices, the figure we obtain approximately

represents some form of the curve y = Ae^{-hx^{2}}, and tends to

become identical with it, as m is increased without limit. In other

words, if we suppose the errors to be produced by a limited number

of finite, equal and independent causes, we have an approximation to

the exponential Law of Error, which merges into identity as the

causes are increased in number and diminished in magnitude without

limit. Jevons has given (_Principles of Science_, p. 381) a diagram

drawn to scale, to show how rapid this approximation is. One point

must be carefully remembered here, as it is frequently overlooked

(by Quetelet, for instance). The coefficients of a binomial of two

equal terms--as (1 + 1)^{m}, in the preceding paragraph--are

symmetrical in their arrangement from the first, and very speedily

become indistinguishable in (graphical) outline from the final

exponential form. But if, on the other hand, we were to consider the

successive terms of such a binomial as (1 + 4)^{m} (which are

proportional to the relative chances of 0, 1, 2, 3, ... failures in

m ventures, of an event which has one chance in its favour to four

against it) we should have an unsymmetrical succession. If however

we suppose m to increase without limit, as in the former

supposition, the unsymmetry gradually disappears and we tend towards

precisely the same exponential form as if we had begun with two

equal terms. The only difference is that the position of the vertex

of the curve is no longer in the centre: in other words, the

likeliest term or event is not an equal number of successes and

failures but successes and failures in the ratio of 1 to 4.

12. 'Law of Error' is the usual technical term for what has been

elsewhere spoken of above as a Law of Divergence from a mean. It is

in strictness only appropriate in the case of one, namely the third,

of the three classes of phenomena mentioned in §4, but by a

convenient generalization it is equally applied to the other two; so

that we term the amount of the divergence from the mean an 'error'

in every case, however it may have been brought about.

13. This however seems to be the purport, either by direct assertion

or by implication, of two elaborate works by Quetelet, viz. his

_Physique Sociale_ and his _Anthropométrie_.

14. He scarcely, however, professes to give these as an accurate

measure of the mean height, nor does he always give precisely the

same measure. Practically, none but soldiers being measured in any

great numbers, the English stature did not afford accurate data on

any large scale. The statistics given a few pages further on are

probably far more trustworthy.

15. This statement will receive some explanation and correction in the

next chapter.

16. I am not speaking here of the now familiar results of

Psychophysics, which are mainly occupied with the measurement of

perceptions and other simple states of consciousness.

17. Perhaps the best brief account of Mr Galton's method is to be

found in a paper in _Mind_ (July, 1880) on the statistics of Mental

Imagery. The subject under comparison here--viz. the relative

power, possessed by different persons, of raising clear visual

images of objects no longer present to us--is one which it seems

impossible to 'measure', in the ordinary sense of the term. But by

arranging all the answers in the order in which the faculty in

question seems to be possessed we can, with some approach to

accuracy, select the middlemost person in the row and use him as a

basis of comparison with the corresponding person in any other

batch. And similarly with those who occupy other relative positions

than that of the middlemost.

18. I refer to the introductory and concluding chapters: the bulk of

the book is, from the nature of the case, mainly occupied with

statistical and biographical details.

19. See Galton's _Hereditary Genius_, pp. 336-350, "On the comparative

worth of different races."

CHAPTER III.

_ON THE CAUSAL PROCESS BY WHICH THE GROUPS OR SERIES OF PROBABILITY ARE

BROUGHT ABOUT._

1. In discussing the question whether all the various

groups and series with which Probability is concerned are of

precisely one and the same type, we made some examination

of the process by which they are naturally produced, but we

must now enter a little more into the details of this process.

All events are the results of numerous and complicated

antecedents, far too numerous and complicated

in fact for it to be possible for us to determine or take

them all into account. Now, though it is strictly true that

we can never determine them all, there is a broad distinction

between the case of Induction, in which we can

make out enough of them, and with sufficient accuracy, to

satisfy a reasonable certainty, and Probability, in which we

cannot do so. To Induction we shall return in a future

chapter, and therefore no more need be said about it here.

We shall find it convenient to begin with a division

which, though not pretending to any philosophical accuracy,

will serve as a preliminary guide. It is the simple division

into objects, and the agencies which affect them. All the

phenomena with which Probability is concerned (as indeed

most of those with which science of any kind is concerned)

are the product of certain objects natural and artificial,

acting under the influence of certain agencies natural and

artificial. In the tossing of a penny, for instance, the objects

would be the penny or pence which were successively

thrown; the agencies would be the act of throwing, and

everything which combined directly or indirectly with this

to make any particular face come uppermost. This is a

simple and intelligible division, and can easily be so extended

in meaning as to embrace every class of objects with

which we are concerned.

Now if, in any two or more cases, we had the same

object, or objects indistinguishably alike, and if they were

exposed to the influence of agencies in all respects precisely

alike, we should expect the results to be precisely similar.

By one of the applications of the familiar principle of the

uniformity of nature we should be confident that exact

likeness in the antecedents would be followed by exact

likeness in the consequents. If the same penny, or similar

pence, were thrown in exactly the same way, we should

invariably find that the same face falls uppermost.

2. What we actually find is, of course, very far removed

from this. In the case of the objects, when they

are artificial constructions, e.g. dice, pence, cards, it is true

that they are purposely made as nearly as possible indistinguishably

alike. We either use the same thing over and

over again or different ones made according to precisely

the same model. But in natural objects nothing of the

sort prevails. In fact when we come to examine them, we

find reproduced in them precisely the same characteristics

as those which present themselves in the final result which

we were asked to explain, so that unless we examine them

a stage further back, as we shall have to do to some extent

at any rate, we seem to be merely postulating again the

very peculiarity of the phenomena which we were undertaking

to explain. They will be found, for instance, to

consist of large classes of objects, throughout all the individual

members of which a general resemblance extends.

Suppose that we were considering the length of life. The

objects here are the human beings, or that selected class

of them, whose lives we are considering. The resemblance

existing among them is to be found in the strength and

soundness of their principal vital organs, together with all

the circumstances which collectively make up what we call

the goodness of their constitutions. It is true that most of

these circumstances do not admit of any approach to actual

measurement; but, as was pointed out in the last chapter,

very many of the circumstances which do admit of such

measurement have been measured, and found to display

the characteristics in question. Hence, from the known

analogy and correlation between our various organs, there

can be no reasonable doubt that if we could arrange human

constitutions in general, or the various elements which compose

them in particular, in the order of their strength, we

should find just such an aggregate regularity and just such

groupings about the mean, as the final result (viz. in this

case the length of their lives) presents to our notice.

3. It will be observed therefore that for this purpose

the existence of natural kinds or groups is necessary.

In our games of chance of course the same die may be

thrown, or a card be drawn from the same pack, as often

as we please; but many of the events which occur to

human beings either cannot be repeated at all, or not often

enough to secure in the case of the single individual any

sufficient statistical uniformity. Such regularity as we trace

in nature is owing, much more than is often suspected,

to the arrangement of things in natural kinds, each of

them containing a large number of individuals. Were each

kind of animals or vegetables limited to a single pair, or

even to but a few pairs, there would not be much scope

left for the collection of statistical tables amongst them.

Or to take a less violent supposition, if the numbers

in each natural class of objects were much smaller than

they are at present, or the differences between their varieties

and sub-species much more marked, the consequent

difficulty of extracting from them any sufficient length of

statistical tables, though not fatal, might be very serious.

A large number of objects in the class, together with that

general similarity which entitles the objects to be fairly

comprised in one class, seem to be important conditions

for the applicability of the theory of Probability to any

phenomenon. Something analogous to this excessive paucity

of objects in a class would be found in the attempt to

apply special Insurance offices to the case of those trades

where the numbers are very limited, and the employment

so dangerous as to put them in a class by themselves. If

an insurance society were started for the workmen in

gunpowder mills alone, a premium would have to be charged

to avoid possible ruin, so high as to illustrate the extreme

paucity of appropriate statistics.

4. So much (at present) for the objects. If we turn

to what we have termed the agencies, we find much the

same thing again here. By the adjustment of their relative

intensity, and the respective frequency of their occurrence,

the total effects which they produce are found to be also

tolerably uniform. It is of course conceivable that this

should have been otherwise. It might have been found

that the second group of conditions so exactly corrected the

former as to convert the merely general uniformity into

an absolute one; or it might have been found, on the

other hand, that the second group should aggravate or

disturb the influence of the former to such an extent

as to destroy all the uniformity of its effects. Practically

neither is the case. The second condition simply varies the

details, leaving the uniformity on the whole of precisely

the same general description as it was before. Or if the

objects were supposed to be absolutely alike, as in the case

of successive throws of a penny, it may serve to bring about

a uniformity. Analysis will show these agencies to be

thus made up of an almost infinite number of different

components, but it will detect the same peculiarity that

we have so often had occasion to refer to, pervading almost

all these components. The proportions in which they are

combined will be found to be nearly, though not quite, the

same; the intensity with which they act will be nearly

though not quite equal. And they will all unite and blend

into a more and more perfect regularity as we proceed to

take the average of a larger number of instances.

Take, for instance, the length of life. As we have seen,

the constitutions of a very large number of persons selected

at random will be found to present much the same feature;

general uniformity accompanied by individual irregularity.

Now when these persons go out into the world, they are

exposed to a variety of agencies, the collective influence

of which will assign to each the length of life allotted to

him. These agencies are of course innumerable, and their

mutual interaction complicated beyond all power of analysis

to extricate. Each effect becomes in its turn a cause, is

interwoven inextricably with an indefinite number of other

causes, and reacts upon the final result. Climate, food,

clothing, are some of these agencies, or rather comprise

aggregate groups of them. The nature of a man's work

is also important. One man overworks himself, another

follows an unhealthy trade, a third exposes himself to infection,

and so on.

The result of all this interaction between what we have

thus called objects and agencies is that the final outcome

presents the same general characteristics of uniformity as may

be detected separately in the two constituent elements. Or

rather, as we shall proceed presently to show, it does so

in the great majority of cases.

5. It may be objected that such an explanation as

the above does not really amount to anything deserving

of the name, for that instead of explaining how a particular

state of things is caused it merely points out that the

same state exists elsewhere. There is a uniformity discovered

in the objects at the stage when they are commonly

submitted to calculation; we then grope about

amongst the causes of them, and after all only discover

a precisely similar uniformity existing amongst these causes.

This is to some extent true, for though part of the objection

can be removed, it must always remain the case that the

foundations of an objective science will rest in the last resort

upon the mere fact that things are found to be of such and

such a character.

6. This division, into objects and the agencies which

affect them, is merely intended for a rough practical arrangement,

sufficient to point out to the reader the immediate

nature of the causes which bring about our familiar

uniformities. If we go back a step further, it might fairly

be maintained that they may be reduced to one, namely,

to the agencies. The objects, as we have termed them,

are not an original creation in the state in which we now

find them. No one supposes that whole groups or classes

were brought into existence simultaneously, with all their

general resemblances and particular differences fully developed.

Even if it were the case that the first parents

of each natural kind had been specially created, instead

of being developed out of pre-existing forms, it would still

be true that amongst the numbers of each that now present

themselves the characteristic differences and resemblances

are the result of what we have termed agencies. Take, for

instance, a single characteristic only, say the height; what

determines this as we find it in any given group of men?

Partly, no doubt, the nature of their own food, clothing,

employment, and so on, especially in the earliest years of

their life; partly also, very likely, similar conditions and

circumstances on the part of their parents at one time or

another. No one, I presume, in the present state of knowledge,

would attempt to enumerate the remaining causes,

or even to give any indication of their exact nature; but

at the same time few would entertain any doubt that

agencies of this general description have been the determining

causes at work.

If it be asked again, Into what may these agencies

themselves be ultimately analysed? the answer to this

question, in so far as it involves any detailed examination

of them, would be foreign to the plan of this essay. In so

far as any general remarks, applicable to nearly all classes

alike of such agencies, are called for, we are led back to

the point from which we started in the previous chapter,

when we were discussing whether there is necessarily one

fixed law according to which all our series are formed. We

there saw that every event might be regarded as being

brought about by a comparatively few important causes, of

the kind which comprises all of which ordinary observation

takes any notice, and an indefinitely numerous group of

small causes, too numerous, minute, and uncertain in their

action for us to be able to estimate them or indeed to take

them individually into account at all. The important ones,

it is true, may also in turn be themselves conceived to be

made up of aggregates of small components, but they are

still best regarded as being by comparison simple and distinct,

for their component parts act mostly in groups collectively,

appearing and disappearing together, so that they

possess the essential characteristics of unity.

7. Now, broadly speaking, it appears to me that the

most suitable conditions for Probability are these: that the

important causes should be by comparison fixed and permanent,

and that the remaining ones should on the average

continue to act as often in one direction as in the other.

This they may do in two ways. In the first place we

may be able to predicate nothing more of them than the

mere fact that they act[1] as often in one direction as the

other; what we should then obtain would be merely the

simple statistical uniformity that is described in the first

chapter. But it may be the case, and in practice generally

is so more or less approximately, that these minor causes

act also in independence of one another. What we then

get is a group of uniformities such as was explained and

illustrated in the second chapter. Every possible combination

of these causes then occurring with a regular degree

of frequency, we find one peculiar kind of uniformity

exhibited, not merely in the mere fact of excess and defect

(of whatever may be the variable quality in question), but

also in every particular amount of excess and defect.

Hence, in this case, we get what some writers term a

'mean' or 'type,' instead of a simple average. For instance,

suppose a man throwing a quoit at a mark. Here

our fixed causes are his strength, the weight of the quoit,

and the intention of aiming at a given point. These we

must of course suppose to remain unchanged, if we are

to obtain any such uniformity as we are seeking. The

minor and variable causes are all those innumerable little

disturbing influences referred to in the last chapter. It

might conceivably be the case that we were only able to

ascertain that these acted as often in one direction as in

the other; what we should then find was that the quoit

tended to fall short of the mark as often as beyond it.

But owing to these little causes being mostly independent

of one another, and more or less equal in their influence,

we find also that every _amount_ of excess and defect presents

the same general characteristics, and that in a large number

of throws the quantity of divergences from the mark, of any

given amount, is a tolerably determinate function, according

to a regular law, of that amount of divergence.[2]

8. The necessity of the conditions just hinted at

will best be seen by a reference to cases in which any of

them happen to be missing. Thus we know that the length

of life is on the whole tolerably regular, and so are the

numbers of those who die in successive years or centuries

of most of the commoner diseases. But it does not seem

to be the case with all diseases. What, for instance, of

the Sweating Sickness, the Black Death, the Asiatic

Cholera? The two former either do not recur, or, if they

do, recur in such a mild form as not to deserve the same

name. What in fact of any of the diseases which are

epidemic rather than endemic? All these have their causes

doubtless, and would be produced again by the recurrence

of the conditions which caused them before. But some of

them apparently do not recur at all. They seem to have

depended upon such rare conditions that their occurrence

was almost unique. And of those which do recur the course

is frequently so eccentric and irregular, often so much dependent

upon human will or want of will, as to entirely

deprive their results (that is, the annual number of deaths

which they cause) of the statistical uniformity of which we

are speaking.

The explanation probably is that one of the principal

causes in such cases is what we commonly call contagion.

If so, we have at once a cause which so far from being fixed is

subject to the utmost variability. Stringent caution may

destroy it, carelessness may aggravate it to any extent. The

will of man, as finding its expression either on the part of

government, of doctors, or of the public, may make of it

pretty nearly what is wished, though against the possibility

of its entrance into any community no precautions can absolutely

insure us.

9. If it be replied that this want of statistical regularity

only arises from the fact of our having confined ourselves

to too limited a time, and that we should find

irregularity disappear here, as elsewhere, if we kept our

tables open long enough, we shall find that the answer will

suggest another case in which the requisite conditions for

Probability are wanting. Such a reply would only be conclusive

upon the supposition that the ways and thoughts of

men are in the long run invariable, or if variable, subject to

periodic changes only. On the assumption of a steady progress

in society, either for the better or the worse, the argument

falls to the ground at once. From what we know of

the course of the world, these fearful pests of the past may

be considered as solitary events in our history, or at least

events which will not be repeated. No continued uniformity

would therefore be found in the deaths which they occasion,

though the registrar's books were kept open for a thousand

years. The reason here is probably to be sought in the

gradual alteration of those indefinitely numerous conditions

which we term collectively progress or civilization. Every

little circumstance of this kind has some bearing upon the

liability of any one to catch a disease. But when a kind of

slow and steady tide sets in, in consequence of which these

influences no longer remain at about the same average

strength, warring on about equal terms with hostile influences,

but on the contrary show a steady tendency to increase

their power, the statistics will, with consequent steadiness

and permanence, take the impress of such a change.

10. Briefly then, if we were asked where the distinctive

characteristics of Probability are most prominently

to be found, and where they are most prominently absent,

we might say that (1) they prevail principally in the properties

of natural kinds, both in the ultimate and in the derivative

or accidental properties. In all the characteristics of

natural species, in all they do and in all which happens to

them, so far as it depends upon their properties, we seldom

fail to detect this regularity. Thus in men; their height,

strength, weight, the age to which they live, the diseases of

which they die; all present a well-known uniformity. Life

insurance tables offer the most familiar instance of the importance

of these applications of Probability.

(2) The same peculiarity prevails again in the force

and frequency of most natural agencies. Wind and weather

are seen to lose their proverbial irregularity when examined

on a large scale. Man's work therefore, when operated on

by such agencies as these, even though it had been made in

different cases absolutely alike to begin with, afterwards

shows only a general regularity. I may sow exactly the

same amount of seed in my field every year. The yield may

one year be moderate, the next year be abundant through

favourable weather, and then again in turn be destroyed by

hail. But in the long run these irregularities will be equalized

in the result of my crops, because they are equalized in the

power and frequency of the productive agencies. The

business of underwriters, and offices which insure the crops

against hail, would fall under this class; though, as already

remarked, there is no very profound distinction between them

and the former class.

The reader must be reminded again that this fixity is

only temporary, that is, that even here the series belong to

the class of those which possess a fluctuating type. Those

indeed who believe in the fixity of natural species will have

the best chance of finding a series of the really permanent

type amongst them, though even they will admit that some

change in the characteristic is attainable in length of time.

In the case of the principal natural agencies, it is of course

incontestable that the present average is referable to the

present geological period only. Our average temperature

and average rainfall have in former times been widely

different from what they now are, and doubtless will be so again.

Any fuller investigation of the process by which, on the

Theory of Evolution, out of a primeval simplicity and uniformity

the present variety was educed, hardly belongs to

the scope of the present work: at most, a few hints must suffice.

11. The above, then, are instances of natural objects

and natural agencies. There seems reason to believe that it

is in such things only, as distinguished from things artificial,

that the property in question is to be found. This is an assertion

that will need some discussion and explanation. Two

instances, in apparent opposition, will at once occur to the

mind of some readers; one of which, from its great intrinsic

importance, and the other, from the frequency of the problems

which it furnishes, will demand a few minutes' separate

examination.

(1) The first of these is the already mentioned case of

instrumental observations. In the use of astronomical and

other instruments the utmost possible degree of accuracy is

often desired, a degree which cannot be reasonably hoped for

in any one single observation. What we do therefore in

these cases is to make a large number of successive observations

which are naturally found to differ somewhat from each

other in their results; by means of these the true value

(as explained in a future chapter, on the Method of Least

Squares) is to be determined as accurately as possible. The

subjects then of calculation here are a certain number of

elements, slightly incorrect elements, given by successive

observations. Are not these observations artificial, or the

direct product of voluntary agency? Certainly not: or rather,

the answer depends on what we understand by voluntary.

What is really intended and aimed at by the observer, is of

course, perfect accuracy, that is, the true observation, or the

voluntary steps and preliminaries on which this observation

depends. Whether voluntary or not, this result only can be

called intentional. But this result is not obtained. What

we actually get in its place is a series of deviations from it,

containing results more or less wide of the truth. Now by

what are these deviations caused? By just such agencies as

we have been considering in some of the earlier sections in

this chapter. Heat and its irregular warping influence,

draughts of air producing their corresponding effects, dust

and consequent friction in one part or another, the slight

distortion of the instrument by strains or the slow uneven

contraction which continues long after the metal was cast;

these and such as these are some of the causes which divert

us from the truth. Besides this group, there are others

which certainly do depend upon human agency, but which

are not, strictly speaking, voluntary. They are such as the

irregular action of the muscles, inability to make our various

organs and members execute precisely the purposes we have

in mind, perhaps different rates in the rapidity of the nervous

currents, or in the response to stimuli, in the same or

different observers. The effect produced by some of these,

and the allowance that has in consequence to be made, are

becoming familiar even to the outside world under the name

of the 'personal equation' in astronomical, psychophysical,

and other observations.

12. (2) The other example, alluded to above, is the stock

one of cards and dice. Here, as in the last case, the result

is remotely voluntary, in the sense that deliberate volition

presents itself at one stage. But subsequently to this stage,

the result is produced or affected by so many involuntary

agencies that it owes its characteristic properties to these.

The turning up, for example, of a particular face of a die is

the result of voluntary agency, but it is not an immediate

result. That particular face was not chosen, though the fact

of its being chosen was the remote consequence of an act of

choice. There has been an intermediate chaos of conflicting

agencies, which no one can calculate before or distinguish

afterwards. These agencies seem to show a uniformity in

the long run, and thence to produce a similar uniformity in

the result. The drawing of a card from a pack is indeed

more directly volitional, as in cutting for partners in a game

of whist. But no one continues to do this long without

having the pack well shuffled in the interval, whereby a host

of involuntary influences are let in.

13. The once startling but now familiar uniformities

exhibited in the cases of suicides and misdirected letters, do

not belong to the same class. The final resolution, or want

of it, which leads to these results, is in each case indeed an

important ingredient in the individual's action or omission;

but, in so far as volition has anything to do with the results

_as a whole_, it instantly disturbs them. If the voice of the

Legislature speaks out, or any great preacher or moralist

succeeds in deterring, or any impressive example in influencing,

our moral statistics are instantly tampered with.

Some further discussion will be devoted to this subject in a

future chapter; it need only be remarked here that (always

excluding such common or general influence as those just

mentioned) the average volition, potent as it is in each

separate case, is on the whole swayed by non-voluntary conditions,

such as those of health, the casualties of employment, &c.,

in fact the various circumstances which influence

the length of a man's life.

14. Such distinctions as those just insisted on may

seem to some persons to be needless, but serious errors have

occasionally arisen from the neglect of them. The immediate

products of man's mind, so far indeed as we can make

an attempt to obtain them, do not seem to possess this

essential characteristic of Probability. Their characteristic

seems rather to be, either perfect mathematical accuracy or

utter want of it, either law unfailing or mere caprice. If,

e.g., we find the trees in a forest growing in straight lines,

we unhesitatingly conclude that they were planted by man

as they stand. It is true on the other hand, that if we find

them not regularly planted, we cannot conclude that they

were not planted by man; partly because the planter may

have worked without a plan, partly because the subsequent

irregularities brought on by nature may have obscured the

plan. Practically the mind has to work by the aid of imperfect

instruments, and is subjected to many hindrances

through various and conflicting agencies, and by these means

the work loses its original properties. Suppose, for instance,

that a man, instead of producing numerical results by imperfect

observations or by the cast of dice, were to select

them at first hand for himself by simply thinking of them

at once; what sort of series would he obtain? It would be

about as difficult to obtain in this way any such series as

those appropriate to Probability as it would be to keep his

heart or pulse working regularly by direct acts of volition,

supposing that he had the requisite control over these organs.

But the mere suggestion is absurd. A man must have an

object in thinking, he must think according to a rule or formula;

but unless he takes some natural series as a copy, he

will never be able to construct one mentally which shall permanently

imitate the originals. Or take another product of

human efforts, in which the intention can be executed with

tolerable success. When any one builds a house, there are

many slight disturbing influences at work, such as shrinking

of bricks and mortar, settling of foundations, &c. But the

effect which these disturbances are able to produce is so inappreciably

small, that we may fairly consider that the result

obtained is the direct product of the mind, the accurate

realization of its intention. What is the consequence?

Every house in the row, if designed by one man and at one

time, is of exactly the same height, width, &c. as its

neighbours; or if there are variations they are few, definite,

and regular. The result offers no resemblance whatever to

the heights, weights, &c. of a number of men selected at

random. The builder probably had some regular design in

contemplation, and he has succeeded in executing it.

15. It may be replied that if we extend our observations,

say to the houses of a large city, we shall then detect

the property under discussion. The different heights of a

great number, when grouped together, might be found to

resemble those of a great number of human beings under

similar treatment. Something of this kind might not improbably

be found to be the case, though the resemblance

would be far from being a close one. But to raise this

question is to get on to different ground, for we were

speaking (as remarked above) not of the work of different

minds with their different aims, but of that of one mind.

In a multiplicity of designs, there may be that variable uniformity,

for which we may look in vain in a single design.

The heights which the _different_ builders contemplated

might be found to group themselves into something of the

same kind of uniformity as that which prevails in most

other things which they should undertake to do independently.

We might then trace the action of the same two

conditions,--a uniformity in the multitude of their different

designs, a uniformity also in the infinite variety of the

influences which have modified those designs. But this is a

very different thing from saying that the work of one man

will show such a result as this. The difference is much like

that between the tread of a thousand men who are stepping

without thinking of each other, and their tread when they

are drilled into a regiment. In the former case there is

the working, in one way or another, of a thousand minds;

in the latter, of one only.

The investigations of this and the former chapter

constitute a sufficiently close examination into the detailed

causes by which the peculiar form of statistical results with

which we are concerned is actually produced, to serve the

purpose of a work which is occupied mainly with the _methods_

of the Science of Probability. The great importance, however,

of certain statistical or sociological enquiries will demand

a recurrence in a future chapter to one particular

application of these statistics, viz. to those concerned with

some classes of human actions.

16. The only important addition to, or modification of,

the foregoing remarks which I have found occasion to make

is due to Mr Galton. He has recently pointed out,--and was

I believe the first to do so,--that in certain cases some

analysis of the causal processes can be effected, and is in fact

absolutely necessary in order to account for the facts observed.

Take, for instance, the _heights_ of the population of

any country. If the distribution or dispersion of these about

their mean value were left to the unimpeded action of those

myriad productive agencies alluded to above, we should certainly

obtain such an arrangement in the posterity of any

one generation as had already been exhibited in the parents.

That is, we should find repeated in the previous stage the

same kind of order as we were trying to account for in the

following stage.

But then, as Mr Galton insists, if such agencies acted

freely and independently, though we should get the same

_kind_ of arrangement or distribution, we should not get the

same _degree_ of it: there would, on the contrary, be a tendency

towards further dispersion. The 'curve of facility' (v. the

diagram on p. 29) would belong to the same class, but would

have a different modulus. We shall see this at once if we

take for comparison a case in which similar agencies work

their way without any counteraction whatever. Suppose, for

instance, that a large number of persons, whose fortunes

were equal to begin with, were to commence gambling or

betting continually for some small sum. If we examine

their circumstances after successive intervals of time, we

should expect to find their fortunes distributed according to

the same general law,--i.e. the now familiar law in question,--but

we should also expect to find that the poorest

ones were slightly poorer, and the richest ones slightly

richer, on each successive occasion. We shall see more

about this in a future chapter (on _Gambling_), but it may

be taken for granted here that there is nothing in the laws

of chance to resist this tendency towards intensifying the extremes.

Now it is found, on the contrary, in the case of vital

phenomena,--for instance in that of height, and presumably

of most of the other qualities which are in any way characteristic

of natural kinds,--that there is, through a number of

successive generations, a remarkable degree of fixity. The

tall men are not taller, and the short men are not shorter,

per cent. of the population in successive generations: always

supposing of course that some general change of circumstances,

such as climate, diet, &c. has not set in. There

must therefore here be some cause at work which tends, so

to say, to draw in the extremes and thus to check the otherwise

continually increasing dispersion.

17. The facts were first tested by careful experiment.

At the date of Mr Galton's original paper on the subject,[3]

there were no available statistics of heights of human beings;

so a physical element admitting of careful experiment (viz.

the size or weight of certain seeds) was accurately estimated.

From these data the actual amount of reversion from the

extremes, that is, of the slight pressure continually put upon

the extreme members with the result of crowding them back

towards the mean, was determined, and this was compared

with what theory would require in order to keep the characteristics

of the species permanently fixed. Since then,

statistics have been obtained to a large extent which deal

directly with the heights of human beings.

The general conclusion at which we arrive is that there

are several causes at work which are neither slight nor independent.

There is, for instance, the observed fact that the

extremes are as a rule not equally fertile with the means,

nor equally capable of resisting death and disease. Hence

as regards their mere numbers, there is a tendency for them

somewhat to thin out. Then again there is a distinct

positive cause in respect of 'reversion.' Not only are the

offspring of the extremes less numerous, but these offspring

also tend to cluster about a mean which is, so to say, shifted

a little towards the true centre of the whole group; i.e. towards

the mean offspring of the mean parents.

18. For a full discussion of these characteristics, and

for a variety of most ingenious illustrations of their mode of

agency and of their comparative efficacy, the reader may be

referred to Mr Galton's original articles. For our present

purpose it will suffice to say that these characteristics tend

towards maintaining the fixity of species; and that though

they do not affect what may be called the general nature of

the 'probability curve' or 'law of facility', they do determine

its precise value in the cases in question. If, indeed, it be

asked why there is no need for any such corrective influence

in the case of, say, firing at a mark: the answer is that there

is no opening for it except where a _cumulative_ influence is introduced.

The reason why the fortunes of our betting party

showed an ever increasing divergency, and why some special

correction was needed in order to avert such a tendency in

the case of vital phenomena, was that the new starting-point

at every step was slightly determined by the results of the

previous step. The man who has lost a shilling one time

starts, next time, worse off by just a shilling; and, but for

the corrections we have been indicating, the man who was

born tall would, so to say, throw off his descendants from a

vantage ground of superior height. The true parallel in the

case of the marksmen would be to suppose that their new

points of aim were always shifted a little in the direction of

the last divergence. The spreading out of the shot-marks

would then continue without limit, just as would the divergence

of fortunes of the supposed gamblers.

1. As stated above, this is really little more than a re-statement, a

stage further back, of the existence of the same kind of uniformity

as that which we are called upon to explain in the concrete details

presented to us in experience.

2. "It would seem in fact that in coarse and rude observations the

errors proceed from a _very few_ principal causes, and in

consequence our hypothesis [as to the Exponential Law of Error] will

probably represent the facts only imperfectly, and the frequency of

the errors will only approximate roughly and vaguely to the law

which follows from it. But when astronomers, not content with the

degree of accuracy they had reached, prosecuted their researches

into the remaining sources of error, they found that not three or

four, but a _great number_ of minor sources of error of nearly

co-ordinate importance began to reveal themselves, having been till

then masked and overshadowed by the graver errors which had been now

approximately removed.... There were errors of graduation, and many

others in the contraction of instruments; other errors of their

adjustments; errors (technically so called) of _observation_; errors

from the changes of temperature, of weather, from slight irregular

motions and vibrations; in short, the thousand minute disturbing

influences with which modern astronomers are familiar." (Extracted

from a paper by Mr Crofton in the Vol. of the _Philosophical

Transactions_ for 1870, p. 177.)

3. _Typical Laws of Heredity_; read before the Royal Institution,

Feb. 9, 1877. See also _Journal of the Anthrop. Inst._ Nov. 1885.

CHAPTER IV.

_ON THE MODES OF ESTABLISHING AND DETERMINING THE EXISTENCE AND

NUMERICAL PROPORTIONS OF THE CHARACTERISTIC PROPERTIES OF OUR SERIES

OR GROUPS._

1. At the point which we have now reached, we are

supposed to be in possession of series or groups of a certain

kind, lying at the bottom, as one may say, and forming the

foundation on which the Science of Probability is to be

erected. We have described with sufficient particularity the

characteristics of such a series, and have indicated the process

by which it is, as a rule, actually brought about in

nature. The next enquiries which have to be successively

made are, how in any particular case we are to establish

their existence and determine their special character and

properties? and secondly,[1] when we have obtained them, in

what mode are they to be employed for logical purposes?

The answer to the former enquiry does not seem difficult.

Experience is our sole guide. If we want to discover what is

in reality a series of _things_, not a series of our own conceptions,

we must appeal to the things themselves to obtain it,

for we cannot find much help elsewhere. We cannot tell

how many persons will be born or die in a year, or how

many houses will be burnt or ships wrecked, without actually

counting them. When we thus speak of 'experience' we

mean to employ the term in its widest signification; we

mean experience supplemented by all the aids which inductive

or deductive logic can afford. When, for instance,

we have found the series which comprises the numbers of

persons of any assigned class who die in successive years, we

have no hesitation in extending it some way into the future

as well as into the past. The justification of such a procedure

must be sought in the ordinary canons of Induction.

As a special discussion will be given upon the connection

between Probability and Induction, no more need be said

upon this subject here; but nothing will be found there at

variance with the assertion just made, that the series we

employ are ultimately obtained by experience only.

2. In many cases it is undoubtedly true that we do not

resort to direct experience at all. If I want to know what is

my chance of holding ten trumps in a game of whist, I do

not enquire how often such a thing has occurred before. If

all the inhabitants of the globe were to divide themselves up

into whist parties they would have to keep on at it for a

great many years, if they wanted to settle the question

satisfactorily in that way. What we do of course is to calculate

algebraically the proportion of possible combinations

in which ten trumps can occur, and take this as the answer

to our problem. So again, if I wanted to know the chance

of throwing six with a die whose faces were unequal, it

would be a question if my best way would not be to calculate

geometrically the solid angle subtended at the centre of

gravity by the opposite face, and the ratio of this to the

whole surface of a sphere would represent sufficiently closely

the chance required.

It is quite true that in such examples as the above, especially

the former one, nobody would ever think of appealing

to statistics. This would be a tedious process to adopt when,

as here, the mechanical and other conditions upon which the

production of the events depend are comparatively few, determinate,

and admit of isolated consideration, whilst the

enormous number of combinations which can be constructed

out of them causes an enormous consequent multiplicity of

ways in which the events can possibly happen. Hence, in

practice, _à priori_ determination is often easy, whilst _à posteriori_

appeal to experience would be not merely tedious but

utterly impracticable. This, combined with the frequent

simplicity and attractiveness of such examples when deductively

treated, has made them very popular, and produced the

impression in many quarters that they are the proper typical

instances to illustrate the theory of chance. Whereas, had

the science been concerned with those kinds of events only

which in practice are commonly made subjects of insurance,

probably no other view would ever have been taken than

that it was based upon direct appeal to experience.

3. When, however, we look a little closer, we find that

there is no occasion for such a sharp distinction as that apparently

implied between the two classes of examples just

indicated. In such cases as those of dice and cards, even, in

which we appear to reason directly from the determining

conditions, or possible variety of the events, rather than from

actual observation of their occurrence, we shall find that this

procedure is only valid by the help of a tacit assumption

which can never be determined otherwise than by direct

experience. It is, no doubt, an exceedingly natural and

obvious assumption, and one which is continually deriving

fresh weight from every-day observation, but it is one which

ought not to be admitted without consideration. As this is a

very important matter, not so much in itself as in connection

with the light which it throws upon the theory of the subject,

we will enter into a somewhat detailed examination of it.

Let us take a very simple example, that of tossing up a

penny. Suppose that I am contemplating a succession of

two throws; I can see that the only possible events are[2]

HH, HT, TH, TT. So much is certain. We are moreover

tolerably well convinced from experience that these

events occur, in the long run, about equally often. This is

of course admitted on all hands. But on the view commonly

maintained, it is contended that we might have known the

fact beforehand on grounds which are applicable to an indefinite

number of other and more complex cases. The form in

which this view would generally be advanced is, that we are

enabled to state beforehand that the four throws above mentioned

are _equally likely_. If in return we ask what is meant

by the expression 'equally likely', it appears that there are

two and only two possible forms of reply. One of these seeks

the explanation in the state of mind of the observer, the

other seeks it in some characteristic of the things observed.

(1) It might, for instance, be said on the one hand, that

what is meant is that the four events contemplated are

equally easy to imagine, or, more accurately, that our expectation

or belief in their occurrence is equal. We could

hardly be content with this reply, for the further enquiry

would immediately be urged, On what ground is this to be

believed? What are the characteristics of events of which

our expectation is equal? If we consented to give an answer

to this further enquiry, we should be led to the second form

of reply, to be noticed directly; if we did not consent we

should, it seems, be admitting that Probability was only a

portion of Psychology, confined therefore to considering states

of mind in themselves, rather than in their reference to facts,

viz. as being true or false. We should, that is, be ceasing

to make it a science of inference about things. This point

will have to be gone into more thoroughly in another chapter;

but it is impossible to direct attention too prominently

to the fact that Logic (and therefore Probability as a branch

of Logic) is not concerned with what men _do_ believe, but

with what they ought to believe, if they are to believe

correctly.

(2) In the other form of reply the explanation of the

phrase in question would be sought, not in a state of mind,

but in a quality of the things contemplated. We might assign

the following as the meaning, viz. that the events really

would occur with equal frequency in the long run. The

ground of this assertion would probably be found in past experience,

and it would doubtless be impossible so to frame

the answer as to exclude the notion of our belief altogether.

But still there is a broad distinction between seeking an

equality in the amount of our belief, as before, and in the

frequency of occurrence of the events themselves, as here.

4. When we have got as far as this it can readily be

shown that an appeal to experience cannot be long evaded.

For _can_ the assertion in question (viz. that the throws of the

penny will occur equally often) be safely made _à priori_?

Those who consider that it can seem hardly to have fully

faced the difficulties which meet them. For when we begin

to enquire seriously whether the penny will really do what

is expected of it, we find that restrictions have to be introduced.

In the first place it must be an ideal coin, with

its sides equal and fair. This restriction is perfectly intelligible;

the study of solid geometry enables us to idealize a

penny into a circular or cylindrical lamina. But this condition

by itself is not sufficient, others are wanted as well. The

penny was supposed to be tossed up, as we say 'at random.'

What is meant by this, and how is this process to be idealized?

To ask this is to introduce no idle subtlety; for it

would scarcely be maintained that the heads and tails would

get their fair chances if, immediately before the throwing,

we were so to place the coin in our hands as to start it

always with the same side upwards. The difference that

would result in consequence, slight as its cause is, would

tend in time to show itself in the results. Or, if we persisted

in starting with each of the two sides alternately

upwards, would the longer repetitions of the same side get

their fair chance?

Perhaps it will be replied that if we think nothing whatever

about these matters all will come right of its own accord.

It may, and doubtless will be so, but this is falling back upon

experience. It is here, then, that we find ourselves resting on

the experimental assumption above mentioned, and which

indeed cannot be avoided. For suppose, lastly, that the

circumstances of nature, or my bodily or mental constitution,

were such that the same side always _is_ started upwards, or

indeed that they are started in any arbitrary order of our

own? Well, it will be replied, it would not then be a fair

trial. If we press in this way for an answer to such enquiries,

we shall find that these tacit restrictions are really nothing

else than a mode of securing an experimental result. They

are only another way of saying, Let a series of actions be

performed in such a way as to secure a sequence of a particular

kind, viz., of the kind described in the previous chapters.

5. An intermediate way of evading the direct appeal

to experience is sometimes found by defining the probability

of an event as being measured by the ratio which the

number of cases favourable to the event bears to the total

number of cases which are possible. This seems a somewhat

loose and ambiguous way of speaking. It is clearly not

enough to _count_ the number of cases merely, they must also

be _valued_, since it is not certain that each is equally potent

in producing the effect. This, of course, would never be

denied, but sufficient importance does not seem to be attached

to the fact that we have really no other way of

valuing them except by estimating the effects which they

actually do, or would produce. Instead of thus appealing to

the proportion of cases favourable to the event, it is far better

(at least as regards the foundation of the science, for we are

not at this moment discussing the practical method of facilitating

our calculations) to appeal at once to the proportion

of cases in which the event actually occurs.

6. The remarks above made will apply, of course, to

most of the other common examples of chance; the throwing

of dice, drawing of cards, of balls from bags, &c. In the

last case, for instance, one would naturally be inclined to

suppose that a ball which had just been put back would

thereby have a better chance of coming out again next time,

since it will be more in the way for that purpose. How is

this to be prevented? If we designedly thrust it to the

middle or bottom of the others, we may overdo the precaution;

and are in any case introducing human design, that

element so essentially hostile to all that we understand by

chance. If we were to trust to a good shake setting matters

right, we may easily be deceived; for shaking the bag can

hardly do more than diminish the disposition of those balls

which were already in each other's neighbourhood, to remain

so. In the consequent interaction of each upon all, the

arrangement in which they start cannot but leave its impress

to some extent upon their final positions. In all such cases,

therefore, if we scrutinize our language, we shall find that

any supposed _à priori_ mode of stating a problem is little else

than a compendious way of saying, Let means be taken for

obtaining a given result. Since it is upon this result that

our inferences ultimately rest, it seems simpler and more

philosophical to appeal to it at once as the groundwork of

our science.

7. Let us again take the instance of the tossing of a

penny, and examine it somewhat more minutely, to see what

can be actually proved about the results we shall obtain.

We are willing to give the pence fair treatment by assuming

that they are perfect, that is, that in the long run they show

no preference for either head or tail; the question then

remains, Will the repetitions of the same face obtain the

proportional shares to which they are entitled by the

usual interpretations of the theory? Putting then, as before,

for the sake of brevity, H for head, and HH for heads twice

running, we are brought to this issue;--Given that the

chance of H is 1/2, does it follow necessarily that the chance of HH

(with two pence) is 1/4? To say nothing of 'H ten times'

occurring once in 1024 times (with ten pence), need it occur

at all? The mathematicians, for the most part, seem to think

that this conclusion follows necessarily from first principles;

to me it seems to rest upon no more certain evidence than a

reasonable extension by Induction.

Taking then the possible results which can be obtained

from a pair of pence, what do we find? Four different

results may follow, namely, (1) HT, (2) HH, (3) TH, (4) TT.

If it can be proved that these four are equally probable, that

is, occur equally often, the commonly accepted conclusions

will follow, for a precisely similar argument would apply to

all the larger numbers.

8. The proof usually advanced makes use of what is

called the Principle of Sufficient Reason. It takes this

form;--Here are four kinds of throws which may happen;

once admit that the separate elements of them, namely, H

and T, happen equally often, and it will follow that the above

combinations will also happen equally often, for no reason can

be given in favour of one of them that would not equally hold

in favour of the others.

To a certain extent we must admit the validity of the

principle for the purpose. In the case of the throws given

above, it would be valid to prove the equal frequency of (1)

and (3) and also of (2) and (4); for there is no difference

existing between these pairs except what is introduced by our

own notation.[3] TH is the same as HT, except in the order

of the occurrence of the symbols H and T, which we do not

take into account. But either of the pair (1) and (3) _is_

different from either of the pair (2) and (4). Transpose the

notation, and there would still remain here a distinction which

the mind can recognize. A succession of the same thing

twice running is distinguished from the conjunction of two

different things, by a distinction which does not depend

upon our arbitrary notation only, and would remain entirely

unaltered by a change in this notation. The principle therefore

of Sufficient Reason, if admitted, would only prove that

doublets of the two kinds, for example (2) and (4), occur

equally often, but it would not prove that they must each

occur once in four times. It cannot be proved indeed in this

way that they need ever occur at all.

9. The formula, then, not being demonstrable _à priori_,

(as might have been concluded,) can it be obtained by experience?

To a certain extent it can; the present experience

of mankind in pence and dice seems to show that the smaller

successions of throws do really occur in about the proportions

assigned by the theory. But how nearly they do so no one

can say, for the amount of time and trouble to be expended

before we could feel that we have verified the fact, even for

small numbers, is very great, whilst for large numbers it

would be simply intolerable. The experiment of throwing

often enough to obtain 'heads ten times' has been actually

performed by two or three persons, and the results are given

by De Morgan, and Jevons.[4] This, however, being only

sufficient on the average to give 'heads ten times' a single

chance, the evidence is very slight; it would take a considerable

number of such experiments to set the matter

nearly at rest.

Any such rule, then, as that which we have just been

discussing, which professes to describe what will take place

in a long succession of throws, is only conclusively proved by

experience within very narrow limits, that is, for small repetitions

of the same face; within limits less narrow, indeed,

we feel assured that the rule cannot be flagrantly in error,

otherwise the variation would be almost sure to be detected.

From this we feel strongly inclined to infer that the same

law will hold throughout. In other words, we are inclined

to extend the rule by Induction and Analogy. Still there

are so many instances in nature of proposed laws which hold

within narrow limits but get egregiously astray when we

attempt to push them to great lengths, that we must give at

best but a qualified assent to the truth of the formula.

10. The object of the above reasoning is simply to

show that we cannot be certain that the rule is true. Let

us now turn for a minute to consider the causes by which

the succession of heads and tails is produced, and we may

perhaps see reasons to make us still more doubtful.

It has been already pointed out that in calculating probabilities

_à priori_, as it is called, we are only able to do so by

introducing restrictions and suppositions which are in reality

equivalent to assuming the expected results. We use words

which in strictness mean, Let a given process be performed;

but an analysis of our language, and an examination of

various tacit suppositions which make themselves felt the

moment they are not complied with, soon show that our real

meaning is, Let a series of a given kind be obtained; it is to

this series only, and not to the conditions of its production,

that all our subsequent calculations properly apply. The

physical process being performed, we want to know whether

anything resembling the contemplated series really will be

obtained.

Now if the penny were invariably set the same side uppermost,

and thrown with the same velocity of rotation and

to the same height, &c.--in a word, subjected to the same

conditions,--it would always come down with the same side

uppermost. Practically, we know that nothing of this kind

occurs, for the individual variations in the results of the

throws are endless. Still there will be an _average_ of these

conditions, about which the throws will be found, as it were,

to cluster much more thickly than elsewhere. We should be

inclined therefore to infer that if the same side were always

set uppermost there would really be a departure from the

sort of series which we ordinarily expect. In a very large

number of throws we should probably begin to find, under

such circumstances, that either head or tail was having a

preference shown to it. If so, would not similar effects be

found to be connected with the way in which we started

each successive _pair_ of throws? According as we chose to

make a practice of putting HH or TT uppermost, might

there not be a disturbance in the proportion of successions of

two heads or two tails? Following out this train of reasoning,

it would seem to point with some likelihood to the conclusion

that in order to obtain a series of the kind we

expect, we should have to dispose the antecedents in a

similar series at the start. The changes and chances produced

by the act of throwing might introduce infinite individual

variations, and yet there might be found, in the very

long run, to be a close similarity between these two series.

11. This is, to a certain extent, only shifting the difficulty,

I admit; for the claim formerly advanced about the

possibility of proving the proportions of the throws in the

former series, will probably now be repeated in favour of

those in the latter. Still the question is very much narrowed,

for we have reduced it to a series of _voluntary_ acts.

A man may put whatever side he pleases uppermost. He

may act consciously, as I have said, or he may think nothing

whatever about the matter, that is, throw at random; if so,

it will probably be asserted by many that he will involuntarily

produce a series of the kind in question. It may be

so, or it may not; it does not seem that there are any

easily accessible data by which to decide. All that I am

concerned with here is to show the likelihood that the commonly

received result does in reality depend upon the fulfilment

of a certain condition at the outset, a condition which

it is certainly optional with any one to fulfil or not as he

pleases. The short successions doubtless will take care of

themselves, owing to the infinite complications produced by

the casual variations in throwing; but the long ones may

suffer, unless their interest be consciously or unconsciously

regarded at the outset.

12. The advice, 'Only try long enough, and you will

sooner or later get any result that is possible,' is plausible,

but it rests only on Induction and Analogy; mathematics do

not prove it. As has been repeatedly stated, there are two

distinct views of the subject. Either we may, on the one

hand, take a series of symbols, call them heads and tails;

H, T, &c.; and make the assumption that each of these, and

each pair of them, and so on, will occur in the long run with

a regulated degree of frequency. We may then calculate

their various combinations, and the consequences that may

be drawn from the data assumed. This is a purely algebraical

process; it is infallible; and there is no limit whatever to the

extent to which it may be carried. This way of looking at

the matter may be, and undoubtedly should be, nothing

more than the counterpart of what I have called the substituted

or idealized series which generally has to be introduced

as the basis of our calculation. The danger to be guarded

against is that of regarding it too purely as an algebraical

conception, and thence of sinking into the very natural

errors both of too readily evolving it out of our own consciousness,

and too freely pushing it to unwarranted lengths.

Or on the other hand, we may consider that we are treating

of the behaviour of _things_;--balls, dice, births, deaths, &c.;

and drawing inferences about them. But, then, what

were in the former instance allowable assumptions, become

here propositions to be tested by experience. Now the whole

theory of Probability as a practical science, in fact as anything

more than an algebraical truth, depends of course upon

there being a close correspondence between these two views

of the subject, in other words, upon our substituted series

being kept in accordance with the actual series. Experience

abundantly proves that, between considerable limits, in the

example in question, there does exist such a correspondence.

But let no one attempt to enforce our assent to every remote

deduction that mathematicians can draw from their formulæ.

When this is attempted the distinction just traced becomes

prominent and important, and we have to choose our side.

Either we go over to the mathematics, and so lose all right

of discussion about the things; or else we take part with the

things, and so defy the mathematics. We do not question

the formal accuracy of the latter within their own province,

but either we dismiss them as somewhat irrelevant, as applying

to data of whose correctness we cannot be certain, or

we take the liberty of remodelling them so as to bring them

into accordance with facts.

13. A critic of any doctrine can hardly be considered

to have done much more than half his duty when he has

explained and justified his grounds for objecting to it. It

still remains for him to indicate, if only in a few words,

what he considers its legitimate functions and position to be,

for it can seldom happen that he regards it as absolutely

worthless or unmeaning. I should say, then, that when

Probability is thus divorced from direct reference to objects,

as it substantially is by not being founded upon experience,

it simply resolves itself into the common algebraical or arithmetical

doctrine of Permutations and Combinations.[5] The

considerations upon which these depend are purely formal

and necessary, and can be fully reasoned out without any

appeal to experience. We there start from pure considerations

of number or magnitude, and we terminate with them,

having only arithmetical calculations to connect them together.

I wish, for instance, to find the chance of throwing

heads three times running with a penny. All I have to do

is first to ascertain the possible number of throws. Permutations

tell me that with two things thus in question (viz.

head and tail) and three times to perform the process, there

are eight possible forms of the result. Of these eight one

only being favourable, the chance in question is pronounced

to be one-eighth.

Now though it is quite true that the actual calculation of

every chance problem must be of the above character, viz. an

algebraical or arithmetical process, yet there is, it seems to

me, a broad and important distinction between a material

science which employs mathematics, and a formal one which

consists of nothing but mathematics. When we cut ourselves

off from the necessity of any appeal to experience, we

are retaining only the intermediate or calculating part of the

investigation; we may talk of dice, or pence, or cards, but

these are really only names we choose to give to our symbols.

The H's and T's with which we deal have no bearing on objective

occurrences, but are just like the x's and y's with

which the rest of algebra deals. Probability in fact, when so

treated, seems to be absolutely nothing else than a system of

applied Permutations and Combinations.

It will now readily be seen how narrow is the range of

cases to which any purely deductive method of treatment can

apply. It is almost entirely confined to such employments

as games of chance, and, as already pointed out, can only be

regarded as really trustworthy even there, by the help of

various tacit restrictions. This alone would be conclusive

against the theory of the subject being rested upon such a

basis. The experimental method, on the other hand, is, in the

same theoretical sense, of universal application. It would

include the ordinary problems furnished by games of chance,

as well as those where the dice are loaded and the pence are

not perfect, and also the indefinitely numerous applications

of statistics to the various kinds of social phenomena.

14. The particular view of the deductive character of

Probability above discussed, could scarcely have intruded

itself into any other examples than those of the nature of

games of chance, in which the conditions of occurrence are

by comparison few and simple, and are amenable to accurate

numerical determination. But a doctrine, which is in reality

little else than the same theory in a slightly disguised form,

is very prevalent, and has been applied to truths of the most

purely empirical character. This doctrine will be best introduced

by a quotation from Laplace. After speaking of

the irregularity and uncertainty of nature as it appears at

first sight, he goes on to remark that when we look closer we

begin to detect "a striking regularity which seems to suggest

a design, and which some have considered a proof of

Providence. But, on reflection, it is soon perceived that this

regularity is nothing but the development of the respective

probabilities of the simple events, which ought to occur more

frequently according as they are more probable."[6]

If this remark had been made about the succession of

heads and tails in the throwing up of a penny, it would

have been intelligible. It would simply mean this: that the

constitution of the body was such that we could anticipate

with some confidence what the result would be when it was

treated in a certain way, and that experience would justify

our anticipation in the long run. But applied as it is in a

more general form to the facts of nature, it seems really to

have but little meaning in it. Let us test it by an instance.

Amidst the irregularity of individual births, we find that the

male children are to the female, in the long run, in about

the proportion of 106 to 100. Now if we were told that

there is nothing in this but "the development of their respective

probabilities," would there be anything in such a

statement but a somewhat pretentious re-statement of the

fact already asserted? The probability _is_ nothing but that

proportion, and is unquestionably in this case derived from

no other source but the statistics themselves; in the above

remark the attempt seems to be made to invert this process,

and to derive the sequence of events from the mere numerical

statement of the proportions in which they occur.

15. It will very likely be replied that by the probability

above mentioned is meant, not the mere numerical

proportion between, the births, but some fact in our constitution

upon which this proportion depends; that just as

there was a relation of equality between the two sides of the

penny, which produced the ultimate equality in the number

of heads and tails, so there may be something in our constitution

or circumstances in the proportion of 106 to 100,

which produces the observed statistical result. When this

something, whatever it might be, was discovered, the observed

numbers might be supposed capable of being determined

beforehand. Even if this were the case, however, it

must not be forgotten that there could hardly fail to be, in

combination with such causes, other concurrent conditions in

order to produce the ultimate result; just as besides the shape

of the penny, we had also to take into account the nature of

the 'randomness' with which it was tossed. What these

may be, no one at present can undertake to say, for the best

physiologists seem indisposed to hazard even a guess upon

the subject.[7] But without going into particulars, one may

assert with some confidence that these conditions cannot well

be altogether independent of the health, circumstances, manners

and customs, &c. (to express oneself in the vaguest way)

of the parents; and if once these influencing elements are

introduced, even as very minute factors, the results cease to

be dependent only on fixed and permanent conditions. We

are at once letting in other conditions, which, if they also

possess the characteristics that distinguish Probability (an

exceedingly questionable assumption), must have that fact

specially proved about them. That this should be the case

indeed seems not merely questionable, but almost certainly

impossible; for these conditions partaking of the nature of

what we term generally, Progress and Civilization, cannot be

expected to show any permanent disposition to hover about

an average.

16. The reader who is familiar with Probability is of

course acquainted with the celebrated theorem of James

Bernoulli. This theorem, of which the examples just adduced

are merely particular cases, is generally expressed

somewhat as follows:--in the long run all events will tend

to occur with a relative frequency proportional to their

objective probabilities. With the mathematical proof of this

theorem we need not trouble ourselves, as it lies outside the

province of this work; but indeed if there is any value in the

foregoing criticism, the basis on which the mathematics rest

is faulty, owing to there being really nothing which we can

with propriety call an objective probability.

If one might judge by the interpretation and uses to

which this theorem is sometimes exposed, we should regard

it as one of the last remaining relics of Realism, which after

being banished elsewhere still manages to linger in the remote

province of Probability. It would be an illustration of

the inveterate tendency to objectify our conceptions, even in

cases where the conceptions had no right to exist at all. A

uniformity is observed; sometimes, as in games of chance, it

is found to be so connected with the physical constitution of

the bodies employed as to be capable of being inferred beforehand;

though even here the connection is by no means

so necessary as is commonly supposed, owing to the fact that

in addition to these bodies themselves we have also to take

into account their relation to the agencies which influence

them. This constitution is then converted into an 'objective

probability', supposed to develop into the sequence which

exhibits the uniformity. Finally, this very questionable objective

probability is assumed to exist, with the same faculty

of development, in all the cases in which uniformity is observed,

however little resemblance there may be between

these and games of chance.

17. How utterly inappropriate any such conception is

in most of the cases in which we find statistical uniformity,

will be obvious on a moment's consideration. The observed

phenomena are generally the product, in these cases, of very

numerous and complicated antecedents. The number of

crimes, for instance, annually committed in any society, is a

function amongst other things, of the strictness of the law,

the morality of the people, their social condition, and the

vigilance of the police, each of these elements being in itself

almost infinitely complex. Now, as a result of all these

agencies, there is some degree of uniformity; but what has

been called above the change of type, which it sooner or

later tends to display, is unmistakeable. The average annual

numbers do not show a steady gradual approach towards

what might be considered in some sense a limiting value,

but, on the contrary, fluctuate in a way which, however it

may depend upon causes, shows none of the permanent uniformity

which is characteristic of games of chance. This

fact, combined with the obvious arbitrariness of singling out,

from amongst the many and various antecedents which produced

the observed regularity, a few only, which should constitute

the objective probability (if we took all, the events

being absolutely determined, there would be no occasion for

an appeal to probability in the case), would have been sufficient

to prevent any one from assuming the existence of

any such thing, unless the mistaken analogy of other cases

had predisposed him to seek for it.

There is a familiar practical form of the same error, the

tendency to which may not improbably be derived from a

similar theoretical source. It is that of continuing to accumulate

our statistical data to an excessive extent. If the

type were absolutely fixed we could not possibly have too

many statistics; the longer we chose to take the trouble

of collecting them the more accurate our results would be.

But if the type is changing, in other words, if some of the

principal causes which aid in their production have, in regard

to their present degree of intensity, strict limits of time or

space, we shall do harm rather than good if we overstep

these limits. The danger of stopping too soon is easily seen,

but in avoiding it we must not fall into the opposite error of

going on too long, and so getting either gradually or suddenly

under the influence of a changed set of circumstances.

18. This chapter was intended to be devoted to a

consideration, not of the processes by which nature produces

the series with which we are concerned, but of the theoretic

basis of the methods by which we can determine the existence

of such series. But it is not possible to keep the two enquiries

apart, for here, at any rate, the old maxim prevails that to

know a thing we must know its causes. Recur for a minute

to the considerations of the last chapter. We there saw

that there was a large class of events, the conditions of production

of which could be said to consist of (1) a comparatively

few nearly unchangeable elements, and (2) a vast number

of independent and very changeable elements. At least if

there were any other elements besides these, we are assumed

either to make special allowance for them, or to omit them

from our enquiry. Now in certain cases, such as games of

chance, the unchangeable elements may without practical

error be regarded as really unchangeable throughout any

range of time and space. Hence, as a result, the deductive

method of treatment becomes in their case at once the most

simple, natural, and conclusive; but, as a further consequence,

the statistics of the events, if we choose to appeal to them,

may be collected _ad libitum_ with better and better approximation

to truth. On the other hand, in all social

applications of Probability, the unchangeable causes can only

be regarded as really unchangeable under many qualifications.

We know little or nothing of them directly; they are

often in reality numerous, indeterminate, and fluctuating;

and it is only under the guarantee of stringent restrictions of

time and place, that we can with any safety attribute to

them sufficient fixity to justify our theory. Hence, as a

result, the deductive method, under whatever name it may

go, becomes totally inapplicable both in theory and practice;

and, as a further consequence, the appeal to statistics has to

be made with the caution in mind that we shall do mischief

rather than good if we go on collecting too many of them.

19. The results of the last two chapters may be

summed up as follows:--We have extended the conception

of a series obtained in the first chapter; for we have found

that these series are mostly presented to us in groups.

These groups are found upon examination to be formed upon

approximately the same type throughout a very wide and

varied range of experience; the causes of this agreement we

discussed and explained in some detail. When, however, we

extend our examination by supposing the series to run to a

very great length, we find that they may be divided into two

classes separated by important distinctions. In one of these

classes (that containing the results of games of chance) the

conditions of production, and consequently the laws of statistical

occurrence, may be practically regarded as absolutely

fixed; and the extent of the divergences from the mean

seem to know no finite limit. In the other class, on the

contrary (containing the bulk of ordinary statistical enquiries),

the conditions of production vary with more or less rapidity,

and so in consequence do the results. Moreover it is often

impossible that variations from the mean should exceed a

certain amount. The former we may term _ideal_ series. It

is they alone which show the requisite characteristics with

any close approach to accuracy, and to make the theory of

the subject tenable, we have really to substitute one of this

kind for one of the less perfect ones of the other class, when

these latter are under treatment. The former class have,

however, been too exclusively considered by writers on the

subject; and conceptions appropriate only to them, and not

always even to them, have been imported into the other

class. It is in this way that a general tendency to an excessive

deductive or _à priori_ treatment of the science has

been encouraged.

1. This latter enquiry belongs to what may be termed the more purely

logical part of this volume, and is entered on in the course of

Chapter VI.

2. For the use of those not acquainted with the common notation

employed in this subject, it may be remarked that HH is simply an

abbreviated way of saying that the two successive throws of the

penny give head; HT that the first of them gives head, and the

second tail; and so on with the remaining symbols.

3. I am endeavouring to treat this rule of Sufficient Reason in a way

that shall be legitimate in the opinion of those who accept it, but

there seem very great doubts whether a contradiction is not involved

when we attempt to extract results from it. If the sides are

absolutely alike, how can there he any difference between the terms

of the series? The succession seems then reduced to a dull

uniformity, a mere iteration of the same thing many times; the

series we contemplated has disappeared. If the sides are not

absolutely alike, what becomes of the applicability of the rule?

4. _Formal Logic_, p. 185. _Principles of Science_, p. 208.

5. The close connection between these subjects is well indicated in

the title of Mr Whitworth's treatise, _Choice and Chance_.

6. _Essai Philosophique_. Ed. 1825, p. 74.

7. An opinion prevailed rather at one time (quoted and supported by

Quetelet amongst others) that the relative ages of the parents had

something to do with the sex of the offspring. If this were so, it

would quite bear out the above remarks. As a matter of fact, it

should be observed, that the proportion of 106 to 100 does not seem

by any means universal in all countries or at all times. For various

statistical tables on the subject see Quetelet, _Physique Sociale_,

Vol. I. 166, 173, 238.

CHAPTER V.

_THE CONCEPTION _RANDOMNESS_ AND ITS SCIENTIFIC TREATMENT._

1. There is a term of frequent occurrence in treatises

on Probability, and which we have already had repeated occasion

to employ, viz. the designation _random_ applied to an

event, as in the expression 'a random distribution'. The

scientific conception involved in the correct use of this term

is, I apprehend, nothing more than that of aggregate order

and individual irregularity (or apparent irregularity), which

has been already described in the preceding chapters. A

brief discussion of the requisites in this scientific conception,

and in particular of the nature and some of the reasons for

the departure from the popular conception, may serve to

clear up some of the principal remaining difficulties which

attend this part of our subject.

The original,[1] and still popular, signification of the term

is of course widely different from the scientific. What it

looks to is the origin, not the results, of the random performance,

and it has reference rather to the single action

than to a group or series of actions. Thus, when a man

draws a bow 'at a venture', or 'at random', we mean only

to point out the aimless character of the performance;

we are contrasting it with the definite intention to hit a

certain mark. But it is none the less true, as already

pointed out, that we can only apply processes of inference to

such performances as these when we regard them as being

capable of frequent, or rather of indefinitely extended repetition.

Begin with an illustration. Perhaps the best typical

example that we can give of the scientific meaning of

random distribution is afforded by the arrangement of the

drops of rain in a shower. No one can give a guess whereabouts

at any instant a drop will fall, but we know that if

we put out a sheet of paper it will gradually become uniformly

spotted over; and that if we were to mark out any

two equal areas on the paper these would gradually tend

to be struck equally often.

2. I. Any attempt to draw inferences from the assumption

of random arrangement must postulate the occurrence

of this particular state of things at some stage or

other. But there is often considerable difficulty, leading occasionally

to some arbitrariness, in deciding the particular

stage at which it ought to be introduced.

(1) Thus, in many of the problems discussed by mathematicians,

we look as entirely to the results obtained, and

think as little of the actual process by which they are obtained,

as when we are regarding the arrangement of the drops of

rain. A simple example of this kind would be the following.

A pawn, diameter of base one inch, is placed at random on

a chess-board, the diameter of the squares of which is one inch

and a quarter: find the chance that its base shall lie across

one of the intersecting lines. Here we may imagine the

pawns to be so to say rained down vertically upon the board,

and the question is to find the ultimate proportion of those

which meet a boundary line to the total of those which fall.

The problem therefore becomes a merely geometrical one,

viz. to determine the ratio of a certain area on the board to

the whole area. The determination of this ratio is all that

the mathematician ever takes into account.

Now take the following. A straight brittle rod is broken

at random in two places: find the chance that the pieces can

make a triangle.[2] Since the only condition for making a

triangle with three straight lines is that each two shall be

greater than the third, the problem seems to involve the

same general conception as in the former case. We must

conceive such rods breaking at one pair of spots after

another,--no one can tell precisely where,--but showing the

same ultimate tendency to distribute these spots throughout

the whole length uniformly. As in the last case, the mathematician

thinks of nothing but this final result, and pays no

heed to the process by which it may be brought about. Accordingly

the problem is again reduced to one of mensuration,

though of a somewhat more complicated character.

3. (2) In another class of cases we have to contemplate

an intermediate process rather than a final result; but the

same conception has to be introduced here, though it is now

applied to the former stage, and in consequence will not in

general apply to the latter.

For instance: a shot is fired at random from a gun whose

maximum range (i.e. at 45° elevation) is 3000 yards: what is

the chance that the actual range shall exceed 2000 yards?

The ultimately uniform (or random) distribution here is

commonly assumed to apply to the various directions in

which the gun can be pointed; all possible directions above

the horizontal being equally represented in the long run.

We have therefore to contemplate a surface of uniform distribution,

but it will be the surface, not of the ground, but

of a hemisphere whose centre is occupied by the man who

fires. The ultimate distribution of the bullets on the spots

where they strike the ground will not be uniform. The

problem is in fact to discover the law of variation of the

density of distribution.

The above is, I presume, the treatment generally adopted

in solving such a problem. But there seems no absolute

necessity for any such particular choice. It is surely open to

any one to maintain[3] that his conception of the randomness of

the firing is assigned by saying that it is likely that a man

should begin by facing towards any point of the compass indifferently,

and then proceed to raise his gun to any angle

indifferently. The stage of ultimately uniform distribution

here has receded a step further back. It is not assigned

directly to the surface of an imaginary hemisphere, but to

the lines of altitude and azimuth drawn on that surface.

Accordingly, the distribution over the hemisphere itself will

not now be uniform,--there will be a comparative crowding

up towards the pole,--and the ultimate distribution over the

ground will not be the same as before.

4. Difficulties of this kind, arising out of the uncertainty

as to what stage should be selected for that of uniform

distribution, will occasionally present themselves. For

instance: let a book be taken at random out of a bookcase;

what is the chance of hitting upon some assigned volume?

I hardly know how this question would commonly be treated.

If we were to set our man opposite the middle of the shelf

and inquire what would generally happen in practice, supposing

him blindfolded, there cannot be much doubt that

the volumes would not be selected equally often. On the

contrary, it is likely that there would be a tendency to increased

frequency about a centre indicated by the height

of his shoulder, and (unless he be left-handed) a trifle to the

right of the point exactly opposite his starting point.

If the question were one which it were really worth

while to work out on these lines we should be led a long

way back. Just as we imagined our rifleman's position (on

the second supposition) to be determined by two independent

coordinates of assumed continuous and equal facility,

so we might conceive our making the attempt to analyse the

man's movements into a certain number of independent

constituents. We might suppose all the various directions

from his starting point, along the ground, to be equally

likely; and that when he reaches the shelves the random

motion of his hand is to be regulated after the fashion of a

shot discharged at random.

The above would be one way of setting about the statement

of the problem. But the reader will understand that

all which I am here proposing to maintain is that in these,

as in every similar case, we always encounter, under this

conception of 'randomness', at some stage or other, this

postulate of ultimate uniformity of distribution over some

assigned magnitude: either time; or space, linear, superficial,

or solid. But the selection of the stage at which this is to

be applied may give rise to considerable difficulty, and even

arbitrariness of choice.

5. Some years ago there was a very interesting discussion

upon this subject carried on in the mathematical part of

the _Educational Times_ (see, especially, Vol. VII.). As not

unfrequently happens in mathematics there was an almost

entire accord amongst the various writers as to the assumptions

practically to be made in any particular case, and therefore

as to the conclusion to be drawn, combined with a very

considerable amount of difference as to the axioms and definitions

to be employed. Thus Mr M. W. Crofton, with the

substantial agreement of Mr Woolhouse, laid it down unhesitatingly

that "at random" has "a very clear and definite

meaning; one which cannot be better conveyed than by Mr

Wilson's definition, '_according to no law_'; and in this sense

alone I mean to use it." According to any scientific interpretation

of 'law' I should have said that where there was

no law there could be no inference. But ultimate tendency

towards equality of distribution is as much taken for granted

by Mr Crofton as by any one else: in fact he makes this a

deduction from his definition:--"As this infinite system of

parallels are drawn according to no law, they are as thickly

disposed along any part of the [common] perpendicular as

along any other" (VII. p. 85). Mr Crofton holds that any

kind of _unequal_ distribution would imply law,--"If the points

[on a plane] tended to become denser in any part of the plane

than in another, there must be some law attracting them

there" (ib. p. 84). The same view is enforced in his paper

on _Local Probability_ (in the _Phil. Trans._, Vol. 158). Surely

if they tend to become _equally_ dense this is just as much

a case of regularity or law.

It may be remarked that wherever any serious practical

consequences turn upon duly securing the desired randomness,

it is always so contrived that no design or awkwardness

or unconscious one-sidedness shall disturb the result. The

principal case in point here is of course afforded by games of

chance. What we want, when we toss a die, is to secure that

all numbers from 1 to 6 shall be equally often represented in

the long run, but that no person shall be able to predict the

individual occurrence. We might, in our statement of a

problem, as easily postulate 'a number _thought of_ at random'

as 'a shot fired at random', but no one would risk his chances

of gain and loss on the supposition that this would be done

with continued fairness. Accordingly, we construct a die

whose sides are accurately alike, and it is found that we may

do almost what we like with this, at any previous stage to

that of its issue from the dice box on to the table, without

interfering with the random nature of the result.

6. II. Another characteristic in which the scientific

conception seems to me to depart from the popular or original

signification is the following. The area of distribution which

we take into account must be a finite or limited one. The

necessity for this restriction may not be obvious at first sight,

but the consideration of one or two examples will serve to

indicate the point at which it makes itself felt. Suppose

that one were asked to choose a number at random, not

from a finite range, but from the inexhaustible possibilities

of enumeration. In the popular sense of the term,--i.e. of

uttering a number without pausing to choose,--there is no

difficulty. But a moment's consideration will show that no

arrangement even tending towards ultimately uniform distribution

can be secured in this way. No _average_ could be

struck with ever increasing steadiness. So with spatial

infinity. We can rationally speak of choosing a point at

random in a given straight line, area, or volume. But if we

suppose the line to have no end, or the selection to be made

in infinite space, the basis of ultimate tendency towards

what may be called the equally thick deposit of our random

points fails us utterly.

Similarly in any other example in which one of the

magnitudes is unlimited. Suppose I fling a stick at random

in a horizontal plane against a row of iron railings and

inquire for the chance of its passing through without touching

them. The problem bears some analogy to that of the

chessmen, and so far as the motion of translation of the

stick is concerned (if we begin with this) it presents no

difficulty. But as regards the rotation it is otherwise. For

any assigned linear velocity there is a certain angular velocity

_below_ which the stick may pass through without contact,

but _above_ which it cannot. And inasmuch as the former

range is limited and the latter is unlimited, we encounter

the same impossibility as before in endeavouring to conceive

a uniform distribution. Of course we might evade this

particular difficulty by beginning with an estimate of the

angular velocity, when we should have to repeat what has

just been said, mutatis mutandis, in reference to the linear

velocity.

7. I am of course aware that there are a variety of

problems current which seem to conflict with what has just

been said, but they will all submit to explanation. For instance;

What is the chance that three straight lines, taken

or drawn at random, shall be of such lengths as will admit of

their forming a triangle? There are two ways in which we

may regard the problem. We may, for one thing, start with

the assumption of three lines not greater than a certain

length n, and then determine towards what limit the chance

tends as n increases unceasingly. Or, we may maintain that

the question is merely one of _relative proportion_ of the three

lines. We may then start with any magnitude we please to

represent one of the lines (for simplicity, say, the longest of

them), and consider that all possible shapes of a triangle

will be represented by varying the lengths of the other two.

In either case we get a definite result without need to make

an attempt to conceive any random selection from the infinity

of possible length.

So in what is called the "three-point problem":--Three

points in space are selected at random; find the chance of

their forming an acute-angled triangle. What is done is to

start with a closed volume,--say a sphere, from its superior

simplicity,--find the chance (on the assumption of uniform

distribution within this volume); and then conceive the continual

enlargement without limit of this sphere. So regarded

the problem is perfectly consistent and intelligible, though I

fail to see why it should be termed a random selection _in

space_ rather than in a sphere. Of course if we started with

a different volume, say a cube, we should get a different

result; and it is therefore contended (e.g. by Mr Crofton in

the _Educational Times_, as already referred to) that infinite

space is more naturally and appropriately regarded as tended

towards by the enlargement of a sphere than by that of a

cube or any other figure.

Again: A group of integers is taken at random; show

that the number thus taken is more likely to be odd than

even. What we do in answering this is to start with any

finite number n, and show that of all the possible combinations

which can be made within this range there are

more odd than even. Since this is true irrespective of the

magnitude of n, we are apt to speak as if we could conceive

the selection being made at random from the true infinity

contemplated in numeration.

8. Where these conditions cannot be secured then it

seems to me that the attempt to assign any finite value to

the probability fails. For instance, in the following problem,

proposed by Mr J. M. Wilson, "Three straight lines are

drawn at random on an infinite plane, and a fourth line is

drawn at random to intersect them: find the probability of

its passing through the triangle formed by the other three"

(_Ed. Times_, Reprint, Vol. V. p. 82), he offers the following

solution: "Of the four lines, two must and two must not pass

within the triangle formed by the remaining three. Since

all are drawn at random, the chance that the last drawn

should pass through the triangle formed by the other three

is consequently 1/2."

I quote this solution because it seems to me to illustrate

the difficulty to which I want to call attention. As the

problem is worded, a triangle is supposed to be assigned by

three straight lines. However large it may be, its size bears

no finite ratio whatever to the indefinitely larger area outside

it; and, so far as I can put any intelligible construction

on the supposition, the chance of drawing a fourth random

line which should happen to intersect this finite area must

be reckoned as zero. The problem Mr Wilson has solved

seems to me to be a quite different one, viz. "Given four

intersecting straight lines, find the chance that we should,

at random, select one that passes through the triangle formed

by the other three."

The same difficulty seems to me to turn up in most other

attempts to apply this conception of randomness to real

infinity. The following seems an exact analogue of the

above problem:--A number is selected at random, find the

chance that another number selected at random shall be

greater than the former;--the answer surely must be that

the chance is unity, viz. certainty, because the range above

any assigned number is infinitely greater than that below it.

Or, expressed in the only language in which I can understand

the term 'infinity', what I mean is this. If the first

number be m and I am restricted to selecting up to n

(n > m) then the chance of exceeding m is n - m : n; if I

am restricted to 2n then it is 2n - m : 2n and so on. That

is, however large n and m may be the expression is always

intelligible; but, _m being chosen first_, n may be made as

much larger than m as we please: i.e. the chance may be

made to approach as near to unity as we please.

I cannot but think that there is a similar fallacy in De Morgan's

admirably suggestive paper on _Infinity_ (_Camb.

Phil. Trans._ Vol. 11.) when he is discussing the "three-point

problem":--i.e. given three points taken at random find the

chance that they shall form an acute-angled triangle. All

that he shows is, that if we start with _one side as given_ and

consider the subsequent possible positions of the opposite

vertex, there are infinitely as many such positions which

would form an acute-angled triangle as an obtuse: but, as

before, this is solving a different problem.

9. The nearest approach I can make towards true indefinite

randomness, or random selection from true indefiniteness,

is as follows. Suppose a circle with a tangent line extended

indefinitely in each direction. Now from the centre draw

radii at random; in other words, let the semicircumference

which lies towards the tangent be ultimately uniformly intersected

by the radii. Let these radii be then produced so as

to intersect the tangent line, and consider the distribution

of these points of intersection. We shall obtain in the result

_one_ characteristic of our random distribution; i.e. no portion

of this tangent, however small or however remote, but will

find itself in the position ultimately of any small portion of

the pavement in our supposed continual rainfall. That is,

any such elementary patch will become more and more closely

dotted over with the points of intersection. But the other

essential characteristic, viz. that of ultimately _uniform_ distribution,

will be missing. There will be a special form of

distribution,--what in fact will have to be discussed in a

future chapter under the designation of a 'law of error',--by

virtue of which the concentration will tend to be greatest at

a certain point (that of contact with the circle), and will thin

out from here in each direction according to an easily

calculated formula. The existence of such a state of things

as this is quite opposed to the conception of true randomness.

10. III. Apart from definitions and what comes of

them, perhaps the most important question connected with

the conception of Randomness is this: How in any given

case are we to determine whether an observed arrangement

is to be considered a random one or not? This question will

have to be more fully discussed in a future chapter, but we

are already in a position to see our way through some of the

difficulties involved in it.

(1) If the events or objects under consideration are supposed

to be continued indefinitely, or if we know enough

about the mode in which they are brought about to detect

their ultimate tendency,--or even, short of this, if they are

numerous enough to be beyond practical counting,--there is

no great difficulty. We are simply confronted with a question

of fact, to be settled like other questions of fact. In the

case of the rain-drops, watch two equal squares of pavement

or other surfaces, and note whether they come to be more

and more densely uniformly and evenly spotted over: if they

do, then the arrangement is what we call a random one. If

I want to know whether a tobacco-pipe really breaks at random,

and would therefore serve as an illustration of the

problem proposed some pages back, I have only to drop

enough of them and see whether pieces of all possible lengths

are equally represented in the long run. Or, I may argue

deductively, from what I know about the strength of materials

and the molecular constitution of such bodies, as to

whether fractures of small and large pieces are all equally

likely to occur.

11. The reader's attention must be carefully directed

to a source of confusion here, arising out of a certain cross-division.

What we are now discussing is a question of fact,

viz. the nature of a certain ultimate arrangement; we are

not discussing the particular way in which it is brought

about. In other words, the antithesis is between what is and

what is not random: it is not between what is random and

what is designed. As we shall see in a few moments it is

quite possible that an arrangement which is the result,--if

ever anything were so,--of 'design', may nevertheless present

the unmistakeable stamp of randomness of arrangement.

Consider a case which has been a good deal discussed,

and to which we shall revert again: the arrangement of the

stars. The question here is rather complicated by the fact

that we know nothing about the actual mutual positions of

the stars, all that we can take cognizance of being their apparent

or visible places as projected upon the surface of a

supposed sphere. Appealing to what alone we can thus

observe, it is obvious that the arrangement, as a whole, is

not of the random sort. The Milky Way and the other resolvable

nebulæ, as they present themselves to us, are as obvious

an infraction of such an arrangement as would be the

occurrence here and there of patches of ground in a rainfall

which received a vast number more drops than the spaces

surrounding them. If we leave these exceptional areas out

of the question and consider only the stars which are visible

by the naked eye or by slight telescopic power, it seems

equally certain that the arrangement _is_, for the most part, a

fairly representative random one. By this we mean nothing

more than the fact that when we mark off any number of

equal areas on the visible sphere these are found to contain

approximately the same number of stars.

The actual arrangement of the stars in space _may_ also

be of the same character: that is, the apparently denser

aggregation may be apparent only, arising from the fact that

we are looking through regions which are not more thickly

occupied but are merely more extensive. The alternative

before us, in fact, is this. If the whole volume, so to say, of

the starry heavens is tolerably regular in shape, then the

arrangement of the stars is not of the random order; if that

volume is very irregular in shape, it is possible that the arrangement

within it may be throughout of that order.

12. (2) When the arrangement in question includes

but a comparatively small number of events or objects, it

becomes much more difficult to determine whether or not it

is to be designated a random one. In fact we have to shift

our ground, and to decide not by what has been actually

observed but by what we have reason to conclude would be

observed if we could continue our observation much longer.

This introduces what is called 'Inverse Probability', viz. the

determination of the nature of a cause from the nature of the

observed effect; a question which will be fully discussed in

a future chapter. But some introductory remarks may be

conveniently made here.

Every problem of Probability, as the subject is here understood,

introduces the conception of an ultimate limit, and

therefore presupposes an indefinite possibility of repetition.

When we have only a finite number of occurrences before us,

_direct_ evidence of the character of their arrangement fails us,

and we have to fall back upon the nature of the agency

which produces them. And as the number becomes smaller

the confidence with which we can estimate the nature of the

agency becomes gradually less.

Begin with an intermediate case. There is a small lawn,

sprinkled over with daisies: is this a random arrangement?

We feel some confidence that it is so, on mere inspection;

meaning by this that (negatively) no trace of any regular

pattern can be discerned and (affirmatively) that if we take

any moderately small area, say a square yard, we shall find

much about the same number of the plants included in it.

But we can help ourselves by an appeal to the known agency

of distribution here. We know that the daisy spreads by

seed, and considering the effect of the wind and the continued

sweeping and mowing of the lawn we can detect causes at

work which are analogous to those by which the dealing of

cards and the tossing of dice are regulated.

In the above case the appeal to the process of production

was subsidiary, but when we come to consider the nature of

a very small succession or group this appeal becomes much

more important. Let us be told of a certain succession of

'heads' and 'tails' to the number of ten. The range here is

far too small for decision, and unless we are told whether the

agent who obtained them was tossing or designing we are

quite unable to say whether or not the designation of 'random'

ought to be applied to the result obtained. The truth

must never be forgotten that though 'design' is sure to

break down in the long run if it make the attempt to produce

directly the semblance of randomness,[4] yet for a short

spell it can simulate it perfectly. Any short succession, say

of heads and tails, may have been equally well brought

about by tossing or by deliberate choice.

13. The reader will observe that this question of

randomness is being here treated as simply one of ultimate

statistical fact. I have fully admitted that this is not the

primitive conception, nor is it the popular interpretation,

but to adopt it seems the only course open to us if we are to

draw inferences such as those contemplated in Probability.

When we look to the producing agency of the ultimate

arrangement we may find this very various. It may prove

itself to be (a few stages back) one of conscious deliberate

purpose, as in drawing a card or tossing a die: it may be

the outcome of an extremely complicated interaction of many

natural causes, as in the arrangement of the flowers scattered

over a lawn or meadow: it may be of a kind of which we

know literally nothing whatever, as in the case of the actual

arrangement of the stars relatively to each other.

This was the state of things had in view when it was

said a few pages back that randomness and design would

result in something of a cross-division. Plenty of arrangements

in which design had a hand, a stage or two back, can

be mentioned, which would be quite indistinguishable in

their results from those in which no design whatever could

be traced. Perhaps the most striking case in point here is

to be found in the arrangement of the digits in one of the

natural arithmetical constants, such as π or e, or in a table

of logarithms. If we look to the process of production of

these digits, no extremer instance can be found of what we

mean by the antithesis of randomness: every figure has its

necessarily pre-ordained position, and a moment's flagging of

intention would defeat the whole purpose of the calculator.

And yet, if we look to results only, no better instance can

be found than one of these rows of digits if it were intended

to illustrate what we practically understand by a chance

arrangement of a number of objects. Each digit occurs

approximately equally often, and this tendency develops as

we advance further: the mutual juxtaposition of the digits

also shows the same tendency, that is, any digit (say 5) is

just as often followed by 6 or 7 as by any of the others. In

fact, if we were to take the whole row of hitherto calculated

figures, cut off the first five as familiar to us all, and contemplate

the rest, no one would have the slightest reason

to suppose that these had not come out as the results of a

die with ten equal faces.

14. If it be asked _why_ this is so, a rather puzzling

question is raised. Wherever physical causation is involved

we are generally understood to have satisfied the demand

implied in this question if we assign antecedents which will

be followed regularly by the event before us; but in geometry

and arithmetic there is no opening for antecedents. What

we then commonly look for is a demonstration, i.e. the resolution

of the observed fact into axioms if possible, or at

any rate into admitted truths of wider generality. I do not

know that a demonstration can be given as to the existence

of this characteristic of statistical randomness in such successions

of digits as those under consideration. But the

following remarks may serve to shift the onus of unlikelihood

by suggesting that the preponderance of analogy is

rather in favour of the existence.

Take the well-known constant π for consideration. This

stands for a quantity which presents itself in a vast number

of arithmetical and geometrical relations; let us take for

examination the best known of these, by regarding it as

standing for the ratio of the circumference to the diameter

of a circle. So regarded, it is nothing more than a simple

case of the measurement of a magnitude by an arbitrarily

selected unit. Conceive then that we had before us a rod

or line and that we wished to measure it with absolute

accuracy. We must suppose--if we are to have a suitable

analogue to the determination of π to several hundred

figures,--that by the application of continued higher magnifying

power we can detect ever finer subdivisions in the

graduation. We lay our rod against the scale and find it,

say, fall between 31 and 32 inches; we then look at the

next division of the scale, viz. that into tenths of an inch.

Can we see the slightest reason why the number of these

tenths should be other than independent of the number of

whole inches? The "piece over" which we are measuring

may in fact be regarded as an entirely new piece, which had

fallen into our hands after that of 31 inches had been

measured and done with; and similarly with every successive

piece over, as we proceed to the ever finer and finer divisions.

Similar remarks may be made about most other incommensurable

quantities, such as irreducible roots. Conceive

two straight lines at right angles, and that we lay off a

certain number of inches along each of these from the point

of intersection; say two and five inches, and join the extremities

of these so as to form the diagonal of a right-angled

triangle. If we proceed to measure this diagonal in terms

of either of the other lines we are to all intents and purposes

extracting a square root. We should expect, rather than

otherwise, to find here, as in the case of π, that incommensurability

and resultant randomness of order in the digits

was the rule, and commensurability was the exception. Now

and then, as when the two sides were three and four, we

should find the diagonal commensurable with them; but

these would be the occasional exceptions, or rather they

would be the comparatively finite exceptions amidst the

indefinitely numerous cases which furnished the rule.

15. The best way perhaps of illustrating the truly

random character of such a row of figures is by appealing to

graphical aid. It is not easy here, any more than in ordinary

statistics, to grasp the import of mere figures; whereas the

arrangement of groups of points or lines is much more

readily seized. The eye is very quick in detecting any

symptoms of regularity in the arrangement, or any tendency

to denser aggregation in one direction than in another.

How then are we to dispose our figures so as to force them

to display their true character? I should suggest that we

set about _drawing a line at random_; and, since we cannot

trust our own unaided efforts to do this, that we rely upon

the help of such a table of figures to do it for us, and then

examine with what sort of efficiency they can perform the

task. The problem of drawing _straight_ lines at random,

under various limitations of direction or intersection, is

familiar enough, but I do not know that any one has suggested

the drawing of a line whose shape as well as position

shall be of a purely random character. For simplicity we

suppose the line to be confined to a plane.

The definition of such a line does not seem to involve

any particular difficulty. Phrased in accordance with the

ordinary language we should describe it as the path (i.e. any

path) traced out by a point which at every moment is as

likely to move in any one direction as in any other. That

we could not ourselves draw such a line, and that we could

not get it traced by any physical agency, is certain. The

mere inertia of any moving body will always give it a

tendency, however slight, to go on in a straight line at each

moment, instead of being instantly responsive to instantaneously

varying dictates as to its direction of motion. Nor can

we conceive or picture such a line in its ultimate or ideal condition.

But it is easy to give a graphical approximation to it,

and it is easy also to show how this approximation may be

carried on as far as we please towards the ideal in question.

We may proceed as follows. Take a sheet of the ordinary

ruled paper prepared for the graphical exposition of curves.

Select as our starting point the intersection of two of these

lines, and consider the eight 'points of the compass' indicated

by these lines and the bisections of the contained

right angles.[5] For suggesting the random selection amongst

these directions let them be numbered from 0 to 7, and

let us say that a line measured due 'north' shall be designated

by the figure 0, 'north-east' by 1, and so on. The

selection amongst these numbers, and therefore directions, at

every corner, might be handed over to a die with eight faces;

but for the purpose of the illustration in view we select the

digits 0 to 7 as they present themselves in the calculated

value of π. The sort of path along which we should

travel by a series of such steps thus taken at random

may be readily conceived; it is given at the end of this

chapter.

For the purpose with which this illustration was proposed,

viz. the graphical display of the succession of digits

in any one of the incommensurable constants of arithmetic

or geometry, the above may suffice. After actually testing

some of them in this way they seem to me, so far as the eye,

or the theoretical principles to be presently mentioned, are

any guide, to answer quite fairly to the description of randomness.

16. As we are on the subject, however, it seems worth

going farther by enquiring how near we could get to the

ideal of randomness of direction. To carry this out completely

two improvements must be made. For one thing,

instead of confining ourselves to eight directions we must

admit an infinite number. This would offer no great difficulty;

for instead of employing a small number of digits we

should merely have to use some kind of circular teetotum

which would rest indifferently in any direction. But in the

next place instead of short finite steps we must suppose them

indefinitely short. It is here that the actual unattainability

makes itself felt. We are familiar enough with the device,

employed by Newton, of passing from the discontinuous

polygon to the continuous curve. But we can resort to this

device because the ideal, viz. the curve, is as easily drawn

(and, I should say, as easily conceived or pictured) as any of

the steps which lead us towards it. But in the case before

us it is otherwise. The line in question will remain discontinuous,

or rather angular, to the last: for its angles do

not tend even to lose their sharpness, though the fragments

which compose them increase in number and diminish in

magnitude without any limit. And such an ideal is not conceivable

as an ideal. It is as if we had a rough body under

the microscope, and found that as we subjected it to higher

and higher powers there was no tendency for the angles to

round themselves off. Our 'random line' must remain as

'spiky' as ever, though the size of its spikes of course

diminishes without any limit.

The case therefore seems to be this. It is easy, in words,

to indicate the conception by speaking of a line which at

every instant is as likely to take one direction as another.

It is easy moreover to draw such a line with any degree

of minuteness which we choose to demand. But it is not

possible to conceive or picture the line in its ultimate form.[6]

There is in fact no 'limit' here, intelligible to the understanding

or picturable by the imagination (corresponding to

the asymptote of a curve, or the continuous curve to the

incessantly developing polygon), towards which we find ourselves

continually approaching, and which therefore we are

apt to conceive ourselves as ultimately attaining. The usual

assumption therefore which underlies the Newtonian infinitesimal

geometry and the Differential Calculus, ceases to

apply here.

17. If we like to consider such a line in one of its

approximate stages, as above indicated, it seems to me that

some of the usual theorems of Probability, where large

numbers are concerned, may safely be applied. If it be

asked, for instance, whether such a line will ultimately tend

to stray indefinitely far from its starting point, Bernoulli's

'Law of Large Numbers' may be appealed to, in virtue of

which we should say that it was excessively unlikely that its

divergence should be relatively great. Recur to our graphical

illustration, and consider first the resultant deviation

of the point (after a great many steps) right or left of the

vertical line through the starting point. Of the eight admissible

motions at each stage two will not affect this relative

position, whilst the other six are equally likely to move us a

step to the right or to the left. Our resultant 'drift' therefore

to the right or left will be analogous to the resultant

difference between the number of heads and tails after a

great many tosses of a penny. Now the well-known outcome

of such a number of tosses is that ultimately the

_proportional_ approximation to the _à priori_ probability, i.e. to

equality of heads and tails, is more and more nearly carried

out, but that the _absolute_ deflection is more and more widely

displayed.

Applying this to the case in point, and remembering

that the results apply equally to the horizontal and vertical

directions, we should say that after any very great number

of such 'steps' as those contemplated, the ratio of our distance

from the starting point to the whole distance travelled

will pretty certainly be small, whereas the actual distance

from it would be large. We should also say that the longer

we continued to produce such a line the more pronounced

would these tendencies become. So far as concerns this test,

and that afforded by the general appearance of the lines

drawn,--this last, as above remarked, being tolerably trustworthy,--I

feel no doubt as to the generally 'random'

character of the rows of figures displayed by the incommensurable

or irrational ratios in question.

As it may interest the reader to see an actual specimen

of such a path I append one representing the arrangement

of the eight digits from 0 to 7 in the value of π. The data

are taken from Mr Shanks' astonishing performance in the

calculation of this constant to 707 places of figures (_Proc. of

R. S._, XXI. p. 319). Of these, after omitting 8 and 9, there

remain 568; the diagram represents the course traced out

by following the direction of these as the clue to our path.

Many of the steps have of course been taken in opposite

directions twice or oftener. The result seems to me to

furnish a very fair graphical indication of randomness. I

have compared it with corresponding paths furnished by

rows of figures taken from logarithmic tables, and in other

ways, and find the results to be much the same.

[Figure: Digits of π as a path.]

1. According to Prof. Skeat (_Etymological Dictionary_) the earliest

known meaning is that of _furious_ action, as in a charge of

cavalry. The etymology, he considers, is connected with the

Teutonic word _rand_ (brim), and implies the furious and irregular

action of a river full to the brim.

2. See the problem paper of Jan. 18, 1854, in the Cambridge

Mathematical Tripos.

3. As, according to Mr H. Godfray, the majority of the candidates

_did_ assume when the problem was once proposed in an

examination. See the _Educational Times_ (Reprint, Vol. VII.

p. 99.)

4. Vide p. 68.

5. It would of course be more complete to take _ten_ alternatives of

direction, and thus to omit none of the digits; but this is much

more troublesome in practice than to confine ourselves to eight.

6. Any more than we picture the shape of an equiangular spiral _at the

centre_.