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THE PRINCIPLES OF SCIENCE:

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METHOD.

BY

W. STANLEY JEVONS, M.A., F.R.S.,

FELLOW OF UNIVERSITY COLLEGE, LONDON ;
PROFESSOR OF LOGIC AND POLITICAL ECONOMY IN THE OWENS COLLEGE, MANCHESTER.

VOL. I.

MACMILLAN AND CO.

1874

[ The Right of Translation and Repruduction is reserved.}

OXFORD:

B. PIOKABD HALL, AND J. H. STACY,

PRINTERS TO THE UNIVERSITY.

PREFACE.

IT may be truly asserted that the rapid progress of
the physical sciences during the last three centuries has
not been accompanied by a corresponding advance in
the theory of reasoning. Physicists speak familiarly of
Scientific Method, but they could not readily describe
what they mean by that expression. Profoundly engaged
in the study of particular classes of natural phenomena,
they are usually too much engrossed in the immense and
ever-accumulating details of their special sciences, to
generalize upon the methods of reasoning which they
unconsciously employ. Yet few will deny that these
methods of reasoning ought to be studied, especially by
those who endeavour to introduce scientific order into less
successful and methodical branches of knowledge.

The application of Scientific Method cannot be re
stricted to the sphere of lifeless objects. We must sooner
or later have strict sciences of those mental and social
phenomena, which, if comparison be possible, are of
more interest to us than purely material phenomena.
But it is the proper course of reasoning to proceed from
the known to the unknown — from the evident to the
obscure — from the material and palpable to the subtle
and refined. The physical sciences may therefore be
properly made the practice-ground of the reasoning

V

vi PREFACE.

powers, because they furnish us with a great body of
precise and successful investigations. In these sciences
we meet with happy instances of unquestionable deductive
reasoning, of extensive generalization, of happy prediction,
of satisfactory verification, of nice calculation of proba
bilities. We can note how the slightest analogical clue
has been followed up to a glorious discovery, how a rash
generalization has at length been exposed, or a conclusive
experimentum crucis has decided the long-continued strife
between two rival theories.

In following out my design of detecting the general
methods of inductive ^investigation, I have found that the
more elaborate and interesting processes of quantitative
induction have their necessary foundation in the simpler
science of Formal Logic. The earlier, and probably by
far the least attractive part of this work, consists, there
fore, in a statement of the so-called Fundamental Laws of
Thought, and of the all-important Principle of Substi
tution, of which, as I think, all reasoning is a develop
ment. The whole procedure of inductive inquiry, in its
most complex cases, is foreshadowed in the combinational
view of Logic, which arises directly from these fundamental
principles. Incidentally I have described the mechanical
arrangements by which the use of the important form
called the Logical Abecedarium, and the whole working
of the combinational system of Formal Logic, may be ren
dered evident to the eye, and easy to the mind and
hand.

The study both of Formal Logic and of the Theory of
Probabilities, has led me to adopt the opinion that there
is no such thing as a distinct method of induction as con
trasted with deduction, but that induction is simply an
inverse employment of deduction. Within the last cen
tury a reaction has been setting in against the purely
empirical procedure of Francis Bacon, and physicists have

PREFACE. vii

learnt to advocate the use of hypotheses. I take the
extreme view of holding that Francis Bacon, although he
correctly insisted upon constant reference to experience,
had no correct notions as to the logical method by which,
from particular facts, we educe laws of nature. I en
deavour to show that hypothetical anticipation of nature
is an essential part of inductive inquiry, and that it is the
Newtonian method of deductive reasoning combined with
elaborate experimental verification, which has led to all
the great triumphs of scientific research.

In attempting to give an explanation of this view of
Scientific Method, I have first to show that the sciences of
number and quantity repose upon and spring from the
simpler and more general science of Logic. The Theory of
Probability, which enables us to estimate and calculate
quantities of knowledge, is then described, and especial
attention is drawn to the Inverse Method of Proba
bilities, which involves, as I conceive, the true principle
of inductive procedure. No inductive conclusions are more
than probable, and I adopt the opinion that the theory of
probability is an essential part of logical method, so that
the logical value of every inductive result must be deter
mined consciously or unconsciously, according to the
principles of the inverse method of probability.

The phenomena of nature are commonly manifested in.
quantities of time, space, force, energy, &c., and the ob
servation, measurement, and analysis of the various quan
titative conditions or results involved, even in a simple
experiment, demand much employment of systematic pro
cedure. I devote a book, therefore, to a simple and
general description of the devices by which exact measure
ment is effected, errors eliminated, a probable mean result
attained, and the probable error of that mean ascertained.
I then proceed to the principal, and probably the most
interesting, subject of the book, illustrating successively

PREFACE.

the conditions and precautions requisite for accurate ob
servation, for successful experiment, and for the sure
detection of the quantitative laws of nature. As it is
impossible to comprehend aright the value of quantitative
laws without constantly bearing in mind the degree of
quantitative approximation to the truth probably attained,
I have devoted a special chapter to the Theory of Ap
proximation, and however imperfectly I may have treated
this subject, I must look upon it as a very essential part
of a work on Scientific Method.

It then remains to illustrate the sound use of hypo
thesis, to distinguish between the portions of knowledge
which we owe to empirical observation, to accidental
discovery, or to scientific prediction. Interesting questions
arise concerning the accordance of quantitative theories
and experiments, and I point out how the successive veri
fication of an hypothesis by distinct methods of experi
ment yields conclusions approximating to but never
attaining certainty. Additional illustrations of the general
procedure of inductive investigations are given in a
chapter on the Character of the Experimentalist, in which
I endeavour to show, moreover, that the inverse use of
deduction was really the logical method of such great
masters of experimental inquiry as Newton, Huyghens,
and Faraday.

In treating Generalization and Analogy, I consider the
precautions requisite in inferring from one case to another,
or from one part of the universe to another part, the
validity of all such inferences resting ultimately upon the
inverse method of probabilities. The treatment of Ex
ceptional Phenomena appeared to afford an interesting
subject for a further chapter illustrating the various modes
in which an outstanding fact may eventually be explained.
The formal part of the book closes with the subject of
Classification, which is, however, very inadequately treated.

PREFACE.

I have, in fact, almost restricted myself to showing that
all classification is fundamentally carried out upon the
principles of Formal Logic and the Logical Abecedarium
described at the outset.

In certain concluding remarks I have expressed the
conviction which the study of Logic has by degrees forced
upon my mind, that serious misconceptions are entertained
by some scientific men as to the logical value of our know
ledge of nature. We have heard much of what has been
aptly called the Reign of Law, and the necessity and uni
formity of natural forces has been not uncommonly inter
preted as involving the non-existence of an intelligent and
benevolent Power, capable of interfering with the course
of natural events. Fears have been expressed that the
progress of Scientific Method must therefore result in dis
sipating the fondest beliefs of the human heart. Even the
' Utility of Religion ' is seriously proposed as a subject of
discussion. It seemed to be not out of place in a work on
Scientific Method to allude to the ultimate results and
limits of that method. I fear that I have very imper
fectly succeeded in expressing my strong conviction that
before a rigorous logical scrutiny the Reign of Law will
prove to be an unverified hypothesis, the Uniformity of
Nature an ambiguous expression, the certainty of our
scientific inferences to a great extent a delusion. The
value of science is of course very high, while the con
clusions are kept well within the limits of the data on
which they are founded, but it is pointed out that our
experience is of the most limited character compared with
what there is to learn, while our mental powers seem to
fall infinitely short of the task of comprehending and
explaining fully the nature of any one object. I draw the
conclusion that we must interpret the results of Scientific
Method in an affirmative sense only. Ours must be a
truly positive philosophy, not that false negative philo-

PREFACE.

sophy which, building on a few material facts, presumes
to assert that it has compassed the bounds of existence,
while it nevertheless ignores the most unquestionable
phenomena of the human mind and feelings.

I have to thank my colleague, Professor Barker, for
carefully revising several of the sheets most abounding in
mathematical considerations. It is approximately certain
that in freely employing illustrations drawn from many
different sciences, I have frequently fallen into errors of
detail. In this respect I must throw myself upon the
indulgence of the reader, who will bear in mind, as I hope,
that the scientific facts are generally mentioned purely for
the purpose of iUustration, so that inaccuracies of detail
wih1 not in the majority of cases affect the truth of the
general principles iUustrated.

December I5th, 1873.

CONTENTS.

BOOK I.

FORMAL LOGIC, DEDUCTIVE AND INDUCTIVE.
CHAPTER I.

INTRODUCTION.

SECTION PAGE

1. Introduction ...... 1

2. The Powers of Mind concerned in the Creation of Science . 4

3. Laws of Identity and Difference . . . .6

4. The Nature and Authority of the Laws of Identity and Dif

ference ....... 7

5. The Process of Inference . . . . .11

6. Deduction and Induction . . . . .13

7. Symbolic Expression of Logical Inference . . .15

8. Expression of Identity and Difference . . .18

9. General Formula of Logical Inference . . .21

10. The Propagating Power of Identity . . . .24

11. Anticipations of the Principle of Substitution . .25

12. The Logic of Relatives . . . . .27

CHAPTER II.

TEBMS.

1. Terms . . ..... 29

2. Twofold Meaning of General Names . . .31

3. Abstract Terms . . . . . .33

4. Substantial Terms . » . . . .34

5. Collective Terms . . . . . .35

6. Synthesis of Terms . . . . . .36

7. Symbolic Expression of the Law of Contradiction . .38

8. Certain Special Conditions of Logical Symbols r .39

CONTENTS.

CHAPTER III.

PROPOSITIONS.

SECTION PAGE

1. Propositions . . . . . . .43

2. Simple Identities . . . . . .44

3. Partial Identities . . . . . .47

4. Limited Identities . . . . . .51

5. Negative Propositions . . . . .52

6. Conversion of Propositions . . . . .55

7. Twofold Interpretation of Propositions . . .57

CHAPTER IV.

DEDUCTIVE REASONING.

1. Deductive Reasoning . . . . .59

2. Immediate Inference . . . . . .60

3. Inference with Two Simple Identities . . .61

4. Inference with a Simple and a Partial Identity . .64

5. Inference of a Partial from Two Partial Identities . . 66

6. On the Ellipsis of Terms in Partial Identities . .69

7. Inference of a Simple from Two Partial Identities . . 70

8. Inference of a Limited from Two Partial Identities . .71

9. Miscellaneous Forms of Deductive Inference . . .72
10. Fallacies . . . . . . .74

CHAPTER V.

DISJUNCTIVE PROPOSITIONS.

1. Disjunctive Propositions . . . . .79

2. Expression of the Alternative Relation . . .81

3. Nature of the Alternative Relation . . . .81

4. Laws of the Disjunctive Relation . . . .85

5. Symbolic Expression of the Law of Duality . . .87

6. Various Forms of the Disjunctive Proposition . .89

7. Inference by Disjunctive Propositions . . .90

CHAPTER VI.

THE INDIRECT METHOD OF INFERENCE.

1. The Indirect Method of Inference . . . .95

2. Simple Illustrations . . . . . .97

3. Employment of the Contrapositive Proposition . . 99

4. Contrapositive of a Simple Identity . . . .101

5. Miscellaneous Examples of the Method . . . 103

6. Abbreviation of the Process ..... 105

7. The Logical Abecedarium . . . . 10£

8. The Logical Slate . . . . . .110

9. Abstraction of Indifferent Circumstances . 112

CONTENTS. xiii

SECTION PAGE

10. Illustrations of the Indirect Method . . . .113

11. Fallacies analysed by the Indirect Method . . .117

12. The Logical Abacus . . . . . .119

13. The Logical Machine . . . . .123

14. The Order of Premises . . . . .131

15. The Equivalency of Propositions . . . .132

16. The Nature of Inference . . . . .136

CHAPTER VII.

INDUCTION.

1. Induction . . . . . . .139

2. Induction an Inverse Operation . . . .140

3. Induction of Simple Identities . . . .146

4. Induction of Partial Identities . . . .149

5. Complete Solution of the Inverse or Inductive Logical Pro

blem . . . . . . .154

6. The Inverse Logical Problem involving Three Terms . 157

7. Distinction between Perfect and Imperfect Induction . 164

8. Transition from Perfect to Imperfect Induction . .168

BOOK II.

NUMBER, VARIETY, AND PROBABILITY.
CHAPTER VIII.

PRINCIPLES OF NUMBER.

1. Principles of Number . . . . .172

2. The Nature of Number . . . . .175

3. Of Numerical Abstraction . . . . .177

4. Concrete and Abstract Numbers . . . .178

5. Analogy of Logical and Numerical Terms . . .180

6. Principle of Mathematical Inference . . . .183

7. Reasoning by Inequalities . . . . .186

8. Arithmetical Reasoning . . . . .188

9. Numerically Definite Reasoning . . . .190

CHAPTER IX.

THE VARIETY OP NATURE, OR THE DOCTRINE OP COMBINATIONS
AND PERMUTATIONS.

1. The Variety of Nature . . . .195

2. Distinction of Combinations and Permutations . . 200

3. Calculation of Number of Combinations . . . 204

4. The Arithmetical Triangle . . . . .206

xiv CONTENTS.

SECTION PAGE

5. Connexion between the Arithmetical Triangle and the

Logical Abecedarium . . . . .214

6. Possible Variety of Nature and Art . . . .216

7. Higher Orders of Variety . . . . .219

CHAPTER X.

THEORY OF PROBABILITY.

1. Theory of Probability . . . . .224

2. Fundamental Principles of the Theory . . . 228

3. Rules for the Calculation of Probabilities . . .231

4. Employment of the Logical Abecedarium in questions of

Probability . . . . . .234

5. Comparison of the Theory with Experience . . . 236

6. Probable Deductive Arguments .... 239

7. Difficulties of the Theory . . . . .243

CHAPTER XL

PHILOSOPHY OF INDUCTIVE INFERENCE.

1. Philosophy of Inductive Inference .... 250

2. Various Classes of Inductive Truths . . . .251

3. The Relation of Cause and Eifect .... 253

4. Fallacious Use of the Term Cause . . . .254

5. Confusion of Two Questions . . . . .256

6. Definition of the Term Cause .... 257

7. Distinction of Inductive and Deductive Results . .260

8. On the Grounds of Inductive Inference . . .262

9. Illustrations of the Inductive Process . . .263

10. Geometrical Reasoning . . . . .268

11. Discrimination of Certainty and Probability in the Inductive

Process ....... 271

CHAPTER XII.

THE INDUCTIVE OR INVERSE APPLICATION OF THE THEORY
OF PROBABILITIES.

1. The Inductive or Inverse Application of the Theory of

Probabilities . . . . . .276

2. Principle of the Inverse Method . . . .279

3. Simple Applications of the Inverse Method . . .281

4. Application of the Theory of Probabilities in Astronomy . 285

5. Statement of the General Inverse Problem . . .289

6. Simple Illustration of the Inverse Problem . . .292

7. General Solution of the Inverse Problem . . . 295

8. Rules of the Inverse Method . . . .297

9. Fortuitous Coincidences ..... 302
10. Summary of the Theory of Inductive Inference . . 307

CONTENTS. xv

BOOK III.

METHODS OF MEASUREMENT.

CHAPTER XIII.

THE EXACT MEASUREMENT OF PHENOMENA.

SECTION PAGE

1. The Exact Measurement of Phenomena . . .313

2. Division of the Subject . . . . .318

3. Continuous Quantity . . . . .318

4. The Fallacious Indications of the Senses . . .320

5. Complexity of Quantitative Questions . . .323

6. The Methods of Accurate Measurement . . , 328

7. Conditions of Accurate Measurement . . . 328

8. Measuring Instruments . . . . .330

9. The Method of Repetition . ... 336

10. Measurements by Natural Coincidence . . .341

11. Modes of Indirect Measurement .... 345

12. Comparative Use of Measuring Instruments . . .349

13. Systematic Performance of Measurements . . .351

14. The Pendulum . . . . . .352

15. Attainable Accuracy of Measurement . . .354

CHAPTER XIV.

UNITS AND STANDARDS OP MEASUREMENT.

1. Units and Standards of Measurement . . .357

2. Standard Unit of Time . . . . .359

3. The Unit of Space and the Bar Standard . . .365

4. The Terrestrial Standard . . . . .367

5. The Pendulum Standard ..... 369

6. Unit of Density . . . . . .371

7. Unit of Mass . . . . . .372

8. Subsidiary Units ...... 374

9. Derived Units . . . . . .375

10. Provisionally Independent Units . . . .377

11. Natural Constants and Numbers .... 380

12. Mathematical Constants . . . . .381

13. Physical Constants . . . . . .383

14. Astronomical Constants . . . . .384

15. Terrestrial Numbers . . . . .385

16. Organic Numbers . . . . . 385

17. Social Numbers . . . . . .386

CHAPTER XV.

ANALYSIS OF QUANTITATIVE PHENOMENA.

1. Analysis of Quantitative Phenomena . . . .387

2. Illustrations of the Complication of Effects . . .388

3. Methods of Eliminating Error . . . .391

xvi CONTENTS.

SECTION PAGE

4. Method of Avoidance of Error .... 393

5. Differential Method . . . . . .398

6. Method of Correction ..... 400

7. Method of Compensation ..... 406

8. Method of Reversal . . . . .410

CHAPTER XVI.

THE METHOD OF MEANS.

1. The Method of Means . . . . .414

2. Several Uses of the Mean Result . . . .416

3. The Significations of the Terms Mean and Average . .418

4. On the Fictitious Mean or Average Result . . .422

5. The Precise Mean Result . . . . .424

6. Determination of the Zero Point by the Method of Means . 428

7. Determination of Maximum Points . . . .431

CHAPTER XVII.

THE LAW OF ERROR.

1. The Law of Error .... .434

2. Establishment of the Law of Error .... 435

3. Herschel's Geometrical Proof . . .437

4. Laplace's and Quetelet's Proof of the Law of Error . . 438

5. Derivation of the Law of Error from Simple Logical Prin

ciples ....... 443

6. Verification of the Law of Error . . 444

7. Remarks on the General Law of Error . .447

8. The Probable Mean Result as defined by the Law of Error . 447

9. Weighted Observations . . .449

10. The Probable Error of Mean Results . 451

11. The Rejection of the Mean Result . . 454

12. Method of Least Squares . .458

13. Works upon the Theory of Probability and the Law of Error 459

14. Detection of Constant Errors . . . 460

THE PRINCIPLES OF SCIENCE,

CHAPTEE I.

INTRODUCTION.

SCIENCE arises from the discovery of Identity amid
Diversity. The process may be described in many dif
ferent words, but our language must always imply the
presence of one common and necessary element. In
every act of inference or scientific method we are engaged
about a certain identity, sameness, similarity, likeness,
resemblance, analogy, equivalence or equality apparent
between two objects. It is doubtful whether an entirely
isolated phenomenon could present itself to our notice,
since there must always be a contrast between object
and object to awaken our consciousness. But in any
case an isolated phenomenon could be studied to no
useful purpose. The whole value of science consists in
the power which it confers upon us of applying to one
object the knowledge acquired from like objects ; and it
is only so far, therefore, as we can discover and register
resemblances or differences that we can turn our obser
vations to account.

Nature is a spectacle continually exhibited to our
senses, in which phenomena are mingled in combina
tions of endless variety and novelty. Wonder fixes the
mind's attention; memory stores up a record of each

B

THE PRINCIPLES OF SCIENCE.

distinct impression ; the powers of association bring forth
the record when the like is felt again. By the higher
faculties of judgment and reasoning the mind compares
the new with the old, recognises essential identity, even
when disguised by diverse circumstances, and expects to
find again what was before experienced. It must be the
ground of all reasoning and inference that what is true
of one thing will be true of its equivalent, and that under
carefully ascertained conditions Nature repeats herself.

Were this indeed a Chaotic Universe, the powers of
mind employed in science would be useless to us. Did
Chance wholly take the place of order, and did all phe
nomena come out, not of one same Infinite Lottery, to
use Condorcet's expression, but out of lotteries ever
changing in their conditions, there could be no reason to
expect the like result in like circumstances. It is possible
to conceive a world in which no two things should be

O

associated more often, in the long run, than any other
two things. The frequent conjunction of any two events
would then be purely fortuitous, and if we expected
conjunctions to recur continually we should be disap
pointed. In such a world we might recognise the same
phenomenon as it appeared from time to time, just as we
might recognise a marked ball as it was occasionally
drawn from a ballot-box ; but the approach of any one
phenomenon would be in no way indicated by what had
gone before, nor would it be at all a sign of what was to
come after. In such a world knowledge would be no
more than the memory of past coincidences, and the
reasoning powers, if they existed at all, would give no
clue to the nature of the present, and no presage of the
future.

Happily the Universe in which we dwell is not the
result of chance, and where chance seems to work it is
our own deficient faculties which prevent us from recog-

INTRODUCTION.

nising the operation of Law and of Design. In the
material framework of this world, substances and forces
present themselves in definite and stable combinations.
All things are not in perpetual flux, as ancient philoso
phers held. Element remains element ; iron changes not
into gold, nor oxygen into hydrogen. With suitable pre
cautions we can calculate upon finding the same thing
again endowed with the same properties. The con
stituents of the globe, indeed, appear in almost endless
combinations ; but each combination bears its fixed cha
racter, and when resolved is found to be the compound of
definite substances. Misapprehensions must continually
occur, owing to the limited extent of our experience.
We can never have examined and registered possible ex
istences so thoroughly as to be sure that no new ones will
occur and frustrate our calculations. The same outward
appearances may cover any amount of hidden differences
which we have not yet suspected. To the variety of
substances and powers diffused through nature at its
creation, we must not suppose that our brief experience
can assign a limit ; and the necessary imperfection of our
knowledge should be ever borne in mind.

Yet there is much to give us confidence in science.
The wider our experience, the more minute our examina
tion of the globe, the greater the accumulation of well-
reasoned knowledge, — the fewer must become the failures
of inference compared with the successes. Exceptions to
the prevalence of Law are gradually reduced to Law
themselves. Certain deep similarities have been detected
among the objects around us, and have never yet been
found wanting. As the means of examining distant parts
of the universe have been acquired, those similarities have
been traced there as here. Other worlds and stellar
systems may be almost incomprehensively different from
ours in magnitude, condition and disposition of parts, and

B 2

THE PRINCIPLES OF SCIENCE.

yet we detect there the same elements of which our
own limbs are composed. The same natural laws can be
detected in operation in every part of the universe within
the scope of our instruments ; and doubtless these laws are
obeyed irrespective of distance, time and circumstance.

It is the prerogative of Intellect to discover what is
uniform and unchanging in the phenomena around us.
So far as object is different from object, knowledge is
useless and inference impossible. But so far as object
resembles object, we can pass from one to the other. In
proportion as resemblance is deeper and more general, the
commanding powers of knowledge become more wonder
ful. Identity in one or other of its phases is thus always
the bridge by which we pass in inference from case to
case ; and it is my purpose in this treatise to trace out the
various forms in which the one same process of reasoning
presents itself in the ever-growing achievements of
Scientific Method.

The Powers of Mind concerned in the Creation of

Science.

It is no part of the purpose of this work to investigate
the nature of mind, except so far as its powers are
requisite to the formation of Science. In this place I
need only point out that the mental powers engaged in
knowledge are probably three in number. They are
substantially as Mr. Bain has stated thema : —

1 . The Power of Discrimination.

2. The Power of Detecting Identity.

3. The Power of Eetention.

We exert the first power in every act of perception.
Hardly can we have a sensation or feeling unless we
discriminate it from something else which preceded.

a 'The Senses and the Intellect,' Second Ed., pp. 5, 325, &c.

INTRODUCTION.

Consciousness would almost seem to consist in the break
between one state of mind and the next, just as an
induced current of electricity arises from the beginning
or the ending of the primary current. We are always
engaged in discrimination ; and the rudiment of thought
which exists in the lower animals probably consists in
their power of feeling difference and being agitated by
its occurrence.

But had we power of discrimination only, Science could
not be created. To know that one feeling differs from
another gives purely negative information. It cannot teach
us what will happen. Each sensation would stand out dis
tinct from any other, and there would be no tie, no bridge
of affinity between them. We want a unifying power by
which the present and the future may be linked to the
past ; and this seems to be accomplished by a different
power of mind. Francis Bacon has pointed out that dif
ferent men possess in very different degrees the powers of
discrimination and identification. It may be said indeed
that discrimination necessarily implies the opposite process
of identification ; and so it doubtless does in superficial
points. But there is a rare property of mind which
consists in penetrating the disguise of variety and seizing
the common elements of sameness ; and it is this pro
perty which furnishes the true measure of intellect. The
very name of intellect (interligo) expresses the action, not
of separating, but of uniting and binding together the
particular and various into the general and like. Logic
is but another name for the same process b, the peculiar
work of reason ; and Plato said of this unifying power,
that if he met the man who could detect the one in the
many, he would follow him as a god.

b Max Miiller, ' Lectures on Language,' Second Series, vol. ii. p. 63.

THE PRINCIPLES OF SCIENCE.

Laws of Identity and Difference.

At the basis of all thought and science must lie the
laws which express the very nature and conditions of the
discriminating and identifying powers of mind. These
are the so-called Fundamental Laws of Thought, usually
stated as follows : —

1 . The Law of Identity. Whatever is, is.

2. The Law of Contradiction. A thing cannot both be

and not be. ±

3. The Law of Duality. A thing must either be or

not be.

The first of these statements may perhaps be regarded
as a description of identity itself, if so fundamental a
notion can admit of description. A thing at any moment
is perfectly identical with itself, and if any person were
unaware of the meaning of the word ' identity' we could
not better describe it than by such an example.

The second law points out that contradictory attri
butes can never be joined together. The same object may
vary in its different parts ; here it may be black, and
there white ; at one time it may be hard and at another
time soft : but at the same time and place an attribute
cannot be both present and absent. Aristotle truly
described this law as the first of all axioms0 — one of
which we need not seek for any demonstration. All
truths cannot be proved, otherwise there would be an
endless chain of demonstration ; and it is in self-evident
truths like this that we find the fittest foundation.

The third of these laws completes the other two. It
asserts that at every step there are two possible alter
natives — presence or absence, affirmation or negation.
Hence I propose to name this law the Law of Duality,

c 'Metaphysics,' Bk. III. chap. iii. 9-12,

INTRODUCTION.

for it gives to all the formulae of reasoning a dual
character. It asserts also that between presence or
absence, existence or non-existence, affirmation or ne
gation, there is no third alternative. As Aristotle said,
there can be no mean between opposite assertions : we
must either affirm or deny. Hence the somewhat incon
venient name by which it has been generally known —
The Law of Excluded Middle.

It may be held that these laws are not three inde
pendent and distinct laws, they rather express three
different aspects of the same truth, and each law doubt
less presupposes and implies the other two. But it has
not hitherto been found possible to state these characters
of identity and difference in less than the three-fold
formula. The reader may perhaps desire some infor
mation as to the mode in which these laws have been
stated, or the way in which they have been regarded,
by philosophers in different ages of the world. Abundant
information on this and many other points of logical
history will be found in Ueberweg's ' System of Logic/
of which an excellent translation has been published by
Mr. T. M. Lindsay d. I must confess however that the
history of logical doctrines has seemed to me one of the
most confusing and least beneficial studies in which a
person can engage ; and over-abundant attention perhaps
has been paid to it by Hamilton, Mansel, and many
German logicians.

The Nature and Authority of the Laws of Identity
and Difference.

I must at least allude to the profoundly difficult
question concerning the nature and authority of these

d Ueberweg's 'System of Logic/ transl. by Lindsay, London, 1871,
pp. 228-281.

THE PRINCIPLES OF SCIENCE.

Laws of Identity or Difference. Are they Laws of Thought
or Laws of Things ? Do they belong to mind or to
material nature 1 On the one hand it may be said
that science is a purely mental existence, and must
therefore conform to the laws of that which formed it.
Science is in the mind and not in the things, and the
properties of mind are therefore all important. It is true
that these laws are verified in the observation of the
exterior world ; and it would seem that they might have
been gathered and proved by generalisation, had they
not already been in our possession. But on the other
hand, it may well be urged that we cannot prove these
laws by any process of reasoning or observation, be
cause the laws themselves are presupposed, as Leibnitz
acutely remarked, in the very notion of a proof. They
are the prior conditions of all thought and all know
ledge, and even to question their truth is to allow
them true. Hartley ingeniously refined upon this argu
ment, remarking that if the fundamental laws of logic
be not certain, there must exist a logic of a second
order whereby we may determine the degree of uncer
tainty : if the second logic be not certain, there must
be a third, and so on ad infinitum. Thus we must sup
pose either that absolutely certain laws of thought exist,
or that there is no such thing as certainty whatever e.

Logicians, indeed, appear to me to have paid insuf
ficient attention to the fact that mistakes in reasoning
are always likely to occur. The Laws of Thought are
often called necessary laws, that is, laws which cannot
but be obeyed. Yet as a matter of fact who is there
that does not often fail to obey them ? They are the
laws which the mind ought to obey rather than what
it always does obey. Our thoughts cannot be the
criterion of truth, for we often have to acknowledge

e Hartley on Man, vol. i. p. 359.

INTRODUCTION.

mistakes in arguments of very moderate complexity, and
we sometimes only discover our mistakes by a collision
between our mental expectations and the events of ob
jective nature.

Mr. Herbert Spencer holds that the laws of logic are
objective lawsf, and he regards the mind as being in
a state of constant education, each act of false reasoning
or miscalculation leading to results which are likely to
prevent similar mistakes from being again committed.
I am quite inclined to accept such ingenious views ; but
at the same time it is necessary to distinguish between
the accumulation of knowledge and experience, and the
constitution of the mind which allows of the acquisition
of knowledge. Before the mind can perceive or reason
at all it must have the conditions of thought impressed
upon it. Before a mistake can be committed, the mind
must clearly distinguish tiie mistaken conclusion from all
other assertions. Are not the Laws of Identity and
Difference the prior conditions of all consciousness and
all existence \ Must they not hold true, alike of things
material and immaterial \ and if so, can we say that
they are only subjectively true or objectively true 1 I
am inclined, in short, to regard them as true both 'in
the nature of thought and things,' as I expressed it in
my first logical essay s, and I hold that they belong to
the common basis of all existence. But this is one of
the most profound and difficult questions of psychology
and metaphysics which can be raised, and it is hardly
one for the logician to decide. As the mathematician does
not inquire into the nature of unity and plurality, but
developes the formal laws of plurality, so the logician,
as I conceive, must assume the truth of the Laws of

f ' Principles of Psychology,' Second Ed., vol. ii. p. 86.
s ' Pure Logic, or the Logic of Quality apart from Quantity,' London
(Stanford), 1864, pp. 10, 16, 22, 29, 36, &c.

10 THE PRINCIPLES OF SCIENCE.

Identity and Difference, and occupy himself in developing
the variety of forms of reasoning in which their truth may
be manifested.

Again, I need hardly dwell upon the question whether
logic treats of language, notions, or things. As reasonably
might we debate whether a mathematician treats of
symbols, quantities, or things. A mathematician certainly
does treat of symbols, but only as the instruments
whereby to facilitate his reasoning concerning quantities ;
and as the axioms and rules of mathematical science must
be verified in concrete objects in order that the calcu
lations founded upon them may have any validity or
utility, it follows that the ultimate objects of mathe
matical science are the things themselves. In like man
ner I conceive that the logician treats of language so far
as it is essential for the embodiment and exhibition of
thought. Even if reasoning can take place in the inner
consciousness of man without the use of any signs, at
any rate it cannot become the subject of discussion until
by some system of material signs it is manifested to other
persons. The logician then uses words and symbols as
instruments of reasoning, and leaves the nature and pe
culiarities of existing language to the grammarian. But
signs again must correspond to the thoughts and things
expressed, in order that they shall serve their intended
purpose. We may therefore say that logic treats ulti
mately of thoughts and things, and immediately of the
signs which stand for them. Signs, thoughts and ex
terior objects may be regarded as parallel and analogous
series of phenomena, and to treat one series is equivalent
to treating either of the other series h.

h See also 'Elementary Lessons in Logic/ Second Ed., p. 10.

INTRODUCTION. \\

TJie Process of Inference.

The fundamental action of our reasoning faculties
consists in inferring or carrying to a new instance of a
phenomenon whatever we have previously known of its
like, analogue, equivalent or equal. Sameness or identity
presents itself in all degrees, and is known under various
names ; but the great rule of inference embraces all
degrees, and affirms that so far as there exists sameness,
identity or likeness, what is true of one thing will be true
of the other. The great difficulty of reasoning doubtless
consists in ascertaining that there does exist a sufficient
degree of likeness or sameness to warrant an intended
inference ; and it will be our main task to investigate the
conditions under which the inference is valid. In this
place I wish to point out that there is something common
to all acts of inference however different their apparent
forms. The one same rule lends itself to the most diverse
applications.

The simplest possible case of inference, perhaps, occurs
in the use of a pattern, example, or, as it is commonly
called, a sample. To prove the exact similarity of two
portions of commodity, we need not bring one portion
beside the other. It is sufficient that we cut a sample
which exactly represents the texture, appearance, and
general nature of one portion, and according as this
sample agrees or not with the other, so will the two
portions of commodity agree or differ. Whatever is true
as regards the colour, texture, density, material of the
sample will be true of the goods themselves. In such
cases likeness of quality is the condition of inference.

Exactly the same mode of reasoning holds true of
magnitude and figure. To compare the size of two
objects, we need not lay them alongside each other. A

12 THE PRINCIPLES OF SCIENCE.

staff, string, or other kind of measure may be employed
to represent the length of one object, and according as it
agrees or not with the other, so must the two objects
agree or differ. In this case the proxy or sample repre
sents length ; but the fact that lengths can be added and
multiplied renders it unnecessary that the proxy should
always be as large as the object. Any standard of con
venient length, such as a common foot-rule, may be made
the medium of comparison. The height of a church in
one town may be carried to that in another, and objects
existing immoveably at opposite sides of the earth may be
vicariously measured against each other. We obviously
employ the rule that whatever is true of a thing as
regards its length, is true of its equal.

To every other simple phenomenon in nature the same
principle of substitution is applicable. We may compare
weights or densities or degrees of hardness, and all other
qualities, in like manner. To ascertain whether two
sounds are in unison we need not compare them directly,
but a third sound may be the go-between. If a tuning-
fork is in unison with the middle C of York Minster
organ, and we afterwards find it to be in unison with the
same note of the organ in Westminster Abbey, then it
follows that the two organs are tuned in unison. The
rule of inference now is that what is true, as regards
pitch, of the tuning-fork, is true of any sound in unison
with it.

The skilful employment of this substitutive process
enables us to make measurements beyond the powers of
our senses. No one can count the vibrations, for instance,
of an organ pipe. But we can construct an instrument
called the syren, so that while producing a sound of any
pitch it shall register the number of vibrations consti
tuting the sound. Adjusting the sound of the syren in
unison with an organ pipe, we measure indirectly the

INTRODUCTION. 13

number of vibrations belonging to a sound of that pitch.
To measure a sound of the same pitch is as good as to
measure the sound itself.

Sir David Brewster, in a somewhat similar manner,
succeeded in measuring the refractive index of irregular
fragments of transparent minerals. It was a troublesome,
and sometimes impracticable work to grind the minerals
into prisms, so that their powers of refracting light could
be directly observed ; but he fell upon the ingenious device
of forming a liquid possessing exactly the same refractive
power as the transparent fragment under examination.
The moment when this equality was attained could be
known by the fragments ceasing to reflect or refract light
when immersed in the liquid, so that they became almost
invisible in it. The refractive power of the liquid being
then measured gives that of the solid ; and a more beau
tiful instance of representative measurement, depending
immediately upon the principle of inference, could not
be found*.

Throughout the various logical processes which we are
about to consider — Deduction, Induction, Generalisation,
Analogy, Classification, Quantitative Reasoning — we shall
find the one same principle operating in a more or less
disguised form.

Deduction and Induction.

The processes of inference always depend on the one
same method of substitution ; but they may nevertheless
be distinguished according as the results are inductive or
deductive. As generally stated, deduction consists in

1 Brewster, 'Treatise on New Philosophical Instruments,' p. 273.
See also "Whewell, ' Philosophy of the Inductive Sciences,' vol. ii. p. 355 ;
Tomlinson, 'Philosophical Magazine,' Fourth Series, vol. xl. p. 328;
Tyndall, in Youman's ' Modern Culture/ p. 1 6.

14 THE PRINCIPLES OF SCIENCE.

passing from more general to less general truths ; induc
tion is the contrary process from less to more general
truths. We may however describe the difference in
another manner. In deduction we are engaged in develop
ing the consequences of a law or identity. We learn
the meaning, contents, results or inferences, which attach
to any given proposition. Induction is the exactly inverse
process. Given certain results or consequences, we are re
quired to discover the general law from which they flow.

In a certain sense all knowledge is inductive. We can
only learn the laws and relations of things in nature by
observing those things. But the knowledge gained from
the senses is knowledge only of particular facts, and we
require some process of reasoning by which we may con
struct out of the facts the laws obeyed by them. Expe
rience gives us the materials of knowledge : induction
digests those materials, and yields us general knowledge.
Only when we possess such knowledge, in the form of
general propositions and natural laws, can we usefully
apply the reverse process of deduction to ascertain the
exact information required at any moment. In its ultimate
origin or foundation, then, all knowledge is inductive — in
the sense that it is derived by a certain inductive
reasoning from the facts of experience.

But it is nevertheless true, — and this is a point to
which insufficient attention has been paid, — that all reason
ing is founded on the principles of deduction. I call in
question the existence of any method of reasoning which
can be carried on without a knowledge of deductive pro
cesses. I shall endeavour to show that induction is really
the inverse process of deduction. There is no mode of
ascertaining the laws which are obeyed in certain pheno
mena, except we previously have the power of determining
what results would follow from a given law. Just as the
process of division necessitates a prior knowledge of multi-

INTRODUCTION. 15

plication, or the integral calculus rests upon the obser
vation and remembrance of the results of the differential
calculus, so induction requires a prior knowledge of
deduction. An inverse process is the undoing of the
direct process. A person who enters a maze must either
trust to chance to lead him out again, or he must carefully
notice the road by which he entered. The facts furnished
to us by experience are a maze of particular results ; we
might by chance observe in them the fulfilment of a law,
but this is scarcely possible, unless we thoroughly learn
the effects which would attach to any particular law.

Accordingly, the importance of deductive reasoning is
doubly supreme. Even when we gain the results of in
duction they would be of little or no use without we
could deductively apply them. But before we can gain
them at all we must understand deduction, since it is the
inversion of deduction which constitutes induction. Our
first task then, in this work, must be to trace out fully the
nature of identity in all its forms of occurrence. Having
given any series of propositions we must be prepared to
develop the whole meaning embodied in them, and the
whole of the consequences which flow from them.

Symbolic Expression of Logical Inference.

In developing the results of the Principle of Inference
we require to use an appropriate language of signs. It
would indeed be quite possible to explain the processes of
reasoning merely by the use of words found in the ordinary
grammar and dictionary. Special examples of reasoning,
too, may seem to be more readily apprehended than general
and symbolic forms. But it has been abundantly proved
in the mathematical sciences that the attainment of truth
depends greatly upon the invention of a clear, brief, and
appropriate system of symbols. Not only is such a

16 THE PRINCIPLES OF SCIENCE.

language convenient, but it is essential to the expression
of those general truths which are the very soul of science.
To apprehend the truth of special cases of inference does
not constitute logic ; we must apprehend them as cases of
more general truths. The object of all science is the
separation of what is common and general from what is
accidental and different. In a system of logic, if anywhere,
we should esteem this generality, and strive to exhibit
clearly what is similar in very diverse cases. Hence the
great value of general symbols by which we can represent
the form and character of a reasoning process, disentangled
from any consideration of the special subject to which it is
applied.

The signs required in logic are of a very simple kind.
As every sameness or difference must exist between two
things or notions, we need signs or terms to indicate the
things or notions compared, and other signs to denote the
relation between them. We shall need, then, (i) symbols
for terms, (2) a symbol for sameness, (3) a symbol for differ
ence, and (4) one or two symbols to take the place of
conjunctions.

Ordinary nouns substantive, such as Iron, Metal, Elec
tricity, Undulation, might serve as terms, but for the
reasons explained above it is better to adopt blank letters,
devoid of special signification, such as A, B, C, D, E, &c.
Each letter must be understood to represent a noun, and,
so far as the conditions of the argument allow, any noun.
Just as in Algebra, x, y, z, p, q, r, &c. are used for any
quantities, undetermined or unknown, except when the
special conditions of the problem are taken into account,
so will our letters stand for undetermined or unknown
things.

These letter-terms will be used indifferently for nouns
substantive and adjective. Between these two kinds of
nouns there may be important differences in a metaphysical

INTRODUCTION. 17

or grammatical point of view. But grammatical usage
readily sanctions the free conversion of adjectives into
substantives, and vice versd ; we may avail ourselves of
this latitude without in any way prejudging the meta
physical difficulties which may be involved. Here, as
throughout this work, I shall devote my attention to
truths which I can exhibit in a clear and formal manner,
believing that, in the present condition of logical science,
this will lead to much greater advantage than discussion
upon the metaphysical questions which may underlie any
part of the subject.

Every noun or term denotes an object, and usually im
plies the possession by that object of certain qualities or
circumstances common to all the objects denoted. There
are certain terms, however, which imply the absence of
qualities or circumstances attaching to other objects. It
will be convenient to employ a special mode of indicating
these negative terms, as they are called. If the general
name A denotes an object or class of objects possessing
certain defined qualities, then the term Not-A will denote
any object which does not possess the whole of those
qualities ; in short, Not-A is the sign for anything which
differs from A in regard to any one or more of the assigned
qualities. If A denote ' transparent object,' Not-A will
denote ' not transparent object.' Brevity and facility of
writing and reading are of no slight importance in a system
of notation, and it will therefore be desirable to substitute
for the negative term Not-A a briefer mode of expression.
The late Prof, de Morgan represented negative terms by
small Eoman letters, or sometimes by small italic letters k,
and as the latter seem to be highly convenient, I shall use
a, 6, c, d, e, . . . p, q, r, &c., as the negative terms corre
sponding to A, B, C, D, E, . . . P, Q, R, &c. Thus if A
means ' fluid,' a will mean ' not-fluid/ and so on.

k 'Formal Logic,' p. 38.
C

18 THE PRINCIPLES OF SCIENCE.

Expression of Identity and Difference.

To denote the relation of sameness or identity I unhesi
tatingly adopt the sign = , so long used by mathematicians
to denote equality. This symbol was originally appropri
ated by Robert Recorde in his 'Whetstone of Wit/ to
avoid the tedious repetition of the words 'is equal to';
and he chose a pair of parallel lines, because no two things
can be more equal l. The meaning of the sign has how
ever been gradually extended beyond that of common
equality ; mathematicians have themselves used it to
indicate equivalency of operations. The force of analogy
has been so great that writers in all other branches of
science have more or less employed the same sign. The
philologist indicates by it equivalency of meaning of words :
chemists adopt it to signify the identity in kind and
equality in weight of the elements which form two different
compounds. Not a few logicians, for instance Ploucquet,
Condillacm, George Benthamn, Boole, have employed it
as the copula of propositions. Prof, de Morgan declined
to use it for this purpose, but still further extended its
meaning so as to include the equivalency of a proposition
with the premises from which it can be inferred °, and
Herbert Spencer has applied it in a like manner P.

Many persons may think that the choice of a symbol is
a matter of slight importance or of mere convenience, but
I hold that the common use of this sign = in so many
different meanings is really founded upon a generalisation

1 Hallam's 'Literature of Europe,' First Ed. vol. ii. p. 444,
m Condillac, ' Langue des Calculs/ p. 157.

n 'Outline of a New System of Logic/ London, 1827, pp. 133, &c.
o 'Formal Logic,' pp. 82, 106. In his later work, 'The Syllabus of a
New System of Logic,' he discontinued the use of the sign.
P ' Principles of Psychology,' Second Ed., vol. ii. pp. 54, 55.

INTRODUCTION. 19

of the widest character and of the greatest importance —
one indeed which it is a principal object of this work to
endeavour to explain. The employment of the same sign
in different cases would be wholly unphilosophical unless
there were some real analogy between its diverse meanings.
If such analogy exist, it is not only allowable, but highly
desirable and even imperative, to use the symbol of equi
valency with a generality of meaning corresponding to the
generality of the principles involved. Accordingly Prof,
de Morgan's refusal to use the symbol in logical proposi
tions indicated his opinion that there was a want of analogy
between logical propositions and mathematical equations.
I use the sign because I hold the contrary opinion.

I conceive that the sign = always denotes some form
or degree of sameness or equivalency, and the particular
form is usually indicated by the nature of the terms joined
by it. Thus '6720 pounds = 3 tons' is evidently a.n
equation of quantities. The formula — x — = + ex
presses the equivalency of operations. ' Exogens — Dico
tyledons ' is a logical identity expressing a profound
truth concerning the character of vegetables.

We have great need in logic of a distinct sign for the
copula, because the little verb is, hitherto used both in
logic and ordinary discourse, is thoroughly ambiguous.
It sometimes denotes identity, as in ' St. Paul's is the
chef-d'oeuvre of Sir Christopher Wren/ but it more
commonly indicates inclusion of class within class, or
partial identity, as in 'Bishops are members of the House
of Lords.' This latter relation involves identity, but re
quires careful discrimination from simple identity, as will
be shown further on.

When with this sign of equality we join two nouns or
logical terms, as in

Hydrogen = The least dense element,
we signify that the object or group of objects denoted by

c 2

20 THE PRINCIPLES OF SCIENCE.

one term is identical with that denoted by the other in
everything except the names. The general formula

A = B

must be taken to mean that A and B are symbols for the
same object or group of objects. This identity may some
times arise from the mere imposition of names, but it
may also arise from the deepest laws of the constitution
of nature ; as when we say

Gravitating matter = Matter possessing inertia,
Exogenous plants = Dicotyledonous plants,
Plagihedral quartz cry stale = Quartz crystals rotating

the plane of polarisation of light.

We shall need carefully to distinguish between relations
of terms which can be modified at our own will and those
which are fixed as expressing the laws of nature ; but at
present we are considering only the mode of expression.

We may sometimes, but much less frequently, require a
symbol to indicate difference or the absence of complete
sameness. For this purpose we may generalise in like
manner the symbol -*• , which was introduced by Wallis to
signify difference between two numbers or quantities.
The general formula

B - C

denotes that B and C are the names of some two objects
or groups of objects which are not identical with each
other. Thus we may say

Acrogens * Flowering plants.

SnowTdon *» The highest mountain in Great Britain.
I shall also occasionally use the sign «» to signify in the
most general manner the existence of any relation between
the two terms connected by it. Thus «» might mean not
only the relations of equality or inequality, sameness or
difference, but any special relation of time, place, size,
causation, &c. in which one thing may stand to another.
By A «o* B I mean, then, any two objects of thought re
lated to each other in any matter whatsoever.

INTRODUCTION. 21

General Formula of Logical Inference.

The one supreme rule of inference consists, as I have
said, in the direction to affirm of anything whatever is
known of its like, equal or equivalent. The Substitution
of Similars is a phrase which seems aptly to express the
power of mutual replacement existing between any two
objects which are to a sufficient degree like or equivalent.
It is a matter for further investigation to point out when
and for what purposes a degree of similarity less than
complete identity is sufficient to warrant substitution.
For the present we think only of the exact sameness
expressed in the form

A = B.

Now if we take the letter C to denote any third con
ceivable object, and use the sign <** in its stated meaning
of indefinite relation, then the general formula of all
inference may be thus exhibited : —

From A = B «» C

we may infer A *» C

or, in words — In ivhatever relation a thing stands to a
second thing, in the same relation it stands to the like or
equivalent of that second thing. The identity between A
and B allows us indifferently to place A where B was or
B where A was, and there is no limit to the variety of
special meanings which we can bestow upon the signs
used in this formula consistently with its truth. Thus if
we first specify only the meaning of the sign «o», we may
say that if C is the weight of B, then C is also the weight
of A. Similarly

If C is the father of B, C is the father of A ;

If C is a fragment of B, C is a fragment of A ;

If C is a quality of B, C is a quality of A ;

If C is a species of B, C is a species of A ;

If C is the equal of B, C is the equal of A ;
and so on ad infinitum.

22 THE PRINCIPLES OF SCIENCE.

We may also endow with special meanings the letter-
terms A, B and C, and the process of inference will never
be false. Thus let the sign ** mean 'is height of,' and
let

A = Snowdon,

B = Highest mountain in England or Wales,
C = 3590 feet;

then it obviously follows that since '3590 feet is the
height of Snowdon/ and ' Snowdon = the highest mountain
in England or Wales/ then '3590 feet is the height of the
highest mountain in England or Wales/

One result of this general process of inference is that
we may in any aggregate or complex whole replace any
part by its equivalent without altering the whole. To
alter is to make a difference, but if in replacing a part I
make no difference, there is no alteration of the whole.
Many inferences which have been very imperfectly in
cluded in logical formulae at once follow. I remember the
late Prof, de Morgan remarking that all Aristotle's logic
could not prove that ' Because a horse is an animal, the
head of a horse is the head of an animal.' I conceive that
this amounts merely to replacing in the complete notion
head of a horse, the term ' horse' by its equivalent some
animal or an animal. Similarly, since

The Lord Chancellor = The Speaker of the House of

Lords,
it follows that

The death of the Lord Chancellor = The death of the

Speaker of the House of Lords ;

and any event, circumstance or thing which stands in a
certain relation to the one will stand in like relation to
the other. Milton reasons in this way when he says, in
his Areopagitica, * Who kills a man, kills a reasonable crea
ture, God's image/ If we may suppose him to mean

God's image = man = some reasonable creature,

INTRODUCTION. 23

it follows that ' The killer of a man is the killer of some
reasonable creature/ and also ' The killer of God's image/

This replacement of equivalents may be repeated over
and over again to any extent. Thus if person is identical
in meaning with individual, it follows that

Meeting of persons = meeting of individuals ;
and if assemblage = meeting, we may make a new replace
ment and show -that

Meeting of persons = assemblage of individuals.
We may in fact found upon this principle of substitution
a most general axiom in the following terms ^ :—

Same parts samely related make same wholes.
If, for instance, exactly similar bricks be used to build
two houses, and they be similarly placed in each house, the
two houses must be similar. There are millions of cells
in a human body, but if each cell of one person were
represented by an exactly similar cell similarly placed in
another body, the two persons would be undistinguishable,
and would be only numerically different. It is upon this
principle, as we shall see, that all accurate processes of
measurement depend. If for a weight in a scale of
a balance we substitute another weight, and the equili
brium remains entirely unchanged, then the weights must
be exactly equal. The general test of equality is substi
tution. Objects are equally bright when on replacing one
by the other the eye perceives no difference. Two objects
are equal in dimensions when tested by the same gauge
they fit in the same manner. Generally speaking, two
objects are alike so far as when substituted one for another
no alteration is produced, and vice versd when alike no
alteration is produced by the substitution.

<i ' Pure Logic, or the Logic of Quality,' p. 14.

24 THE PRINCIPLES OF SCIENCE.

The Propagating Power of Identity.

The relation of identity or sameness in all its degrees
is reciprocal. So far as things are alike, either may be
substituted for the other ; and this may perhaps be con
sidered the very meaning of the relation. But it is well
worth notice that there is in identity a peculiar power of
extending itself among all the things which are identical.
To render a number of things similar to each other we
need only render them similar to one standard object.
Each coin struck from a pair of dies not only exactly
resembles the matrix or original pattern from which the
dies were struck, but exactly resembles every other coin
manufactured from the same original pattern. Among a
million such coins there are not less than 499,999,500,000
of pairs of coins exactly resembling each other. Similars
to the same are similars to all. It is one great advantage
of printing that all copies of a document taken from the
same type are necessarily identical each with each, and
whatever is true of one copy will be true of every copy.
Similarly, if fifty rows of pipes in an organ be tuned in
perfect unison with one row, usually the Principal, they
must be in unison with each other. Identity can also
reproduce or propagate itself ad infinitum ; for if a
number of tuning-forks be adjusted in perfect unison
with one standard fork, all instruments tuned to any one
fork will agree with any instrument tuned to any other
fork. Standard measures of length, capacity, or weight,
or any other measureable quality, are propagated in the
same manner. So far as copies of the original standard,
or copies of copies, or copies again of those copies, are
accurately executed, they must all agree each with every
other.

It is the power of mutual substitution which gives

IN TROD UCTION. 25

such great value to the modern methods of mechanical
construction, according to which all the parts of a
machine are exact facsimiles of a fixed pattern. All
the rifles used in the British army are constructed on
the interchangeable system, so that any one part of any
one rifle can be substituted indifferently for the same
part of another. A bullet fitting one rifle wih1 fit all
others of the same bore. Sir J. Whitworth has extended
the same system to the screws and screw-bolts used in
connecting together the parts of machines, by establishing
a series of standard screws.

Anticipations of the Principle of Substitution.

In such a subject as logic it is hardly possible to put
forth any opinions or principles which have not been
in some degree previously entertained. The germ at
least of every doctrine will be found in earlier writers,
and novelty must arise chiefly in the mode of harmonising
and developing ideas. When I first proposed to employ
the process and name of substitution in logic r, I believe
that I was led to do so from analogy with the familiar

o«/

mathematical process of substituting for a symbol its value
as given in an equation. In writing my first logical essay
I had a most imperfect conception of the importance
and generality of the process, and I described, as if they
were of equal importance, a number of other laws which
now seem to be but particular cases of the one general
rule of substitution.

My second essay, the Substitution of Similars, was
written shortly after I had become aware of the great
simplification which may be effected by a proper appli
cation of the principle of substitution. I was not then
acquainted with the fact that the German logician

r 'Pure Logic,' pp. 18-19.

26 THE PRINCIPLES OF SCIENCE.

Beneke had employed the principle of substitution, and
had used the word itself in forming a theory of the
syllogism. My imperfect acquaintance with the German
language had prevented me from acquiring a complete
knowledge of Beneke's views, but there is no doubt
that Mr. Lindsay is right in saying that he, and probably
other previous logicians, were in some degree familiar
with the principle8. Even Aristotle's dictum may be
regarded as an imperfect statement of the principle of
substitution ; and, as I have pointed out, we have only
to modify that dictum in accordance with the quantifi
cation of the predicate in order to arrive at the complete
process of substitution *. The Port-Royal logicians appear
to have entertained nearly equivalent views, for they
considered that all moods of the syllogism might be
reduced under one general principle11. Of two premises
they regard one as the containing proposition (propositio
continens), and the other as the applicative proposition.
The latter proposition must always be affirmative, and
represents that by which a substitution is made ; the
former may or may not be negative, and is that in
which a substitution is effected. They also show that
this method will embrace certain cases of complex reason
ing which had no place in the Aristotelian syllogism.
Their views probably constitute the greatest improvement
in logical doctrine made up to that time since the days
of Aristotle. But a true reform in logic must consist,
not in explaining the syllogism in one way or another,
but in doing away with all the narrow restrictions of
the Aristotelian system, and in showing that there exists

6 Ueberweg's 'System of Logic,' transl. by Lindsay, pp. 442-446,
571,572.

fc ' Substitution of Similars,' p. 9.

u 'Port-Royal Logic,' transl. by Spencer Baynes, pp. 212-219.
Part III. chap. x. and xi.

INTRODUCTION. 27

an indefinite extent of logical arguments immediately
deducible from the principle of substitution of which
the ancient syllogism forms but a small and not even
the most important part.

The Logic of Relatives.

There is a difficult and important branch of logic
which may be called the Logic of Relatives. If I argue,
for instance, that because Daniel Bernoulli was the son
of John, and John the brother of James, therefore Daniel
was the nephew of James, it is not possible to prove
this conclusion by any simple logical process. We re
quire at any rate to assume that the son of a brother is
a nephew. A simple logical relation is that which exists
between properties and circumstances of the same object
or class. But objects and classes of objects may also be
related according to all the properties of time and space.
I believe it may be shown, indeed, that where an inference
concerning such relations is drawn, a process of substi
tution is really employed and an identity must exist ;
but I will not undertake to prove the assertion in this
work. The relations of time and space are logical
relations of a complicated character demanding much
abstract and difficult investigation. The subject has been
treated with such great ability by Professors Peircex,
De Morgan y, Ellis2, and Harley, that I will not in the

x ' Description of a Notation for the Logic of Relatives, resulting from
an Amplification of the Conceptions of Boole's Calculus of Logic.' By
C. S. Peirce. ' Memoirs of the American Academy,' vol. ix. Cam
bridge, U.S., 1870.

y ' On the Syllogism, No. IV, and on the Logic of Relations.' By
Augustus De Morgan. 'Transactions of the Cambridge Philosophical
Society,' vol. x. part ii. 1860.

z ' Observations on Boole's Laws of Thought.' By the late R.
Leslie Ellis; communicated by the Rev. Robert Harley, F.R.S, 'Report

28 THE PRINCIPLES OF SCIENCE.

present work attempt any review of their writings, but
merely refer to the publications in which they are to
be found.

of the British Association,' 1870. 'Report of Sections/ p. 12. Also,
< On Boole's Laws of Thought.' By the Rev. Robert Harley, F.R.S.,
ibid. p. 14.

CHAPTEE II.

TERMS.

EVERY proposition expresses the resemblance or differ
ence of the things denoted by its terms. As reasoning
or inference treats of the relation between two or more
propositions, so a proposition consists in a relation
between two or more terms. In the portion of this
work which treats of deduction it will be convenient
to follow the usual order of exposition, and consider in
succession the various kinds of terms, propositions, and
arguments, and we commence in this chapter with terms.

The simplest and most palpable meaning which can
belong to a term consists of some single material object,
such as Westminster Abbey, the Sun, Sirius, Stonehenge,
&c. It is probable that in the earliest stages of intellect
only concrete and palpable things are the objects of
thought. The youngest child knows the difference
between a hot and a cold body. The dog can recognise
his master among a hundred other persons, and animals
of much lower intelligence know and discriminate their
haunts. In all such acts there is judgment concerning
the likeness or unlikeness of physical objects, but there
is little or no power of analysing each object and re
garding it as a group of qualities or circumstances.

The dignity of intellect begins with the power of
separating points of agreement from those of difference.
Comparison of two objects may lead us to perceive that

30 THE PRINCIPLES OF SCIENCE.

they are at once like and unlike. Two fragments of
rock may differ entirely in outward form, yet they may
have the same colour, hardness, and texture. Flowers
which agree in colour may differ in odour. The mind
learns to regard each object as an aggregate of qualities,
and acquires the power of dwelling at will upon one or
other of those qualities to the exclusion of the rest.
Logical abstraction, in short, comes into play, and the
mind becomes capable of reasoning, not merely about
objects which are physically complete and concrete, but
about things which may be thought of separately in
the mind though they exist not separately in nature.
We can think of the hardness of a rock, or the colour
of a flower, and thus produce abstract notions, denoted
by abstract terms which will form a subject for further
consideration.

At the same time arise general notions and classes of
objects. We cannot fail to observe that the quality
hardness exists in many objects, for instance in many
fragments of rock ; and mentally joining these we create
the class hard object, which will include, not only the
actual objects examined, but all others which may
happen to agree with them as they agree with each
other. As our senses cannot possibly report to us all
the contents of space, we cannot usually set any limits
to the number of objects which may fall into any such
class. At this point we begin to perceive the power and
generality of thought which enables us at once to treat
of indefinitely or even infinitely numerous objects. We
can safely assert that whatever is true of any one object
coming under a general notion or class is true of any of
the other objects so far as they possess the common
qualities implied in their belonging to the class. We
must not place an individual thing in a class unless we
are prepared to believe of it all that is believed of the

TERMS. 31

class in general ; but it remains as a matter of important
consideration how far and in what manner we can safely
undertake thus to assign the place of objects in that
general system of classification which constitutes the
whole body of science.

Twofold Meaning of General Names.

Etymologically the meaning of a name is what we are
caused to think of when the name is used. Now every
general name causes us to think of some one or more of
the objects belonging to a class ; it may also cause us to
think of the common qualities possessed by those objects.
A name is said to denote the distinct object of thought
to which it may be applied ; it implies at the same time
the possession of certain qualities or circumstances. The
number of objects denoted forms the extent of meaning
of the term ; the number of qualities implied forms
the intent of meaning. Crystal is the name of any sub
stance of which the molecules are arranged in a regular
geometrical manner. The substances or objects in ques
tion form the extent of meaning ; the circumstance of
having the molecules so arranged forms the intent of
meaning.

When we compare a variety of general terms it may
often be found that the meaning of one is included in
the meaning of another. Thus all crystals are included
among material substances, and all opaque crystals are in
cluded among crystals: here the inclusion is in extension.
We may also have inclusion of meaning in regard to
intension. For as all crystals are material substances,

«/

the qualities implied by the term material substance
must be among those implied by crystal. Again, it is
obvious that while in extension of meaning opaque
crystals are but a part of crystals, in intension of meaning

32 THE PRINCIPLES OF SCIENCE.

crystal is but part of opaque crystal. We increase the
intent of meaning of a term by joining adjectives, or
phrases equivalent to adjectives, to it, and the removal
of such adjectives of course decreases the intensive
meaning. Now concerning such changes of meaning
the following all-important law holds universally true.
When the intent of meaning of a term is increased the
extent is decreased; and vice versa, ivhen the extent is
increased the intent is decreased. In short, as one is
increased the other is decreased.

This law refers only to logical changes. The number
of steam engines in the world may be undergoing a
rapid increase without the intensive meaning of the
name being altered. The law will only be verified again
when there is a real change in the intensive meaning,
and an adjective may often be joined to a noun without
making a change. Elementary metal is identical with
metal ; mortal man with man ; it being a property of all
metals to be elements, and all men to be mortals.

There is no limit to the amount of meaning which
a term may have. A term may denote one object, or
many, or an infinite number; it may imply a single
quality, if such there be, or a group of any number of
qualities, and yet the law connecting the extension and
intension will infallibly apply. Taking the general
name planet, we increase its intension and decrease its
extension by prefixing the adjective exterior ; and if we
further add nearest to the earth, there remains but one
planet Mars, to which the name can then be applied.
Singular terms, which denote a single individual only,
come under the same law of meaning as general names.
They may be regarded as general names of which the
meaning in extension is reduced to a minimum. Logi
cians have erroneously asserted, as it seems to me, that
singular terms are devoid of meaning in intension, the

TERMS. 33

fact being that they exceed all other terms in that kind
of meaning, as I have elsewhere tried to show a.

Abstract Terms.

Comparison of different objects, and analysis of the
complex resemblances and differences which they present,
lead us to the conception of abstract qualities. We learn
to think of one object as not only different from another,
but as differing in some particular point, such as colour,
or weight, or size. We may then convert points of
agreement or difference into separate objects of thought
called qualities, and denoted by abstract terms. Thus
the term redness means something in which a number
of objects agree as to colour, and in virtue of which they
are called red. Eedness forms, in fact, the intensive
meaning of the term red.

Abstract terms are strongly distinguished from general
terms by possessing only one kind of meaning ; for as
they denote qualities there is nothing which they can in
addition imply. The adjective 'red' is the name of red
objects, but it implies the possession by them of the
quality redness ; but this latter term has one single
meaning — the quality alone. Thus it arises that abstract
terms are incapable of number or plurality. Red objects
are numerically distinct each from each, and there are a
multitude of such objects ; but redness is a single exis
tence which runs through all those objects, and is the
same in one as it is in another. It is true that we may
speak of rednesses, meaning different kinds or tints of
redness, just as we may speak of colours, meaning dif
ferent kinds of colours. But in distinguishing kinds,

a J. S. Mill, 'System of Logic,' Book I. chap. ii. section 5. Jevons'
'Elementary Lessons in Logic/ pp. 41-43; 'Pure Logic,' p. 6. See
also Shedden's 'Elements of Logic/ London, 1864, pp. 14, &c.

D

decrees or other differences, we render the terms so for
concrete. In that they are merely red there is but a
single nature in red objects, and so far as things are
merely coloured, colour is a single indivisible quality.
Redness, so far as it is redness merely, is one and
same everywhere, and possesses absolute oneness or unity.
In virtue of this unity we acquire the power of treating
all instances of such quality as we may treat any one.
We possess, in short, general knowledge.

Substantial Terms.

Logicians appear to have taken very little notice of a
large class of terms which partake in certain respects of
the character of abstract terms and yet are undoubtedly
the names of concrete existing things. These terms are
the names of substances, such as gold, carbonate of lime,
nitrogen, &c. We cannot speak of two golds, twenty
carbonates of lime, or a hundred nitrogens. There is no
such distinction between the parts of a uniform sub
stance as will allow of a discrimination of numerous
individuals. The qualities of colour, lustre, malleability,
density, &c., by which we recognise gold, extend through
its substance irrespective of particular size or shape. So
far as a substance is gold, it is one and the same every
where ; so that terms of this kind, which I propose to call
substantial terms, possess the peculiar unity of abstract
terms. Yet they are not abstract ; for gold is of course
a tangible visible body, entirely concrete, and existing
physically independent of other bodies.

It is only when we break up, by actual mechanical
division, the uniform whole which forms the meaning of
a substantial term, that we introduce the notion of
number. Piece of gold is a term capable of plurality ;
for there may be an endless variety of pieces discriminated

TERMS. 35

from each other, either by their various shapes and sizes,
or, in the absence of such marks, by occupying simul
taneously different parts of space. In substance they are
one ; as regards the properties of space they are many.
We need not further pursue this distinction between
unity and plurality until we come to consider the prin
ciples of number in a subsequent chapter.

Collective Terms.

We must clearly distinguish between the collective and
the general meaning of terms. The same name may be
used to denote the whole body of existing objects of a
certain kind, or any one of those objects taken separately.
' Man ' may mean the aggregate of existing men, which we
sometimes describe as mankind ; it is also the general
name applying to any man. The vegetable kingdom is
the name of the whole aggregate of plants, but ' plant '
itself is a general name applying to any one or other
plant. Every material object may be conceived as divi
sible into parts, and is therefore collective as regards
those parts. The animal body is made up of cells and
fibres, a crystal of molecules ; wherever physical division,
or as it has been called partition, is possible, there we
deal in reality with a collective whole. Thus the greater
number of general terms are at the same time collective
as regards each individual whole which they denote.

It need hardly be pointed out that we must not infer
of a collective whole what we know of the parts, nor of
the parts what we know only of the whole. The relation
of whole and part is not one of identity, and does not
allow of substitution. There may nevertheless be qualities
or circumstances which are true alike of the whole and its
parts. Thus a number of organ pipes tuned in unison
produce an aggregate of sound which is of exactly the same

D 2

pitch SB each separate sound. In the case of substantial
terms, certain qualities may be present equally m each
minutest part as in the whole. The chemical nature of
the largest mass of pure carbonate of lime in existence is
the same as the nature of the smallest particle In the
case of abstract terms, again, we cannot draw a distinction
between whole and part ; what is true of redness in any
case is always true of redness, so far as it is merely red.

Synthesis of Terms.

We continually combine simple terms together so as to
form new terms of more complex meaning. Thus, to
increase the intension of meaning of a term we write it
with an adjective or a phrase of adjectival nature. By
joining 'brittle' to 'metal/ we obtain a combined term,
'brittle metal,' which denotes a certain portion of the
metals, namely such as are selected on account of pos
sessing the quality of brittleness. As we have already seen,
'brittle metal' possesses less extension and greater in
tension than metal. Nouns, prepositional phrases, parti
cipial phrases and subordinate propositions may also be
added to terms so as to increase their intension and
decrease their extension.

In our symbolic language we need some mode of
indicating this junction of terms, and the most convenient
device will be the simple juxtaposition of the distinct
letter-terms. Thus if A mean brittle, and B mean metal,
then AB will mean brittle metal. Nor need there be any
limit to the number of letters thus joined together, or the
complexity of the notions which they may represent.

Thus if we take the letters
P = metal,
Q = white,
B = monovalent,

TERMS. 37

S = of specific gravity io-5,

T = melting above iooo°C,

Y = good conductor of heat and electricity,
then we can form a combined term PQRSTY, which will
denote ' a white monovalent metal, of specific gravity
10*5, melting above iooo°C, and a good conductor of heat
and electricity/

There are many grammatical rules or usages concerning
the junction of words and phrases to which we need pay
no attention in logic. We can never say in ordinary
language 'of wood table/ meaning 'a table of wood/
but we may consider ' of wood ' as logically an exact
equivalent of ' wooden ' ; so that if
X = of wood,
Y = table,

there is no reason why, in our symbols, XY should not be
the correct term for ' table of wood/ In this case indeed
we might substitute the corresponding adjective 'wooden/
but we should often fail to find any adjective answering
exactly to a phrase. There is no single word which could
express the notion ' of specific gravity 10*5 ': but logically
we may consider these words as forming an adjective ; and
denoting this by S and metal by P, we may say that SP
means 'metal of specific gravity 10*5.' It is one of many
advantages in these blank letter- symbols that they enable
us completely to abstract all grammatical peculiarities and
fix our attention solely on the purely logical relations
involved. Investigation will probably show that the rules
of grammar are mainly founded upon traditional usage
and have little logical signification. This indeed is suffi
ciently proved by the wide grammatical differences which
exist between languages where the logical foundation must
be the same.

38

TlIK PRINCIPLES OF SCIENCE.

Symbolic Expression of the Law of Contradiction.

The synthesis of terms is subject to the all-important
Law of Thought, described in a previous section (p. 6)
and called the Law of Contradiction. It is self-evident
that no quality or circumstance can be both present and
absent at the same time and place. This fundamental
condition of all thought and all existence is expressed
symbolically by a rule that a term and its negative shall
never be allowed to come into combination. Such com
bined terms as Ao, B&, Cc, &c. are self-contradictory and
devoid of all meaning. If they represented anything, it
would be what cannot exist, and cannot even be imagined
in the mind. They can therefore only enter into our con
sideration to suffer immediate exclusion. The criterion
of false reasoning, as we shall find, is that it involves
self-contradiction, the affirming and denying of the same
statement. Thus we might represent the object of all
reasoning as the separation of the consistent and possible
from the inconsistent and impossible ; and we cannot make
any inference without implying that certain combinations
of terms are contradictory and excluded from thought.
To conclude that 'all A's are BV is equivalent to the
assertion that ' A's which are riot B's cannot exist.'

It will be convenient to have the means of indicating
this exclusion of the self-contradictory ; and we may use
the familiar sign for nothing, the cipher o. Thus the
second law of thought may be symbolised in the forms

Aa = o AB6 = o ABCa — o.
We may variously describe the meaning of o in logic as
the non-existent, the impossible, the self-inconsistent, the
inconceivable. Close analogy exists between this meaning
and its mathematical signification.

^

TERMS. 39

Certain Special Conditions of Logical Symbols.

In order that we may argue and infer truly we must
treat our logical symbols according to the fundamental
laws of Identity and Difference. But in thus using our
symbols we shall frequently meet with combinations of
which the meaning will not at first be apparent. In some
cases, for instance, we may learn that an object is ' yellow
and round/ in other cases that it is ' round and yellow ' :
there arises the question whether these two descriptions
are identical in meaning or not. Or again, if we proved
that an object was ' round round' the meaning of such an
expression would be open to doubt. Accordingly we must
take notice, before proceeding further, of certain special
laws which govern the combination of logical terms.

In the first place the combination of a logical term
with itself is without effect, just as the repetition of a
statement does not alter the meaning of the statement :
' a round round object' is simply 'a round object.' What
is yellow yellow is merely yellow ; metallic metals cannot
differ from metals, nor elementary elements from elements.
In our symbolic language we may similarly hold that AA
is identical with A, or

A = AA = AAA = &c.

The late Professor Boole is the only logician in modern
times who has drawn attention to this remarkable property
of logical termsb ; but in place of the name which he gave
to the law, I have proposed to call it The Law of Sim
plicity c. Its high importance will only become apparent
when we attempt to determine the relations of logical and
mathematical science. Two symbols of quantity, and only

b 'Mathematical Analysis of Logic,' Cambridge, 1847, P- J7- ' An
Investigation of the Laws of Thought,' London, 1854, p. 29.
c ' Pure Logic/ p. 1 5.

40 THE PRINCIPLES OF SCIENCE.

two, seem to obey this law ; we may say that 1.1 = i, and
o.o = o (taking o to mean absolute zero or i — i) ; there
is apparently no other number which combined with itself
gives an unchanged result. I shall point out, however, in
the chapter upon Number, that in reality all numerical
symbols obey this logical principle.

It is curious that this Law of Simplicity, though almost
unnoticed in modern times, was known to Boethius, who
makes a singular remark in his treatise ' De Trinitate
et Unitate Dei' (p. 959). He says, ' If I should say sun,
sun, sun, I should not have made three suns, but I should
have named one sun so many times d.' Ancient discussions
concerning the doctrine of the Trinity drew more atten
tion to subtle questions concerning the nature of unity
and plurality than has ever since been given to them.

It is a second law of logical symbols that order of
combination is a matter of indifference. 'Rich and rare
gems ' are the same as ' rare and rich gems,' or even as
' gems, rich and rare.' Grammatical, rhetorical or poetic
usage may give considerable significance to order of ex
pression. The limited power of our minds prevents our
grasping many ideas at once, and thus the order of
statement may produce some effect, but not in a strictly
logical mariner. All life proceeds in the succession of
time, and we are obliged to write, speak, or even think of
things and their qualities one after the other ; but be
tween the things and their qualities there need be no such
relation of order in time or space. The sweetness of sugar
is neither before nor after its weight and solubility. The
hardness of a metal, its colour, weight, opacity, mallea
bility, electric and chemical properties, are all coexistent
and coextensive, pervading the metal and every part of it

• \ dut si dicam Sol, Sol, Sol, non tres soles eff'ecerim, seel uno totics
pnodicaverim.1

TERMS. 41

in perfect community, none before nor after the others.
In our words and symbols we cannot observe this natural
condition ; we must name one quality first and another
second, just as some one must be the first to sign a petition,
or to walk foremost in a procession. In nature there is
no such precedence.

A little reflection will show that knowledge in the
highest perfection would consist in the simultaneous
possession of a multitude of facts. To comprehend a
science perfectly we should have every fact present with
every other fact. We must write a book and we must
read it successively word by word, but how infinitely
higher would be our powers of thought if we could
grasp the whole in one collective act of consciousness.
Compared with the brutes we do possess some slight
approximation to such power, and it is just conceivable
that in the indefinite future mind may acquire a vast
increase of capacity, and be less restricted to the piece
meal examination of a subject. But I wish here to
make plain that there is no logical foundation for the
successive character of thought and reasoning unavoidable
under our present mental conditions. The fact that we
must think of one thing first, and another second, is a
logical weakness and imperfection. We must describe
metal as ' hard and opaque/ or ' opaque and hard,' but
in the metal itself there is no such difference of order ;
the properties are simultaneous and coextensive in
existence.

Setting aside all grammatical peculiarities which render
a substantive less moveable than an adjective, and dis
regarding any meaning indicated by emphasis or marked
order of words, we may state, as a general law of logic,
that AB is identical with BA.

THE PRINCIPLES OF SCIENCE.

The late Professor Boole first drew attention, so far as
I know, to this property of logical terms, and he called
it the property of Commutativenesse. He not only stated
the law with the utmost clearness, but pointed out that
it is a Law of Thought rather than a Law of Things.
I shall have in various parts of the following pages to
show how the necessary imperfection of our symbols
expressed in this law clings to our modes of expression,
and introduces complication into the whole body of
mathematical formulae, which are really founded on a
logical basis.

It is of course apparent that the power of commutation
belongs only to terms related in the simple logical mode
of synthesis. No one can confuse 'a house of bricks/
with ' bricks of a house/ ' twelve square feet' with ' twelve
feet square/ 'the water of crystallization' with 'the
crystallization of water.' All relations which involve
differences of time and space are inconvertible; the
higher must not be made to change place with the
lower, or the first with the last. For the parties con
cerned there is all the difference in the world between A
killing B and B kiUing A. The law of commutativeness
simply asserts that difference of order does not attach to
the connection between the properties and circumstances
of a thing— to what I shall call simple logical relations.

c 'Laws of Thought/ p. 29.

CHAPTER III.

PROPOSITIONS.

WE now proceed to consider the variety of forms of
propositions in which the truths of science must be
expressed. I shall endeavour to show that, however
diverse these forms may be, they all admit the application
of the one same principle of influence, that what is true
of one thing or circumstance is true of the like or same.
This principle holds true whatever be the kind or manner
of the likeness, provided proper regard be had to its
degree. Propositions may assert an identity of time,
space, manner, quantity, degree, or any other circumstance
in which things may agree or differ.

We find an instance of a proposition concerning time
in the following : — ' The year in which Newton was born,
was the year in which Galileo died/ This proposition
expresses an approximate identity of time between two
events ; hence whatever is true of the year in which
Galileo died is true of that in which Newton was born,
and vice versd. ' Tower Hill is the place where Raleigh
was executed' expresses an identity of place ; and what
ever is true of the one spot is true of the spot otherwise
defined, but in reality the same. In ordinary language
we have many propositions obscurely expressing identities
of number, quantity, or degree. ' So many men, so many
minds/ is a proposition concerning number or an equa
tion ; whatever is true of the number of men is true of
the number of minds, and vice versd. ' The density of
Mars is (nearly) the same as that of the Earth/ ' The force

44 THE PRINCIPLES OF SCIENCE.

of gravity is directly as the product of the masses, and
inversely as the square of the distance,' are propositions
concerning magnitude or degree. Logicians have not paid
adequate attention to the great variety of propositions
which can be stated by the use of the little conjunction
as, together with so. ' As the home so the people,' is a
proposition expressing identity of manner ; and a great
number of similar propositions all indicating some kind of
resemblance might be quoted. Whatever be the special
kind or form of identity, all such expressions of identity
are subject to the great principle of inference ; but as we
shall in later parts of this work treat more particularly
of inference in cases of number and magnitude, we will
here confine our attention to the logical propositions
which involve only notions of quality.

Simple Identities.

The most important class of propositions consists of
those which fall under the formula

A=B,

and may be called simple identities. I may instance, in
the first place, those most elementary propositions which
express the exact similarity of a quality encountered in
two or more objects. I may compare by memory or
otherwise the colour of the Pacific ocean with that of
the Atlantic, and declare them identical. I may assert
that 'the smell of a rotten egg is that of hydrogen
sulphide,' ' the taste of silver hyposulphite is that of
cane sugar,' ' the sound of an earthquake is that of distant
artillery.' Such are propositions stating, accurately or
otherwise, the identity or non-identity of simple physical
sensations. Judgments of this kind are necessarily pre
supposed in more complex judgments. If I declare that
' this coin is made of gold/ I must base the judgment upon

PROPOSITIONS. 45

the exact likeness of the substance in several qualities to
other pieces of substance which are undoubtedly of gold.
I must make a judgment of the colour, the specific
gravity, the hardness, sound, and chemical properties ;
and each of these judgments might be expressed in an
elementary proposition, 'the colour of this coin is the
colour of gold/ and so on. Even when we establish
the identity of a thing with itself under a different
name or aspect, it is by distinct judgments concerning
single circumstances. To prove that the Homeric xa^K °'?
is copper we must show the identity of each quality
recorded of ^aX«-o? with a quality of copper. To establish
Deal as the landing-place of Caesar, every circumstance
must be shown to agree. If the modern Wroxeter is
the ancient Uriconium, there must be the like agreement
of all features of the country not subject to alteration by
time.

All such identities may be expressed in the form A = B.
We may say

Colour of Pacific Ocean = Colour of Atlantic Ocean.
Smell of rotten egg = Smell of hydrogen sulphide.
In these and similar propositions we assert identity of
single qualities or sensations. But in the same form
we may express identity of any group of qualities, as in

^ccA/co? = Copper.

Deal = Landing-place of Csesar.

A multitude of propositions involving singular terms fall
into the same form, as in

The Pole star = The slowest-moving star.

Jupiter = The greatest of the planets.

The ringed planet = The planet having seven satel
lites.

The Queen of England = The Queen of India.

The number two = The even prime number.

Honesty = The best policy.

46 THE PRINCIPLES OF SCIENCE.

In mathematical and scientific theories we often meet
with simple identities capable of expression in the same
form. Thus in mechanical science ' The process for finding
the resultant of forces = the process for finding the re
sultant of simultaneous velocities a/ Theorems in geometry
often give results in this form, as-
Equilateral triangles = Equiangular triangles.
Circle = Finite plane curve of constant curvature.
Circle = Curve of least perimeter.

The more profound and important laws of nature are
often expressible in the form of identities ; in addition to
some instances which have already been given I may
suggest —

Crystals of cubical system = Crystals incapable of

double refraction.

All definitions are necessarily of this form of simple
identity, whether the objects defined be many, few, or sin
gular. Thus we may say —

Common salt = Sodium chloride.

Chlorophyl = Green colouring matter of leaves.

Square = Equal-sided rectangle.

It is an extraordinary fact that propositions of this
elementary form, all-important and very numerous as
they are, had no recognised place in Aristotle's system of
Logic. Accordingly their importance was overlooked until
very recent times, and logic was the most deformed of
sciences. But it is quite impossible that Aristotle or any
other person should avoid constantly using them ; not a
term could be defined without their use. In one place at
least Aristotle actually notices a proposition of the kind.
He observes:— 'We sometimes say that that white thing
is Socrates, or that the object approaching is CalliasV
Here we certainly have simple identity of terms ; but he

1 Thomson and Tait, 'Treatise on Natural Philosophy,' vol. i. p. 182.
b.' Prior Analytics,' I. cap. xxvii. 3.

PROPOSITIONS. 47

considered such propositions purely accidental, and came
to the extraordinary conclusion, that ' Singulars cannot be
predicated of other terms.'

Propositions may also express the identity of extensive
groups of objects taken collectively or in one connected
whole ; as when we say —

' The Queen, Lords, and Commons = The Legislature

of the United Kingdom/

When Blackstone asserts, ' The only true and natural
foundation of society are the wants and fears of indi
viduals/ we must interpret him as meaning that the whole
of the wants and fears of individuals in the aggregate form
the foundation of society. But many propositions which
might seem to be collective are but groups of singular pro
positions or identities. When we say ' Potassium andsodium
are the metallic bases of potash and soda,' we obviously
mean-
Potassium = Metallic base of potash ;
Sodium = Metallic base of soda.

It is the work of grammatical analysis to separate the
various propositions often combined in a single sentence.
Logic cannot be properly required to interpret the forms
and devices of language, but to treat the meaning or
information when clearly exhibited.

Partial Identities.

However numerous and important may be propositions
expressing simple identity of one term or class with
another, there is an almost equally important kind of
proposition which I propose to call a partial identity.
When we say that ' All mammalia are vertebrata,' we do
not mean that mammalian animals are identical with
vertebrate animals, but only that the mammalian form a
part of the class vertebrata. Such a proposition was
regarded in the old logic as asserting the inclusion of one

48 777^ PRINCIPLES OF SCIENCE.

class in another, or of an object in a class. It was called
a universal affirmative proposition, because the attribute
vertebrate was affirmed of the whole subject mammalia ;
but the attribute was said to be undistributed, because
not all vertebrata were of necessity involved in the propo
sition. Aristotle, overlooking the importance of simple
identities, and indeed almost denying their existence, un
fortunately founded his system upon the notion of inclusion
in a class, in place of identity. He regarded inference as
resting upon the rule that what is true of the containing
class is true of the contained, instead of the vastly more
general rule that what is true of a class or thing is true
of the like. Thus he not only reduced logic to a fragment
of its proper self, but destroyed the deep analogies which
bind together logical and mathematical reasoning. Hence
a crowd of defects, difficulties and errors which will long
disfigure the first and simplest of the sciences.

It is surely evident that the relation of inclusion rests
upon a relation of identity. Mammalian animals cannot
be included among vertebrates unless they be identical
with part of the vertebrates. Cabinet Ministers are in
cluded almost always in the class Members of Parlia
ment, because they are identical with some who sit in
Parliament. We may indicate this identity with a part
of the larger class in various ways ; as for instance —

Mammalia = part of the vertebrata

Diatoms = species of plants.

Cabinet Ministers = some Members of Parliament.

Iron = a metal.

In ordinary language the verbs is or are express mere
inclusion more often than not. Men are mortals, means
that men form a part of the class mortal, but great con
fusion exists between this sense of the verb and that in
which it expresses identity, as in ' The sun is the centre of
the planetary system.' The introduction of the indefinite

PROPOSITIONS. 49

article a often seems to express partiality, as when we say
' Iron is a metal ' we clearly mean one only of several
metals.

Certain eminent recent logicians have proposed to avoid
the indefiniteness in question by what is called the Quan
tification of the Predicate, and they have generally used
the little word some to show that only a part of the
predicate is identical with the subject0. Some is an in
determinate adjective ; it implies unknown qualities by
which we might select the part in question if they were
known, but it gives no hint as to their nature. I might
make extensive use of such an indeterminate sign to
express partial identities in this work. Thus, taking the
special symbol V = some, the general form of a partial
identity would be A = VB, and in Boole's Logic expres
sions of the kind were freely used. But I .find that
indeterminate symbols only introduce complexity, and
destroy the beauty and simple universality of the system
which may be created without their use. A vague word
like some is only used in ordinary language by ellipsis,
and to avoid the trouble of attaining accuracy. We can
always substitute for it more definite expressions if we
like ; but when once the indefinite some is introduced we
cannot replace it by the special description. We do not
know whether some colour is red, yellow, blue, or what it
is ; but on the other hand red colour is certainly some
colour ; as is also yellow, blue, &c.

Throughout this system of logic I shall usually dispense
with all such indefinite expressions ; and this can readily
be done by substituting one of the other terms. To
express the proposition ' All A's are some B's ' I shall not
use the form A = VB, but

A = AB.

c 'Elementary Lessons in Logic,' p. 183. ' Substitution of Similars,'
p. 7.

E

50 THE PRINCIPLES OF SCIENCE.

This formula expresses that the class A is identical
with the class AB ; and as the latter must be a part at
least of the class B, it implies the inclusion of the class A
in that of B. Thus we might represent our former ex
ample thus—

Mammalia = Mammalian vertebrata.
This proposition asserts identity between a part of the
vertebrata and the mammalia. If it is asked What part 1
the proposition affords no answer except that it is the
part which is mammalian ; but the assertion ' mammalia =
some vertebrata' tells us no more.

It is quite likely that some readers may think this
mode of representing the universal affirmative proposition
of the old logic artificial and complicated. I will not
undertake to convince them of the opposite at this point
of the system. My justification for it will be found, not
in the immediate treatment of this proposition, but in
the general harmony which it enables us to discover
between all parts of reasoning. I have no doubt that
this is the point of critical difficulty in the relation of
logical to other forms of reasoning. Grant this mode
of denoting that ' all A's are B's/ and I fear no further
difficulties ; refuse it, and we find want of analogy and
endless complication in every direction. For instance
—Aristotle, in accepting inclusion of class in class as
the fundamental relation of logic, was at once obliged
to ignore the existence of the very extensive and all-
important class of propositions denoting the similarity
of one thing with another. It is on general grounds
that I hope to show overwhelming reasons for seeking
to reduce every kind of proposition to the form of an
identity.

I may add that not a few previous logicians have
accepted this view of the universal affirmative proposition.
Boole often employed this mode of expression, and

PROPOSITIONS. 51

Spalding d distinctly says that the proposition ' all metals
are minerals ' might be described as an assertion of partial
identity between the two classes. Hence the name which
I have adopted for the proposition.

Limited Identities.

A highly important class of propositions have the
general form

AB = AC,

expressing the identity of the class AB with the class AC.
In other words, ' Within the sphere of the class of things
A, all the B's are all the C's,J or 'The B's and C's, which
are A's, are identical/ But it will be observed that nothing
is asserted concerning things which are outside of the
class A ; and thus the identity is of limited extent. It is
the proposition B = C limited to the sphere of the class A.
Thus if we say ' Plants are devoid of locomotive power/
we must limit the statement to large plants, since minute
microscopic plants often have very remarkable powers of
motion. When we say ' Metals possess metallic lustre,' we
mean in their uncombined state.

A barrister may make numbers of most general state
ments concerning the relations of persons and things in
the course of an argument, but it is of course to be under
stood that he speaks only of persons and things under the
English Law. Even mathematicians make statements
which are not true with absolute generality. They say
that imaginary roots enter into equations by pairs ; but
this is only true under the tacit condition that the
equations in question shall not have imaginary coefficients.*3

d 'Encyclopaedia Britannica,' Eighth Ed. art. Logic, sect. 37, note.
8vo reprint, p. 79.

e De Morgan ' On the Root of any Function.' Cambridge Philosophical
Ti-ansactions, 1867, vol.xi. p. 25.

E 2

52

THE PRINCIPLES OF SCIENCE.

The universe, in short, within which they habitually dis
course is that of equations with real coefficients. These
implied limitations form part of that great mass of tacil
knowledge which accompanies all special arguments.

It is worthy of inquiry whether almost all identities
are not really limited to an implied sphere of meaning.
When we make such a plain statement as ' Gold is mal
leable' we obviously speak of gold only in its solid state ;
when we say that ' Mercury is a liquid metal ' we must
be understood to exclude the frozen condition to which it
may be reduced in the Arctic regions. Even when we
take such a fundamental law of nature as ' All substances
gravitate,3 we must mean by substance, material sub
stance, not including that basis of heat, light and electrical
undulations which occupies space and possesses many
mechanical properties, but not gravity. The proposition
then is really of the form

Material substance = Material gravitating substance.

To De Morgan is due the remark, that we do usually
think and argue in a limited universe or sphere of notions
even when it is not expressly stated f.

Negative Propositions.

In every act of intellect, as we have seen, we are en
gaged with a certain degree of identity or difference
between certain things or sensations compared together.
Hitherto I have treated only of identities ; and yet
it might seem that the relation of difference must be
infinitely more common than that of likeness. One
thing may resemble a great many other things, but
then it differs from all remaining things in the world.
Difference or diversity may almost be said to constitute
life, being to thought what motion is to a river. The

f 'Syllabus of a Proposed System of Logic,' §§ 122, 123.

PROPOSITIONS. 53

very perception of an object involves its discrimination
from all other objects. But we may nevertheless be said
to detect resemblance as often as we detect difference.
We cannot, in fact, assert the existence of a difference,
without at the same time implying the existence of an
agreement.

If I compare mercury, for instance, with other metals,
and decide that it is not solid, here is a difference between
mercury and solid things, expressed in a negative propo
sition ; but there must be implied, at the same time, an
agreement between mercury and the other substances
which are not solid. As it is impossible in the alphabet
to separate the vowels from the consonants without at
the same time separating the consonants from the vowels,
so I cannot select as the object of thought solid things,
without thereby throwing together into another class all
things which are not solid. The very fact of not possess
ing a quality, constitutes a new quality or circumstance
which may equally be the ground of judgment and classi
fication. In this point of view, agreement and difference
are ever the two sides of the same act of intellect, and it
becomes equally possible to express the same judgment in
the one or other aspect.

Between affirmation and negation there is accordingly
a perfect balance or equilibrium. Every affirmative propo
sition implies a negative one, and vice versd. It is even
a matter of indifference, in a logical point of view, whether
a positive or negative term be used to denote a given
quality and the class of things possessing it. If the
ordinary state of man's body be called good health, then in
other circumstances he is said not to l)e in good health ;
but we might equally describe him in the latter state as
sickly, and in his normal condition he would be not sickly.
Animal and vegetable substances are now called organic,
so that the other substances, forming an immensely greater

54 THE PRINCIPLES OF SCIENCE.

part of the globe, are described negatively as inorganic.
But we might, with at least equal logical correctness,
have described the preponderating class of substances as
mineral, and then vegetable and animal substances would
have been non-mineral.

It is plain that any positive term, and its corresponding
negative divide between them the whole universe of
thought : whatever does not fall into one must fall into
the other, by the third fundamental Law of Thought,
the Law of Duality. It follows at once that there are
two modes of representing a difference. Suppose that
the things or classes represented by A and B are found
to differ, we may indicate the result of the judgment by
the notation (see p. 20)

A-B.

But we may now represent the same judgment by the
assertion that A agrees with those things which differ from
B, or that A agrees with the not-B's. Using our notation
for negative terms (see p. 17), we obtain

A = A6

as the expression of the ordinary negative proposition.
Thus if we take A to mean quicksilver, and B solid, then
we have the following proposition :—

Quicksilver = Quicksilver not-solid.
There may also be several other classes of negative
propositions, of which no notice was taken in the old logic.
We may have cases where all A's are not-B's, and at the
same time all not-B's are A's ; there may, in short, be a
simple identity between A and not-B^ which may be
expressed in the form

A=6.
An example of this form would be

Conductors of electricity = non-electrics.
We shall also frequently have to deal as results of

PROPOSITIONS. 55

deduction, with simple, partial, or limited identities be
tween negative terms, in the forms

a = b, a = ab, aC = bC.

It would be equally possible to represent affirmative
propositions in the negative form. Thus ' Iron is solid,'
might be expressed as ' Iron is not not-solid,' or ' Iron is not
fluid'; or, taking A and 6 for the terms 'iron,' and 'not -solid,'
the form would be

A -6.

But there are very strong reasons why we should em
ploy all propositions in their affirmative form. All infer
ence proceeds by the substitution of equivalents, and a
proposition expressed in the form of an identity is ready
to yield all its consequences in the most direct manner.
As will be more fully shown, we can infer in a negative
proposition, but not by it. Difference is incapable of
becoming the ground of inference ; it is only the implied
agreement with other differing objects, which admits of
deduction ; and it will always be found advantageous to
employ propositions in the form which exhibits clearly all
the implied agreements.

Conversion of Propositions.

The old books of logic contain many rules concerning
the conversion of propositions, that is, the transposition
of the subject and predicate in such a way as to obtain
a new proposition which will be equally true with the
original. The reduction of every proposition to the
form of an identity renders all such rules and processes
needless. Identity is essentially reciprocal. If the colour
of the Atlantic Ocean is the same as that of the Pacific
Ocean, that of the Pacific must be the same as that of
the Atlantic. Sodium chloride being identical with
common salt, common salt must be identical with sodium

56

chloride. If the number of windows in Salisbury
Cathedral equals, the number of days in the year, the
number of days in the year must equal the number of
the windows. Lord Chesterfield was not wrong when
he said, ' I will give anybody their choice of these two
truths, which amount to the same thing ; He who loves
himself best is the honestest man ; or, The honestest man
loves himself best * .' Scotus Erigena exactly expresses this
reciprocal character of identity in saying, ' There are not
two studies, one of philosophy and the other of religion ;
true philosophy is true religion, and true religion is true
philosophy.'

A mathematician would not think it worth mention
that if x = y then also y = x. He would not consider
these to be two equations at all, but one same equation
accidentally written in two different manners. In written
symbols one of two names must come first, and the other
second, and a like succession must perhaps be observed in
our thoughts : but in the relation of identity there is no
need for succession in order ; each is simultaneously equal
and identical to the other. These remarks will hold true
equally of logical and mathematical identity; so that I
shall consider the two forms

A = B and B = A

to express exactly the same identity differently written.
All need for rules of conversion disappears, and there
will be no single proposition in the system which may
not be written with either term foremost. Thus A = AB
is the same as AB = A, AB = AC as AC = AB, and so on.

The same remarks are partially true of differences or
inequalities, which are also reciprocal to the extent that
one thing cannot differ from a second without the second
differing from the first. Mars differs in colour from

£ Chesterfield's Letters, 8vo, 1744; vol. i. p. 302.

PROPOSITIONS. 57

Venus, and Venus must differ from Mars. The Earth
diifers from Jupiter in density ; therefore Jupiter must
differ from the .Earth. Speaking generally, if A ~ B we
shall also have B -»- A, and these two forms may be con
sidered expressions of the same difference. But the
reader will notice that the relation of differing things
is not wholly reciprocal. The density of Jupiter does
not differ from that of the Earth in the same way that
that of the Earth differs from that of Jupiter. The change
of sensation which we experience in passing from Venus
to Mars is not the same as what we experience in passing
back to Venus, but just the opposite in nature. The
colour of the sky is lighter than that of the ocean ;
therefore that of the ocean cannot be lighter than that
of the sky, but darker. In these and all similar cases
we gain a notion of direction or character of change,
and results of immense importance may be shown to
rest on this notion. For the present we shall be
concerned with the mere fact of identity existing or
not existing.

Twofold Interpretation of Propositions.

Terms, as we have seen (p. 31), may have a meaning
either in extension or intension ; and according as one
or the other meaning is attributed to the terms of a
proposition, so may a different interpretation be assigned
to the proposition itself: When the terms are abstract
we must read them in intension, and a proposition con
necting such terms must denote the identity or non-
identity of the qualities respectively denoted by the
terms. Thus if we say

Equality = Identity of magnitude,
the assertion means that the circumstance of being equal

58 THE PRINCIPLES OF SCIENCE.

exactly corresponds with the circumstance of being iden
tical in magnitude. Similarly in

Opacity = Incapability of transmitting light,
the quality of being incapable of transmitting light is
declared to be the same as the intended meaning of the

word opacity.

When general names form the terms of a proposition
we may apply a double interpretation. Thus

Exogens = Dicotyledons

means either that the qualities which belong to all exo-
gens are the same as those which belong to all dicotyle
dons, or else that every individual falling under one name
falls equally under the other. Hence it may be said that
there are two distinct fields of logical thought. We may
argue either by the qualitative meaning of names or
by the quantitative, that is, the extensive meaning.
Every argument involving concrete plural terms might
be converted into one involving only abstract singular
terms, and vice versa. But there are many reasons for
believing that the intensive or qualitative form of reason
ing is the primary and fundamental one. It is sufficient
to point out that we may use abstract terms which contain
no reference to an extensive meaning ; and when there
is a mode which we must sometimes and may always
adopt, it is higher in importance than a mode which we
never need adopt necessarily.

CHAPTER IV.

DEDUCTIVE REASONING.

THE general principle of inference having been ex
plained in the previous chapters, and a suitable system
of symbols provided, we have now before us the com
paratively easy task of tracing out the most common and
important forms of deductive reasoning. The general
problem of deduction is as follows : — From one or more
propositions called premises to draw such other proposi
tions as ivill necessarily be true when the premises are
true. By deduction we investigate and unfold the in
formation contained in the premises ; and this we can do
by one single rule — For any term occurring in any pro
position or expression substitute the expression which is
asserted in any premise to be identical with it. To obtain
certain deductions, especially those involving negative
conclusions, we shall require to bring into use the
second and third Laws of Thought, and the process of
reasoning will then be called Indirect Deduction. In the
present chapter, however, I shall confine my attention to
those results which can be obtained by the process of
Direct Deduction, that is, by applying to the premises
themselves the rule of substitution. It will be found
that we can combine in one harmonious system, not only
the various moods of the ancient syllogism, but a great
number of equally important forms of reasoning, which had
no distinct place in the old logic. We can at the same
time dispense entirely with the elaborate apparatus of
logical rules and mnemonic lines, which were requisite

GO THE PRINCIPLES OF SCIENCE.

so long as the vital principle of reasoning was not clearly
expressed.

Immediate Inference.

Probably the simplest of all forms of inference is that
which has been called Immediate Inference, because it
can be performed upon a single proposition. It consists
in joining an adjective, or other qualifying clause of the
same nature, to both sides of an identity, and asserting
the equivalence of the terms thus produced. For instance,
since

Conductors of electricity = Non-electrics,
it follows that

Liquid conductors of electricity = Liquid non-electrics.

If we suppose that

Plants = Bodies decomposing carbonic acid,

it follows that

Microscopic plants = Microscopic bodies decomposing

carbonic acid.
In general symbols, from the identity

A = B
we can infer the identity

AC=BC.

This is but a case of plain substitution ; for by the first
Law of Thought it must be admitted that

and if in the second side of this identity we substitute
for A its equivalent B, we obtain

AC = BC.

In like manner from the partial identity

A = AB

we may obtain

AC = ABC

by an exactly similar form of substitution ; and in every

DEDUCTIVE REASONING. 61

other case the rule will be found capable of verification by
the principle of inference. The process when performed as
here described will be found free from the liability to
error which I have shown a to exist in Immediate Inference
by added Determinants, as described by Dr. Thomson b.

Inference with Tivo Simple Identities.

One of the most common forms of inference, and one to
which I shall especially direct attention, is practised with
two simple identities. From the two statements that
'London is the capital of England' and 'London is the
most populous city in the world/ we instantaneously draw
the conclusion that ' The capital of England is the most
populous city in the world/ Similarly, from the identities
Hydrogen = Substance of least density
Hydrogen = Substance of least atomic weight,
we infer

Substance of least density = Substance of least atomic

weight.

The general form of the argument is exhibited in the
symbols

B = A (i)

B=C (2)

hence A = C. (3)

We may describe the result by saying that terms
identical with the same term are identical with each
other ; and it is impossible to overlook the analogy to the
first axiom of Euclid that ' things equal to the same thing
are equal to each other.' It has been very commonly sup
posed that this was a fundamental principle of thought
incapable of reduction to anything simpler. But I enter
tain no doubt that this form of reasoning is only one case

a ' Elementary Lessons in Logic,' p. 86.
b < Outline of the Laws of Thought,' § 87.

G2 THE PRINCIPLES OF SCIENCE.

of the general rule of inference. We have two propo
sitions, A = B and B = C, and we may for a moment con
sider the second one as affirming a truth concerning B
while the former one informs us that B is identical with
A ; hence by substitution we may affirm the same truth
of A. It happens in this particular form that the truth
affirmed is identity to C, and we might, if we had preferred,
have considered the substitution as made by means of the
second identity in the first. Having two identities we
have a choice of the mode in which we will make the
substitution, though the result is exactly the same in
either case.

Now compare the three following formulas

(1) A = B = C hence A^C

(2) A = B-C hence A-C

(3) A -»- B ** C, no inference.

In the second formula we have an identity and a differ
ence, and we are able to infer a difference ; in the third
we have two differences and are unable to make any
inference at all. Because A and C both differ from B, we
cannot tell whether they will or will not differ from each
other. The flowers and leaves of a plant may both differ
in colour from the earth in which the plant grows, and
yet they may differ from each other ; in other cases the
leaves and stem may both differ from the soil and yet agree
with each other. Where we have difference only we can
make no inference ; where we have identity we can infer.
This fact gives great countenance to my assertion that
inference proceeds always through an identity, but may
be indifferently effected in a difference or an identity.

Deferring a more complete discussion of this point, I
will only mention now that arguments from double
identity occur very frequently, and are usually taken
for granted owing to their extreme simplicity. In the
equivalency of words it must be constantly employed. If

DEDUCTIVE REASONING. G3

the ancient Greek ^aX/co? is our copper, tlien it must be
the French cuivre, the German kupfer, the Latin cuprum,
because ^these are words, in one sense at least, equivalent
to copper. Whenever we can give two definitions or
expressions for the same term, the formula applies ; thus
Senior defined wealth as ' whatever is transferable, limited
in supply, and productive of pleasure or preventive of
pain ;' it is also equivalent to ' whatever has value in
exchange ;' hence obviously ' Whatever has value in ex
change' = ' Whatever is transferable, limited in supply, and
productive of pleasure or preventive of pain/ Two ex
pressions for the same term are often given in the same
sentence, and their equivalency implied. Thus Thomson
and Tait sayc, ' The naturalist may be content to know
matter as that which can be perceived by the senses, or as
that which tcan be acted upon by or can exert force.' I
take this to mean —

Matter = what can be perceived by the senses ;

Matter -. what can be acted upon by or can exert force.
For the term ' matter' in either of these identities we
may substitute its equivalent given in the other definition.
Elsewhere they often employ sentences of the form exem-
plified'in the following d; ' The integral curvature, or whole
change of direction of an arc of a plane curve, is the angle
through which the tangent has turned as we pass from
one extremity to the other.' This sentence is certainly of
the form —

The integral curvature = the whole change of direction,
&c. = the angle through which the tangent has
turned, &c.

Disguised cases of the same kind of inference occur
throughout ah1 sciences, and a remarkable instance is
found in algebraic geometry. Mathematicians readily

c ' Treatise on Natural Philosophy,' vol. i. p. 1 6 1 .
d Ibid. vol. i. p. 6.

s

G4 THE PRINCIPLES OF SCIENCE.

^7~tuT^ery equation of the form y =
equivalent to or represented by a straight line ; it is also
easily proved that the same equation is equivalent to one
of the form Az + B?/ + C = o, and vice versa. Hence it
follows that every equation of the first degree is equivalent
to or represents a straight line e.

Inference with a Simple and. a Partial Identity.

A form of reasoning somewhat different from that last
considered consists in inference between a simple and a
partial identity. If we have two propositions of the

form

A = B,

B = BC,

we may then substitute for B in either proposition its
equivalent in the other, getting in both cases A — BC ;
in this we may if we like make a second substitution for

B, getting

A = AC.

Thus, since 'Mont Blanc is the highest mountain in
Europe, and Mont Blanc is deeply covered with snow/ we
infer by an obvious substitution that * The highest moun
tain in Europe is deeply covered with snow.' These pro
positions when rigorously stated fall into the form above
exhibited.

This form of inference is constantly employed when for
a term we substitute its definition, or vice versa. The
very purpose of a definition is in fact to allow a single
term to be employed in place of a long descriptive phrase.
Thus when we say ' Circles are curves of the second
degree,' we may substitute the definition of a circle,
getting ' A plane curve, all points of whose perimeter are
at equal distances from a certain fixed point, is a curve of

c Todhunter's ' Plane Co-ordinate Geometry,' chap. ii. pp. 11-14.

DEDUCTIVE REASONING. 65

the second degree.' The real forms of the propositions
here given are exactly those shown in the symbolic state
ment, but in this and many other cases it will be sufficient
to state them in ordinary elliptical language for sake of
brevity. In scientific treatises a term and its definition
are often both given in the same sentence, as in ' The
weight of a body in any given locality, or the force with
which the earth attracts it, is proportional to its mass.'
The conjunction or in this statement gives the force of
equivalence to the parenthetic definition, so that the
propositions really are

Weight of a body = force with which the earth at
tracts it.

Weight of a body = weight, &c. proportional to its
mass.

A slightly different case of inference consists in sub
stituting in a proposition of the form A = AB a defi
nition of the term B. Thus from A = AB and B = C
we get A = AC. For instance, we may say that ' Metals
are elements' and 'Elements are incapable of decompo
sition.'

Metal = metal element.

Element = what is incapable of decomposition.

Hence

Metal = metal incapable of decomposition.
It is almost needless to point out that the form of these
arguments would not suffer any real modification if some
of the terms happened to be negative ; indeed in the last
example * incapable of decomposition ' may be treated as
a negative term. Taking

A = metal

B = element

C = what is capable of decomposition
c =~what is incapable of decomposition (p. 17);

66 THE PRINCIPLES OF SCIENCE.

the propositions are of the form

A = AB

B = c;
whence, by substitution,

A = Ac.

Inference of a Partial from Two Partial Identities.

However common be the cases of inference already
noticed, there is a form occurring almost more frequently,
and which deserves much attention because it occupied a
prominent place in the ancient syllogistic system. That
system strangely overlooked all the kinds of argument we
have as yet considered, and selected as the type of all
reasoning one which employs two partial identities as
premises. Thus from the propositions

Sodium is a metal (i)

Metals conduct electricity, (2,)

we may conclude that

Sodium conducts electricity. (3)

Taking A, B, C, respectively to represent the three terms,

the premises are of the form

A = AB (i)

B = BC. (2)

Now for B in (i) we can substitute its description as

given in (2), obtaining

A = ABC, (3)

or, in words, from

Sodium = sodium metal (i)

Metal = metal conducting electricity, (2)

we infer

Sodium = sodium metal conducting electricity, (3)
which in the elliptical language of common life becomes

' Sodium conducts electricitv.'

DEDUCTIVE REASONING. 67

The above is a syllogism in the mood called Barbara f
in the truly barbarous language of ancient logicians ; and
the first figure of the syllogism alone contained three other
moods which were esteemed distinct forms of argument.
But it is worthy of notice that without any real change
in our form of inference we readily include these three
other moods under it. The negative mood Celarent will
be represented by the example

Neptune is a planet (i)

No planet has retrograde motion, (2)

hence Neptune has riot retrograde motion. (3)

If we put A for Neptune, B for planet,and C for ' having
retrograde motion,' then by the corresponding negative
term c, we denote ' not having retrograde motion.' The
premises now fall into the form

A = AB (i)

B = Be, (2)

and by substitution for B, exactly as before, we obtain

A = ABc. (3)

What is called in the old logic a particular conclusion
may be deduced without any real variation in the sym
bols. Particular quantity is indicated, as before mentioned
(p. 49), by joining to the term an indefinite adjective of
quantity, such as some, a part, certain, &c., meaning that
an unknown part of the term enters into the proposition
as subject. Considerable doubt and ambiguity arise out
of the question whether the part may not in some cases
be the whole, and in the syllogism at least it must be
understood in this senses. Now if we take a letter to
represent this indefinite part, we need make no change in

f An explanation of this and other technical terms of the old logic
will be found in my 'Elementary Lessons in Logic,' Second Ed. 1871.
Macmillan & Co.

f 'Elementary Lessons in Logic,' pp. 67, 79.

F 2

68 THE PRINCIPLES OF SCIENCE.

our formulae to express either of the syllogisms Darii or
Ferio. Consider the example —

Some metals are of less density than water (i)
All bodies of less density than water will float

upon its surface (2)

Some metals will float upon its surface, (3)

Let A = some metals

B = body of less density than water
C = floating on the surface of water ;
then the propositions are evidently as before,

A = AB (i)

B = BC; (2)

hence A = ABC. (3)

Thus the syllogism Darii does not really differ from Bar
bara. If the reader prefer it, we can readily employ a
distinct symbol for the indefinite sign of quantity.
Let P = some

Q = metal,

B and C having the same meanings as before. Then the
premises become

PQ == PQB (i)

B = BC; (2)

hence, by substitution, as before,

PQ == PQBC. (3)

Except that the formulae look a little more complicated
there is no difference whatever.

The mood Ferio is of exactly the same character as
Darii or Barbara, except that it involves the use of a
negative term. Take the example —

Bodies which are equally elastic in all directions do

not doubly refract light,

Some crystals are bodies equally elastic in all direc
tions; therefore some crystals do not doubly
refract light.
Assigning the letters as follows —

DEDUCTIVE REASONING. 69

A = some crystals.

B = bodies equally elastic in all directions

C = doubly refracting light

c = not doubly refracting light.

Our argument is of the same form as before, and may
be concisely stated in one line

A = AB = ABc.
If we take PQ for the indefinite some crystals, we have

PQ = PQB = PQBc.

The only difference is that the negative term c occurs
instead of C in the mood Darii (p. 68).

On the Ellipsis of Terms in Partial Identities-.

The reader will probably have noticed that the conclu
sion which we obtain from premises is often more full
than that drawn by the old Aristotelian processes. Thus
from ' Sodium is a metal/ and ' Metals conduct electricity,'
we inferred (p. 66) that ' Sodium = sodium metal, con
ducting electricity/ whereas the old logic simply concludes
that ' Sodium conducts electricity.' Symbolically, from
A = AB, and B = BC, we get A = ABC, whereas the old
logic gets at the most A = AC. It is therefore well to
show that without employing any other principles of
inference than those already described, we may infer
A = AC from A = ABO, though we cannot infer the
latter more full and accurate result from the former.
We may show this most simply as follows : —

By the first law of thought it is evident that

AA = AA ;

and if we have given the proposition A = ABC, we may
substitute for both the A's in the second side of the above,
obtaining

AA = ABC . ABC.
But from the property of logical symbols expressed in the

70 THE PRINCIPLES OF SCIENCE.

Law of Simplicity (p. 39) some of the repeated letters may
be made to coalesce, and we have

A = ABC . C.

Substituting again for ABC its equivalent A, we obtain

A = AC,
the desired result.

By a similar process of reasoning it may be shown that
we can always drop out any term appearing in one member
of a proposition, provided that we substitute for it the
whole of the other member. This process was described
in my first logical Essay h, as Intrinsic Elimination, but it
might perhaps be better entitled the Ellipsis of Terms.
It enables us to get rid of needless terms by strict sub-
stitutive reasoning.

Inference of a Simple from Two Partial Identities.

Two terms may be connected together by two partial
identities in yet another manner, and a case of inference
then arises which is of the highest importance. In the
two premises

A = AB (i)

B - AB, (2)

the second member of each is the same ; so that we can
by obvious substitution obtain

A = B.

Thus in plain geometry we readily prove that ' Every
equilateral triangle is also an equiangular triangle,' and
we can with equal ease prove that ' Every equiangular
triangle is an equilateral triangle.' Thence by substitu
tion, as explained above, we pass to the simple identity-
Equilateral triangle = equiangular triangle.
We thus prove that one class of triangles is entirely
identical with another class ; that is to say, they differ
only m our way of naming and regarding them.
h ' l\uc Logic ,' p. 19.

DEDUCTIVE REASONING. 71

The great importance of this process of inference arises
from the fact that the conclusion is more simple and
general than either of the premises, and contains as much
information as both of them put together. It is on this
account constantly employed in inductive investigation,
as will afterwards be more fully explained, and it is the
natural mode by which we arrive at a conviction of the
truth of simple identities as existing between classes of
numerous objects.

Inference of a Limited from Two Partial Identities.

We have just considered arguments which are of the
type treated by Aristotle in the first figure of the
syllogism. But there are two other types of argument
which employ a pair of partial identities. If our premises
are, as shown in these symbols,

B = AB (i)

B = CB, (2)

we may substitute for B either by (i) in (2) or by (2) in
(i), and by both modes we obtain the conclusion

AB = CB, (3)

a proposition of the kind which we have called a limited
identity (p. 51). Thus, for example,

Potassium = potassium metal (i)

Potassium = potassium floating on water ; (2)

hence

Potassium metal = potassium floating on water. (3)
Now this is really a syllogism of the mood Darapti in the
third figure, except that we obtain a conclusion of a much
more exact character than the old syllogism gives. From
the premises ' Potassium is a metal ' and ' Potassium floats
on water,' Aristotle would have inferred that ' Some
metals float on water.' But if inquiry were made what
the some metals are, the answer would certainly be ' Metal
which is potassium/ Hence Aristotle's conclusion simply

72 THE PRINCIPLES OF SCIENCE.

leaves out some of the information afforded in the premises ;
it even leaves us open to interpret the some metals in a
wider sense than we are warranted in doing. From these
distinct defects of the syllogism the process of substitution
is free, and it only incurs the possible objection of being
tediously minute and accurate.

Miscellaneous Forms of Deductive Inference.

The more simple and common forms of deductive
reasoning having been exhibited and demonstrated on
the principle of substitution, there remain many, in fact
an indefinite number, which may be explained with nearly
equal ease. Such as involve the use of disjunctive propo
sitions will be deferred to a later chapter, and several of
the syllogistic moods which include negative terms will be
more conveniently treated after we have introduced the
symbolic use of the second and third laws of thought.

We sometimes meet with a chain of propositions which
allow of repeated substitution and form an argument called
in the old logic a Sorites. Take, for instance, the premises
Iron is a metal (i)

Metals are good conductors of electricity (2)

Good conductors of electricity are useful for

telegraphic purposes. (3)

It obviously follows that

Iron is useful for telegraphic purposes. (4)

Now if we take our letters thus —

A = Iron, B = metal, C = good conductor of
electricity, D = useful for telegraphic purposes,
the premises will assume the form —

A = AB (i)

B = BC (a)

C = CD (3)

For B in (i) we can substitute its equivalent in (2), and

DEDUCTIVE REASONING. 73

for C in (2) we can substitute its equivalent in (3). We
shall obtain as an intermediate result,

A = ABC,
and from this the complete conclusion

A = ABCD. (4)

The full interpretation is that Iron is iron, metal, good
conductor of electricity, use/id for telegraphic purposes,
which is abridged in common language by the ellipsis of
the circumstances which are not of immediate importance.
Instead of all the propositions being of one type, as in
the last example, we may have a series of premises of
various character ; for instance

Common salt is sodium chloride (i)

Sodium chloride crystallizes in a cubical form (2)
What crystallizes in a cubical form does not

possess the power of double refraction ; (3)
it will follow that

Common salt does not possess the power of

double refraction. (4)

Taking our letter-terms thus —
A = Common salt,
B = Sodium chloride,
C = Crystallizing in a cubical form,
D = Possessing the power of double refraction,
we may state the premises in the form

A = B, (i)

B = BC, (2)

C = Cd. (3)

Substituting by (2) in (i) and by (3) in (2) we obtain

A = BCd, (4)

which is a more precise version of the common conclusion.
We often meet with a series of propositions describing
the qualities or circumstances of one same thing, and we
may if we like combine them all into one proposition
by the process of substitution. This case is, in fact,

74 THE PRINCIPLES OF SCIENCE.

that which Archbishop Thomson has called ' immediate
inference by the sum of several predicates,' and his
example will serve my purpose well1. He describes
copper as 'A metal, of a red colour, and disagreeable
smell and taste, all the preparations of which are
poisonous, which is highly malleable, ductile, and tena
cious, with a specific gravity of about 8.83.' If we
assign the letter A to copper, and the succeeding letters
of the alphabet in succession to the series of predicates,
we have nine distinct statements, of the form

A = AB(i) A = AC(2) A = AD(3) A = AK (9).

We can readily combine these propositions into one bv
substituting for A in the second side of (i) its expression
in (2). We thus get

A - ABC,

and by repeating the process over and over again we
obtain the single proposition

A = ABCDEFGHIJK.

But Dr. Thomson is mistaken in supposing that we can
obtain in this manner a definition of copper. Strictly
speaking, the above proposition is only a description of
copper, and all the ordinary descriptions of substances
in scientific works may be summed up in this form.
Thus we may assert of the organic substances called
Paraffins that they are all saturated hydrocarbons, in
capable of uniting with other substances, produced by
heating the alcoholic iodides with zinc, and so on. It
may be shown that no amount of ordinary description
can be equivalent to definition.

Fallacies.

I have hitherto been engaged in showing that all the
forms of reasoning of the old syllogistic logic, and an
Jfimte number of other forms in addition, may be
' Au Outline of the Laws of Thought/ Fifth Ed. p. 161.

DEDUCTIVE REASONING. 75

readily and clearly explained on the single principle of
substitution. It is now desirable to show that the same
principle would prevent us falling into fallacies. So long
as we exactly observe the one rule of substitution of
equivalents it will be impossible to commit a paralogism,
or to break any one of the elaborate rules of the ancient
system. One rule is thus proved to be as powerful as
the six, eight, or more rules by which the correctness of
syllogistic reasoning was guarded.

It was a fundamental rule, for instance, that two nega
tive premises could give no conclusion. If we take the
propositions-
Granite is not a sedimentary rock, (i)
Basalt is not a sedimentary rock, (2)
we ought not to be able to draw any inference concerning
the relation of granite and basalt. Taking our letter-
terms thus

A = granite
B = sedimentary rock
C = basalt,
the premises may be expressed in the form

A - B (i)

C - B. (2)

We have in this form two statements of difference ; but
the principle of inference can only work with a statement
of agreement or identity (p. 62). Thus our rule gives
us no power whatever of drawing any inference.

It is to be remembered, indeed, that we claim the
power of always turning a negative proposition into an
affirmative one ; and it might seem that the old rule of
negative premises would be thus circumvented. Let us
try. The premises (i) and (2) when affirmatively stated
(see p. 54), will take the form

A = A6 (i)

C = 06. (2)

76 THE PRINCIPLES OF SCIENCE.

The reader will find it impossible by the rule of substitu
tion to discover a relation between A and C. Three terms
occur in these premises, namely A, b, and C ; but they
are so combined that no term occurring in one has its
exact equivalent stated in the other. No substitution
can therefore be made, and the principle holds true.
Fallacy is impossible.

It would be a mistake to suppose that the mere
occurrence of negative terms in both premises render
them incapable of yielding a conclusion. The old rules
of logic informed us that from two negative premises no
conclusion could be drawn, but it is a fact that the rule
in this bare form does not hold universally true ; and I
am not aware that any precise explanation has been
given of the conditions under which it is or is not
imperative. Consider the following example —

Whatever is not metallic is not capable of power
ful magnetic influence, (i)

Carbon is not metallic, (2)

Therefore, carbon is not capable of powerful mag
netic influence. (3)
Here we have two distinctly negative premises (i) and
(2), and yet they yield a perfectly valid negative con
clusion (3). The syllogistic rule is actually falsified in
its bare and general statement. In this and many other
cases we can convert the propositions into affirmative
ones which yield a conclusion. To show this let
A = carbon, B = metallic,
C = capable of powerful magnetic influence.
The premises readily take the form

b = be (i)

A = A6, (a)

and substitution for b in (2) by means of (i), gives the
conclusion

A = Abe (3)

DEDUCTIVE REASONING. 77

Our principle of inference then includes the rule of nega
tive premises whenever it is true, and discriminates correctly
between the cases where it does and does not apply.

The paralogism, anciently called Undistributed Middle,
is also easily exhibited and infallibly avoided by our
system. Let the premises be

Hydrogen is an element, (i)

All metals are elements. (2)

According to the syllogistic rules the middle term element
is here undistributed, and no conclusion can be obtained ;
we cannot tell then whether hydrogen is or is not a
metal. Eepresent the terms as follows —
A = hydrogen
B = element
C = metal.
The premises then become

A = AB (i)

C = CB. (2)

The reader will here, as in a former page (p. 75), find
it impossible to make any substitution. The only term
which occurs in both premises is B, but it is combined
with different letters. For CB we cannot substitute the
equivalent of AB. We have no right to decompose
combinations ; and if we adhere rigidly to the rule given,
that if two terms are stated to be equivalent we may
substitute one for the other, we cannot commit the
fallacy. It is apparent that the form of premises given
above is the same as that which we obtained by trans
lating two negative premises into the affirmative form.

The old fallacy, technically called the Illicit Process of
the Major Term, is more easy to commit and more diffi
cult to detect than anv other breach of the syllogistic rules.

*• I/O

In our system it could hardly occur. From the premises
All planets are subject to gravity, (i)

Fixed stars are not planets, (2)

78 THE PRINCIPLES OF SCIENCE.

we might inadvertently but fallaciously infer that, ' Fixed
stars are not subject to gravity.' To reduce the premises
to symbolic form, let

A = planet

B = fixed star

C = subject to gravity ;
then we have the propositions

A = AC (i)

B = Ba. (2)

The reader will try in vain to produce from these
premises by legitimate substitution any relation between
B and C ; he could not then commit the fallacy of
asserting that B is not C.

There remain two other kinds of paralogism, com
monly known as the fallacy of Four Terms and the Illicit
Process of the Minor term. They are so evidently impos
sible while we obey the rule of the substitution of equi
valents, that it is not necessary to give any illustrations.
When there are four distinct terms in two propositions
there could be no opening for a substitution. As to the
Illicit Process of the Minor it consists in a flagrant sub
stitution for a term of another wider term which is not
known to be equivalent to it, and which is therefore
forbidden by our rule to be substituted for it.

CHAPTER V.

DISJUNCTIVE PROPOSITIONS.

IN the previous chapter I have exhibited various forms
of deductive reasoning by the process of substitution, so
far as they can be treated without the use of disjunctive
propositions ; but we cannot long defer the consideration
of this more complex class of identities. General terms
arise, as we have seen (p. 29), from classifying or men
tally uniting together all objects which agree in certain
qualities, the value of this union consisting in the fact
that the power of knowledge is multiplied thereby. In
forming such classes or general notions, we overlook or
abstract the points of difference which exist between the
objects joined together, and fix our attention only on the
points of agreement. But every process of thought may
be said to have its inverse process, which consists in
undoing the effects of the direct process. Just as division
undoes multiplication, and evolution undoes involution,
so we must have a process which undoes abstraction, or
the operation of forming general notions. This inverse
process will consist in distinguishing the separate objects
or minor classes which are the constituent parts of any
wider class. When we mentally unite together certain
objects visible in the sky and call them planets, we shall
afterwards need to distinguish the contents of this general
notion, which we do in the disjunctive proposition —

A planet is either Mercury or Venus or the Earth or

or Neptune.

Having formed the very wide class 'vertebrate animal/

80 THE PRINCIPLES OF SCIENCE.

we may specify its subordinate classes thus :— ' A verte
brate animal is either a mammalian, bird, reptile, or
fish.' Nor is there any limit to the number of possible
alternatives. ' An exogenous plant is either a ranun
culus, a poppy, a crucifer, a rose, or it belongs to some
one of the other seventy natural orders of exogens at
present recognised by botanists/ A cathedral church in
England must be either that of London, Canterbury,
Winchester, Salisbury, Manchester, or of one of about
twenty-four cities possessing such churches. And if we
were to attempt to specify the meaning of the term
'star/ we should require to enumerate as alternatives,
not only the many thousands of stars recorded in cata
logues, but the many millions yet unnamed.

Whenever we thus distinguish the parts of a general
notion we employ a disjunctive proposition, in at least
one side of which are several alternatives joined by the
so called disjunctive conjunction or, a contracted form of
other. There must be some relation between the parts
thus connected in one proposition ; we may call it the
disjunctive or alternative relation, and we must carefully
inquire into its nature and results. This relation is that
of doubt and ignorance, giving rise to choice or uncer
tainty. Whenever we classify and abstract we must open
the way to such uncertainty. By fixing our attention on
certain attributes to the exclusion of others we necessarily
leave it doubtful what those other attributes are. The
term ' molar tooth ' bears upon the face of it that it is a
part of the wider term ' tooth/ But if we meet with the
simple term 'tooth' there is nothing to indicate whether it
is an incisor, a canine, or a molar tooth. This doubt,
however, may be resolved by other information, and we
have to consider what are the appropriate logical pro
cesses for treating disjunctive propositions in connection
with other propositions disjunctive or otherwise.

DISJUNCTIVE PROPOSITIONS. 81

Expression of the Alternative Relation.

In order to represent disjunctive propositions with
convenience we require a sign of the alternative or dis
junctive relation, equivalent to one meaning at least of
the little conjunction or so frequently used in common
language. I propose to use for this purpose the sym
bol | . In my first logical Essay I followed the example
of Dr. Boole and adopted the common sign + ; but this sign
should not be employed unless there exists exact analogy
between mathematical addition and logical alternation.
We shall find that the analogy is of a very partial cha
racter, and that there is such profound difference between
a logical and a mathematical term as should prevent our
uniting them by the same symbol. Accordingly I have
chosen a sign -|- , which seems aptly to suggest whatever
degree of analogy may exist without implying more.
The exact meaning of the symbol we will now proceed to
investigate and determine.

Nature of the Alternative Relation.

Before treating disjunctive propositions it is indis
pensable to decide whether the alternatives shall be
considered exclusive or unexclusive. By exclusive alter
natives we mean those which cannot contain the same
things. Thus

Matter is solid, or liquid, or gaseous ;

but the same portion of matter cannot be at once solid and
liquid, properly speaking ; still less can we suppose it to
be solid and gaseous, or solid, liquid and gaseous all at
the same time. Many examples on the other hand can
readily be suggested in which two or more alternatives
may hold true of the same object. Thus

Luminous bodies are self-luminous or luminous by
reflection.

G

82 Tin: ritixciPLES OF SCIENCE.

It is undoubtedly possible by the laws of optics, that the
same surface may at one and the same moment give off
light of its own and reflect the light from other bodies.
We speak familiarly of deaf or dumb persons, knowing
that the majority of those who are deaf from birth are
also dumb.

There can be no doubt that in a great many cases,
perhaps the greater number of cases, alternatives are
exclusive as a matter of fact. Any one number is incom
patible with any other; one point of time or place is
exclusive of all others. Eoger Bacon died either in 1284
or 1292 ; it is certain that he could not die in both years.
Henry Fielding was born either in Dublin or Somerset
shire ; he could not be born in both places. There is so
much more precision and clearness in the use of exclusive
alternatives that we ought doubtless to select them
when possible. Old works on logic accordingly contained
a rule directing that the Membra dividentia, the parts of
a division or the constituent species of a genus should be
exclusive of each other.

It is no doubt owing to the great prevalence and
convenience of exclusive divisions that the majority of
logicians have held it necessary to make every alternative
in a disjunctive proposition exclusive of every other one.
Aquinas considered that when this was not the case the
proposition was actuallyyafoe, and Kant adopted the same
opinion a. A multitude of statements to the same effect
might readily be quoted, and if the question were to be
determined by the weight of historical evidence, it would
.certainly go against my view. Among recent logicians
Sir W. Hamilton, as well as Dr. Boole, took the exclusive
side. But there are authorities to the opposite effect.
Whately, Mansel, and J. S. Mill, have all pointed out that

a Hansel's ' Aldrich,' p. 103, and 'Prolegomena Loglea,' p. 221.

DISJUNCTIVE PROPOSITIONS. 83

we may often treat alternatives as Compossible, or true at
the same time. Whately gives as an example b, 'Virtue
tends to procure us either the esteem of mankind, or the
favour of God/ and he adds, ' Here both members are
true, and consequently from one being affirmed we are not
authorized to deny the other. Of course we are left to
conjecture in each case, from the context, whether it is
meant to be implied that the members are or are not
exclusive.' Mansel says c, ' We may happen to know that
two alternatives cannot be true together, so that the
affirmation of the second necessitates the denial of the first ;
but this, as Boethius observes, is a material, not & formal
consequence/ Mr. J. S. Mill has also pointed out the
absurdities which would arise from always interpreting
alternatives as exclusive. ' If we assert/ he says d , ' that
a man who has acted in some particular way must be
either a knave or a fool, we by no means assert, or intend
to assert, that he cannot be both.' Again, 'to make an
entirely unselfish use of despotic power, a man must be
either a saint or a philosopher Does the dis
junctive premise necessarily imply, or must it be construed
as supposing, that the same person cannot be both a
saint and a philosopher ? Such a construction would be
ridiculous.'

I discuss this subject fully because it is really the point
which separates my logical system from that of the late
Dr. Boole. In his 'Laws of Thought' (p. 32) he expressly
says, 'In strictness, the words "and/' "or," interposed
between the terms descriptive of two or more classes of
objects, imply that those classes are quite distinct, so that
no member of one is found in another/ This I altogether

b ' Elements of Logic,' Book II. chap. iv. sect. 4.
c Aldrich, 'Artis Logicse Rudimenta,' p. 104.
a 'Examination of Sir W. Hamilton's Philosophy,' pp. 452-454-

G 2

84 TEE PRINCIPLES OF SCIENCE.

dispute. In the ordinary use of these conjunctions we do
not necessarily join distinct terms only ; and when terms
so joined do prove to be logically distinct, it is by virtue
of a tacit premise, something in the meaning of the names
and our knowledge of them, which teaches us they are
distinct. And when our knowledge of the meanings of the
words joined is defective it will often be impossible to
decide whether terms joined by conjunctions are exclusive
or not.

Take, for instance, the proposition 'A peer is either
a duke, or a marquis, or an earl, or a viscount, or a baron/
If expressed in Professor Boole's symbols, it would be
implied that a "peer cannot be at once a duke and marquis,
or marquis and earl. Yet many peers do possess two or
more titles, and the Prince of Wales is Duke of Cornwall,
Earl of Dublin, and Baron Renfrew. If it were enacted by
parliament that no peer should have more than one title,
this would be the tacit premise which Professor Boole
assumes to exist. Nor is the restriction true of more
common terms.

In the sentence 'Repentance is not a single act, but a
habit or virtue/ it cannot be implied that a virtue is not a
habit ; by Aristotle's definition it is.

Milton has the expression in one of his sonnets,
1 Unstain'd by gold or fee/ where it is obvious that if
the fee is not always gold, the gold is meant to be a fee
or bribe.

Tennyson has the expression 'wreath or anadem/ Most
readers would be quite uncertain whether a wreath may
be an anadem, or an anadem a wreath, or whether they
are quite distinct or quite the same.

From Darwin's * Origin/ I take the expression, ' When
we see any part or organ developed in a remarkable
degree or manner.' In this, or is used twice, and neither
time disjunctively. For if part and organ are not

DISJUNCTIVE PROPOSITIONS. 85

synonymous, at any rate an organ is a part. And it is
obvious that a part may be developed at the same time
both in an extraordinary degree and manner, although
such cases may be comparatively very rare.

From a careful examination of ordinary writings, it
will thus be found that the meanings of terms joined by
' and ' * or ' vary from absolute identity up to absolute
contrariety. There is no logical condition of distinctness
at all, and when we do choose exclusive expressions, it is
because our subject demands it. The matter, not the form
of an expression, points out whether terms are exclusive e.

The question, as we shall afterwards see, is one of the
greatest theoretical importance, because it furnishes the
true distinction between the sciences of Logic and Ma
thematics. It is the very foundation of number that every
unit shall be distinct from every other unit ; but Dr. Boole
imported the conditions of number into the science of
Logic, and produced a system which, though wonderful in
its results, was not a system of logic at all.

Laws of the Disjunctive Relation.

In considering the combination or synthesis of terms
(p- 39)' we fcavnd that certain laws, those of Simplicity and
Commutativeness, must be observed. In uniting terms by
the disjunctive symbol we shall find that the same or
closely similar laws hold true. The alternatives of either
member of a disjunctive proposition are certainly commu
tative. Just as we cannot properly distinguish between
rich and rare gems and rare and rich gems, so we must
consider as identical the expression rich or rare gems, and
rare or rich gems. In our symbolic language we may say

generally

A ! B = B | A.

e ' Pure Logic,.' pp. 76, 77.

86

THE PRINCIPLES OF SCIENCE.

The order of statement, in short, has no effect upon the
meaning of an aggregate of alternatives, so that the Law
of Commutativeness holds true of the disjunctive symbol.

As we have admitted the possibility of joining as alter
natives terms which are not really different, the ques
tion arises, How shall Ave treat two or more alternatives
when they are clearly shown to be the samel If we
have it asserted that P is Q or R, and it is afterwards
proved that Q is but another name for R, the result is
that P is either R or R. How shall we interpret such a
statement \ What would be the meaning, for instance, of
' wreath or anadem ' if, on referring to a dictionary, we
found anadem described as a wreath ? I take it to be
self-evident that the meaning would then become simply
' wreath.' Accordingly we may affirm the general law

A .|. A ~ A.

Any number of identical alternatives may always be
reduced to, and are logically equivalent to, any one of
those alternatives. This is a law which distinguishes
mathematical terms from logical terms, because it ob
viously does not apply to the former. I propose to call
it the Law of Unity, because it must really be involved
in any definition of a mathematical unit. This law is
closely analogous to the Law of Simplicity, A A = A ; and
the nature of the connection is worthy of attention.

I am not aware that logicians have in any adequate way
noticed the close relation between combined and dis
junctive terms, namely that every disjunctive term is the
negative of a corresponding combined term, and vice versd.
Consider the term

Malleable dense metal.

How shall we describe the class of things which are not

o

malleable-dense-metals ? Whatever is included under that
term must have all the qualities of malleability, denseness,
and metallic nature. Wherever any one or more of the

DISJUNCTIVE PROPOSITIONS, 87

qualities is wanting, the combined term will not apply.
Hence the negative of the whole term is

Not-malleable or not-dense or not-metallic.
In the above the conjunction or must clearly be inter
preted as unexckuave ; for there may readily be objects
which are both not-malleable, and not-dense, and perhaps
not-metallic at the same time. If in fact we were required
to use or in a strictly exclusive manner, it would be
requisite to specify seven distinct alternatives in order to
describe the negative of a combination of three single
terms. The negatives of four or five terms would consist

o

of fifteen or thirty-one alternatives. This consideration
alone is sufficient to prove that the meaning of or can
not 'be always exclusive in common language.

Expressed symbolically, we may say that the negative

of

ABC

is not- A or not-B or not-C ;

that is, a { b •{• c.

Eeciprocally the negative of

P I Q I R

is pqr.

Every disjunctive term, then, is the negative of a
combined term, and vice versa.

Apply this result to the combined term A A A, and its
negative is

a •[ a •[ a.

Now since AAA is by the Law of Simplicity equivalent to
A, so a | a -|- a must be by the Law of Unity equivalent
to a. Each law thus necessarily presupposes the other.

Symbolic expression of the Law of Duality.

We may now employ our symbol of alternation to express
in a clear and formal manner the third Fundamental Law

88 THE PRINCIPLES OF SCIENCE.

of Thought, which I have called the Law of Duality.
Taking A to represent any class or object or quality, and
B any other class, object or quality, we may always
assert that A either agrees with B, or does not agree.

Thus we may say

A - AB -I- A6.

This is a formula which will henceforth be constantly
employed, and it lies at the basis of reasoning.

The reader may perhaps wish to know why A is inserted
in both alternatives of the second member of the identity,
and why the law is not stated in the form

A=B + 2>.

But if he will consider the contents of the last section
(p. 87), he will see that the latter expression cannot be
correct, otherwise no term would have any negative.
For the negative of B -|- b is &B, or a self-contradictory
term ; so that if A were identical with B -|- b, its nega
tive a would be non-existent. This result would generally
be an absurd one, and I see much reason to think that in
a strictly logical point of view it would always be absurd.
In all probability we ought to assume as a fundamental
logical axiom that every term has its negative in thought.
We cannot think at all without separating what we think
about from other different things, and these things neces
sarily form the negative notion f. If so, it follows that
any term of the form B f b is just as self-contradictory
as one of the form B6.

It will be convenient to recapitulate in this place the
three great Laws of Thought in their symbolic form, thus

Law of Identity A = A.

Law of Contradiction Aa = o.

Law of Duality A = AB | A.b.

'Pure Logic,' p. 65. See also the criticism of this point by De
Morgan in the ' Athenseum,' No. 1892, soth January, 1864 ; p. 155.

DISJUNCTIVE PROPOSITIONS. 89

Various Forms of the Disjunctive Proposition.

Disjunctive propositions may occur in a great variety of
forms, of which the old logicians took very insufficient

O I/

notice. There may be any number of alternatives each of
which may be a combination of any number of simple
terms. A proposition, again, may be disjunctive in one
or both members. The proposition

Solids or liquids or gases are electrics or conductors of

electricity

is an example of the doubly disjunctive form. The mean
ing of any such proposition is that whatever falls under
any one or more alternatives on one side must fall under
one or more alternatives on the other side. From what
has been said before, it is apparent that the proposition

A | B = C | D
will correspond to

ab = cd,

each member of the latter being the negative of a
member of the former proposition.

As an instance of a complex disjunctive proposition
I may give Senior's definition of wealth, namely ' Wealth
is what is transferable, limited in supply, and either
productive of pleasure or preventive of pain &'.'
Let A = wealth

B = transferable
C = limited in supply
D = productive of pleasure
E = preventive of pain.
The definition takes the form

A = BC(D* E);

but if we develop the alternatives by a method to be
afterwards more fully considered, it becomes
A = BCDE | BCDe I- BCdE.

s Boole's 'Laws of Thought,' p. 106. Jevoiis' 'Pure Logic,' p. 69.

90 THE PRINCIPLES OF SCIENCE.

An example of a still more complex proposition may
be found in De Morgan's writings11, and is as follows :—
' He must have been rich, and if not absolutely mad was
weakness itself, subjected either to bad advice or to most
unfavourable circumstances/

If we assign the letters of the alphabet in succession,

thus,

A - he

B = rich

C = absolutely mad

D = weakness itself

E = subjected to bad advice

F = subjected to most unfavourable circumstances,
the proposition will take the form

and if we develop the alternatives, expressing some of
the different cases which may happen, we obtain
A- ABC | ABcDEF |- ABcDE/ |- ABcDeF.

Inference by Disjunctive Propositions.

Before we can make a free use of disjunctive propositions
in the processes of inference \ve must consider how dis
junctive terms can be combined together or with simple
terms. In the first place, to combine a simple term with
a disjunctive one, we must combine it with every alter
native of the disjunctive term. A vegetable, for instance,
is either a herb, a shrub, or a tree. Hence an exogenous
vegetable is either an exogenous herb, or an exogenous
shrub, or an exogenous tree. Symbolically stated this
process of combination is as follows —

A(B-| C) = AB-| AC.

Secondly, to combine two disjunctive terms with each
other, combine each alternative of one separately with each

h 'On the Syllogism,' No. iii. p. 12. Camb. Phil. Trans., vol. x.
part i.

DISJUNCTIVE PROPOSITIONS. 91

alternative of the other. Since flowering plants are
either exogens or endogens, and are at the same time either
herbs, shrubs or trees, it follows that there are altogether
six alternatives — namely, exogenous herbs, exogenous
shrubs, exogenous trees, endogenous herbs, endogenous
shrubs, endogenous trees. This process of combination is
shown in the general form

(A I B) (C -|- D) - AC | AD + BC \ BD.
It is hardly necessary to point out that, however numerous
the terms combined, or the alternatives in those terms, we
may effect the combination provided each alternative is
combined with each alternative of the other terms, as in
the algebraic process of multiplication.

Some processes of deduction may at once be exhibited.
We may always, for instance, unite the same qualifying
term to each side of an identity even though one or both
members of the identity be disjunctive. Thus let

A = B | - C.
Now it is self-evident that

AD -AD,

and in one side of this identity we may for A substitute
its equivalent B -|- C obtaining

AD = BD | CD.

Since ' a gaseous element is either hydrogen, or oxygen,
or nitrogen, or chlorine, or fluorine/ it follows that ' a free
gaseous element is either free hydrogen, or free oxygen,
or free nitrogen, or free chlorine, or free fluorine.'

This process of combination will lead to most useful
inferences when the qualifying adjective combined with
both sides of the proposition is a negative of one or more
alternatives. Since chlorine is a coloured gas, we may
infer that ' a colourless gaseous element is either (colour
less) hydrogen, oxygen, nitrogen, or fluorine.' The alter
native chlorine disappears because colourless chlorine does
not exist. Again, since 'a tooth is either an incisor,

92 THE PRINCIPLES OF SCIENCE.

canine, bicuspid, or molar,' it follows that 'a not-incisor
tooth is either canine, bicuspid, or molar.' The general
rule is that from the denial of any of the alternatives the
affirmation of the remainder can be inferred. Now this
result clearly follows from our process of substitution ; for
if we have the proposition

A = B |C|D,

and insert this expression for A on one side of the self-
evident identitv

A6 = A6,

we obtain A.b = AB6 -I- A6C I AbD ;.

and, as the first of the three alternatives is self-contra
dictory, we strike it out according to the law of contra
diction : there remains

Ab = AbC -I- A6D.

Thus our system fully includes and explains that mood of
the Disjunctive Syllogism technically called the modus
tollendo ponens.

But the reader must carefully observe that the Dis
junctive Syllogism of the mood ponendo tollens, which af
firms one alternative, and thence infers the denial of the rest,
cannot be held true in this system. If I say, indeed, that

Water is either salt or fresh water,

it seems evident that ' water which is salt is not fresh/
But this inference really proceeds from our knowledge
that water cannot be at once salt and fresh. This incon
sistency of the alternatives, as I have fully shown, will
not always hold. Thus, if I say

Gems are either rare stones or beautiful stones, (i)
it will obviously not follow that

A rare gem is not a beautiful stone, (2)

nor that

A beautiful gem is not a rare stone. (3)

Our symbolic method gives only true conclusions ; for if
we take

DISJUNCTIVE PROPOSITIONS. 93

A = gem
B = rare stone
C = beautiful stone,
the proposition ( i ) is of the form

A - B | C

hence AB = B -|- BC

and AC - BC I C ;

but these inferences are not equivalent to the false ones
(2) and (3).

We can readily represent such disjunctive reasoning, when
it is valid, by expressing the inconsistency of the alterna
tives explicitly. Thus if we resort to our instance of

Water is either salt or fresh,
and take A = Water

B .-= salt
C = fresh,
then the premise is apparently of the form

A = AB I AC ;

but in reality there are the unexpressed conditions that
' what is salt is not fresh/ and ' what is fresh is not salt ; '

or, in letter-terms,

B = Be

C = 60.

Now, if we substitute these descriptions in the original
proposition, we obtain

A = ABc |A6C;
uniting B to each side we infer

AB = ABc | AB6C
or AB = ABc ;

that is,

Water which is salt is water salt and not fresh.

I should weary the reader if I attempted to illustrate
the multitude of forms which disjunctive reasoning may
take ; and as in the next chapter we shall be constantly
treating the subject, I must here restrict myself to a single

94 THE PRINCIPLES OF SCIENCE.

instance. A very common process of reasoning consists in
the determination of the name of a thing by the successive
exclusion of alternatives, a process called by the old name
abscissio infiniti. Take the case : —

Red-coloured metal is either copper or gold ( i )
Copper is dissolved by nitric acid (2)

This specimen is red-coloured metal (3)

This specimen is not dissolved by nitric acid (4)
Therefore this specimen consists of gold. (5)

Assigning our letter-symbols thus —
A = this specimen
B = red-coloured metal
C = copper
D = gold

E = dissolved by nitric acid,
the premises may be stated in the form

B = BCcM-BcD (i)

C = CE (2)

A = AB (3)

A = Ae. (4)

Substituting for C in (i) by means of (2) we get

B = BCdE ! BcD.
From (3) and (4) we may infer likewise

A = ABe,

and if in this we substitute for B its equivalent just
stated, it follows that

A = ABCcZEe -I ABcDe.

The first of the alternatives being contradictory, the result
is A = ABcDe

which contains a full description of ' this specimen/ as
furnished in the premises, but by ellipsis indicates that
it is gold. It will be observed that in the symbolic
expression (i) I have explicitly stated what is certainly
implied, that copper is not gold, and gold not copper,
without which condition the inference would not hold good.

CHAPTEK VI.

THE INDIRECT METHOD OF INFERENCE.

THE forms of deductive reasoning as yet considered, are
mostly cases of Direct Deduction as distinguished from
those which we are now about to treat. The method of
Indirect Deduction may be described as that which points
out what a thing is, by showing that it cannot be anything
else. We can define a certain space upon a map, either by
colouring that space, or by colouring all except the space ;
the first mode is positive, the second negative. The dif
ference, it will be readily seen, is exactly analogous to that
between the direct and indirect proof in geometry. Euclid
often shows that two lines are equal, by showing that they
cannot be unequal, and the proof rests upon the known num
ber of alternatives, greater, equal or less, which are alone
conceivable. In other cases, as for instance in the seventh
proposition of the first book, he shows that two lines must
meet in a particular point, by showing that they cannot
meet elsewhere.

In logic we can always define with certainty the utmost
number of alternatives which are conceivable. The Law
of Duality (pp. 6, 88) enables us always to assert that any
quality or circumstance whatsoever is either present or
absent in anything. Whatever may be the meaning and
nature of the terms A and B it is certainly true that

A = AB -i- Kb
B = AB I aB.

These are universal though tacit premises which may
be employed in the solution of every problem, and which

96 THE PRINCIPLES OF SCIENCE.

are such invariable- and necessary conditions of all thought,
that they need not be specially laid down. The Law of
Contradiction is a further condition of all thought and of
all logical symbols ; it enables, and in fact obliges, us to
reject from further consideration all terms which imply
the presence and absence of the same quality. Now,
whenever we bring both these Laws of Thought into ex
plicit action by the method of substitution, we employ the
Indirect Method of Inference. It will be found that we
can treat not only those arguments already exhibited
according to the direct method, but we can also include an
infinite multitude of other arguments which are incapable
of solution by any other means.

Some philosophers, especially those of France, have
held that the Indirect Method of Proof has a certain infe
riority to a direct method, which should prevent our using
it except when obliged. But there are an unlimited
number of truths which we can prove only indirectly.
We can prove that a number is a prime only by the
purely indirect method of showing that it is not any of the
numbers which have divisors, and the remarkable process
known as Eratosthenes' Sieve is the only mode by which
we can select the prime numbers a. It bears a strong
analogy to the indirect method here to be described. We
can also prove that the side and diameter of a square are
incommensurable, but only in the negative or indirect
manner, by showing that the contrary supposition con
stantly and inevitably leads to contradiction b. Many other
demonstrations in various branches of the mathematical
sciences rest upon a like method. Now if there is only
one important truth which must be, and can only be

"See Horsley, ' Philosophical Transactions,' 1772; vol. Ixii. p. 327.
Hontucla, ' Histoire des Mathematiques,' vol. i. p. 239. ' Penny
Cyclopaedia, ' article Eratosthenes.

b Euclid, Book x. Prop. 117.

THE INDIRECT METHOD OF INFERENCE. 97

proved indirectly, we may say that the process is a
necessary and sufficient one, and the question of its com
parative excellence or usefulness is not worth discussion.
As a matter of fact I believe that nearly half our logical
conclusions rest upon its employment.

Simple Illustrations.

In tracing out the powers and results of this method, we
will begin with the simplest possible instance. Let us take
a proposition of the very common form, A = AB, say,

A Metal is an Element,

and let us investigate its full meaning. Any person who
has had the least logical training, is aware that we can
draw from the above proposition an apparently different
one, namely,

A Not-element is a Not-metal.

While some logicians, as for instance De Morgan,6 have
considered the relation of these two propositions to be
purely self-evident, and neither needing nor allowing
analysis, a great many more persons, as I have observed
while teaching logic, are at first unable to perceive the
close connection between them. I believe that a true and
complete system of logic will furnish a clear analysis of
this process which has been called Contrapositive Con
version ; the full process is as follows : —

Firstly, by the Law of Duality we know that

Not-element is either Metal or Not-metal.
Now if it be metal, we know that it is by the premise
an element ; we should thus be supposing that the very
same thing is an element and a not-element, which is
in opposition to the Law of Contradiction. According to
the only other alternative, then, the not-element must be
a not-metal.

c 'Philosophical Magazine,' December 1852, Fourth Series, vol. iv.
p. 435, 'On Indirect Demonstration,'

H

98 THE PRINCIPLES OF SCIENCE.

To represent this process of inference symbolically we
take the premise in the form

A - AB. (i)

We observe that by the Law of Duality the term not-B is

thus described

I = A6 -I- ab. (2)

For A in this proposition we substitute its description
as given in (i), obtaining

1} - AB6 -I- ab.

But according to the Law of Contradiction the term
AB6 must be excluded from thought or

AB6 = o.

Hence it results that 6 is either nothing at all, or it is
ab ; and the conclusion is

b = ab.

As it will often be necessary to refer to a conclusion
of this kind I shall call it, as is usual, the Contrapositive
Proposition of the original. The reader need hardly be
cautioned to observe that from all A's are B's it does not
follow that all not-A's are not-B's. For by the Law
of Duality we have

a = aB | ab,

and it will not be found possible to make any substitu
tion in this by our original premise A = AB. It still
remains doubtful, therefore, whether not-metal is element
or not-element.

The proof of the Contrapositive Proposition given above
is exactly the same as that which Euclid applies in the
case of geometrical notions. De Morgan describes Euclid's
process as follows d : — ' From every not-B is not-A he pro
duces every A is B, thus — If it be possible, let this A be
not-B, but every not-B is not-A, therefore this A is not-A,
which is absurd : whence every A is B.' Now De Morgan
thinks that this proof is entirely needless, because common

d 'Philosophical Magazine,' Dec. 1852 ; p. 437.

THE INDIRECT METHOD OF INFERENCE. 99

logic gives the inference without the use of any geo
metrical reasoning. I conceive however that logic gives
the inference only by an indirect process. De Morgan
claims 'to see identity in every A is B and every not-B
is not-A, by a process of thought prior to syllogism/
But whether prior to syllogism or not, I claim that it
is not prior to the laws of thought and the process of
substitutive inference by which it may be undoubtedly
demonstrated.

Employment of the Contrapositive Proposition.

We can frequently employ the contrapositive form of
a proposition by the method of substitution ; and certain
moods of the ancient syllogism, which we have hitherto
passed over, may thus be satisfactorily comprehended
in our system. Take for instance the following syllogism
in the mood Camestres :—

' Whales are not true fish : for they do not respire
water, whereas true fish do respire water.'

Let us take

A = whales,
B = true fish,
C = respiring water.
The premises are of the form

A = Ac, (i)

B = BC. (2)

Now, by the process of contraposition we obtain from (2)

c = be,

and we can substitute this expression for c in (i), ob
taining

A = A&c,

or ' Whales are not true fish, not respiring water/

The mood Cesare does not really differ from Camestres
except in the order of the premises, and it could be
exhibited in an exactly similar manner.

H 2

100 THE PRINCIPLES OF SCIENCE.

The mood Baroko gave much trouble to the old lo
gicians who could not reduce it to the first figure in
the same manner as the other moods, and were obliged
to invent, specially for it and for Bokardo, a method of
Indirect Eeduction closely analogous to the Indirect proof
of Euclid. Now these moods require no exceptional
processes, in this system. Let us take as an instance of
Baroko, the argument

All heated solids give continuous spectra, (i)

Some nebulaB do not give continuous spectra ; (2)
Therefore some nebulae are not heated solids. (3)
Treating the little word some as an indeterminate
adjective of selection, to which we assign a symbol like
any other adjective, let

A = some
B = nebulas

C = giving continuous spectra
D = heated solid,
The premises then become

D = DC (i)

AB = ABc. (2)

Now from (i) we obtain by the Indirect method the
Contrapositive

c = cd,
and if we substitute this expression for c in (2) we have

AB = ABcrf;

the full meaning of which is that ' some nebulae do not
give continuous spectra and are not solids/

We might similarly apply the contrapositive in many
other instances. Take the argument — ' All fixed stars
are self-luminous ; but some of the heavenly bodies are
not self-luminous, an,d are therefore not fixed stars.'
Taking our terms

A = fixed stars
B = self-luminous

THE INDIRECT METHOD OF INFERENCE. 101

C = some

D = heavenly bodies,
we have the premises

A - AB, (i)

CD = bCV. (2)

Now from (i) we can draw the Contrapositive

b = ab,
and substituting this expression for b in (2) we obtain

CD = a&CD,

which expresses the conclusion of the argument that
* some heavenly bodies are not fixed stars.'

Contrapositive of a Simple Identity.

The reader should carefully note that when we apply
the process of Indirect Inference to a simple identity
of the form

A = B,

we may obtain further results. If we wish to know
what is the term not-B, we have as before, by the Law of
Duality,

b = A6 -I- ab,
and substituting for A we obtain

b = B6 | ab = ab.
But we may now also draw a second Contrapositive ; for

we have

a = aB -|- ab,
and substituting for B its equivalent A we have

a = aA. •[ ab = ab.

Hence from the single identity A = B we can draw
the two propositions

a = ab
b = ab,

and observing that these propositions have a common
term we can make a new substitution, getting

a = b.

102 THE PRINCIPLES OF SCIENCE.

This result is in strict accordance with the fundamental
principles of inference, and it may be a question whether
it is not a self-evident result, independent of the steps of
deduction by which we have reached it. For where two
classes are coincident like A and B, whatever is true
of the one is true of the other ; what is excluded from
the one must be excluded from the other similarly.
Now as a bears to A exactly the same relation that b
bears to B, the identity of either pair follows from the
identity of the other pair. In every identity, equality,
or similarity, we may argue from the negative of the
one side to the negative of the other. Thus at ordinary
temperatures

Mercury = liquid-metal,
hence obviously

Not-mercury = not-liquid-metal ;
or since

Sirius = brightest fixed star,

it follows that whatever star is not the brightest is not
Sirius, and vice versd. Every correct definition is of the
form A = B, and may often require to be applied in the
equivalent negative form.

Let us take as an illustration of the mode of using this
result the argument following : —

Vowels are letters which can be sounded alone, (i)
The letter w cannot be sounded alone ; (2)

Therefore the letter w is not a vowel. (t\

Here we have a definition (i), and a comparison of a
thing with that definition (2), leading to exclusion of the
thing from the class defined.
Taking the terms
A = vowel,

3 = letter which can be sounded alone,
C = letter w,

THE INDIRECT METHOD OF INFERENCE. 103

the premises are plainly of the form

A = B, (i)

C = 60. (2)

Now by the Indirect method we obtain from (i) the
Contrapositive

b = a,
and inserting in (2) the equivalent for b we have

C = aO, (3)

or ' the letter w is not a vowel/

Miscellaneous Examples of the Method.

We can apply the Indirect Method of Inference how
ever many may be the terms involved or the premises
containing those terms. As the working of the method
is best learnt from examples, I will take a case of two
premises forming the syllogism Barbara : thus

Iron is a metal (i)

Metal is element. (2)

If we want to ascertain what inference is possible con
cerning the term Iron, we develop the term by the Law
of Duality. Iron must be either metal or not-metal ; iron
which is metal must be either element or not-elemerit ;
and similarly iron which is not-metal must be either
element or not-element. There are then altogether four
alternatives among which the description of iron must be
contained ; thus

Iron, metal, element, (a)

Iron, metal, not-element, (/3)

Iron, not-metal, element, (7)

Iron, not-metal, not-element. (§)

Our first premise informs us that iron is a metal, and if
we substitute this description in (7) and (3) we shall have
self -contradictory combinations. Our second premise

104 THE PRINCIPLES OF SCIENCE.

likewise informs us that metal is element, and applying
this description to (/3) we again have self-contradiction,
so that there remains only (a) as a description of iron—
our inference is

Iron = iron, metal, element.
To represent this process of reasoning in general

symbols, let

A = iron

B = metal
C = element.

The premises of the problem take the form

A = AB (i)

B - BC. (2)

By the Law of Duality we have

A = AB -I- A6 (3)

A = AC -I- Ac. (4)

Now, if we insert for A in the second side of (3) its

description in (4), we obtain what I shall call the

development of A

A = ABO -|- ABc -I- A6C I Abe. (5)

Wherever the letters A or B appear in the second side

of (5) substitute their equivalents given in (i) and (2) and

the results at full length are

A = ABC | ABCc ! AB6C | AB&Cc.

The last three alternatives break the Law of Contradic
tion, so that

A = ABC -I- o I o -I- o

A = ABC.

This conclusion is, indeed, no more than we could obtain by
the direct process of substitution ; it is the characteristic
of the Indirect process that it gives all possible logical
conclusions, both those which we have previously obtained,
and an almost infinite number of others of which the
ancient logic took little or no account. From the same
premises, for instance, we can obtain a description of the

THE INDIRECT METHOD OF INFERENCE. 105

class not-element or c. By the Law of Duality we can
develop c into four alternatives, thus —

c = ABc -I- A6c | aBc -|- abc.
Now if we substitute for A and B as before, we get

c = ABO I AB&c -I- aBCc -|- abc,

and striking out the terms which break the Law of Contra
diction there remains

c = abc,

or what is not element is also not iron and not metal.
This Indirect Method of Inference thus furnishes a
complete solution of the following problem — Given any
number of logical premises or conditions, required the
description of any class of objects, or any term> as governed
by those conditions.

The steps of the process of inference may thus be
concisely stated : —

1. By the Law of Duality develop the utmost number
of alternatives which may exist in the description of the
required class or term as regards the terms involved in
the premises.

2. For each term in these alternatives substitute its
description as given in the premises.

3. Strike out every alternative which is then found to
break the Law of Contradiction,

4. The remaining terms may be equated to the term in
question as the desired description or inference.

Abbreviation of the Process.

Before proceeding to illustrations of the use of this
method, I must point out how much its practical em
ployment can be simplified, and. how much more easy it
is than would appear from the description. When we
want to effect at all a complete solution of a logical

106 THE PRINCIPLES OF SCIENCE.

problem it is best to form, in the first place, a complete
series of all the combinations of terms involved in it.
If there be two terms A and B, the utmost variety of
combinations in which they can appear are

AB

Ab

aB

ab.

The term A appears in the first and second ; B in the
first and third ; a in the third and fourth ; and b in the
second and fourth. Now if we have any premise, say

A = B,

we must ascertain which of these combinations would be
rendered self-contradictory by substitution ; the second
and third would have to be struck out, and there would

remain

AB

ab.
Hence we draw the following inferences

A = AB, B = AB, a - ab, b = ab.
Exactly the same method must be followed where a
question involves a greater number of terms. Thus by
the Law of Duality the three terms A, B, C, give rise to
eight conceivable combinations, namely

ABC . («)

ABc (0)

AbC (7)

Abe (S)

aBC (e)

aBc (0

abC W

abc. (0)

The development of the term A is formed by the first four
of these ; for B we must select (a), (/3), (e), (£) ; C consists
of («), (?), W M ; b of (7), (S), (,), (<9), and so on.

THE INDIRECT METHOD OF INFERENCE. 107

Now if we want to investigate completely the meaning
of the premises

A - AB (i)

B = EC, , (2)

we examine each of the eight combinations as regards
each premise; (7) and ($) are contradicted by (i), and (/3)
and (£) by (2), so that there remain only

ABC (a)

aBC (e)

abC W

ale. (0)

To describe any term under the conditions of the premises
(i) and (2), we have only to draw out the proper com
binations from this list ; thus — A is represented only by

ABC or

A = ABC,

similarly c — ale.

For B we have two alternatives thus stated,

B = ABC I aBC ;
and for I we have

I — abC •[ ale.

When we have a problem involving four distinct terms
we need to double the number of combinations, and as
we add each new term the combinations become twice as
numerous. Thus

A, B produce four combinations

A, B, C, „ eight

A, B, C, D „ sixteen „

A, B, C, D, E „ thirty-two „

A, B, C, D, E, F „ sixty-four „

and so on.

I propose to call any such series of combinations the
Logical Alecedarium. It holds in logical science a posi
tion of importance which cannot be exaggerated. As we
proceed from logical to mathematical considerations it will

108 THE PRINCIPLES OF SCIENCE.

become apparent that there is a close connection between
these combinations and the most fundamental theorems of
mathematical science. For the convenience of the reader
who may wish to employ the abecedarium in logical
questions, I have had printed on the next page a complete
series of the combinations up to those of six terms. At
the very commencement in the first column is placed a
single letter X which might seem to be superfluous. This
letter serves to denote that it is always some higher class
which is divided up. Thus the combination AB really
means ABX, or that part of some larger class, say X,
which has the qualities of A and B present. The letter
X is omitted in the greater part of the table merely for
the sake of brevity and clearness. In a later chapter on
Combinations it will become apparent that the intro
duction of this unit class is requisite in order to com
plete the analogy with the Arithmetical Triangle there
described.

The reader ought to bear in mind that though the
abecedarium seems to give mere lists of combinations,
these combinations are intended in every case to con
stitute the development of a term of a proposition.
Thus the four combinations AB, Afr, aB, ab really mean
that any class X is described by the following proposition,

X = X (AB -I- A6 ! aB \ ab).
If we select the A's, we obtain the following proposition

AX = X(AB I A6).

Thus whatever group of combinations we treat must be
conceived as part of a higher class, summum genus or
universe symbolised in the term X ; but bearing this in
mind, it is needless to complicate our formulas by always
introducing the letter. All inference consists in passing
from propositions to propositions, and combinations per se
have no meaning. They are consequently to be regarded
in all cases as forming parts of propositions.

THE LOGICAL ABECEDAEIUM.

i. ii. in, ly. y.

VI.

X AX AB ABC ABCD

ABODE

a X Ab A Be ABCd

ABCDe

a B Ab C ABcD

A B C d E

a b Abe A B c d

A B C d e

aB C AbC D

AB cDE

a B c A b C d

A B c D e

a b C A & c D

AB c d E

a b c Abed

A B c c? e

oBC D

A6CDE

aB C d

Ab C D e

a B c D

A bC d E

a B c d

A 6 C d e

ab G D

Ab cD E

a b C d

A b c D e

a b c D

Ab c d E

a b c d

A b c d e

aBC D E

aB C D e

aBC d E

a B C d e

a B c D E

a B c D e

a B c d E

a B c d e

a b C DE

a b C D e

0 b C d E

a b C d e

a b c D E

a b c D e

a b c d E

a b c d e

ABCDEF
ABCDE/
ABCD eF
ABCDe/
ABCdEF
AB CdE/
A BCdeF
ABC d e f
ABcDE F
ABc DE/
A BcD eF
ABcD e f
ABcdE F
A B cdE/
ABcd e F
A B c d e/
AfcCDEF
A&CDE/
AfeCDeF
A&CD e f
A b G d E F
A&Crf E/
A bC de F
A b C d e f
A&cDEF
A&cD E/
A&c D e F
A b c D e/
A&cdE F
Abe d E /
Ab c d e F
A b c d e f
aBCDEF
oBCDE/
a B C D e F
aB C D e f
a B C d E F
aBCd E/
a B C d e F
aB C d e f
aB cD E F
aBcD E /
aB c D e F
a B c D e f
aB cd E F
aB cd E /
a B c d e F
a B c d e f
a&CDE F
afeC D E /
a&C De F
ab C D e f
a bC dE F
ab C d E/
ab C d e F
a b C d e f
ab c D E F
ab c D E /
a & c D e F
a b c D e f
a b c d E F
a b c d E f
a b c d e F
a b c d e f

110 THE PRINCIPLES OF SCIENCE.

In a theoretical point of view we may conceive that the
abecedarium is always extended indefinitely. Every new
quality or circumstance which can belong to an object,
subdivides each combination or class, so that the number
of such combinations when unrestricted by logical con
ditions is represented by an indefinitely high power of
two. The extremely rapid increase in the number of sub
divisions obliges us to confine our attention to a few
circumstances at a time.

When contemplating the properties of this abecedarium,
I am often inclined to think that Pythagoras perceived
the deep logical importance of duality ; for while unity
was the symbol of identity and harmony, he described the
number two as the origin of contrasts, or the symbol of
diversity, division and separation. The number four or
the Tetractys was also regarded by him as one of the chief
elements of existence, for it represented the generating
virtue whence come all combinations.

In one of the golden verses ascribed to Pythagoras, he
conjures his pupil to be virtuous6 :

'By him who stampt The Four upon the Mind,
Tlie Four, the fount of Nature's endless stream.'

Now four and the higher powers of duality do represent
in this logical system the variety of combinations which
can be generated in the absence of logical restrictions. The
followers of Pythagoras may have shrouded their master's
doctrines in mysterious and superstitious notions, but in
many points these doctrines seem to have some basis in
logical philosophy.

The Logical Slate.

To a person who has once comprehended the extreme
significance and utility of the Logical Abecedarium, the

e Whewell, 'History of the Inductive Sciences,' vol. i. p. 222.

THE INDIRECT METHOD OF INFERENCE. Ill

indirect process of inference becomes reduced to the repe
tition of a few uniform operations of classification, selection,
and elimination of contradictories. Logical deduction even
in the most complicated questions becomes a matter of
mere routine,, and the amount of labour required is the
only impediment when once the meaning of the premises
is rendered clear. But the amount of labour is often
found to be considerable. The mere writing down of
sixty-four combinations of six letters each is no small
task, and, if we had a problem of five premises, each of
the sixty-four combinations would have to be examined
in connection with each premise. The requisite com
parison is often of a very tedious character and consider
able chance of errors thus arises.

I have given much attention therefore to reducing both
the manual and mental labour of the process, and I shall
describe several devices which may be adapted for saving
trouble and risk of mistake.

In the first place, as the same sets of combinations
occur over and over again in different problems, we may
avoid the labour of writing them out by having the sets
of letters ready printed upon small sheets of writing paper.
It has also been suggested by a correspondent that, if any
one series of combinations were marked upon the margin
of a sheet of paper, and a slit cut between each pair of
combinations, it would be easy to fold down any particular
combination, and thus strike it out of view. The combi
nations consistent with the premises would then remain
in a broken series. This method answers sufficiently well
for occasional use.

A more convenient mode, however, is to have the series
of letters shown on p. 109, engraved upon a common
school writing slate, of such a size, that the letters may
occupy only about a third of the space on the left hand
side of the slate. The conditions of the problem can then

112 THE PRINCIPLES OF SCIENCE.

be written down on the unoccupied part of the slate, and
the proper series of combinations being chosen, the contra
dictory combinations can be struck out with the pencil.
I have used a slate of this kind, which I call a Logical
Slate, for more than ten years, and it has saved me much
trouble. It is hardly possible to apply this process to
problems of more than six terms, owing to the large num
ber of combinations which would require examination ;
thus seven terms would give 128 combinations, eight
terms 256, nine terms 512, ten terms 1024, eleven terms
2048, twelve terms 4096, and so on in geometrical pro
gression.

Abstraction of Indifferent Circumstances.

There is a simple but highly important process of
inference which enables us to abstract, eliminate or disre
gard all circumstances indifferently present and absent.
Thus if I were to state that 'a triangle is a figure of
three sides, with or without equal angles/ the latter
qualification would be superfluous, because by a law of
thought I know that angles must be either equal or
unequal. To add the qualification gives no new know
ledge since the existence of the two alternatives will be
understood in the absence of any information to the
contrary. Accordingly, when two alternatives differ only
as regards a single component term which is positive in
one and negative in the other, we may always reduce
them to one term by striking out their indifferent part.
It is really a process of substitution which enables us
to do this ; for having any proposition of the form

A = ABC -|- ABc, (i)

we know by the Law of Duality that

B = BC -|- Be. (2)

Hence AB = ABC | ABc. (3)

THE INDIRECT METHOD OF INFERENCE. 113

And as the second member of this is identical with the
second member of ( i ) we may substitute, obtaining

A = AB.

This process of reducing useless alternatives, may be
applied again and again ; for it is plain that

A = AB (CD | Cd -I- cD ! cd)

communicates no more information than that A is B.
This abstraction of indifferent terms is in fact the con
verse process to that of development described in p. 104 ;
and it is one of the most important operations in the
whole sphere of reasoning.

The reader should observe that in the proposition

we cannot abstract C and infer

A = B;
but from

AC | Ac = BC I Be

we may abstract all reference to the term C.

Illustrations of the Indirect Method.

An infinite variety of arguments and logical problems
might be introduced here to show the comprehensive
character and powers of the Indirect Method. We can
treat either a single premise or a series of premises.

Take in the first place a simple definition, such as ' a
triangle is a three-sided rectilinear figure.' Let
A = triangle
B — three-sided
C = rectilinear figure,
then the definition is of the form

A- BC.

If we take the series of eight combinations of three
letters (see p. 106) and strike out those which are

I

114 THE PRINCIPLES OF SCIENCE.

inconsistent mth the definition, we have the following

result : —

ABC -*%&-

«Bc.

Abe a&c.

For the description of the class C we have

C - ABC -I- a&C,

that is, ' a rectilinear figure is either a triangle and three-
sided, or not a triangle and not three-sided.'
For the class & we have

I = a&C I a&c.

To the second side of this we may apply the process of
simplification by abstraction described in the last section ;
for by the Law of Duality

a& = a&C I a&c ;

and as we have two propositions identical in the second
side of each we may substitute, getting

& = a&,

or what is not three-sided is not a triangle (whether it be
rectilinear or not).

Let us treat by this method the following argument :—
* Blende is not an elementary substance ; elementary
substances are those which are undecomposable ;
blende, therefore, is decomposable.'
Taking our letters thus —
A = blende,

B = elementary substance,
C = undecomposable,
the premises are of the form

A = A&, (i)

B = C. (2)

No immediate substitution can be made ; but if we take
the contrapositive of (2), namely

& = c, (3)

THE INDIRECT METHOD OF INFERENCE. 115

we can substitute in ( i ) obtaining the conclusion

A - Ac.

But the same result may be obtained by taking the
eight combinations of A, B, 0, of the abecedarium ; it will
be found that only three combinations, namely

Kbc
aBC
abc,
are consistent with the premises, whence it results that

A — Abc,
or by the process of Ellipsis before described (p. 69)

A - Ac.

As a somewhat more complex example I take the
argument thus stated, one which could not be thrown
into the syllogistic form.

' All metals except gold and silver are opaque ; there
fore what is not opaque is either gold or silver
or is not-metal/

There is more implied in this statement than is dis
tinctly asserted, the full meaning being as follows :

All metals not gold or silver are opaque, ( i )
Gold is not opaque but is a metal. (2)

Silver is not opaque but is a metal, (3)

Gold and silver are distinct substances. (4)

Taking our letters thus—

A = metal C = silver

B = gold D = opaque,

we may state the premises in the form

Kbc = A&cD (i)

B - ABd (2)

C = ACd (3)

B - Be. (4)

To obtain a complete solution of the question we take
the sixteen combinations of A, B, C, D, and striking out

i 2

116 THE PRINCIPLES OF SCIENCE.

those which are inconsistent with the premises, there

remain only

ABcd

AbCd

abcD
abed.

The expression for not-opaque things consists of the
three combinations containing d, thus

d = ABcd -I- AbCd •[ abed,
or d = Ad (Be -|- bC) •{• abed.

In ordinary language, what is not-opaque is either
metal which is gold, and then not-silver, or silver and then
not gold, or else it is not-metal and neither gold nor silver.
A good example for the illustration of the Indirect
Method is to be found in De Morgan's Formal Logic (p.
123), the premises being substantially as follows :—

From A follows B, and from C follows D ; but B and
D are inconsistent with each other ; therefore A and C
are inconsistent.

The meaning no doubt is that where A is, B will be
found, or that every A is a B, and similarly every C is a D ;
but B and D cannot occur together. The premises there
fore appear to be of the form

A = AB, (i)

C = CD, (2)

B = Brf. (3)

On examining the series of sixteen combinations, but five
are found to be consistent with the above conditions,
namely,

ABcd
dBcd
abCD
abcD
abed.

THE INDIRECT METHOD OF INFERENCE. 117

In these combinations the only A which appears is
joined to c, and similarly C is joined to a, or A is incon
sistent with C.

A more complex argument, also given by De Morgan f,
contains five terms, and is as stated below, except that I
have altered the letters.

' Every A is one only of the two B or C ; D is both B
and C, except when B is E, and then it is
neither ; therefore no A is D/

A little reflection will show that these premises are
capable of expression in the following symbolic forms —
A = ABc -|- A30, (i)

DP - DeBC, (2)

' \*j/

As five letters, A, B, C, D, E3 enter into these premises it

is requisite to treat their thirty-two combinations, and it
will be found that fourteen of them remain consistent with
the premises, namely

ABcc?E aBCDe a&Cc£E

ABccfe aBC<iE abGde

A6Cc£E ciBCde a&cDE

A&.Cc^ • aBccffi abcd^t

aRcde abode.

Now if we examine the first four combinations, all of
which contain A, we find that they none of them contain
D ; or again if we select those which contain D, we have

only two, thus —

D - aBCDe I a&dDE.

Hence it is clear that no A is D, and vice versd no D is A.
We might also draw many other conclusions from the
premises ; for instance—

DE — a&cDE,
or D and E never meet but in the absence of A, B, and C.

f ' Formal Logic,' p. 124.

118 THE PRINCIPLES OF SCIENCE.

Fallacies analysed by the Indirect Method.

It has been sufficiently shown, perhaps, that we can by
the Indirect Method of Inference extract the whole truth
from any series of propositions, and exhibit it anew in any
required form of conclusion. But it may also need to be
shown by examples that so long as we follow correctly
the almost mechanical rules of the method, we cannot fall
into any of the common fallacies or paralogisms which are
not seldom committed in ordinary discussion. Let us
take the example of a fallacious argument, previously
treated by the Method of Direct Inference (p. 75),

Granite is not a sedimentary rock, ( i )

Basalt is not a sedimentary rock, (2)

and let us ascertain whether any precise conclusion can be
drawn concerning the relation of granite and basalt.

Taking as before

A = granite,

B = sedimentary rock,
C = basalt,

the premises become A = Ab, (i)

C = Cb. (2)

Of the eight conceivable combinations of A, B, C, five agree
with these conditions, namely

A6C aBc

Abc a&C

abc ;
the description of granite is found to be

A - A&C I Abc = Ab(G | c),

that is, granite is not a sedimentary rock but is either
basalt or not-basalt. If we want a description of basalt
the answer is of like form

C = A&C! a&C = bC (A | a).

Basalt is a sedimentary rock, and either granite or not-
granite. As it is already perfectly evident that basalt

THE LOGICAL ABACUS. 119

must be either granite or not, and vice versd, the premises
fail to give us any information on the point, that is to say
the Method of Indirect Inference saves us from falling
into any fallacious conclusions. This example sufficiently
illustrates both the fallacy of Negative premises and that
of Undistributed Middle of the old logic (pp. 75-77).

The fallacy called the Illicit Process of the Major Term
is also incapable of commission in following the rules of
the method. Our example was (p. 77)

All planets are subject to gravity, (i)

Fixed stars are not planets. (2)

The false conclusion is that ' fixed stars are not subject to
gravity.' The terms are

A — planet

B = fixed star

C = subject to gravity.

And the premises are A = AC, (i)

B = «B. (2)

The combinations which remain uncontradicted on com
parison with these premises are

aBC a&O

abc.

For fixed star we have the description
B = aBC -I- aBc,

that is, ' a fixed star is not a planet, but is either subject
or not, as the case may be, to gravity/

The Logical Abacus.

The Indirect Method of Inference has now been suffi
ciently described, and a careful examination of its powers
will show that it is capable of giving a full analysis and
solution of every question involving any simply logical
relations. The chief difficulty of the method consists in
the great number of combinations which may have to be

120 THE PRINCIPLES OF SCIENCE.

examined; not only may the requisite labour become
formidable, but a considerable chance of mistake may
arise. I have therefore given much attention to modes
of facilitating the work, and have succeeded in reducing
the method to an almost mechanical form. It soon
appeared obvious that if the conceivable combinations
of the abecedarium, for any number of letters, instead
of being printed in fixed order on a piece of paper or
slate, were marked upon light moveable pieces of wood,
mechanical arrangements could readily be devised for

O v

selecting the combinations in any required order. The
labour of comparison and rejection might thus be im
mensely reduced. This idea was first carried out in the
Logical Abacus, which I have found useful in the lecture-
room for exhibiting the complete solution of logical
problems. A minute description of the construction and
use of the abacus, together with figures of the parts, has
already been given in my essay called The Substitution of
Similars s, and I will here give only a general description.

The abacus consists of a common school black-board
placed in a sloping position and furnished with four
horizontal and equi-distant ledges. The combinations of
the letters shown in the first four columns of the abece
darium (see p. 109), are printed in somewhat large type,
so that each letter is about an inch from the neighbour
ing one, but the letters are placed one above the other
instead of being in horizontal lines as in p. 109. Each
combination of letters is separately fixed to the surface
of a thin slip of wood one inch broad and about one-
eighth inch thick. Short steel pins are then driven in an
inclined position into the wood. When a letter is a large
capital representing a positive term, the pin is fixed in
the upper part of its space ; when the letter is a small

g Pp- 55-59.81-86.

THE LOGICAL ABACUS.

121

italic representing a negative term, the pin is fixed in
the lower part of the space. Now, if one of the series of
combinations be ranged upon a ledge of the black-board,
the sharp edge of a flat rule can be inserted beneath the
pins belonging to any one letter — say A, so that all the
combinations marked A can be lifted out and placed upon
a separate ledge. Thus we have represented the act of
thought which separates the class A from what is not- A.
The operation can be repeated ; out of the A's we can in
like manner select those which are B, obtaining the AB's ;
and in like manner we might select any other class such
as the aB's, the ab's or the abcs.

If now we take the series of eight combinations of the
letters A, B, C, a, b, c, and wish to analyse the argument
anciently called Barbara, having the premises

A = AB (i)

B = BC, (2)

•we proceed as follows : — Firstly we raise the combinations
marked a, leaving the A's behind ; out of these A's we
move to a lower ledge such as are not-B's, and to the
remaining AB's we join the a's which have been raised.
The result is that we have divided all the combinations
into two classes, namely, the A&'s which are incapable of
existing consistently with premise (i), and the combina
tions which are consistent with the premise. Turning
now to the second premise, we raise out of those which
agree with (i) the b's, then we lower the Bc's ; lastly we
join the b's to the BC's. We should now find our com
binations arranged as below.

A

a

a

a

B

B

b

b

C

C

C

c

A

A

A

a

B

b

6

B

c

C

c

c

122 THE PRINCIPLES OF SCIENCE.

The lower line contains all the combinations which are
inconsistent with either premise ; we have carried out in
a mechanical manner that exclusion of self-contradictories
which was formerly done upon the slate or paper. Ac
cordingly, from the remaining combinations in the upper
line we can draw any inference which the premises yield.
If we raise the A's we find only one, and that is C, so that
A must be C. If we select the c's we again find only
one which is a and also &, so that we prove that not-C is
not-A and not-B.

When a disjunctive proposition occurs among the
premises the requisite movements become rather more
complicated. Take the disjunctive argument

A is either B or C or D,

A is not C and not D,

Therefore A is B.
The premises are represented accurately as follows :-—

A = ABI AC I AD (i)

A = Ac (2)

A - Ad, (3)

As there are four terms we choose the series of sixteen
combinations and place them on the highest ledge of the
board but one. We raise the as and lower the 6's. But
we are not to reject all the Afr's as contradictory, because
by the first premise A's may be either B's or C's or D's.
Accordingly out of the A&'s we must select the c's, and
out of these again the d's, so that only Abed will remain
to be rejected finally. Joining all the other fifteen com
binations together again we raise the a's and lower the
AC's, and thus reject the combinations inconsistent with
(2) ; similarly we reject the AD's which are inconsistent
with (3). It will be found that there remain in addition
to all the eight combinations containing a only one con
taining A, namely

ABcc/,

THE LOGICAL MACHINE.

whence it is apparent that A must be B, the true conclusion
of the argument.

In my previous Essay11 I have described the working of
two other logical problems upon the abacus, which it
would be tedious to repeat in this place.

The Logical Machine.

Although the Logical Abacus considerably reduced the
labour of using the Indirect Method, it was not free from
the possibility of error. I thought moreover that it would
afford a conspicuous proof of the generality and power of
the method if I could reduce it to a purely mechanical
form. Logicians had long been accustomed to speak of
Logic as an Organoii or Instrument, and even Bacon, while
he rejected the old syllogistic logic, had insisted, in the
second aphorism of his ' New Instrument/ that the mind
required some kind of systematic aid. In the kindred
science of mathematics mechanical assistance of one kind
or another had long been employed. Orreries, globes,
mechanical clocks, and such like instruments, are really
aids to calculation and are of considerable antiquity. The
arithmetical machine of Pascal is more than two centuries
old, having been constructed in 1642-45. M. Thomas of
Colmar has recently manufactured an arithmetical machine
on Pascal's principles which is extensively employed by
engineers and others who need frequently to multiply or
divide. To Babbage, however, was entirely due the
merit of embodying the Calculus of Differences in a
machine, which thus became capable of calculating the
most complicated tables of figures. It seemed strange
that in the more intricate science of quantity mechanism
could be applicable, whereas in the simple science of

11 ' Substitution of Similars,' pp. 56-59.

124 THE PRINCIPLES OF SCIENCE.

qualitative reasoning, the syllogism was only by analogy
or simile called an Instrument. Swift satirically described
the Professors of Laputa as in possession of a thinking
machine, and in 1851 Mr. Alfred Smee actually proposed
the construction of a Relational machine and a Differential
machine, the first of which would be a mechanical dic
tionary and the second a mode of comparing ideas ; but
with these exceptions I have not yet met with so much
as a suggestion of a reasoning machine. It may be added
that Mr. Smee's designs, though highly ingenious, appear
impracticable, and in any case do not attempt the per
formance of logical inference1.

The Logical Abacus soon suggested the notion of a

o oo

Logical Machine, which, after two unsuccessful attempts,
I succeeded in constructing in a comparatively simple and
effective form. The details of the Logical Machine have
been fully described by the aid of plates in the Phi
losophical Transactions k, and it would be both tedious
and needless to repeat the account of the somewhat
intricate movements of the machine in this place.

The general appearance of the machine is shown in a
plate facing the title-page of this volume. It somewhat
resembles a very small upright piano or organ, and has
a keyboard containing twenty-one keys. These keys are
of two kinds, sixteen of them representing the terms or
letters A, a, B, -b, C, c, D, d, which have so often been
employed in our logical notation. When letters occur
on the left-hand side of a proposition, formerly called
the subject, each is represented by a key on the left-hand
half of the keyboard ; but when they occur on the right-

1 See his work called ' The Process of Thought adapted to Words and
Language, together with a description of the Relational and Differential
Machines.' Also 'Philosophical Transactions/ [1870] vol. 160, p. 518.

k ' Philosophical Transactions,' [1870] vol. 160, p. 497. 'Proceedings
of the Royal Society,' vol. xviii. p. 166, Jan. 20, 1870. 'Nature,' vol. i.
T- 343-

THE LOGICAL MACHINE.

125

hand side, or as it used to be called the predicate of the
proposition, the letter keys on the right-hand side of the
keyboard are the proper representatives. The five other
keys may be called operation keys, to distinguish them
from the letter or term keys. They stand for the stops,
copula, and disjunctive conjunctions of a proposition. The
middle key of all is the copula, to be pressed when the
verb is or the sign = is met. The extreme right-hand
key is called the Full Stop, because it should be pressed
when a proposition is completed, in fact in the proper
place of the full stop. The extreme left-hand key is
used to terminate an argument or to restore the machine
to its initial condition ; it is called the Finis key. The
last key but one on the right and left complete the
whole series, and represent the conjunction or in its mi-
exclusive meaning, or the sign •(• which I have employed,
according as it occurs in the right or left hand side
of the proposition. The whole keyboard is arranged
as shown below —

.°°

1

Left-hand side of Proposition,
or Subject.

,5

0.
O
O

Right-hand side of Proposition,
or Predicate.

PH
O

02

"5
&

•1-
Or

d

D

c

C

b

B

a

A

A

a

B

b

C

c

D

d

•1-
Or

To work the machine it is only requisite to press the
keys in succession as indicated by the letters and signs
of a symbolical proposition. All the premises of an ar
gument are supposed to be reduced to the simple notation
which has been employed in the previous pages. Taking-
then such a simple proposition, as

A = AB,

we press the keys A (subject), copula, A (predicate),
B (predicate), and full stop.

126

THE PRINCIPLES OF SCIENCE.

If there be a second premise, for instance

we press in like manner the keys —

B (subj.), copula, B (pred.), C (pred.), full stop.
The process is exactly the same however numerous the
premises may be. When they are completed the operator
will see indicated on the face of the machine the exact
combinations of letters which are consistent with the
premises according to the principles of thought.

As shown in the figure opposite the title-page, the
machine exhibits in front an abecedarium of sixteen com
binations, exactly like that of the abacus, except that the
letters of each combination are separated by a certain
interval. After the above problem has been worked upon
the machine the abecedarium will present the following
appearance —

A

A

a

a

a

a

a

a

B B

B B

6 b I b

C

G

C

C

C

C

C

c

D

d

D

d

D

d

D

d

The operator will collect the various conclusions, as for
instance that A is always C, that not-C is not-B and
not- A ; that not-B is not- A but either C or not-C, as
in the use of the Logical Slate or Abacus.

THE LOGICAL MACHINE. 127

Disjunctive propositions are to be treated in an exactly-
similar manner. Thus, to work the premises

A = AB | AC
B -l- C = BD | CD,
it is only necessary to press in succession the keys

A (subj.), copula, A (pred.), B, { A,C, full stop.
B (subj.), | C, copula, B (pred.), D, \ , C, D, full stop.
The combinations then remaining will be as follows
ABCD aBOD
ABcD aBcD
A&CD a&CD
abcD
abed.

On pressing the subject key A, all the possible com
binations which do not contain A will disappear, and the
description of A may be gathered from what remains,
namely that it is always D. The full-stop key restores
all combinations consistent with the premises and any
other selection may be made, as say not-D, which will
be found to be always not- A, not-B, and not-C.

At the end of every problem, when no further questions
need be addressed to the machine, it is desirable to press
the Finis key, which has the effect of bringing into view
the whole of the conceivable combinations of the abece-
darium. This key in fact obliterates the conditions im
pressed upon the machine by moving back into their
ordinary places those combinations which had been re
jected as inconsistent with the premises. Before begin
ning any new problem it is requisite to observe that
the whole sixteen combinations are visible. After the
Finis key has been used the machine represents a mind
endowed with powers of thought, but wholly devoid
of knowledge. It would not in that condition give any
answer but such as would consist in the primary laws
of thought themselves. But when any proposition is

128 THE PRINCIPLES OF SCIENCE.

worked upon the keys, the machine analyses or digests
the meaning of it and becomes charged with the know
ledge embodied in that proposition. Accordingly it is
able to return as an answer any description of a term
or class so far as furnished by that proposition in ac
cordance with the Laws of Thought. The machine is
thus the embodiment of a true logical system. The com
binations are classified, selected or rejected just as they
should be by a reasoning mind, so that at each step in
a problem, the abecedarium represents the proper con
dition of a mind exempt from mistake. It cannot be
asserted indeed that the machine entirely supersedes the
agency of conscious thought ; mental labour is required
in interpreting the meaning of grammatical expressions
and in correctly impressing that meaning on the machine ;
it is further required in gathering the conclusion from
the remaining combinations. Nevertheless the true pro
cess of logical inference is actually accomplished in a
purely mechanical manner.

It is worthy of remark that the machine can detect
any self-contradiction existing between the premises pre
sented to it, for it will then be found that one or more
of the terms disappear entirely from the abecedarium.
Thus if we worked the two propositions, A is B, and
A is not-B, and then inquired for a description of A,
the machine would refuse to give it by exhibiting no
combination at all containing A. This result is in agree
ment with the law which I have explained that every
term must have its negative (p. 88). Accordingly when
ever any one of the letters A, B, C, D, a, I, c, d wholly
disappears from the abecedarium, it may be safely inferred
that some self-contradiction has been committed in the
premises.

It ought to be carefully observed that the logical
machine cannot receive a simple identity of the form

THE LOGICAL MACHINE. 129

A — B except in the double form of A = AB and B = AB.
To work the proposition A = B it is therefore necessary to
press the keys — A (subj.), Copula, A (pred.), B (pred.), Full
stop, B (subj.), Copula, A (pred.), B (pred.), Full stop.
The same double operation will be necessary whenever
the proposition is not of the kind called a partial
identity (p. 47). Thus AB - CD, AB - AC, A = B I C
A I B = C I D, all require to be read from both ends
separately. This is a remarkable fact which some per
sons may consider as militating against the equational
form of proposition, but I do not think this is really
the case.

Before leaving the subject I may remark that these
mechanical devices are not likely to possess great prac
tical utility. We do not require in common life to be
constantly solving complex logical questions. Even in
mathematical calculation the ordinary rules of arithmetic
are generally sufficient, and a calculating machine could
only be used with advantage in peculiar cases. But the
machine and abacus have nevertheless two important
uses.

i. I trust that the time is not very far distant when
the predominance of the ancient Aristotelian Logic will
be a matter of history, and the teaching of logic will
be placed on a footing more worthy of its supreme
importance. It will then be found that the solution of
logical questions is an exercise of mind at least as valu
able and necessary as mathematical calculation. I believe
that these mechanical devices, or something of the same
kind, will then become useful for exhibiting to a class
of students a clear and visible analysis of logical problems
of any degree of complexity, the nature of each step
being rendered plain to the eye. For this purpose I
have already often used the machine or abacus in my
class lectures at the Owens College.

130 THE PRINCIPLES OF SCIENCE.

2. The more immediate importance of the machine
seems to consist in the unquestionable proof which it
affords that most comprehensive views of the principles
of reasoning have now been attained, although they were
almost wholly unknown to Aristotle and his followers.
The time must come when the inevitable results of the
admirable writings of the late Dr. Boole must be re
cognised at their true value, and the plain and palpable
form in which the machine presents those results will,
I hope, hasten the time. Undoubtedly his life marks
an era in the high science of human reason. It may
seem strange that it had remained for him first to set
forth in its full extent the problem of logic, but I am
not aware that any one before him had treated logic
as a symbolic method for evolving from any premises
the description of any class whatsoever as defined by
those premises. His quasi-mathematical system indeed
could not be regarded as a final and complete solution
of the problem. Not only did it require the manipula
tion of mathematical symbols in a very intricate arid
perplexing manner, but the results when obtained were
devoid of demonstrative force, because they turned upon
the employment of unintelligible symbols, acquiring mean
ing only by analogy. I have also pointed out 'that he
imported into his system a condition concerning the
exclusive nature of alternatives (p. 83), which is not
necessarily true of logical terms. I shall have to show
in the next chapter that logic is really the basis of
the whole science of mathematical reasoning, so that
Boole completely inverted the true order of proof
when he proposed to infer logical truths by algebraic
processes. It is a wonderful evidence of his mental
power that by methods fundamentally false he should
have succeeded in reaching true conclusions and widen
ing the sphere of reason.

THE ORDER OF PREMISES. 131

The mechanical performance of logical inference affords
a demonstration both of the truth of Boole's results and
of the mistaken nature of his mode of deducing them.
Conclusions which he could only obtain by pages of
intricate calculation, are exhibited by the machine after
one or two minutes of manipulation. And not only are
those conclusions easily reached, but they are demon
stratively true, because every step of the process involves
nothing more obscure that the Laws of Thought.

The Order of Premises.

Before quitting the subject of deductive reasoning, I
may remark that the order in which the premises of
an argument, or any propositions whatsoever, are placed,
is a matter of logical indifference. Much discussion has
taken place at various times concerning the arrangement
of the premises of a syllogism ; and it has been generally
held, in accordance with the opinion of Aristotle, that
the so-called major premise, containing the major term,
or the predicate of the conclusion, should stand first.
This distinction however falls to the ground in our system,
since the proposition is reduced to an identical form in
which there is no distinction of subject and predicate.
In a strictly logical and philosophic point of view the
order of statement is whoUy devoid of significance. The
premises are simultaneously coexistent, and are not related
to each other according to any of the properties of space
or time. Just as the qualities of the same object are
neither before nor after each other in nature (p. 40),
and are only thought of in some one order owing to
our limited capacity of mind, so the premises of an
argument are neither before nor after each other, and
are only thought of in succession because the mind can
not grasp many ideas at once. The logical combinations

K 2

132 THE PRINCIPLES OF SCIENCE.

of the Abecedarian are exactly the same in whatever
order the premises be treated on the logical slate or

machine.

Some difference may doubtless exist as regards con
venience to human memory. The mind may take in
the results of an argument more easily in one mode of
statement than another, although there is no real differ
ence in the logical results. But in this point of view
I think that Aristotle and the old logicians were clearly
wrong. It is more easy to conclude that ' all A's are C's '
from ' all A's are B's and all B's are C's/ than from
the same propositions in inverted order, 'all B's are C's
and all A's are B's.'

The Equivalency of Propositions.

One great advantage which arises from the study of
this Indirect Method of Inference consists in the clear
notion which we thus gain of the Equivalency of Propo
sitions. The older logicians showed how from certain
simple premises we might draw an inference, but they
failed to point out whether that inference contained the
whole, or only a part, of the information embodied in the
premises. Now any one proposition or group of propo
sitions may be classed with respect to another proposition
or group of propositions, as

1. Equivalent,

2. Inferrible,

3. Consistent,

4. Contradictory.

Taking the proposition ' All men are mortals' as the
original, ' All immortals are not men ' is its equivalent ;
' Some mortals are men ' is inferrible, or capable of infe
rence, but is not equivalent ; ' All not men are not
mortals ' cannot be inferred, but is consistent, that is, may

THE EQUIVALENCY OF PROPOSITIONS. 133

be true at the same time ; ' All men are immortals' is of
course contradictory.

One sufficient test of equivalency is the capability of
mutual inference. Thus from

All electrics = all non-conductors,
I can infer

All non-electrics = all conductors,

and vice versa from the latter I can pass back to the
former. In short A = B is equivalent to a — b. Again,
from the union of the two propositions, A = AB and
B = AB, I get A — B, and from this I might as easily
deduce the two with which I started. In this case one
proposition is equivalent to two other propositions. There
are indeed no less than four modes in which we may
express the identity of two classes A and B, namely,

FIRST MODE. SECOND MODE. THIRD MODE. FOURTH MODE.

1 a = ab

A -D 7

A = D a = o T) A T> r 1

J3 — AB J 6 =

The Indirect Method of Inference furnishes an universal
and clear criterion as to the relationship of propositions.
The import of a statement is always to be measured by
the combinations of terms which it destroys. Hence two
propositions are exactly equivalent when they remove
exactly the same combinations from the Abecedarium,
and neither more nor less. A proposition is inferrible
but not equivalent to another when it removes some but
riot all the combinations which the other removes. Again,
propositions are consistent provided that they leave some
one combination containing each term, and the negative
of each term. If after all the combinations inconsistent
with two propositions are struck out, there still appears
in the Abecedarium each of the letters A, a, B, b, C, c, D, d,
which were there before, then no inconsistency between
the propositions exists, although they may not be equiva
lent or even inferrible. Finally, contradictory propositions

134 THE PRINCIPLES OF SCIENCE.

are those which altogether remove any one or more letter-
terms from the Abecedariurn.

What is true of single propositions applies also to
groups of propositions, however large or complicated ;
that is to say, one group may be equivalent, inferrible,
consistent, or contradictory as regards another, and we
may similarly compare one proposition with a group of
propositions.

To give in this place illustrations of all the four kinds
of relation would require much space : as the examples
given in previous sections or chapters may serve more or
less to explain the relations of inference, consistency, and
contradiction, I will only add a, few instances of equivalent
propositions or groups.

In the following list each proposition or group of propo
sitions is exactly equivalent in meaning to the correspond
ing one in the other column, and the truth of this state
ment may be tested by working out the combinations of
the Abecedarium, which ought to be found exactly the
same in the case of each pair of equivalents.
A = A6 B = aB

A = b a = B

A = BC a = b\c

A = AB I AC b = ab I A&C

A-|-B = C-|-D ab = cd

A I c = B I- d aC = bV

A = ABI AC

A = ABc I A6C A .~

AB = ABc

A = Bl A = B

B = CJ A- C

A - ABl A = AC

B - BCJ B = AlaBC

A = ABi

A - AC I A = ABCD.

A = AD-I

THE EQUIVALENCY OF PROPOSITIONS. 135

Although in these and many other cases the equivalents
of certain propositions can readily be given, yet I believe
that no uniform and infallible process can be pointed out
by which the exact equivalents of premises can be ascer
tained. Ordinary deductive inference usually gives us
only a portion of the contained information. It is true
that the combinations consistent with a set of propositions
are logically equivalent to them, but the difficulty consists
in passing back from the combinations to a new set of
propositions. The task is here of a different character
from any which we have yet attempted. It is in reality
an inverse process, and is just as much more troublesome
and uncertain than the direct process, as seeking is com
pared with hiding. Not only may several different answers
equally apply, but there is no method of discovering any
of those answers except by repeated trial. The problem
which we have here met is really that of induction, the
inverse of deduction ; and, as I shall soon show, induction
is always tentative, and unless conducted with peculiar
skill and insight must be exceedingly laborious in cases of
any considerable complexity.

The late Professor de Morgan was unfortunately led
by this equivalency of propositions into the most serious
error of his ingenious system of Logic. He held that
because the proposition 'All A's are all B's,' was but
another expression for the two propositions 'All A's are
B's ' and ' All B's are As,' it must be a composite and not
really an elementary form of proposition1. But on taking
a general view of the equivalency of propositions such an
objection seems to have no weight. Logicians have, with
few exceptions, persistently upheld the original error of
Aristotle in rejecting from their science the one simple

1 ( Syllabus of a proposed system of Logic,' §§ 57, 121, &c. 'Formal
Logic,' p. 66.

136 THE PRINCIPLES OF SCIENCE.

relation of identity on which all more complex logical -
relations must really rest.

The Nature of Inference.

The question, What is Inference I is involved, even to
the present day, in as much uncertainty as that ancient
question, What is Truth \ I shall in more than one part
of this work endeavour to show that inference never does
more than explicate, unfold, or develop the information
contained in certain premises or facts. Neither in deduc
tive nor inductive reasoning can we add a tittle to our
implicit knowledge, which is like that contained in an
unread book or a sealed letter. Sir W. Hamilton has well
said, ' Reasoning is the showing out explicitly that a pro
position not granted or supposed, is implicitly contained
in something different which is granted or supposed m/

Professor Bo wen has explained11 with much clearness
that the conclusion of an argument states explicitly what
is virtually or implicitly thought. ' The process of reasoning
is not so much a mode of evolving a new truth, as it is of
establishing or proving an old one, by showing how much
was admitted in the concession of the two premises taken
together.' It is true that the whole meaning of these
statements rests upon that of such words as 'explicit/
'implicit,' 'virtual.' That is implicit which is wrapped up,
and we render it explicit when we unfold it. Just as the
conception of a circle involves a hundred important geome
trical properties, all following from what we know, if we
have acuteness to unfold the results, so every fact and
statement involves more meaning than seems at first
sight. Reasoning explicates or brings to conscious posses
sion what was before unconscious. It does not create, nor

m Lectures on Metaphysics, vol. iv. p. 369.

n Bowen, ' Treatise on Logic/ Cambridge, U. S., 1866; p. 362.

THE NATURE OF INFERENCE. 137

does it destroy, but it transmutes and throws the same
matter into a new form.

The difficult question still remains, Where does novelty
of form begin 1 Is it a case of inference when we pass
from c Sincerity is the parent of truth ' to ' The parent of
truth is sincerity'?' The old logicians would have called
this change conversion one case of immediate inference.
But as all identity is necessarily reciprocal, and the very
meaning of such a proposition is that the two terms are
identical in their signification, I fail to see any difference
between the statements whatever. As well might we say
that a = b and b = a are different equations.

Another point of difficulty is to decide when a change
is merely grammatical and when it involves a real logical
transformation. Between a table of wood and a wooden
table there is no logical difference (p. 37), the adjective
being merely a convenient substitute for the prepositional
phrase. But it is uncertain to my mind whether the
change from ' All men are mortal ' to ' No men are not
mortal' is purely grammatical. Logical change may
perhaps be best described as consisting in the determina-
tjon of a relation between certain classes of objects from
a relation between certain other classes. Thus I consider
it a truly logical inference when we pass from ' All men
are mortal ' to ' All immortals are not-men/ because the
classes immortals and not-men are different from mortals
and men, and yet the propositions contain at the bottom
the very same truth, as shown in the combinations of the
Abecedarium.

From logical inference we must discriminate the passage
from the qualitative to the quantitative form of a pro
position. We state the same truth when we say that
' mortality belongs to all men,' as when we assert that
* all men are mortals.' Here we do not pass from class to
class, but from one kind of term, the abstract, to another

138 THE PRINCIPLES OF SCIENCE.

kind, the concrete. But inference probably enters when
we pass from either of the above propositions to the
assertion that the class of immortal men is zero, or con
tains no objects.

It is really a question of words to what processes we
shall or shall not apply the name ' inference/ and I have
no wish to continue the trifling discussions which have
already taken place upon the subject. We shall not
commit any serious error, provided that we always bear
in mind that two propositions may be connected together
in four different ways. They may be—

1 . Tautologous or identical, involving the same relation
between the same terms and classes, and only differing in
the order of statement ; thus ' Victoria is the Queen of
England' is tautologous with 'The Queen of England
is Victoria/

2. Grammatically equivalent, in which the classes or
objects are the same and similarly related, and the only
difference is in the words ; thus ' Victoria is the Queen
of England' is grammatically equivalent to 'Victoria is
England's Queen.'

3. Equivalent in qualitative and quantitative form,
the classes being the same, but viewed in a different
manner.

4. Logically equivalent, when the classes and relations
are different, but involve the same knowledge of the
possible combinations.

CHAPTEE VII.

INDUCTION.

WE enter in this chapter upon the second great de
partment of logical method, that of Induction or the
Inference of general from particular truths. It cannot
be said that the Inductive process is of greater importance
than the Deductive process already considered, because the
latter process is absolutely essential to the existence of
the former. Each is the complement and counterpart of
the other. The principles of thought and existence which
underlie them are at the bottom the same, just as subtrac
tion of numbers necessarily rests upon the same principles
as addition. Induction is, in fact, the inverse operation
to deduction, and cannot be conceived to exist without
the corresponding operation, so that the question of re
lative importance cannot arise. Who thinks of asking
whether addition or subtraction is the more important
process in arithmetic 1 But at the same time much
difference in difficulty may exist between a direct and
inverse operation ; the integral calculus, for instance, is
almost infinitely more difficult than the differential cal
culus of which it is the inverse. It must be allowed
that in logic inductive investigations are of a far higher
degree of difficulty, variety, and complexity than any
questions of deduction ; and it is this fact no doubt which
has led some logicians to erroneous opinions concerning
the exclusive importance of induction.

Hitherto we have been engaged in considering how

140 THE PRINCIPLES OF SCIENCE.

from certain conditions, laws, or identities governing the
combinations of qualities, we may deduce the nature of
the combinations agreeing with those conditions. Our
work has been to unfold the results of what is contained
in any statements, and the process has been one of Syn
thesis. The terms or combinations of which the character
has been determined have usually, though by no means
always, involved more qualities, and therefore, by the
relation of extension and intension, fewer objects than
the terms in which they were described. The truths
inferred were thus usually less general than the truths
from which they were inferred.

In induction all is inverted. The truths to be ascer
tained are more general than the data from which they
are drawn. The process by which they are reached is
analytical, and consists in separating the complex com
binations in which natural phenomena are presented to
us, and determining the relations of separate qualities.
Given events obeying certain unknown laws, we have to
discover the laws obeyed. Instead of the comparatively
easy task of finding what effects will follow from a given
law, the effects are now given and the law is required.
We have to interpret the will by which the conditions of
creation were laid down.

Induction an Inverse Operation.

I have already asserted that induction is the inverse
operation of deduction, but the difference is one of such
great importance that I must dwell upon it. There are
many cases where we can easily and infallibly do a certain
thing but may have much trouble in undoing it. A per
son may walk into the most complicated labyrinth or the
most extensive catacombs, and turn hither and thither at
his will ; it is when he wishes to return that doubt and

INDUCTION. 141

difficulty commence. In entering, any path served him ;
in leaving, he must select certain definite paths, and in
this selection he must either trust to memory of the way
he entered or else make an exhaustive trial of all possible
ways. The explorer entering a new country makes sure
his line of return by barking the trees.

The same difficulty arises in many scientific processes.
Given any two numbers, we may by a simple and infallible
process obtain their product, but it is quite another matter
when a large number is given to determine its factors.
Can the reader say what two numbers multiplied together
will produce the number 8,616,460,799 1 I think it
unlikely that any one but myself will ever know ; for
they are two large prime numbers, and can only be re
discovered by trying in succession a long series of prime
divisors until the right one be fallen upon. The work
would probably occupy a good computer for many weeks,
but it did not occupy me many minutes to multiply the
two factors together. Similarly there is no direct process
for discovering whether any number is a prime or not ;
it is only by exhaustingly trying all inferior numbers
which could be divisors, that we can show there is none,
and the labour of the process would be intolerable were it
not performed systematically once for all in the process
known as the Sieve of Eratosthenes, the results being
registered in tables of prime numbers.

The immense difficulties which are encountered in the
solution of algebraic equations are another illustration.
Given any algebraic factors, we can easily and infallibly
arrive at the product, but given a product it is a matter
of infinite difficulty to resolve it into factors. Given any
series of quantities however numerous, there is very little
trouble in making an equation which shall have those
quantities as roots. Let a, 1), c, d, &c., be the quantities ;
then (x — a) (x — b) (x — c) (x — d) = o

142 THE PRINCIPLES OF SCIENCE.

is the equation required, and we only need to multiply
out the expression on the left hand by ordinary rules.
But having given a complex algebraic expression equated
to zero, it is a matter of exceeding difficulty to dis
cover all the roots. Mathematicians have exhausted
their highest powers in carrying the complete solution
up to the fourth degree. In every other mathematical
operation the inverse process is far more difficult than
the direct process, subtraction than addition, division
than multiplication, evolution than involution ; but the
difficulty increases vastly as the process becomes more
complex. The differentiation, the direct process, is always
capable of performance by certain fixed rules, but as these
produce considerable variety of results, the inverse process
of integration presents immense difficulties, and in an
infinite majority of cases surpasses the present resources
of mathematicians. There are no infallible and general
rules for its accomplishment ; it must be done by trial,
by guesswork, by remembering the results of differentia
tion, and using them as a guide.

Coming more nearly to our own immediate subject,
exactly the same difficulty exists in determining the law
which certain numbers obey. Given a general mathe
matical expression, we can infallibly ascertain its value
for any required value of the variable. But I am not
aware that mathematicians have ever attempted to lay
down the rules of a process by which, having given cer
tain numbers, one might discover a rational or precise
formula from which they proceed. The problem is always
indeterminate, because an infinite number of formulae
agreeing with certain numbers, might always be dis
covered with sufficient trouble.

The reader may test his power of detecting a law, by
contemplation of its results, if he, not being a mathema
tician, will attempt to point out the law obeyed by the

INDUCTION. 143

following numbers :

11 ii i 5 691 7 3617

, — , , - , , — — j , -, — , CM..

20 30 42 30 66 2730 6 510
These numbers are sometimes negative, more often posi
tive ; sometimes in low terms, but unexpectedly spring
ing up to high terms ; in absolute magnitude they
are very variable. They seem to set all regularity and
method at defiance, and it is hardly to be supposed that
any one could, from contemplation of the numbers, have
detected the relation between them. Yet they are derived
from the most regular and symmetrical laws of relation,
and are of the highest importance in mathematical analysis,
being known as the numbers of Bernouilli.

Compare again the difficulty of decyphering with that
of cyphering. Any one can invent a secret language, and
with a little steady labour can translate the longest letter
into the character. But to decypher the letter having no
key to the signs adopted, is a wholly different matter.
As the possible modes of secret writing are infinite in
number and exceedingly various in kind, there is no direct
mode of discovery whatever. Repeated trial, guided
more or less by knowledge of the customary form of cypher,
and resting entirely on the principles of probability, is
the only resource. A peculiar tact or skill is requisite for
the process, and a few men, such as Wall is or Mr. Wheat-
stone, have attained great success.

Induction is the decyphering of the hidden meaning of
natural phenomena. Given events which happen in certain
definite combinations, we are required to point out the
laws which have governed those combinations. Any laws
being supposed, we can, with ease and certainty, decide
whether the phenomena obey those laws. But the laws
which may exist are infinite in variety, so that the chances
are immensely against mere random guessing. The dif
ficulty is much increased by the fact that several laws will

144 THE PRINCIPLES OF SCIENCE.

usually be in operation at the same time, the effects of
which are complicated together. The only modes of dis
covery consist either in exhaustively trying a great number
of supposed laws, a, process which is exhaustive in more
senses than one, or else by carefully contemplating the
effects, endeavouring to remember cases in which like
effects followed from known laws. However we accom
plish the discovery, it must be done by the more or less
apparent application of the direct process of deduction.

The Logical Abecedarium illustrates induction as well as
it does deduction. In the Indirect process of Inference we
found that from certain propositions we could infaUibly
determine the combinations of terms agreeing with those
premises. The inductive problem is just the inverse.
Having given certain combinations of terms, we need to
ascertain the propositions with which they are consistent,
and from which they may have proceeded. Now if the
reader contemplates the following combinations

ABC abC

aBC abc,

he will probably remember at once that they belong to the
premises A = AB, B = BC. If not, he will require a few
trials before he meets with the right answer, and every
trial will consist in assuming certain laws and observing
whether the deduced results agree with the data. To test
the facility with which he can solve this inductive pro
blem, let him casually strike out any of the combinations,
say of the fourth column of the Abecedarium (p. 109), and
say what laws the remaining combinations obey, observing
that every one of the letter-terms and their negatives
ought to appear in order to avoid self-contradiction in the
premises (pp. 88, 128). Let him say, for instance, what
laws are embodied in the combinations
ABC aBC

INDUCTION. . 145

The difficulty becomes much greater when more terms
enter into the combinations. It would be no easy matter
to point out the complete conditions fulfilled in the com
binations

ACe

aECe

abCe
abcE.

After some trouble the reader may discover that the
principal lawTs are C = e, and A = Ae ; but he would hardly
discover the remaining law, namely that BD = BDe.

The difficulties encountered in the inductive investi
gations of nature, are of an exactly similar kind.

We seldom observe any great law in uninterrupted and
undisguised operation. The acuteness of Aristotle and
the ancient Greeks, did not enable them to detect that all
terrestrial bodies tend to fall towards the centre of the
earth. A very few nights of observation would have con
vinced an astronomer viewing the solar system from its
centre, that the planets travelled round the sun ; but the
fact that our place of observation is one of the travelling
planets, so complicates the apparent motions of the other
bodies, that it required all the industry and sagacity of
Copernicus to prove the real simplicity of the planetary
system. It is the same throughout nature ; the laws may
be simple, but their combined effects are not simple, and
we have no clue to guide us through their intricacies. ' It
is the glory of God/ said Solomon, ' to conceal a thing, but
the glory of a king to search it out.' The laws of nature
are the invaluable secrets which God has hidden, and it is
the kingly prerogative of the philosopher to search them
out by industry and sagacity.

146 THE PRINCIPLES OF SCIENCE.

Induction of Simple Identities.

Many of the most important laws of nature are ex
pressible in the form of simple identities, and I can at once
adduce them as examples to illustrate what I have said
of the difficulty of the inverse process of induction. There
are many cases in which two phenomena are usually con
joined. Thus all gravitating matter is exactly coincident
with all matter possessing inertia ; where one property
appears, the other likewise appears. All crystals of the
cubical system, are all the crystals which do not doubly
refract light. All exogenous plants are, with some ex
ceptions, those which have two cotyledons or seed-leaves.

A little reflection will show that there is no direct and
infallible process by which such complete coincidences may
be discovered. Natural objects are aggregates of many
qualities, and any one of those qualities may prove to be
in close connection with some others. If each of a
numerous group of objects is endowed with a hundred
distinct physical or chemical qualities, there will be no
less than J (100x99) or 4950 pairs of qualities, which
may be connected, and it will evidently be a matter of
great intricacy and labour to ascertain exactly which
qualities are connected by any simple law.

One principal source of difficulty is that the finite powers
of the human mind are not sufficient to compare by a
single act any large group of objects with another large
group. We cannot hold in the conscious possession of the
mind at any one moment more than five or six different
ideas. Hence we must treat any more complex group by
successive acts of attention. The reader will perceive by
an almost individual act of comparison that the words
Roma and Mora contain the same letters. He may
perhaps see at a glance whether the same is true of
Caused and Casual, and of Logica and Caligo. To assure

INDUCTION. 147

himself that the letters in Astronomers make No more
stars, that Serpens in cikuleo is an anagram of Joannes
Keplerus, or Great gun do us a sum an anagram of Au
gustus de Morgan, it will certainly be necessary to break
up the act of comparison into several successive acts. The
process will acquire a double character, and will consist in
ascertaining that each letter of the first group is among
the letters of the second group, and vice versd, that each
letter of the second is among those of the first group.
In the same way we can only prove that two long lists of
names are identical, by showing that each name in one
list occurs in the other, and vice versd.

This process of comparison really consists in establish
ing two partial identities, which are, as already shown
(P- T33)» equivalent in conjunction to one simple iden
tity. We first ascertain the truth of the two propositions
A = AB, B = AB, and we then rise by substitution to the
single law A = B.

There is another process, it is true, by which we may
get to exactly the same result, for the two propositions
A = AB, a = ab are also equivalent to the simple identity
A = B (p. 133). If then we can show that all objects
included under A are included under B, and also that all
objects not included under A are not included under B,
our purpose is effected. By this process we should
usually compare two lists if we are allowed to mark them.
For each name in the first list we should strike off one in
the second, and if, when the first list is exhausted the
second list is also exhausted, it follows that all names
absent from the first must be absent from the second,
and the coincidence must be complete.

The two modes of proving a simple identity are so
closely allied that it is doubtful how far we can detect
any difference in their powers and instances of application.
The first method is perhaps more convenient where the

L 2

148 THE PRINCIPLES OF SCIENCE.

phenomena to be compared are rare. Tims we prove
that all the musical concords coincide with all the more
simple numerical ratios, by showing that each concord
arises from a simple ratio of undulations, and then show
ing that each simple ratio gives rise to one of the con
cords. To examine all the possible cases of discord or
complex ratio of undulation would be impossible. By a
happy stroke of induction Sir John Herschel discovered
that all crystals of quartz which rotate the plane of polar
ization of light are precisely those crystals which have
plagihedral faces, that is, oblique faces on the corners
of the prism unsymmetrical with the ordinary faces.
This singular relation would be proved by observing that
all plagihedral crystals possessed the power of rotation,
and vice versd all crystals possessing this power were
plagihedral. But it might at the same time be noticed
that all ordinary crystals were devoid of the power.
There is no reason why we should not observe any of the
four propositions A = AB, B = AB, a = ab, b = ab, all of
which follow from A = B (see p. 133).

Sometimes the terms of the identity may be singular
objects ; thus we observe that diamond is a combustible
gem, and being unable to discover any other that is, we
affirm

Diamond = combustible gem,
In a similar manner we ascertain that

Mercury = metal liquid at ordinary temperatures,

Substance of least density = substance of least atomic

weight.

Two or three objects may occasionally enter into the
induction, as when we learn that

Sodium I potassium — metal of less density than
water,

Venus I Mercury | Mars = major planet devoid of
satellites.

INDUCTION. H9

Induction of Partial Identities.

We found in the last section that the simple identity of
two classes is almost always discovered not by direct
observation of the fact, but by first establishing two
partial identities. There are also a great multitude of
cases in which the partial identity of one class with an
other is the only relation to be discovered. Thus the most
common of all inductive inferences consists in establishing
the fact that all objects having the properties of A have
also those of B, or that A = AB. To ascertain the truth
of a proposition of this kind it is merely necessary to
assemble together, mentally or physically, all the objects
included under A, and then observe whether B is present
in each of them, or, which is the same, whether it would
be impossible to select from among them any not-B.
Thus, if we mentally assemble together all the heavenly
bodies which move with apparent rapidity, that is to say
the planets, we find that they all possess the property of
not scintillating. We cannot analyse any vegetable sub
stance without discovering that it contains carbon and
hydrogen, but it is not true that all substances containing
carbon and hydrogen are vegetable substances.

The great mass of scientific truths consists of propo
sitions of this form A — AB. Thus in astronomy we
learn that all the planets are spheroidal bodies ; that
they all revolve in one direction round the sun ; that
they aU shine by reflected light ; that they all obey
the law of gravitation. But of course it is not to be
asserted that all bodies obeying the law of gravitation,
or shining by reflected light, or revolving in a particular
direction, or being spheroidal in form, are planets. In
other sciences we have immense numbers of propositions
of the same form, as for instance that all substances in

150 THE PRINCIPLES OF SCIENCE.

becoming gaseous absorb heat ; that all metals are
elements ; that they are all good conductors of heat and
electricity ; that all the alkaline metals are monad
elements ; that all foraminifera are marine organisms ;
that all parasitic animals are non- mammalian ; that
lightning never issues from stratous clouds a ; that pumice
never occurs where only Labrador felspar is present b :
and scientific importance may attach even to such ap
parently trifling observations as that ' white cats having
blue eyes are deaf c.'

The process of inference by which all such truths are
obtained may readily be exhibited in a precise symbolic
form. We must have one premise specifying in a dis
junctive form all the possible individuals which belong
to a class ; we resolve the class, in short, into its con
stituents. We then need a number of propositions each
of which affirms that one of the individuals possesses a
certain property. Thus the premises must be of the
form

A = B! 01 D|. .:...! PI Q

B- BX

C = CX

Q = QX.

Now if we substitute for each alternative of the first
premise its description as found among the succeeding
premises we obtain

A = BXI CXI ......... IPXIQX

or

A = (B!C|

a Arago's Meteorological Essays, p. 10.

b Lyell's Elements of Geology, Fourth ed. p. 373.

c Darwin's Variation of Animals, &c.

INDUCTION. 151

But for the aggregate of alternatives we may now
substitute their equivalent as given in the first premise,
namely A, so that we get the required result

A = AX.

It may be remarked that we should have reached the
same final result if our original premise had been of the
form

A = ABI AC-I- IAQ.

The difference of meaning is that all B's need not now
be A's, nor all C's, &c. But we should still have
A = ABX I ACX I I AQX = AX.

We can always prove a proposition, if we find it more
convenient, by proving its equivalent. To assert that all
not-B's are not- A's, is exactly the same as to assert that all
A's are B's. Accordingly we may ascertain that A = AB
by first ascertaining that b = ab. If we observe, for in
stance, that all substances which are not solids are also
not capable of double refraction, it follows necessarily
that all double refracting substances are solids. We may
convince ourselves that all electric substances are noncon
ductors of electricity, by reflecting that all good conduc
tors do not, and in fact cannot, retain electric excitation.
When we come to questions of probability it will be found
desirable to prove, as far as possible, both the original
proposition and its equivalent, as there is then an increased
area of observation.

The number of alternatives which may arise in the
division of a class varies greatly, and may be any number
from two upwards. Thus it is probable that every sub
stance is either magnetic or diamagnetic, and no substance
can be both at the same time. The division then must be
made in the form

A = ABcl-A&C.

If now we can prove that all magnetic substances
are capable of polarity, say B = BC, and also that all

152 TEE PRINCIPLES OF SCIENCE.

diamagnetic substances are capable of polarity C = CD, it
follows by substitution that all substances are capable of
polarity, or A = AD. We may divide the class substance
again into the three subclasses, solid, liquid, and gas;
and if we can show that in each of these forms it obeys
Carnot's thermodynamic law, it follows that all substances
obey that law. Similarly we may show that all verte
brate animals possess red blood, if we can show separately
that fish, reptiles, birds, marsupials, and mammals possess
red blood, there being, as far as is known, only five
principal subclasses of vertebrata.

Our inductions will often be embarrassed by exceptions,
real or apparent. We might affirm that all gems are
incombustible were not diamond undoubtedly combustible.
Nothing seems more evident than that all the metals are
opaque until we examine them in fine films, when gold
and silver are found to be transparent. All plants absorb
carbonic acid except certain fungi ; all the bodies of the
planetary system have a progressive motion from west to
east, except the satellites of Uranus and Neptune. Even
some of the profoundest laws of matter are not quite
universal ; all solids expand by heat except india-rubber,
and possibly a few other substances ; all liquids which
have been tested expand by heat except water below 4°C
and fused bismuth ; all gases have a coefficient of expan
sion increasing with the temperature except hydrogen.
In a later chapter I shall consider how such anomalous
cases may be regarded and classified ; here we have only
to express them in a consistent manner in our nota
tion.

Let us take the case of the transparency of metals, and
assign the terms thus

A = metal D = iron

B = gold E, F &c. = copper, lead, &c.

C = silver X = opaque.

INDUCTION. 153

Our premises will be

A = BIG! DIE, &c.

B = Baj

C = Cx

D = DX

E = EX,
and so on for the rest of the metals. Now evidently

Kbc = (D-I-EIFI )lc,

ar.d by substitution as before we shall obtain

Abe = A&cX,

or in words, ' All metals not gold nor silver are opaque ; '
at the same time we have

A(BI C) = ABI AC - ABaH- ACa = A(BI C)x,
or ' Metals which are either gold or silver are not opaque.'
In some cases the problem of induction assumes a much
higher degee of complexity. If we examine the properties
of crystallized substances we may find some properties
which are common to all, as cleavage or fracture in definite
planes ; but it would soon become requisite to break up
the class into several minor ones. We should divide
crystals according to the seven accepted systems — and
we should then find that crystals of each system possess
many common properties. Thus crystals of the Eegular
or Cubical system expand equally by heat, conduct heat
and electricity with uniform rapidity, and are of like
elasticity in all directions ; they have but one index of
refraction for light ; and every facet is repeated in like
relation to each of the three axes. Crystals of the system
which possess one principal axis will be found to possess
the various physical powers of conduction, refraction,
elasticity, &c., uniformly in directions perpendicular to
the principal axis, but in other directions their properties
vary according to complicated laws. The remaining systems
in which the crystals possess three unequal axes, or have
inclined axes, exhibit still more complicated results, the

154 THE PRINCIPLES OF SCIENCE.

effects of the crystal upon light, heat, electricity, &c.,
varying in all directions. But when we pursue induction
into the intricacies of its application to Nature we really
enter upon the subject of classification which we must
take up again in a later part of this work.

Complete Solution of the Inverse or Inductive Logical

Problem.

It is now plain that Induction consists in passing back
from a series of combinations to the laws by which such
combinations are governed. The natural law that all
metals are conductors of electricity really means that in
nature we find three classes of objects, namely —

1 . Metals, conductors ;

2. Not-metals, conductors ;

3. Not-metals, not-conductors.

It comes to the same thing if we say that it excludes the
existence of the class, 'metals not-conductors.' In the
same way every other law or group of laws will really
mean the exclusion from existence of certain combinations
of the things, circumstances or phenomena governed by
those laws. Now in logic we treat not the phenomena
and laws but, strictly speaking, the general forms of the
laws ; and a little consideration will show that for a finite
number of things the possible number of forms or kinds
of law governing them must also be finite. Using general
terms we know that A and B can be present or absent in
four ways and no more — thus

AB, Aft, aB, ab •

therefore every possible law which can exist concerning
the relation of A and B must be marked by the exclusion
of one or more of the above combinations. The number
of possible laws then cannot exceed the number of selec
tions which we can make from these four combinations,

INDUCTION.

155

and we arrive at this utmost number of cases by omitting
any one or more of the four. The number of cases to
be considered is therefore 2x2x2x2 or sixteen, since
each may be present or absent ; and these cases are all
shown in the following table, in which the sign o indicates
absence or non-existence of the combination shown at the
left-hand column in the same line, and the mark i its
presence :—

1

2

3

4

5

6

7

*

8

*

9

V

11

12

*

13

14

*

15

*

16

AB

o

o

o

0

o

o -

0

o

I

i

I

i

i

i

i

i

A6

o

0

o

0

I

I

I

I

o

o

o

o

i

i

i

i

aB

o

o

I

I

o

o

I

I

o

o

I

I

o

o

i

i

ab

0

I

0

I

o

I

o

I

0

I

o

I

0

I

o

i

Thus in column sixteen we find that all the conceivable
combinations are present, which means that there are no
special laws in existence in such a case, and that the
combinations are governed only by the universal Laws of
Identity and Difference. The example of metals and
conductors of electricity would be represented by the
twelfth column ; and every other mode in which two
things or qualities might present themselves is shown in
one or other of the columns. More than half the cases
may indeed be at once rejected, because they involve the
entire absence of a term or its negative. It has been
shown to be a necessary logical principle that every term
must have its negative (p. 88), and where this is not the
case some inconsistency between the laws or conditions of
combinations must exist. Thus if we laid down the two
following propositions, * Graphite conducts electricity,'
and ' Graphite does not conduct electricity/ it would
amount to asserting the impossibility of graphite existing
at all ; or in general terms, A is B and A is not B result
in destroying altogether the combinations containing A.

156 THE PRINCIPLES OF SCIENCE.

We therefore restrict our attention to those cases which
may be represented in natural phenomena where at least
two combinations are present, and which correspond to
those columns of the table in which each of A, a, B, b
appears. These cases are shown in the columns marked
with an asterisk.

We find that seven cases remain for examination, thus
characterised—

Four cases exhibiting three combinations,
Two cases exhibiting two combinations,
One case exhibiting four combinations.
It has already been pointed out that a proposition of the
form A = AB destroys one combination A6, so that this
is the form of law applying to the twelfth case. But
by changing one or more of the terms in A = AB into
its negative, or by interchanging A and B, a and b, we
obtain no less than eight different varieties of the one form ;
thus—

1 2th case. 8th case. i^tb case. i4th case.

A = AB A — A.b « = «B a = ab

b = ab B = aB b = A.b B = AB.

But the reader of the preceding sections will at once
see that each proposition in the lower line is logically equi
valent to, and is in fact the contrapositive of, that above
it (p. 98). Thus the propositions A = A& and B = aB
both give the same combinations, shown in the eighth
column of the table, and trial shows that the twelfth,
eighth, fifteenth and fourteenth cases are thus fully ac
counted for. We come to this conclusion then — The
general form of proposition A = AB admits of four
logically distinct varieties, each capable of expression in
two different modes.

In two columns of the table, namely the seventh and
tenth, we observe that two combinations are missing. Now
a simple identity A = B renders impossible both Kb and «B,

INDUCTION. 157

accounting for the tenth case ; and if we change B into b
the identity A = b accounts for the seventh case. There
may indeed be two other varieties of the simple identity,
namely a = b and a = B ; but it has already been shown
repeatedly that these are equivalent respectively to A = B
and A.~b (pp. 133, 134). As the sixteenth column has
already been accounted for as governed by no special
conditions, we come to the following general conclusion : — •
The laws governing the combinations of two terms must
be capable of expression either in a partial identity
(A — AB), or a simple identity (A = B) ; the partial
identity is capable of only four logically distinct varieties,
and the simple identity of two. Every logical relation
between two terms must be expressed in one of these
six laws, or must be logically equivalent to one of them.

In short, we may conclude that in treating of partial
and complete identity, we have exhaustively treated the
modes in which two terms or classes of objects can be
related. Of any two classes it may be said that one must
either be included in the other, or must be identical with
it, or some similar relation must exist between one class
and the negative of the other. We have thus completely
solved the inverse logical problem concerning two terms d.

The Inverse Logical Problem involving Three Terms.

No sooner do we introduce into the problem a third
term C, than the investigation assumes a far more com
plex character, so that some readers may prefer to pass
over this section. Three terms and their negatives may be
combined, as we have frequently seen, in eight different

d The contents of this and the following section nearly correspond
with those of a paper read before the Manchester Literary and Philosophical
Society on December 26th, 1871. See Proceedings of the Society, vol. xi.
pp. 65-68, and Memoirs, Third Series, vol. v. pp. 119-130.

158 THE PRINCIPLES OF SCIENCE.

combinations, and the effect of laws or logical conditions
is to destroy any one or more of these combinations.
Now we may make selections from eight things in 28 or
256 ways ; so that we have no less than 256 different cases
to treat, and the complete solution is at least fifty times
as troublesome as with two terms. Many series of com
binations, indeed, are contradictory, as in the simpler
problem, and may be passed over. The test of consistency
is that each of the letters A, B, C, a, b, c shall appear
somewhere in the series of combinations ; but I have
not been able to discover any mode of calculating the
number of cases in which inconsistency would happen.
The logical complexity of the problem is so great that
the ordinary modes of calculating numbers of combinations
in mathematical science fail to give any aid, and ex
haustive examination of the combinations in detail is
the only method applicable.

My mode of solving the problem was as follows : —
Having written out the whole of the 256 series of com
binations, I examined them separately and struck out
such as did not fulfil the test of consistency. I then chose
some common form of proposition involving two or three
terms, and varied it in every possible manner, both by
the circular interchange of letters (A, B, C into B, C, A
and then C, A, B), and by the substitution for any one or
more of the terms of the corresponding negative terms.
For instance, the proposition AB = ABC can be first varied
by circular interchange, so as to give BC = BCA and then
C A = CAB. Each of these three can then be thrown
into eight varieties by negative change. Thus AB = ABC
gives aB = aBC, Afr = A6C, AB = ABc, db = abC, and so on.
Thus there may possibly exist no less than twenty-four
varieties of the law having the general form AB = ABC,
meaning that whatever has the properties of A and B has
those also of C. It by no means follows that some of the

INDUCTION. 159

varieties may not be equivalent to others ; and trial
shows, in fact, that AB = ABC is exactly the same in
meaning as Ac = A5c or Be = Bca. Thus the law in
question has but eight varieties of distinct logical mean
ing. I now ascertain by actual deductive reasoning which
of the 256 series of combinations result from each of
these distinct laws, and mark them off as soon as found.
I now proceed to some other form of law, for instance
A = ABC, meaning that whatever has the qualities of A has
those also of B and C. I find that it admits of twenty-
four variations, all of which are found to be logically
distinct ; the combinations being worked out, I am able
to mark off twenty-four more of the list of 256 series. I
proceed in this way to work out the results of every form of
law which I can find or invent. If in the course of this
work I obtain any series of combinations which had been
previously marked off, I learn at once that the law is
logically equivalent to some law previously treated. It
may be safely inferred that every variety of the ap
parently new law will coincide in meaning with some
variety of the former expression of the same law. I
have sufficiently verified this assumption in some cases
and have never found it lead to error. Thus just as
AB = ABC is equivalent to Ac = A6c, so we find that
ab = ab(j is equivalent to ac = acB.

Among the laws treated were the two A = AB and
A = B which involve only two terms, because it may of
course happen that among three things two only are
in special logical relation, and the third independent ; and
the series of combinations representing such cases of
relation are sure to occur in the complete enumeration.
All single propositions which I could invent having been
treated, pairs of propositions were next investigated.
Thus we have the relations, 'All A's are B's and all
B's are C's,' of which the old logical syllogism is the

160 THE PRINCIPLES OF SCIENCE.

development. We may also have ' all A's are all B's,
and all B's are C's,' or even ' all A's are all B's, and all
B's are all C's.' All such premises admit of variations,
greater or less in number, the logical distinctness of which
can only be determined by trial in detail. Disjunctive
propositions either singly or in pairs were also treated,
but were often found to be equivalent to other propo
sitions of a simpler form ; thus A — ABC I Kbc is exactly
the same in meaning as AB = AC.

This mode of exhaustive trial bears some analogy to
that ancient mathematical process called the Sieve of
Eratosthenes. Having taken a long series of the natural
numbers, Eratosthenes is said to have calculated out in
succession all the multiples of every number, and to have
marked them off, so that at last the prime numbers alone
remained, and the factors of every number were exhaus
tively discovered. My problem of 2 56 series of combinations
is the logical analogue, the chief points of difference being
that there is a limit to the number of cases, and that prime
numbers have no analogue in logic, since every series of
combinations corresponds to a law or group of conditions.
But the analogy is perfect in the point that they are
both inverse processes. There is no mode of ascertaining
that a number is prime but by showing that it is not
the product of any assignable factors. So there is no
mode of ascertaining what laws are embodied in any
series of combinations but trying exhaustively the laws
which would give them. Just as the results of Erato
sthenes' method have been worked out to a great extent
and registered in tables for the convenience of other
mathematicians, I have endeavoured to work out the
inverse logical problem to the utmost extent which is
at present practicable or useful.

I have thus found that there are altogether fifteen
conditions or series of conditions which may govern

IND UCTION.

161

the combinations of three terms, forming the premises
of fifteen essentially different kinds of arguments. The
following table contains a statement of these conditions,
together with the number of combinations which are
contradicted or destroyed by each, and the number of
logically distinct variations of which the law is capable.
There might be also added, as a sixteenth case, that case
where no special logical condition exists, so that all the
eight combinations remain.

Reference
Number.

Propositions expressing the general
type of the logical conditions.

Number of dis
tinct logical
variations.

Number of
combinations
contradicted
by each.

I.

A = B

6

4

II.

A = AB

12

2

III.

A = B, B = C

4

6

IV.

A = B, B = BC

24

5

V.

A = AB, B = BC

24

4

VI.

A = BC

24

4

VII.

A = ABC

24

3

VIII.

AB=ABC

8

i

IX.

A = AB, «B = aBc

24

3

X,

A = ABC, ab = a&C

8

4

XI.

AB = ABC, ab = abc

4

2

XII.

AB = AC

12

2

XIII.

A = BC-i-A6c

8

3

XIV.

A = BC-|-6c

2

4

XV.

A = ABC, a = Bc-|-b<7

8

5

192

There are sixty-three series of combinations derived from
self-contradictory premises, which with the above 192 series
and the one case where there are no conditions or laws at
all, make up the whole conceivable number of 256 series.

We learn from this table, for instance, that two pro
positions of the form A = AB, B-BC, which are such
as constitute the premises of the old syllogism Barbara,
negative or render impossible four of the eight combi
nations in which three terms may be united, and that
these propositions are capable of taking twenty-four vari
ations by transpositions of the terms or the introduction

M

162 THE PRINCIPLES OF SCIENCE.

of negatives. This table then presents the results of
a complete analysis of all the possible logical relations
arising in the case of three terms, and the old syllogism
forms but one out of fifteen typical forms. Generally
speaking every form can be converted into apparently
different propositions ; thus the fourth type A = B, B = BO
may appear in the form A = ABC, a = ab, or again in the
form of three propositions A = AB, B = BC, aB = aBc ; but
all these sets of premises yield identically the same series
of combinations, and are therefore of exactly equivalent
logical meaning. The fifth type, or Barbara, can also be
thrown into the equivalent forms A = ABC, aB = aBC and
A = AC, B = A I aBC. In other cases I have obtained the
very same logical conditions in four modes of statement
As regards mere appearance and mode of statement, the
number of possible premises would be almost unlimited.

The most remarkable of all the types of logical condition
is the fourteenth, namely A = BC I be. It is that which
expresses the division of a genus into two doubly marked
species, and might be illustrated by the example — ' Com
ponent of the physical universe = matter, gravitating, or
not-mat'ter (ether), not-gravitating.'

It is capable of only two distinct logical variations,
namely, A = BC I be and A = Be I bC. By transposition
or negative change of the letters we can indeed obtain
six different expressions of each of these propositions ;
but when their meanings are analysed, by working' out
the combinations, they are found to be logically equiva
lent to one or other of the above two. Thus the proposi
tion A = BC-|-&c can be written in any of the following
five other modes,

a = KM- Be, B = CA-|-ca, 6=cA-|-Ca,
C = AB •!•«&, c=aB-\-Ab.

I do not think it needful at present to publish the
complete table of 193 series of combinations and the

INDUCTION.

1G3

premises corresponding to each. Such a table enables us
by mere inspection to learn the laws obeyed by any set of
combinations of three things, and is to logic what a table
of factors and prime numbers is to the theory of numbers,
or a table of integrals to the higher mathematics. The
table already given (p. 1 6 1 ) would enable a person with
but little labour to discover the law of any combinations.
If there be seven combinations (one contradicted) the law
must be of the eighth type, and the proper variety will be
apparent. If there be six combinations (two contradicted),
either the second, eleventh, or twelfth type applies, and a
certain number of trials will disclose the proper type and
variety. If there be but two combinations the law must
be of the third type, and so on.

The above investigations are complete as regards the
possible logical relations of two or three terms. But
when we attempt to apply the same kind of method to
the relations of four or more terms, the labour becomes im
practicably great. Four terms give sixteen combinations
compatible with the laws of thought, and the number of
possible selections of combinations is no less than 21" or
6o3536. The following table shows the extraordinary
manner in which the number of possible logical relations
increases with the number of terms involved.

Number of
terms.

Number of
possible com
binations.

Number of possible selections of combi
nations corresponding to consistent or in
consistent logical relations.

2

3

4
5
6

4
8
16

32
64

16
256

65,536
4,294,967,296
18,446,744,073,709,551,616

Some years of continuous labour would be required to
ascertain the precise number of types of laws which may
govern the combinations of only four things, and but a
small part of such laws would be exemplified or capable

M 2

164 THE PRINCIPLES OF SCIENCE.

of practical application in science. The purely logical
inverse problem, whereby we pass from combinations to
their laws, is solved in the preceding pages, as far as it
is likely to be for a long time to come ; and it is almost
impossible that it should ever be carried more than a
single step further.

Distinction between Perfect and Imperfect Induction.

We cannot proceed further with advantage, before
noticing the extreme difference which exists between
cases of perfect and those of imperfect induction. We
call an induction perfect when all the objects or combi
nations of events which can possibly come under the class
treated have been examined. But in the majority of
cases it is impossible to collect together, or in any way to
investigate, the properties of all portions of a substance or
of all the individuals of a race. The number of objects
would often be practically infinite, and the greater part of
them might be beyond our reach, in the interior of the
earth, or in the most distant parts of the Universe. In all
such cases induction is said to be imperfect, and affected
by more or less uncertainty. As some writers have fallen
into much error concerning the functions and relative
importance of these two branches of reasoning, I shall
have to point out that —

1. Perfect Induction is a process absolutely requisite,

both in the performance of imperfect induction and
in the treatment of large bodies of facts of which
our knowledge is complete.

2. Imperfect Induction is founded on Perfect Induction,

but involves another process of inference of a
widely different character.

It is certain that if I can draw any inference at all
concerning objects not examined, it must be done on the

INDUCTION.

data afforded by the objects which have been examined.
If I judge that a distant star obeys the law of gravity,
it must be because all other material objects sufficiently
known to me obey that law. If I venture to assert that
all ruminant animals have cloven hoofs, it is because all
ruminant animals which have come to my notice have
cloven hoofs. On the other hand I cannot safely say
that all cryptogamous plants possess a purely cellular
structure, because some such plants have a partially
vascular structure. The probability that a new crypto
gam will be cellular only can be estimated, if at all, on the
ground of the comparative numbers of known cryptogams
which are and are not cellular. Thus the first step in
every induction will consist in accurately summing up
the number of instances of a particular object or pheno
menon which have fallen under our observation. Adams
and Leverrier, for instance, must have inferred that the
undiscovered planet Neptune would obey Bode's law,
because all the planets known at that time obeyed it. On
what principles and on what circumstances the passage
from the known to the apparently unknown is warranted,
must be carefully discussed in the next section, and in
various parts of this work.

It would be a great mistake, however, to suppose that
Perfect Induction is in itself useless. Even when the
enumeration of objects belonging to any class is complete,
and admits of no inference to unexamined objects, the
enumeration of our knowledge in a general proposition is
a process of so much importance that we may consider it
practically necessary. In many cases we may render our
investigations exhaustive ; all the teeth or bones of an
animal ; all the cells in a minute vegetable organ ; all the
caves in a mountain side ; all the strata in a geological
section ; all the coins in a newly found hoard, may be so
completely scrutinized that we may make some general

166 THE PRINCIPLES OF SCIENCE.

assertion concerning them without fear of mistake. Every
bone might be proved to consist of phosphate of lime ;
every cell to enclose a nucleus; every cave to contain
remains of extinct animals ; every stratum to exhibit signs
of marine origin ; every coin to be of Roman manufacture.
These are cases where our investigation is limited to a
definite portion of matter, or a definite area on the earth's
surface.

There is another class of cases where induction is
naturally and necessarily limited to a definite number of
alternatives. Of the regular solids we can say without
the least doubt that no one has more than twenty faces,
thirty edges, and twenty corners ; for by the principles
of geometry we learn that there cannot exist more than
five regular solids, of each of which we easily observe
that the above statements are true. In the theory of
numbers, an endless variety of perfect inductions might
be made ; we can show that no number less than sixty
possesses so many divisors, and the like is true of 360 e,
for it does not require any very great amount of labour to
ascertain and count all the divisors of numbers up to sixty
or 360. Similarly I can assert that between 60,041 and
60,077 no prime number occurs, because the exhaustive
examination of those who have constructed tables of prime
numbers proves it to be so.

In matters of human appointment or history, we can
frequently have a complete limitation to the numbers of
instances to be included in an induction. We might show
that none of the other kings of England reigned so long as
George III ; that Magna Charta has not been repealed by
any subsequent statute ; that the propositions of the third
book of Euclid treat only of circles ; that no part of the
works of Galen mentions the fourth figure of the syl-

« Wallie's 'Treatise of Algebra' (1685), p. 22.

INDUCTION. 167

logism ; that the price of corn in England has never been
so high since 1847 as ^ was m that year ; that the price
of the English funds has never been lower than it was on
the 23rd of January, 1798, when it fell to 47 J.

It has been urged against this process of Perfect In
duction that it gives no new information, and is merely a
summing up in a brief form of a multitude of particulars.
But mere abbreviation of mental labour is one of the
most important aids we can enjoy in the acquisition of
knowledge. The powers of the human mind are so limited
that multiplicity of detail is alone sufficient to prevent its
progress in many directions. Thought would be prac
tically impossible if every separate fact had to be separately
thought and treated. Economy of mental powTer may be
considered one of the main conditions on which our ele
vated intellectual position depends. Most mathematical
processes are but abbreviations of the simpler acts of
addition and subtraction. The invention of logarithms
was one of the most striking additions ever made to
human power : yet it was a mere abbreviation of oper
ations which could have been done before had a sufficient
amount of labour been available. Similar additions to
our power will, it is hoped, be made from time to time,
for the number of mathematical problems hitherto solved
is but an indefinitely small portion of those which await
solution, because the labour they have hitherto demanded
renders them impracticable. So it is really throughout
all regions of thought. The amount of our knowledge
depends upon our powers of bringing it within prac
ticable compass. Unless we arrange and classify facts,
and condense them into general truths, they soon sur
pass our powers of memory, and serve but to confuse.
Hence Perfect Induction, even as a process of abbrevi
ation, is absolutely essential to any high degree of
mental achievement.

168 THE PRINCIPLES OF SCIENCE.

Transition from Perfect to Imperfect Induction.

It is a question of profound difficulty on what grounds
we are warranted in inferring the future from the present,
or the nature of undiscovered objects from those which we
have examined with our senses. We pass from Perfect to
Imperfect Induction when once we allow our conclusion to
pass, at all events apparently, beyond the data on which it
was founded. In making such a step we seem to gain a
nett addition to our knowledge ; for we learn the nature
of what was unknown. We reap where we have never
sown. We appear to possess the divine power of creating
knowledge, and reaching with our mental arms far beyond
the sphere of our own observation. I shall, indeed, have
to point out certain methods of reasoning in which we
do pass altogether beyond the sphere of the senses, and
acquire accurate knowledge which observation could never
have given ; but it is not imperfect induction that ac
complishes such a task. Of imperfect induction itself, I
venture to assert that it never makes any real addition to
our knowledge, in the meaning of the expression sometimes
accepted. As in other cases of inference it merely unfolds
the information contained in past observations or events ;
it merely renders explicit what was implicit in previous
experience. It transmutes knowledge, but certainly does
not create knowledge.

There is no fact which I shall more constantly keep
before the reader's mind in the following pages than that
the results of imperfect induction, however well authenti
cated and verified, are never more than probable. We
never can be sure that the future will be as the present.
We hang ever upon the Will of the Creator : and it is only
so far as He has created two things alike, or maintains
the framework of the world unchanged from moment to

INDUCTION. 160

moment, that our most careful inferences can be fulfilled.
All predictions, all inferences which reach beyond their
data, are purely hypothetical, and proceed on the assump
tion that new events will conform to the conditions
detected in our observation of past events. No experience
of finite duration can be expected to give an exhaustive
knowledge of all the forces which are in operation. There
is thus a double uncertainty ; even supposing the Uni
verse as a whole to proceed unchanged, we do not really
know the Universe as a whole. Comparatively speaking
we know only a point in its infinite extent, and a moment
in its infinite duration. We cannot be sure, then, that our
observations have not escaped some fact, which will cause
the future to be apparently different from the past ; nor
can we be sure that the future really will be the outcome
of the past. We proceed then in all our inferences to
unexamined objects and times on the assumptions —

1. That our past observation gives us a complete know

ledge of what exists.

2. That the conditions of things which did exist will

continue to be the conditions of things which will

exist.

We shall often need to illustrate the character of our
knowledge of nature by the simile of a ballot-box, so
often employed by mathematical writers in the theory of
probability. Nature is to us like an infinite ballot-box,
the contents of which are being continually drawn, ball
after ball, and exhibited to us. Science is but the careful
observation of the succession in which balls of various
character usually present themselves ; we register the
combinations, notice those which seem to be excluded from
occurrence, and from the proportional frequency of those
which usually appear we infer the probable character of
future drawings. But under such circumstances certainty
of prediction depends on two conditions : —

170 THE PRINCIPLES OF SCIENCE.

1. That we acquire a perfect knowledge of the compara

tive numbers of -balls of each kind within the box.

2. That the contents of the ballot-box remain unchanged.
Of the latter assumption, or rather that concerning the

constitution of the world which it illustrates, the logician
or physicist can have nothing to say. As the Creation of
the Universe is necessarily an act passing all experience
and all conception, so any change in that Creation, or, it
may be, a termination of it, must likewise be infinitely be
yond the bounds of our mental faculties. No science, no
reasoning upon the subject, can have any validity ; for
without experience we are without the basis and materials
of knowledge. It is the fundamental postulate accordingly
of all inference concerning the future, that there shall be
no arbitrary change in the subject of inference ; of the pro
bability or improbability of such a change I conceive that
our faculties can give no estimate.

The other condition of inductive inference — that we

acquire an approximately complete knowledge of the

combinations in which events do occur, is at least in some

degree within the bounds of our perceptive and mental

powers. There are many branches of science in which

phenomena seem to be governed by conditions of a most

fixed and general character. We have much ground in

such cases for believing that the future occurrence of

such phenomena may be calculated and predicted. But

the whole question now becomes one of probability and

improbability. We leave the region of pure logic to enter

one in which the number of events is the ground of

inference. We do not leave the region of logic ; we only

leave that where certainty, affirmative or negative, is the

result, and the agreement or disagreement of qualities the

means of inference. For the future, number and quantity

will enter into most of our processes of reasoning ; but then

I hold that number and quantity are but portions of the

INDUCTION. 171

great logical domain. I venture to assert that number is
wholly logical, both in its fundamental nature and in all
its complex developments. Quantity in all its forms is but
a development of number. That which is mathematical
is not the less logical ; if anything it is the more logical,
in the sense that it presents logical results in the highest
degree of complexity and variety.

Before proceeding then from Perfect to Imperfect In
duction I break off in some degree the course of the work,
to treat of the logical conditions of number. I shall then
employ number to estimate the variety of combinations
in which natural phenomena may present themselves, and
the probability or improbability of their occurrence under
definite circumstances. It is in later parts of the work
that I must endeavour to establish, in a complete manner,
the notions which I have set forth upon the subject of
Imperfect Induction, as applied in the investigation of
Nature, which notions may be thus briefly stated:—

1. Imperfect Induction entirely rests upon Perfect In

duction for its materials.

2. The logical process by which we seem to pass directly

from examined to unexamined cases consists in an
inverse and complex application of deductive in
ference, so that all reasoning may be said to be
either directly or inversely deductive.

3. The result is always of a hypothetical character, and

is never more than probable.

4. No nett addition is ever made to our knowledge by

reasoning ; what we know of future events or unex
amined objects is only the unfolded contents of
our previous knowledge, and it becomes less and
less probable as it is more boldly extended to re
mote cases.

BOOK II,

CHAPTER VIII.

PRINCIPLES OF NUMBER.

NOT without much reason did Pythagoras represent the
world as ruled by number. Into almost all our acts of
clear thought number enters, and in proportion as we can
define numerically we enjoy exact and useful knowledge
of the Universe. The science of numbers, too, the study of
the principles and methods of reasoning in number, has
hitherto presented the widest and most practicable train
ing in logic. So free and energetic has been the study of
mathematical forms, compared with the forms and laws of
logic, that mathematicians have passed far in advance of
any pure logicians. Occasionally, in recent times, they have
condescended to apply their great algebraic instruments
to a reflex advancement of the primary logical science. It
is thus that we chiefly owe to profound mathematicians,
such as Sir John Herschel, Dr. Whewell, Professor De
Morgan or Dr. Boole, the regeneration of logic in the
present century, and I entertain no doubt that it is in
maintaining a close alliance with the extensive branches of
quantitative reasoning that we must look for still further
progress in our comprehension of qualitative inference.

I cannot assent, indeed, to the common notion that

PRINCIPLES OF NUMBER. 173

certainty begins and ends with numerical determination.
Nothing is more certain and accurate than logical truth.
The laws of identity and difference are the tests of all
that is true and certain throughout the range of thought,
and mathematical reasoning is cogent only when it con
forms to these conditions, of which logic is the first
development. And if it be erroneous to suppose that all
certainty is mathematical, it is equally an error to imagine
that all which is mathematical is certain. Many processes
of mathematical reasoning are of most doubtful validity.
There are many points of mathematical doctrine which are
and must long remain matter of opinion ; for instance, the
best form of the definition and axiom concerning parallel
lines, or the true nature of a limit or a ratio of infinitesimal
quantities. In the use of symbolic reasoning questions
occur at every point on which the best mathematicians
may differ, as Bernouilli and Leibnitz differed irreconcile-
ably concerning the existence of the logarithms of ne
gative quantities a. In fact we no sooner leave the simple
logical conditions of number, than we find ourselves in
volved in a mazy and mysterious science of symbols.

Mathematical science enjoys no monopoly, and not even
a supremacy in certainty of results. It is the boundless
extent and variety of quantitative questions that surprises
and delights the mathematical student. When simple
logic can give but a bare answer Yes or No, the algebraist
raises a score of subtle questions, and brings out a score
of curious results. The flower and the fruit, all that is
attractive and delightful, fall to the share of the mathe
matician, who too often despises the pure but necessary
stem from which all has arisen. But in no part of human
thought can a reasoner cast himself free from the prior
conditions of logical correctness. The mathematician is

a Montucla, ' Histoire cles Mathematiques,' vol. iii. p. 373.

174 THE PRINCIPLES OF SCIENCE.

only strong and true as long as he is logical, and if
numbers rule the world, it is the laws of logic which rule
number.

Nearly all writers have hitherto been strangely content
to look upon numerical reasoning as something wholly
apart from logical inference. A long divorce has existed
between quality and quantity, and it has not been un
common to treat them as contrasted in nature and re
stricted to independent branches of human thought. For
my own part, I have a profound belief that all the sciences
meet somewhere upon common ground. No part of know
ledge can stand wholly disconnected from other parts of
the great universe of thought ; it is incredible, above all,
that the two great branches of abstract science, interlac
ing and co-operating in every discourse, should rest upon
totally distinct foundations. I assume that a connection
exists, and care only to inquire, What is its nature 1 Does
the science of quantity rest upon that of quality ; or, vice
versd, does the science of quality rest upon that of
quantity I There might conceivably be a third view, that
they both rest upon some still deeper set of principles yet
undiscovered, but there is an absence of any sugges
tions to this effect. The late Dr. Boole adopted the second
view, and treated logic as a kind of algebra, — a special
case of analytical reasoning which admits but the two
quantities — unity and zero. He proved beyond doubt
that a deep analogy does exist between the forms of
algebraic and logical deduction ; and could this analogy
receive no other explanation we must have accepted his
opinion, however strange. But I shall attempt to show
that just the reverse explanation is the true one.

I hold that algebra is a highly developed logic, and
number but logical discrimination. Logic resembles al-

o o

gebra, as the mould resembles that which is cast in it.
Logic has imposed its own laws upon every branch of

PRINCIPLES OF NUMBER. 175

mathematical science, and it is no wonder that we ever
meet with the traces of those laws from the domain of
which we can never emerge.

The Nature of Number.

Number is but another name for diversity. Exact
identity is "unity, and with difference arises plurality.
An abstract notion, as was pointed out (p. 33), possesses
a certain oneness. The quality of justice, for instance, is
one and the same in whatever just acts it be manifested.
In justice itself there are no marks of difference by which
to discriminate justice from justice. But one just act can
be discriminated from another just act by many circum
stances of time and place, and we can count and number
many acts each thus discriminated from every other. In
like manner pure gold is simply pure gold, and is so far
one and the same throughout. But besides its intrinsic
and invariable qualities, gold occupies space and must
have shape or size. Portions of gold are always mutually
exclusive and capable of discrimination, at least in respect
that they must be each without the other. Hence they
may be numbered.

Plurality arises when and only when we detect differ
ence. For instance, in counting a number of gold coins
I must count each coin once, and not more than once. Let
C denote a coin, and the mark above it the position in
the order of counting. Then I must count the coins

I should make the third coin into two, and should imply
the existence of difference where there is not difference5.
C'" and C'" are but the names of one coin named twice

b 'Pure Logic,' Appendix, p. 82, § 192.

176 THE PRINCIPLES OF SCIENCE.

over. But according to one of the conditions of logical
symbols, which I have called the Law of Unity (p. 86),
the same name repeated has no effect, and

Al A = A.

We must apply the Law of Unity, and must reduce all
identical alternatives before we can count with certainty
and use the processes of numerical calculation. Identical
alternatives are harmless in logic, but produce deadly
error in number. Thus logical science ascertains the
nature of the mathematical unit, and the definition may
be given in these terms — A unit is any object of thought
which can be discriminated from every other object treated
as a unit in the same problem.

It has often been said that units are units in respect of
being perfectly similar to each other ; but though they
may be perfectly similar in some respects, they must be
different in at least one point, otherwise they would be
incapable of plurality. If three coins were so similar
that they occupied the same space at the same time, they
would not be three coins, but one. It is a property of
space that every point is discriminable from every other
point, and in time every moment is necessarily distinct
from any other moment before or after. Hence we fre
quently count in space or time, and Locke, with some
other philosophers, has even held that number arises from
repetition in time. Beats of a pendulum might be so
perfectly similar that we could discover no difference
except that one beat is before and another after. Time
alone is here the ground of difference and is a sufficient
foundation for the discrimination of plurality ; but it is
by no means the only foundation. Three coins are three
coins, whether we count them successively or regard them
all simultaneously. In many cases neither time nor space
is the ground of difference, but pure quality alone enters.
AVe can discriminate for instance the weight, inertia, and

PRINCIPLES OF NUMBER. 177

hardness of gold as three qualities, though none of these
is before or after the other, either in space or time.
Every means of discrimination may be a source of
plurality.

Our logical notation may be used to express the rise of
number. The symbol A stands for one thing or one class,
and in itself must be regarded as a unit, because no differ
ence is specified. But the combinations AB and A.b are
necessarily tivo, because they cannot logically coalesce, and
there is a mark B which distinguishes one from the other.
A logical definition of the number four is given in the
combinations ABC, ABc1, A.bC, Abe., where there is a double
difference, and as Puck says —

' Yet but three ? Come one more ;
Two of both kinds makes up four.'

I conceive that all numbers might be represented as
arising out of the combinations of the Abecedarium, more
or less of each series being struck out by various logical
conditions. The number three, for instance, arises from
the condition that A must be either B or C, so that the
combinations are ABC, ABc, A&C.

Of Numerical Abstraction.

There will now be little difficulty in forming a clear
notion of the nature of numerical abstraction. It consists
in abstracting the character of the difference from which
plurality arises, retaining merely the fact. When I speak
of three men I need not at once specify the marks by
which each may be known from each. Those marks must
exist if they are really three men and not one and the
same, and in speaking of them as many I imply the
existence of the requisite differences. Abstract number,
then, is the empty form of difference; the abstract

N

178 THE PRINCIPLES OF SCIENCE.

number three asserts the existence of marks without
specifying their kind.

Numerical abstraction is then a totally different process
from logical abstraction (see p. 33), for in the latter
process we drop out of notice the very existence of
difference and plurality. In forming the abstract notion
hardness, for instance, I drop out of notice altogether the
diverse circumstances in which the quality may appear.
It is the concrete notion three hard objects, which asserts
the existence of hardness along with sufficient other
undefined qualities, to mark out three such objects.
Numerical thought is indeed closely interwoven with
logical thought. We cannot use a concrete term in the
plural, as men, without implying that there are marks of
difference. Only when we use a term in the singular
and abstract sense man, do we deal with unity, unbroken
by difference.

The origin of the great generality of number is now
apparent. Three sounds differ from three colours, or
three riders from three horses ; but they agree in respect
of the variety of marks by which they can be discriminated.
The symbols i + i + i are thus the empty marks asserting
the fact of discrimination which may apply to objects
wholly independently of their peculiar nature.

Concrete and Abstract Numbers.

The common distinction between concrete and ab
stract numbers can now be easily stated. In proportion
as we specify the logical character of the things num
bered, we render them concrete. In the abstract num
ber three there is no statement of the points in which
the three objects agree ; but in three coins, three men, or
three horses, not only are the variety of objects defined,

PRINCIPLES OF NUMBER. 179

but their nature is restricted. Concrete number thus
implies the same consciousness of difference as abstract
number, but it is mingled with a groundwork of similarity
expressed in the logical terms. There is similarity or
identity so far as logical terms enter ; difference so far as
the terms are merely numerical.

The reason of the important Law of Homogeneity
will now be apparent. This law asserts that in every
arithmetical calculation the logical nature of the things
numbered must remain unaltered. The specified logical
agreement of the things numbered must not be affected by
the unspecified numerical differences. A calculation would
be palpably absurd which, after commencing with length,
gave a result in hours. It is in reality equally absurd in
a purely arithmetical point of view to deduce areas from
the calculation of lengths, masses from the combination of
volume and density, or momenta from mass and velocity.
It must remain for subsequent consideration in what sense
we may truly say that two linear feet multiplied by two
linear feet give four superficial feet, but arithmetically it
is absurd, because there is a change of unit.

As a general rule we treat in each calculation only
objects of one nature. We do not, and cannot properly
add, in the same sum yards of cloth and pounds of sugar.
We cannot even conceive the result of adding area to velo
city, or length to density, or weight to value. The unit
numbered and added must have a basis of homogeneity,
or must be reducible to some common denominator.
Nevertheless it is quite possible, and in fact common, to
treat in one complex calculation the most heterogeneous
quantities, on the condition that each kind of object is
kept distinct, and treated numerically only in conjunction
with its own kind. Different units, so far as their
logical differences are specified, must never be substituted
one for the other. Chemists continually use equations

N 2

180 THE PRINCIPLES OF SCIENCE.

which assert the equivalence of groups of atoms. Ordinary
fermentation is represented by the formula

C H1206= 2C2H60 + 2C02.

Three kinds of units, the atoms respectively of Carbon,
Hydrogen, and Oxygen, are here intermingled, but there is
really a separate equation in regard to each kind. Mathe
maticians also employ compound equations of the same
kind ; for in a + I N/ — i = c + d ^/ — i, it is impossible by
ordinary addition to add a to b ^/ — i . Hence we really
have the separate equations a = c, and b = dc. Similarly
an equation between two quaternions is equivalent to
four equations between ordinary quantities, whence in
deed the origin of the name quaternion.

Analogy of Logical and Numerical Terms.

If my assertion is correct that number arises out of
logical conditions, we ought to find number obeying all
the laws and conditions of logic. It is almost super
fluous to point out that this is the case with the funda
mental laws of identity and difference, and it only remains
for me to show that mathematical symbols do really obey
the special conditions of logical symbols which were
formerly pointed out (p. 39). Thus the Law of Com-
mutativeness, is equally true of quality and quantity. As
in logic we have

AB = BA,

so in mathematics it is familiarly known that
2 x 3 = 3 x 2, or x *y = y xx.

The properties of space, in short, are as indifferent in
pure multiplication as we found them in pure logical
thought.

c De Morgan's ' Trigonometry and Double Algebra,' p. 126.

PRINCIPLES OF NUMBER. 181

Similarly, just as in logic

triangle or square = square or triangle,
or generally A I B = B I A,

so in quantity 2+3 = 3 + 2,

or generally x + iy = y + x.

The symbol I is not identical with + , but it is so far
analogous.

How far, now, is it true that mathematical symbols

v

obey the law of simplicity expressed in the form

AA = A,
or the example

Round round = round ?

Apparently there are but two numbers which obey this
law ; for it is certain that

x x x — x
is true only in the two cases when x = i or o.

In reality all numbers obey the law, for 2 x 2 = 2 is not
really analogous to AA = A. According to the definition
of a unit already given, each unit is discriminated from
each other in the same problem, so that in 2' x 2 ', the
first two involves a different discrimination from the
second ttvo. I get four kinds of things, for instance, if I
first discriminate ' heavy and light ' and then ' cubical and
spherical/ for we now have the following classes-
heavy, cubical. light, cubical,
heavy, spherical. light, spherical.
But suppose that my two classes are in both cases
discriminated by the same difference of light and heavy,
then we have

hearvy heavy = heavy,
heavy light - o,
light heavy = o,
light light = light.

In short, twice two is two unless we take care that the
second two has a different meaning from the first. But

182 THE PRINCIPLES OF SCIENCE.

under similar circumstances logical terms would give
exactly the like result, and it is not true that A' A" = A',
identically where A" is different in meaning from A'.
In an exactly similar manner it may be shown that

the Law of Unity

A-I-A = A

holds true alike of logical and mathematical terms. It is
absurd indeed to say that

X + £C = X

except in the one case when x — absolute zero. But this
contradiction x + x = x arises from the fact that we have
already defined the unit in one x as differing from those in
the other. Under such circumstances the Law of Unity
does not apply. For if in

A'IA" = A'

we mean that A" is in any way different from A' the
assertion of identity is evidently false.

The contrast then which seems to exist between logical
and mathematical symbols is only apparent. It is because
the Law of Simplicity and Unity must always be ob
served in the operation of counting that those laws can
no longer be operative. This is the understood condition
under which we use all numerical symbols. Whenever
I use the symbol 5 I really mean

i + i + i + i + i,

and it is perfectly understood that each of these units is
distinct from each other. If requisite I might mark them
thus

i' + i" + i'" + i"" + i""'.
Were this not the case and were the units really

I'+i' + f' + i"^!"",
the Law of Unity would, as before remarked, apply, and

i"+l"=l".
Mathematical symbols then obey all the laws of logical

PRINCIPLES OF NUMBER. 183

symbols, but two of these laws seein to be inapplicable
simply because they are presupposed in the definition of
the mathematical unit. Logic thus lays down the con
ditions of number, and the science of arithmetic developed
as it is into all the wondrous branches of mathematical
calculus is but an outgrowth of logical discrimination.

Principle of Mathematical Inference.

As I have asserted, the universal principle of all
reasoning is that which allows us to substitute like for
like. I have now to point out that in the mathema
tical sciences this principle is involved in each step of
reasoning. It is in these sciences indeed that we meet
with the clearest cases of substitution, and it is the
simplicity with which the principle can be applied which
probably led to the comparatively early perfection of the
sciences of geometry and arithmetic. Euclid, and the
Greek mathematicians from the first, recognised equality
as the fundamental relation of quantitative thought, but
Aristotle rejected the exactly analogous, but far more
general relation of identity, and thus crippled the formal
science of logic as it has descended to the present day.

Geometrical reasoning starts from the Axiom that
'things equal to the same thing are equal to each other/
Two equalities enable us to infer a third equality ; and this
is true not only of lines and angles, but of areas, volumes,
numbers, intervals of time, forces, velocities, degrees of
intensity, or, in short, anything which is capable of being
equal or unequal. Two stars equally bright with the
same star must be equally bright with each other, and two
forces equally intense with a third force are equally
intense with each other. It is remarkable that Euclid
has not expressly stated two other axioms, the truth of
which is necessarily implied. The second axiom should

184 THE PRINCIPLES OF SCIENCE.

be that ' Two things of which one is equal and the other
unequal to a third common thing, are unequal to each
other/ An equality and inequality, in short, may give an
inequality, and this is equally true with the first axiom of
all kinds of quantity. If Venus, for instance, agrees with
Mars in density, but Mars differs from Jupiter, then Venus
differs from Jupiter. A third axiom must exist to the
effect that 'Things unequal to the same thing may or
may not be equal to each other.' Two inequalities give
no ground of inference tvhatever. If we only know, for
instance, that Mercury and Jupiter differ in density from
Mars, we cannot say whether or not they agree between
themselves. As a fact they do not agree ; but Venus and
Mais on the other hand both differ from Jupiter and yet
closely agree with each other. The force of the axioms
can be most clearly illustrated by drawing lines d.

The general conclusion must be then that where there is
equality there may be inference, but where there is not
equality there cannot be inference A plain induction will
lead us to believe that equality is the condition of inference
concerning quantity. All the three axioms may in fact
be summed up in one, to the effect, that ' in whatever
relation one quantity stands to another, it stands in the
same relation to the equal of that other.'

The active power is always the substitution of equals,
and it is an accident that in a pair of equalities we can
make the substitution in two ways. From a = b = cwe
can infer a = c, either by substituting in a = b the value of
b as given in b — c, or else by substituting in b = c the
value of b as given in a = b. In a — b ^ d we can make
but the one substitution of a for b. In e ^f^ g we can
make no substitution and get no inference.

In mathematics the relations in which terms may stand
to each other are far more varied than in pure logic, yet

'Elementary Lessons in Logic' (Macmillan), p. 123.

PRINCIPLES OF NUMBER. 185

our principle of substitution always holds true. We may
say in the most general manner that In whatever relation
one quantity stands to another, it stands in the same relation
to the equal of that other. In this axiom we sum up a
number of axioms which have been stated in more or less
detail by algebraists e. Thus. ' If equal quantities be
added to equal quantities, the sums will be equal.' To
explain this, let

a = b, c = d.

Now a + c, whatever it means, must be identical with

itself, so that

a + c — a + c.

In one side of this equation substitute for the quantities
their equivalents, and we have the axiom proved

a + c — l> + d.

The similar axiom concerning subtraction is equally evi
dent, for whatever a — c may mean it is equal to a — c,
and therefore by substitution to b — d. Again, ' if equal
quantities be multiplied by the same or equal quantities,
the products will be equal/ For evidently

ac = ac,
arid if for c in one side we substitute its equal d, we have

ac — ad,
and a second similar substitution gives us

ac — bd.

We might prove a like axiom concerning division in an
exactly similar manner. I might even extend the list of
axioms and say that ' Equal powers of equal number are
equal.' For certainly, whatever a x a x a may mean, it is
equal to a x a x a ; hence by our usual substitution

a*axa = bxbxb,
or a3 = Z>3.

The truth will hold of roots, that is to say,

y^~= */T,

e Todhunter's 'Algebra,' 3rd ed. p. 40.

186 THE PRINCIPLES OF SCIENCE.

provided that the same roots are taken, that is that the
root of a shall really be related to a as the root of b is
to I. The ambiguity of meaning of an operation thus fails
in any way to shake the universality of the principle.

We may go further and assert that, not only the above
common relations, but all other known or conceivable
mathematical relations obey the same principle. Let Pa
denote in the most general manner that we do something
with the quantity a ; then if a = b it follows that

Pa = P6.
Let us make Pa, for instance, mean

a3 -3 a2 + 2 a + 5 ;

then it necessarily follows that this quantity is exactly
equal to b3 - 3 b2 + 2 I + 5.

The reader will also remember that one of the most
frequent operations in mathematical reasoning is to sub
stitute for a quantity its equal, as known either by
assumed, natural, or self-evident condition. Whenever a
quantity appears twice over in a problem, we may apply
what we learn of its relations in one place to its relations
in the other. All reasoning in mathematics, as in other
branches of science, thus involves the principle of treating
equals equally, or similars similarly. In whatever way we
employ quantitative reasoning in the remaining parts of
this work, we never can desert the simple principle on
which we first set out.

Reasoning by Inequalities.

I have stated that all the processes of mathematical
reasoning may be deduced from the principle of substitution.
Exceptions to this assertion may seem to exist in the use
of inequalities. The greater of a greater is undoubtedly a
greater, and what is less than a less is certainly less.
Snowdon is higher than the Wrekin, and Ben Nevis than

PRINCIPLES OF NUMBER. 187

Snowdon ; therefore Ben Nevis is higher than the Wrekin.
But a little consideration discloses much reason for be
lieving that even in such cases, where equality does not
apparently enter, the force of the reasoning entirely
depends upon underlying and implied equalities.

In the first place, two statements of mere difference do
not give any ground of inference. We learn nothing
concerning the comparative heights of St. Paul's and
Westminster Abbey from the assertions that they both
differ in height from St. Peter's at Rome. Thus we need
something more than mere inequality ; we require one
identity in addition, namely the identity in direction of
the two differences. Thus we cannot employ inequalities
in the simple way in which we do equalities, and, when
we try to express exactly what other conditions are
requisite, we shall find ourselves lapsing into the use of
equalities or identities.

In the second place, every argument by inequalities may
be represented with at least equal clearness and force in
the form of equalities. Thus we clearly express that a
is greater than b by the equation

a = b + p, (i)

where p is an intrinsically positive quantity, denoting the
difference of a and b. Similarly we express that b is
greater than c by the equation

b = c + q, (2)

and substituting for b in (i) its value in (2) we have

a = c + q + p. (3)

Now as p and q are both positive, it follows that a is
greater than c, and we have the exact amount of excess
specified. It will be easily seen that the reasoning con
cerning that which is less than a less will result in an

equation of the form

c — a — q — p.
Every argument by inequalities may then be thrown

) 88 THE PRINCIPLES OF SCIENCE.

into the form of an equality ; but the converse is not true.
We cannot possibly prove that two quantities are equal by
merely asserting that they are both greater or both less
than another quantity. From e >f and g >f, or e <f and
g <f, we can infer no relation between e and g. And if the
reader take the equations x = y = 3 and attempt to prove
that therefore x = 3, by throwing them into inequalities, he
will find it impossible to do so.

From these considerations I gather that reasoning in
arithmetic or algebra by so-called inequalities is only an
imperfectly expressed reasoning by equalities, and when
we want to exhibit exactly and clearly the conditions of
reasoning, we are obliged to use equalities explicitly. Just
as in pure logic a negative proposition, as expressing mere
difference, cannot be the means of inference, so inequality
can never really be the true ground of inference. I do riot
deny that affirmation and negation, agreement and differ
ence, equality and inequality, are pairs of equally funda
mental relations, but I assert that inference is possible only
where affirmation, agreement, or equality, some species of
identity in fact, is present, explicitly or implicitly.

Arithmetical Reasoning.

It might seem somewhat inconsistent that I assert
number to arise out of difference or discrimination, and
yet hold that no reasoning can be grounded on difference.
Number, of course, opens a most wide sphere for inference,
and a little consideration shows that this is due to the
unlimited series of identities which spring up out of
numerical abstraction If six people are sitting on six
chairs, there is no resemblance between the chairs and the
people in logical character. But if we overlook all the
qualities both of a chair and a person, and merely re
member that there are marks bv which each of six chairs

PRINCIPLES OF NUMBER. 189

may be discriminated from each other, and similarly with
the people, then there arises a resemblance between the
chairs and people, and this resemblance in number may be
the ground of inference. If on another occasion the chairs
are filled by people again, we may infer that these people
must resemble the others in number, though they need not
resemble them in any other points.

Groups of units are what we really treat in arithmetic.
The number Jive is really i + i + i + i + i, but for the sake
of conciseness we substitute the more compact sign 5, or
the name five. These names being arbitrarily imposed in
any one manner, an indefinite variety of relations spring
up between them which are not in the least arbitrary. If
we define four asi + i+i + i, and five asi + i + i + i + i,
then of course it follows th&ijive - four + i ; but it would
be equally possible to take this latter equality as a defi
nition, in which case one of the former equalities would
become an inference. It is hardly requisite to decide how
we define the names of numbers, provided we remember
that out of the infinitely numerous relations of one number
to others, some one relation expressed in an equality
must be a definition of the number in question and the
other relations immediately become necessary inferences.

In the science of number the variety of classes which
can be formed is altogether infinite, and statements of
perfect generality may be made subject only to difficulty
or exception at the lower end of the scale. Every existing
number for instance belongs to the class

m + 7 ;

that is, every number must be the sum of another number
and seven, except of course the first six or seven numbers,
negative quantities not being here taken into account.
Every number is the half of some other, and so on. The
subject of generalization, as exhibited in arithmetical or
mathematical truths, is an indefinitely wide one. In

190 THE PRINCIPLES OF SCIENCE.

number we are only at the first step of an extensive
series of generalizations. A number is general as compared
with the particular things numbered, so we may have
general symbols for numbers, or general symbols not for
numbers, but for the relations between undetermined num
bers. There is, in fact, an unlimited hierarchy of successive
generalizations.

O

Numerically Definite Reasoning.

It was first discovered by Prof, de Morgan that many
arguments are valid which combine logical and numerical
reasoning, although they could in no way be included in
the ancient logical formulas. He developed the doctrine
of the 'Numerically Definite Syllogism,' fully explained
in his 'Formal Logic' (pp. 141-170). Dr. Boole also
devoted considerable attention to the determination of
what he called 'Statistical Conditions/ meaning the
numerical conditions of logical classes. In a paper pub
lished among the Memoirs of the Manchester Literary and
Philosophical Society, Third Series, vol. IV. p. 33°
(Session 1869-70), I have pointed out that we can apply
arithmetical calculation to the Logical Abecedarium.
Having given certain logical conditions and the numbers of
objects in certain classes, we can either determine the
number of objects in other classes governed by those con
ditions, or can show what further data are required to
determine them. As an example of the kind of questions
treated in numerical logic, and the mode of treatment, I
give the following problem suggested by De Morgan, with
my mode of representing its solution f .

f It has been pointed out to me by Mr. A. J. Ellis, F.R.S., that my
solution, as given in the Memoirs of the Manchester Philosophical Society,
does not exactly answer to the conditions of the problem, and I therefore
substitute above a more satisfactory solution.

PRINCIPLES OF NUMBER. 191

' For every man in the house there is a person who is
aged ; some of the men are not aged. It follows that
some persons in the house are not men?/
Now let A = person in house,

B = male,
C = aged.

By enclosing logical symbols in brackets, let us denote
the number of objects belonging to the class indicated by
the symbol. Thus let

(A) = number of persons in house,
(AB) — number of male persons in house,
(ABC) = number of aged male persons in house,
and so on. Now if we use w and w' to denote unknown
and indefinite numbers, the conditions of the problem may
be thus stated according to my interpretation of the
words —

(AB) = (AC) -w, (i)

that is to say, the number of persons in the house who are
aged is at least equal to, and may exceed, the number of
male persons in the house ;

(ABc) - u/, (2)

that is to say, the number of male persons in the house
who are not aged is some unknown positive quantity.

If we develop the terms in (i) by the Law of Duality
(pp. 87, 95, 97), we obtain

(ABC) + (ABc) = (ABC) + (A6C) - w.
Subtracting the common term (ABC) from each side and
substituting for (ABc) its value as given in (2), we get at
once

(A&C) - 10 + u/,
and adding (Abe) to each side, we have

(Ab) = Abe + 10 4 wf.

The meaning of this result is that the number of persons
in the house who are not men is at least equal to w + iu',

S 'Syllabus of a proposed System of Logic,' p. 29.

192 THE PRINCIPLES OF SCIENCE.

and exceeds it by the number of persons in the house who
are neither men nor aged (Abe).

It should be understood that this solution applies only
to the terms of the example quoted above, and not to the
general problem for which De Morgan intended it to serve
as an illustration.

As a second instance, let us take the following ques
tion : — The whole number of voters in a borough is a ; the
number against whom objections have been lodged by
liberals is I; and the number against whom objections
have been lodged by conservatives is c; required the
number, if any, who have been objected to on both sides.

Taking

A — voter,

B = objected to by liberals,

C — objected to by conservatives,

then we require the value of (ABC). Now the following
equation in identically true—

(ABC) - (AB) + (AC) + (A&c) - (A). (i)
For if we develop all the terms on the second side we
obtain
(ABC) - (ABC) + (ABc) + (ABC) + (A&C) + (Abe)

- (ABC) - (ABc) - (A&C) -- (Abe) ;

and striking out the corresponding positive and negative
terms, we have only left (ABC) = (ABC). Since then (i) is
necessarily true, we have only to insert the known values,
and we have

(ABC) = b + c - a + (Abe).

Hence the number who have received objections from both
sides is equal to the excess, if any, of the whole number
of objections over the number of voters together with the
numbers of voters who have received no objections (Abe).

In many cases classes of objects may exist under special
logical conditions, and we must consider how these con
ditions must be interpreted numerically.

PRINCIPLES OF NUMBER. 193

Every logical proposition or equation now gives rise
to a corresponding numerical equation. Sameness of
qualities occasions sameness of numbers. Hence if

A = B

denotes the identity of the qualities of A and B, we may
conclude that

(A) = (B).

It is evident that exactly those objects, and those objects
only, which are comprehended under A must be compre
hended under B. It follows that wherever we can draw
an equation of qualities, we can draw a similar equation of
numbers. Thus, from

A = B = C
we infer

A = C;
and similarly from

(A) = (B) = (C),

meaning the numbers of A's and C's are equal to the
number of B's, we can infer

(A) = (C).

But, curiously enough, this does not apply to negative
propositions and inequalities. For if

A = B - D

means that A is identical with B, which differs from D, it
does not follow that

(A) = (B) , (D).

Two classes of objects may differ in qualities, and yet they
may agree in number. This is a point which strongly
confirms me in the opinion I have already expressed,
that all inference really depends upon equations, not
differences (p. 186).

The Logical Abecedarium thus enables us to make a
complete analysis of any numerical problem, and though
the symbolical statement may sometimes seem prolix, I
conceive that it really represents the course which the

o

194 THE PRINCIPLES OF SCIENCE.

mind must follow in solving the question. Although
thought may seem to outstrip the rapidity with which the
symbols can be written down, yet the mind does not really
follow a different course from that indicated by the sym
bols. For a fuller explanation of this natural system of
Numerically Definite Reasoning, with more abundant
illustrations and an analysis of De Morgan's Numerically
Definite Syllogism, I must refer the reader to the paper in
the Memoirs of the Manchester Literary and Philosophical
Society, as already referred to, portions of which, how
ever, have been embodied in the present section.

The reader may be referred, also, to Boole's writings
upon the subject in the ' Laws of Thought,' chap. xix.
p. 295, and in a paper on ' Propositions Numerically De
finite,' communicated by De Morgan, in 1868, to the
Cambridge Philosophical Society, and printed in their
' Transactions/ vol. xi. part ii. Mr. Alexander J. Ellis
treats the same subject in his ' Contributions to Formal
Logic,' read to the Royal Society, in March, 1872, but
as yet published only in the form of a brief abstract, in
the Proceedings of the Society, vol. xx. p. 307.

CHAPTEE IX.

THE VARIETY OF NATURE, OR THE DOCTRINE OF
COMBINATIONS AND PERMUTATIONS.

NATURE may be said to be evolved from the monotony
of non-existence by the creation of diversity. It is plau
sibly asserted that we are conscious only so far as we
experience difference. Life is change, and perfectly uni
form existence would be no better than non-existence.
Certain it is that life demands incessant novelty, and that
nature though it probably never fails to obey the same
fixed laws, yet presents to us an apparently unlimited
series of varied combinations of events. It is the work of
science to observe and record the kinds and comparative
numbers of such combinations of phenomena, occurring
spontaneously or produced by our interference. Patient
and skilful examination of the records may then disclose
the laws imposed on matter at its creation, and enable us
more or less successfully to predict, or even to regulate,
the future occurrence of any particular combination.

The Laws of Thought are the first and most important
of all the laws which govern the combinations of pheno
mena ; and, even though they be binding on the rnind,
they may also be regarded as verified in the external
world. The Logical Abecedarium develops the utmost
variety of things and events which may occur, and it
is evident that as each new quality is introduced, the
number of combinations is doubled. Thus four qualities
may occur in 16 combinations; five qualities in 32 ; six
qualities in 64 ; and so on. In general language, if n be-

o 2

196 THE PRINCIPLES OF SCIENCE.

the number of qualities, 2n is the number of varieties of
things which may be formed from them, if there be no con
ditions but those of logic. This number, it need hardly
be said, increases after the first few terms, in an extra
ordinary manner, so that it would require 302 figures,
even to express the number of combinations in which 1000
qualities might conceivably present themselves.

If all the combinations allowed by the Laws of Thought
occurred in nature, then science would begin and end with
those laws. To observe nature would give us no ad
ditional knowledge, because no two qualities would in the
long run be oftener associated than any other two. We
could never predict events with more certainty than we
now predict the throws of dice, and experience would be
without use. But the universe, as actually created, pre
sents a far different and much more interesting problem.
The most superficial observation shows that some things
are habitually associated with other things. The more
mature our examination, the more we become convinced
that each event depends upon the prior occurrence of
some other series of events. Action and reaction are
gradually discovered to underlie the whole scene, and an
independent or casual occurrence does not exist except in
appearance. Even dice as they fall are surely determined
in their course by prior conditions and fixed laws. Thus
the combinations of events which can really occur are
found to be very restricted, and it is the work of science
to detect these restricting conditions.

In the English alphabet, for instance, we have twenty-
six letters. Were the combinations of such letters per
fectly free, so that any letter could be indifferently
sounded with any other, the number of words which
could be formed without any repetition would be 226- i,
or 67,108,863, equal in number to the combinations of
the twenty-seventh column of the Abecedarium, excluding

COMBINATIONS AND PERMUTATIONS. 197

one for the case in which all the letters would be
absent. But the formation of our vocal organs prevents
our using the far greater part of these conjunctions of
letters. At least one vowel must be present in each word;
more than two consonants cannot usually be brought to
gether ; and to produce words capable of smooth utterance
a number of other rules must be observed. To determine
exactly how many words might exist in the English
language under these circumstances, would be an exceed
ingly complex problem, the solution of which has never
been attempted. The number of existing English words
may perhaps be said not to exceed one hundred thousand,
and it is only by investigating the combinations presented
in the dictionary, that we can learn the Laws of Euphony
or calculate the possible number. In this example we
have an epitome of the work and method of science. The
combinations of natural phenomena are limited by a great
number of conditions which are in no way brought to our
knowledge except so far as they are disclosed in the ex
amination of nature.

It is often a very difficult matter to determine the
numbers of permutations or combinations which may
exist under various restrictions. Many learned men
puzzled themselves in former centuries over what were
called Protean verses, or Latin verses admitting many
variations in accordance with the Laws of Metre. The
most celebrated of these verses was that invented by
Bernard Bauhusius, as folio wsa: —

| 'Tot tibi sunt dotes, Virgo, quot sidera ccelo.'

One author, Ericius Puteanus, filled forty-eight pages of
a work in reckoning up its possible transpositions, making
them only 1022. Other calculators gave 2196, 3276, 2580
as their results. Dr. Wallis assigned 3096, but without

a Montucla, ' Histoire,' &c., vol. iii. p. 388.

198 TEE PRINCIPLES OF SCIENCE.

much confidence in the accuracy of his result.b It required
the skill of James Bernouilli to decide the number of
transpositions to be 3312, under the condition that the
sense and metre of the verse shall be perfectly preserved.
In approaching the consideration of the great Inductive
problem, it is very necessary that we should acquire correct
notions as to the comparative number of combinations
which may exist under different circumstances. The
doctrine of combinations is that part of mathematical
science which applies numerical calculation to determine
the number of combinations under various conditions.
It is a part of the science which really lies at the base
not only of other sciences, but of other branches of mathe
matical science. The forms of algebraical expressions are
determined by the principles of combination, and Hinden-
burg recognised this fact in his Combinatorial Analysis.
The greatest mathematicians have, during the last three
centuries, given their best powers to the treatment of
this subject ; it was the favourite study of Pascal ; it
early attracted the attention of Leibnitz, who wrote his
curious essay, De Arte Conibinatoria, at twenty years
of age ; James Bernouilli, one of the very profoundest
mathematicians, devoted no small part of his life to the
investigation of the subject as connected with that of
Probability ; and in his celebrated work, De Arte Con-
jectandi, he has so finely described the importance of
the doctrine of combinations, that I need offer no excuses
for quoting his remarks at full length. ' It is easy to
perceive that the prodigious variety which appears both
in the -works of nature and in the actions of men, and
which constitutes the greatest part of the beauty of the
universe, is owing to the multitude of different ways
in which its several parts are mixed with, or placed
near, each other. But, because the number of causes

b Wallis, 'Of Combinations,' &c., p. 119.

COMBINATIONS AND PERMUTATIONS. 199

that concur in producing a given event, or effect, is
oftentimes so immensely great, and the causes them
selves are so different one from another, that it is ex
tremely difficult to reckon up all the different ways in
which they may be arranged or combined together, it
often happens that men, even of the best understandings
and greatest circumspection, are guilty of that fault in
reasoning which the writers on logic call the insufficient
or imperfect enumeration of parts or cases : insomuch
that I will venture to assert, that this is the chief, and
almost the only, source of the vast number of erroneous
opinions, and those too very often in matters of great
importance, which we are apt to form on all the subjects
we reflect upon, whether they relate to the knowledge of
nature or the merits and motives of human actions. It
must therefore be acknowledged, that that art which
affords a cure to this weakness, or defect, of our under
standings, and teaches us so to enumerate all the possible
ways in which a given number of things may be mixed
and combined together, that we may be certain that we
have not omitted any one arrangement of them that can
lead to the object of our inquiry, deserves to be con
sidered as most eminently useful and worthy of our
highest esteem and attention. And this is the business
of the art or doctrine of combinations. Nor is this art
or doctrine to be considered merely as a branch of the
mathematical sciences. For it has a relation to almost
every species of useful knowledge that the mind of man
can be employed upon. It proceeds indeed upon mathe
matical principles, in calculating the number of the com
binations of the things proposed : but by the conclusions
that are obtained by it, the sagacity of the natural
philosopher, the exactness of the historian, the skill and
judgment of the physician, and the prudence and fore
sight of the politician may be assisted ; because the business

200 THE PRINCIPLES OF SCIENCE.

of all these important profes io^s is but to form reasonable
conjectures concerning the several objects which engage
their attention, and all wise conjectures are the results of
a just and careful examination of the several different
effects that may possibly arise from the causes that are
capable of producing them.' c

Distinction of Combinations and Permutations.

We must at once consider the deep difference which
exists between Combinations and Permutations ; a dif
ference involving important logical principles, and in
fluencing the form of all our mathematical expressions.
In permutation we recognise varieties of order or arrange
ment, treating AB as a different group from BA. In
combination we take notice only of the presence or
absence of a certain thing, and pay no regard to its
place in order of time or space. Thus the four letters
a, e, m, n can form but one combination, but they occur
in language in several permutations, as name, amen,
mean, mane.

We have hitherto been dealing with purely logical
questions, involving only combination of qualities. I have
fully pointed out in more than one place that, though our
symbols could not but be written in order of place and
read in order of time, the relations expressed had no
regard to place or time (pp. 40, 131). The Law of Com-
mutativeness, in fact, expresses the condition that in logic
we deal with combinations, and the same law is true
of all the processes of algebra. In nature and art, order
may be a matter of indifference ; it makes no difference,
for instance, whether gunpowder is a mixture of sulphur,
carbon and nitre, or carbon, nitre and sulphur, or nitre, sul
phur and carbon, provided that the substances are present in

c James Bernoulli!, « De Arte Conjectandi,' translated by Baron
Maseres. London, 1795, pp. 35-36.

COMBINATIONS AND PERMUTATIONS. 201

proper proportions and well mixed. But this indifference
of order does not usually extend to the events of physical
science or the operations of art. The change of mechanical
energy into heat is not exactly the same as the change
from heat into mechanical energy ; thunder does not in
differently precede and follow lightning ; it is a matter of
some importance that we load, cap, present, and fire a rifle
in this precise order. Time is the condition of all our
thoughts, space of all our actions, and therefore both in
art and science we are to a great extent concerned with
permutations. All language, for instance, treats different
permutations of letters as having different meanings.

Permutations of certain things are far more numerous
than combinations of those things, for the obvious reason
that each distinct thing is regarded differently according
to its place. Thus the letters A, B, C, will make different
permutations according as A stands first, second, or third ;
having decided the place of A, there are two places
between which we may choose for B ; and then there
remains but one place for C. Accordingly the permuta
tions of these letters will be altogether 3 x 2 x i or 6 in
number. With four things or letters, A, B, C, D, we
shall have four choices of place for the first letter, three
for the second, two for the third, and one for the fourth,
so that there will be altogether 4 x 3 x 2 x i, or 24
permutations. The same simple rule applies in all cases ;
beginning with the whole number of things we multiply
at each step by a number decreased by a unit. In general
language, if n be the number of things in a combination, the

number of permutations is n (n — i) (n — 2) 4.3.2. i.

Thus, if we were to re-arrange the names of the days of the
week, the possible arrangements out of which we should
have to choose the new order, would be no less than
7.6.5.4.3.2. i, or 5040, or, excluding the existing
order, 5039.

202

THE PRINCIPLES OF SCIENCE.

The reader will see that the numbers which we reach in
questions of permutation, increase in a more extraordinary
manner even than in combination. Each new object or
term doubles the number of combinations (p. 195), but
increases the permutations by a factor continually
growing. Instead of2x2x2x2x ..... we have
2x3x4x5x ..... and the products of the latter
expression indefinitely exceed those of the former. These
products of continually increasing factors are constantly
employed, as we shall see, in questions both of permu
tation and combination. They are technically called
factorials, that is to say, the product of all integer
numbers, from unity up to any number n, is the factorial
of n, and is often indicated symbolically by (_»*_. I give
below the factorials up to that of fifteen : —

6=1.2.3
24 = i . 2 . 3 . 4
1 20 = i . 2 .... 5
720 = i . 2 .... 6
5,040 = |_7_
40,320 = I A
362,880 = |_9_
3,628,800 = |i°
39,916,800 = |rj.

479,OOI,6OO =

6,227,020,800 =

87,178,291,200 = |£4

1,307,674,368,000 = |i5

The factorials up to [36 are given in Rees' ' Cyclopaedia/
art. Cipher, and the logarithms of products up to [265
are given at the end of the table of logarithms published
under the superintendence of the Society for the Diffusion
of Useful Knowledge (p. 215). To express the factorial
I265 would require 529 places of figures.

Many writers have from time to time remarked upon

12

COMBINATIONS AND PERMUTATIONS. 203

the extraordinary magnitude of the numbers with which
we deal in this subject. Tacquet calculated d that the
twenty-four letters of the alphabet may be arranged in
more than 620 thousand trillions of orders ; and Schottus
estimated e that if a thousand millions of men were em
ployed for the same number of years in writing out these
arrangements, and each man filled each day forty pages
with forty arrangements in each, they could not have ac
complished the task, as they would have written only 584
thousand trillions instead of 620 thousand trillions.

In some questions the number of permutations may be
restricted and reduced by various conditions. Some
tilings in a group may be undistinguishable from others,
so that change of order will produce no difference. Thus
if we were to permutate the letters of the name Ann,
according to our previous rule, we should obtain 3x2x1,
or 6 orders ; but half of these arrangements would be
identical with the other half, because the interchange of

o

the two n's has no effect. The really different orders will
therefore be ~^-~- or 3, namely Ann, Nan, Nna. In

the word utility there are two i's and two t's, in respect
of both of which pairs the number of permutations must
be halved. Thus we obtain — — • 5 • 4 • 3 • 2 • r or I26o, as

I . 2 . I . 2

the number of permutations. The simple rule evidently
is that when some things or letters are undistinguished,
proceed in the first place to calculate all the possible
permutations as if all were different, and then divide by
the number of possible permutations of those series of
things which are not distinguished, and of which the
permutations have therefore been counted in excess.
Thus since the word Utilitarianism contains fourteen

d ' Aritlmieticse Theoria.' Ed. Amsterd. 1704, p. 517.
e Rees' ' Cyclopaedia/ art. Cipher.

204 THE PRINCIPLES OF SCIENCE.

letters, of which four are i's, two a's, and two t'a, the
number of distinct arrangements will be found by
dividing the factorial of 14, by the factorials of 4, 2,
and 2, the result being 908,107,200. From the letters
of the word Mississippi we can get in like manner

!= or 34,6 50 permutations, or not one-thousandth

[4. X|_4_X|_2_

part of what we should obtain were all the letters
different.

Calculation of Number of Combinations.

Although in many questions both of art and science
we need to calculate the number of permutations on
account of their own interest, it far more frequently
happens in scientific subjects that they possess but an
indirect interest. As I have already pointed out, we
almost always deal in the logical and mathematical
sciences with combinations, and variety of order enters
only through the inherent imperfections of our symbols
and modes of calculation. Signs must be used in some
order, and we must withdraw our attention from this order
before the signs correctly represent the relations of things
which exist neither before nor after each other. Now, it
often happens that we cannot choose all the combinations
of things, without first choosing them subject to the
accidental variety of order, and we must then divide by
the number of possible variations of order, that we may
get to the true number of pure combinations.

Suppose that we wish to determine the number of
ways in which we can select three letters out of the
alphabet, without allowing the same letter to be repeated.
At the first choice we can take any one of 26 letters ; at
the next step there remain 25 letters, any one of which
may be joined with that already taken ; at the third step
there will be 24 choices, so that apparently the whole

COMBINATIONS AND PERMUTATIONS. 205

number of ways of choosing is 26x25x2 4. But the fact
that one choice succeeded another has caused us to obtain
the same combinations of letters in different orders ; we
should get, for instance, a, p, r at one time, and p, r, a at
another, and every three distinct letters will appear six
times over, because three things can be arranged in six
permutations. Thus the true number of combinations

.,-, , 24 x 23 x 22
will be -1- — , or 2024.
1x2x3

It is apparent that we need the doctrine of permuta
tions in order that we may in many questions counteract
the exaggerating effect of successive selection. If out of
a senate of 30 persons we have to choose a committee of 5,
we may choose any of 30 first, any of 29 next, and so on,
in fact there will be 30 x 29 x 28 x 27 x 26 selections ;
but as the actual character of the members of the committee
will not be affected by the accidental order of their selec
tion, we divide by 1x2x3x4x5, and the possible num
ber of different committees will be 142,506. Similarly
if we want to calculate the number of ways in which the
eight major planets may come into conjunction, it is evi
dent that they may meet either two at a time or three at
a time, or four or more at a time, and as nothing is said as to
the relative order or place in the conjunction, we require
the number of combinations. Now a selection of 2 out of 8

& H fi h f\

is possible in — or 28 ways : of 3 out of 8 in -J-^-
1.2 •f 1.2.3

or 56 ways ; of 4 out of 8 in -— -— or 70 ways ; and it

may be similarly shown that for 5, 6, 7, and 8 planets,
meeting at one time, the number of ways is 56, 28, 8
and i. Thus we have solved the whole question of the
variety of conjunctions of eight planets; and adding all the
numbers together, we find that 247 is the utmost possible
number of modes of meeting.

In general algebraic language, we may say that a group

206 THE PRINCIPLES OF SCIENCE.

of m things may be chosen out of a total number of n
things, in a number of combinations denoted by the formula

n.(n-i) (n-2) (n-%) .... (n-m+i)
1.2 . 3 .4 .... m

The extreme importance and significance of this formula
seems to have been first adequately recognised by Pascal,
although its discovery is attributed by him to a friend,
M. de Ganieres/ We shall find it perpetually recurring
in questions both of combinations and probability, and
throughout the formulae of mathematical analysis traces of
its influence will be noticed.

The Arithmetical Triangle.

The Arithmetical Triangle is a name long since given to
a series of remarkable numbers connected with the subject
we are treating. According to Montucla K ' this triangle is
in the theory of combinations and changes of order, almost
what the table of Pythagoras is in ordinary arithmetic,
that is to say, it places at once under the eyes, the numbers
required in a multitude of cases of this theory.' As early
as 1544 Stifels had noticed the remarkable properties of
these numbers and the mode of their evolution.11 Briggs,
the inventor of the common system of logarithms, was so
struck with their importance that he called them the
Abacus Panchrestus. Pascal, however, was the first who
wrote a distinct treatise on these numbers, and gave them
the name by which they are still known. But Pascal did
not by any means exhaust the subject, and it remained for
James Bernouilli to demonstrate fully the importance of
the figurate numbers, as they are also called. In his
treatise De Arte Conjectandi, he points out their appli-

f ' CEuvres Completes de Pascal' (1865), vol. iii. p. 302. Montucla states
the name as De Gruieres, 'Histoire cles Math^matiques/ vol. iii. p. 389.

g 'Histoire des Mathe'matiques/ vol. iii. p. 387.

h Leslie, ' Dissertation on the Progress of Mathematical and Physical
Science,' Encyclopaedia Britannica.

COMBINATIONS AND PERMUTATIONS. 207

cation in the theory of combinations and probabilities, and
remarks of the Arithmetical Triangle, ' It not only contains
the clue to the mysterious doctrine of combinations, but it
is also the ground or foundation of most of the important
and abstruse discoveries that have been made in the other
branches of the mathematics/ l

The numbers of the triangle can be calculated in a very
easy manner by successive additions. We commence with
unity at the apex ; in the next line we place a second
unit to the right of this ; to obtain the third line of
figures we move the previous line one place to the right,
and add them to the same figures as they were before
removal, and we can then repeat the same process ad
infinitum. The fourth line of figures, for instance, con
tains i, 3, 3, i ; moving them one place and adding as
directed we obtain : —

Fourth line .
Fifth line . .
Sixth line . .

Seventh line. . . i 6 15 20 15 6 i

Carrying out this simple process through ten more steps
we obtain the first seventeen lines of the Arithmetical
Triangle as printed on the next page. Theoretically
speaking the Triangle must be regarded as infinite in
extent but the numbers increase so rapidly that it soon
becomes almost impracticable to continue the table. The
longest table of the numbers which I have found is given
in Fortia's ' Traitd des Progressions ' (p. 80), where they
are given up to the fortieth line and the ninth column.

» Bernouilli, ' De Arte Conjectandi,' translated by Francis Maseres,
London, 1795, p. 75.

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i

i

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10

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5

208

THE PRINCIPLES OF SCIENCE.

THE ARITHMETICAL TRIANGLE.

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COM BIN A TIONS A ND PERM UTA TIONS. 209

On carefully examining these numbers, we shall find
that they are connected with each other by an almost
unlimited series of relations, a few of the more simple
of which may be noticed.

1. Each vertical column of numbers exactly corre
sponds with an oblique series descending from left to
right, so that the triangle is perfectly symmetrical in its
contents.

2. The first column contains only units; the second
column contains the natural numbers, i, 2, 3, &c. ; the
third column contains a remarkable series of numbers,
i, 3, 6, 10, 15, &c., which have long been called the tri
angular numbers, because they correspond with the
numbers of balls which may be arranged in a triangular
form, thus —

o

O 00

o o o o o o
o oo ooo oooo
o oo ooo oooo ooooo

These numbers evidently differ each from the previous
one by the series of natural numbers. Their employment
has been explained, and the first 20,000 of the numbers
calculated and printed by E. de Joncourt in a small
quarto volume, which was published at the Hague, in
1762.

The fourth column contains the pyramidal numbers,
so called because they correspond to the number of equal
balls which can be piled in regular triangular pyramids.
Their differences are the triangular numbers.

The numbers of the fifth column have the pyramidal
numbers for their differences, but as there is no regular
figure of which they express the contents, they have been
arbitrarily called the trianguli-triangular numbers. The
succeeding columns have, in a similar manner, been said to

p

210 THE PRINCIPLES OF SCIENCE.

contain the tria ngul i-pyr amidol, the pyramidi-pyramidal
numbers, and so on.k

3. From the mode of formation of the table, it follows
that the differences of the numbers in each column will
be found in the preceding column to the left. Hence
the second differences, or the differences of differences will
be in the second column to the left of any given column,
the third differences in the third column, and so on.
Thus we may say that unity which appears in the first
column is the first difference of the numbers in the
second column ; the second difference of those in the third
column ; the third difference of those in the fourth,
and so on. The triangle is thus seen to be a complete
classification of all numbers according as they have unity
for any of their differences.

4. Every number in the table is equal to the sum of
the numbers which stand higher in the next column to
the left, beginning with the next line above ; thus 84 is
equal to the sum of 28, 21, 15, 10, 6, 3, i.

5. Since each line is formed by adding the previous
line to itself, it is evident that the sum of the numbers
in each horizontal line must be double that of the line
next above. Hence we know, without making any ad
ditions, that the successive sums must be i, 2, 4, 8, 16,
32, 64, &c., the same as the numbers of combinations in
the Logical Abecedarium. Speaking generally, the sum
of the numbers in the nth line will be 2""1.

6. If the whole of the numbers down to any line be
added together, we shall obtain a number less by unity
than some power of 2 ; thus, the first line gives i or
21— i ; the first two lines give 3 or 22— i ; the first three
lines 7 or 2s— i ; the first six lines give 63 or 26 — i ;
or, speaking in general language, the sum of the first
?i lines is 2"— i.

k Wallis's 'Algebra,' Discourse of Combinations, &c. p. 109.

COMBINATIONS AND PERMUTATIONS. 211

7. It follows that the sum of the numbers in any one
line is equal to the sum of those in all the preceding lines
diminished by a unit. For the sum of the nth line is, as
already shewn, 2n~1, and the sum of the first n — i lines
is 2n~l— i, or less by a unit.

This enumeration of the properties of the figurate
numbers does not approach completeness ; a considerable,
perhaps an unlimited, number of less simple and obvious
relations might be traced out. Pascal, after giving many
of the properties, exclaims1 : ' Mais j'en laisse bien plus
que je n'en donne ; c'est ime chose etrange combien il est
fertile en proprietes ! Chacun peut s'y exercer/ The
arithmetical triangle may be considered a natural classifi
cation of numbers, exhibiting, in the most complete
manner, their evolution and relations in a certain point
of view. It is obvious that in an unlimited extension of
the triangle, each number will have at least two places.

Though the properties above explained are highly
curious, the greatest value of the triangle arises from the
fact that it contains a complete statement of the values
of the formula (p. 206), for the number of combinations
of m things out of n, for all possible values of m and n.
Out of seven things one may be chosen in seven ways,
and seven occurs in the eighth line of the second column.
The combinations of two things chosen out of seven

7x6

are - — - or 21, which is the third number in the eighth

line. The combinations of three things out of seven are

^ x 6 x i"

3- or 35, which appears fourth in the eighth line.
1x2x3

In a similar manner, in the fifth, sixth, seventh, and eighth
columns of the eighth line I find it stated in how many
ways I can select combinations of 4, 5, 6, and 7 things
out of 7. Proceeding to the ninth line, I find in succession

1 ' (Euvres Completes,' vol. iii. p. 251.
1' 2

212 THE PRINCIPLES OF SCIENCE.

the number of ways in which I can select i, 2, 3, 4, 5, 6,
7, and 8 things, out of 8 things. In general language, if
I wish to know in how many ways m things can be
selected in combinations out of n things, I must look
in the n + Ith line, and take the m + ith number, counting
from the left, as the answer. In how many ways, for
instance, can a sub-committee of five be chosen out of a
committee of nine. The answer is 126, and is the sixth
number in the tenth line ; it will be found equal to

— , which our previous formula (p. 206) would

7.6

1.2.3.4.5
give.

The full utility of the figurate numbers will be more
apparent when we reach the subject of probabilities, but I
may give an illustration or two in this place. In how
many ways can we arrange four pennies as regards head
and tail 1 The question amounts to asking in how
many ways we can select o, i, 2, 3, or 4 heads out of 4
heads, and the fiftli line of the triangle gives us the
complete answer, thus—

We can select No head and 4 tails in i way.
„ i • head and 3 tails in 4 ways.

„ 2 heads and 2 tails in 6 ways.

„ 3 heads and i tail in 4 ways.

„ 4 heads and o tail in i way.

The total number of different cases is 16, or 24, and
when we come to the next chapter, it will be found that
these numbers give us the respective probabilities of all
throws with four pennies.

I gave in p. 205 a calculation of the number of ways in
which eight planets can meet in conjunction ; the reader
will find all the numbers detailed in the ninth line of the
arithmetical triangle. The sum of the whole line is 28 or
256 ; but we must subtract a unit for the case where no
planet appears, and 8 for the 8 cases in which only one

COMBINATIONS AND PERMUTATIONS. 213

planet appears ; so that the total variety of conjunctions is

28— i _g or 247.

If an organ has twelve stops, we find in the thirteenth
line the numbers of combinations which we can draw,

0, i, 2, 3, &c., at a time; the total number of modes of
varying the sound is no less than 212— i or 4095 m. If a
number be the product of n prime factors, we find in the
n+ Ith line the numbers of divisors, being the product of

1, 2, 3, or more of the prime factors ; and the whole
number of divisors of tli3 number is the sum of the
numbers in the line, subtracting unity, or 2"— i.

One of the most important scientific uses of the arith
metical triangle, consists in the information which it gives
concerning the comparative frequency of divergencies from
an average. Suppose, for the mere sake of argument,
that all persons were naturally of equal stature of five
feet, but enjoyed during youth seven independent chances
of growing one inch in addition. Of these seven chances,
one, two, three, or more, may happen favourably to any
individual, but as it does not matter what the chances
are, so that the inch is gained, the question really turns
upon the number of combinations of o, i, 2, 3, &c.,
things out of seven. Hence the eighth line of the triangle
give us a complete answer to the question, as follows :—
Out of every 128 people-
Feet. Inches.

One person would have the stature of 5 o

7 persons „ ,, 51

21 persons „ „ 52

35 persons „ „ 53

35 persons „ „ 54

21 persons „ „ 5 5

7 persons „ „ 56

i person „ „ 57

m Bernoulli!, 'De Arte Conjectandi,' trans, by Mascrcs, p. 64.

214 THE PRINCIPLES OF SCIENCE.

By taking a proper line of the triangle, an answer
may be had under any more natural supposition. This
theory of comparative frequency of divergence from an
average, was first adequately noticed by M. Quetelet, and
has lately been employed in a very interesting and bold
manner by Mr. Galton, in his work on ' Hereditary Genius/
We shall afterwards find that the theory of error, to which
is made the ultimate appeal in cases of quantitative in
vestigation, is founded upon the comparative numbers of
combinations as displayed in the triangle.

Connection between the Arithmetical Triangle and the
Logical Abecedarium.

There exists a close connection between the arith
metical triangle described in the last section, and the
series of combinations of letters called the Logical Abece-
darium. The one is to mathematical science what the
other is to logical science. In fact the figurate numbers,
or those exhibited in the triangle, are obtained by
summing up the logical combinations. Accordingly, just
as the total of the numbers in each line of the triangle
was twice as great as that for the preceding line (p. 210),
so each column of the Abecedarium (p. 109) contained
twice as many combinations as the preceding one. The
like correspondence would also exist between the sums
of all the lines of figures down to any particular line, and
of the combinations down to any particular column.

By examining any one column of the Abecedarium, we
shall also find that the combinations naturally group
themselves according to the figurate numbers. Take the
combinations of the letters A, B, C, D ; they consist of
all the ways in which I can choose four, three, two, one,
or none of the four letters, filling up the vacant spaces

COMBINATIONS AND PERMUTATIONS.

215

with negative terms. I may arrange the combinations as
follows : —

ABCD . Four out of four . . i combination.
ABCc/

- Three out of four . 4 combinations.
A5CD

aBCD

• Two out of four . . 6 combinations.

aECd
aBcD
a&CD

> One out of four ... 4 combinations.

J
abed None out of four . . i combination.

The numbers, it will be noticed, are exactly the same
as those in the fifth line of the arithmetical triangle, and
an exactly similar correspondence would be found to
exist in the case of each other column of the Abece-
darium.

Numerical abstraction, it has been asserted, consists in
overlooking the kind of difference, and retaining only a
consciousness of its existence (p. 177). While in logic,
then, we have to deal with each combination as a separate
kind of thing, in arithmetic we can distinguish only the
classes which depend upon more or less positive terms
being present, and the numbers of these classes imme
diately produce the numbers of the arithmetical triangle.

It may here be pointed out that there are two modes

216 THE PRINCIPLES OF SCIENCE.

in which we can calculate the whole number of com
binations of certain things. Either we may take the
whole number at once as shown in the Abecedarium, in
which case the number will be some power of two, or
else we may calculate successively, by aid of permutations,
the number of combinations of none, one, two, three, and
so on. Hence we arrive at a necessary identity between
two series of numbers. In the case of four things we
shall have

24 = i + - + 4 ' 3 + 4 • 3 • 2 + 4 -3-2.1

I 1.2 1.2.3 I . 2 '. 3 . 4'

In a general form of expression we shall have

n n n . (n—i) n (n-i\ (n—z\
2 =i+- + - / 4. — 1 LL _J- + &e

I 1.2 1.2-3

the terms being continued until they cease to have any
value. Thus we have arrived at a proof of simple cases
of the Binomial Theorem, of which each column of the
Abecedarium is an exemplification. It may be shown
that all other mathematical expansions likewise arise out
of simple processes of combination, but the more complete
consideration of this subject must be deferred.

Possible Variety of Nature and Art.

We cannot adequately understand the difficulties which
beset us in certain branches of science, unless we gain
a clear idea of the vast number of combinations or per
mutations which may be possible under certain conditions.
Thus only can we learn how hopeless it would be to
attempt to treat nature in detail, and exhaust the whole
number of events which might arise. It is instructive to
consider, in the first place, how immensely great are the
numbers of combinations with which we deal in many
arts and amusements.

COMBINATIONS AND PERMUTATIONS. 217

In dealing a pack of cards, the number of hands, of
thirteen cards each, which can be produced is

52 . 51 • 50 ..... 40

or 635,013,559,600. But in whist four hands are simul
taneously held, and the number of distinct deals becomes
so vast that it would require twenty-eight figures to express
it. If the whole population of the world, say one hundred
thousand millions of persons, were to deal cards day and
night, for a hundred million of years, they would not in
that time have exhausted one hundred-thousandth part of
the possible deals.0 Now, even with the same hands the
play may be almost infinitely varied, so that the complete
variety of games which may exist is almost incalculably
great. It is in the highest degree improbable that any
one game of whist was ever exactly like another, except
by intention.

The end of novelty in art might well be dreaded, did
we not find that nature at least has placed no attainable
limit, and that the deficiency will lie in our inventive
faculties. It would be a cheerless time indeed when all
possible varieties of melody were exhausted, but it is
readily shown that if a peal of twenty-four bells had been
rung continuously from the so-called beginning of the
world to the present day, no approach could have been
made to the completion of the possible changes.. Nay,
had every single minute, been prolonged to 10,000 years,
still the task would have been unaccomplished. P As
regards ordinary melodies, the eight notes of a single
octave give more than 40,000 permutations, and two
octaves more than a million millions. If we were to take

0 ' Essay on Probability/ by Lubbock and Drinkwater, Useful Know
ledge Society, 1833, p. 6.

P Wallis ' Of Combinations,' p. 116, quoting Vossius.

218 THE PRINCIPLES OF SCIENCE.

into account the semitones, it would become apparent that
it is practically impossible to exhaust the variety of
music.

Similar considerations apply to the possible number
of natural substances, though we cannot always give
precisely numerical results. It was recommended by
Hatchett^ that a systematic examination of all alloys
of metals should be carried out, proceeding from the
most simple binary ones to more complicated ternary
or quaternary ones. He can hardly have been aware
of the extent of his proposed inquiry. If we operated
only upon thirty of the known metals, the number of
possible selections of binary alloys would be 435, of
ternary alloys 4060, of quaternary 27,405, without
paying any regard to the varying proportions of the
metals, and only regarding the kind of metal. If we
varied all the ternary alloys by quantities not less than
one per cent., the number of these alloys only would
be 11,445,060. An exhaustive investigation of the sub
ject is therefore out of the question, and unless some
laws connecting the properties of the alloy and its
components can be discovered, it is not apparent how
our knowledge of them can be ever more than most
incomplete.

The possible variety of definite chemical compounds,
again, is enormouslv great. Chemists have already ex
amined many thousands of inorganic substances, and a
still greater number of organic • compounds ; r they have
nevertheless made no appreciable impression on the
number which may exist. Taking the number of ele
ments at sixty- one, the number of compounds contain
ing different selections of four elements each would
be more than half a million (521,855). As the same

i 'Philosophical Transactions' (1803), vol. xciii. p. 193.
r Hofmann's ' Introduction to Chemistry,' p. 36.

COM BIN A TIONS A ND PERM VTA TIOXS. 2 1 9

elements often combine in many different proportions,
and some of them, especially carbon, have the power of
forming an almost endless number of compounds, it
would hardly be possible to assign any limit to the
number of chemical compounds which may be formed.
There are branches of physical science, therefore, of which
it is unlikely that scientific men, with all their industry,
can ever obtain a knowledge in any appreciable degree
approaching to completeness.

Higher Orders of Variety.

The consideration of the facts already given in this
chapter will not produce an adequate notion of the pos
sible variety of existence, unless we consider the com
parative numbers of combinations of different orders. By
a combination of a higher order, I mean a combination
of groups, which are themselves combinations of simpler
groups. The almost unlimited number of compounds
of carbon, hydrogen, and oxygen, described in organic
chemistry, are combinations of a second order, for the
atoms are groups of groups. The wave of sound pro
duced by a musical instrument may be regarded as a
combination of motions ; the body of sound proceeding
from a large orchestra is therefore a complex aggregate
of sounds each in itself a complex combination of move
ments. All literature may be said to be developed
out of the difference of white paper and black ink.
From the almost unlimited number of marks which
might be chosen we select twenty-six customary letters.
The pronounceable combinations of letters are probably
some trillions in number. Now, as a sentence is a cer
tain selection of words, the possible sentences must be
indefinitely more numerous than the words of which
it may be composed. A book is a combination of

220 THE PRINCIPLES OF SCIENCE.

sentences, and a library is a combination of books. A
library, therefore, is a combination of the fifth order, and
the powers of numerical expression would be almost
exhausted in attempting to express the number of dis
tinct libraries which might be constructed. The calcu
lation would not be possible, because the union of letters
in words, of words in sentences, and of sentences in books,
are governed by conditions so' complex as to defy calcu
lation. I wish only to point out that there is no limit
to the multitude of different sentences which may be de
veloped out of the one difference of ink and paper. Galileo
is said to have remarked that all truth is contained in
the compass of the alphabet. We might add that it is all
contained in the difference of ink and paper.

One consequence of this power of successive combi
nation is that the simplest signals or marks will suffice
to express any information. Francis Bacon proposed
for secret writing a biliteral cipher, which resolves all
letters of the alphabet into permutations of the two
letters a and b. Thus A. was aaaaa, B aaaab,
X babal), and so on.s And in a similar way, as Bacon
clearly saw, any one difference can be made the ground
of a code of signals ; we can express, as he says,
omnia per omnia. The Morse alphabet uses only a
succession of long and short marks, and other systems
of telegraphic language employ right and left strokes.
A single lamp obscured at various intervals, long or
short, may be made to spell out any words, and with
two lamps, distinguished by colour or position, we could
at once represent Bacon's biliteral alphabet. Mr. Bab-
bage ingeniously suggested that every lighthouse in
the world should be made to spell out its own name
or number perpetually, by flashes or obscurations of

'Works,' edited by Shaw, vol. i. pp. 141-145, quoted in Rees'
* Encyclopaedia,' art. Cipher,

COMBINATIONS AND PERMUTATIONS. 221

various duration and succession, and the scheme would be
easy of execution if needed.

Let us calculate the number of combinations of dif
ferent orders which may arise out of the presence or
absence of a single mark, say A. Thus in

A [AT I A

we have four distinct varieties. Form them into a group
of a higher order, and consider in how many ways we
may vary that group by omitting one or more of the
component parts. Now, as there are four parts, and any
one may be present or absent, the possible varieties will
be 2 x 2 x 2 x 2, or 1 6 in number. Form these into a new
whole, and proceed again to create variety by omitting
any one or more of the sixteen. The number of pos
sible changes will now be 2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2,
or 216, and we can repeat the process again and again if
we wish. It will be easily seen that we are imagining
the creation of objects, whose numbers are represented
in the series of expressions —

2

2 2

222

2222, and so on.
At the first step we have 2; -at the next 22, or 4;

2

at the third 22 , or 16, numbers of very moderate amount.
Let the reader calculate the next term, and he will
be surprised to find it leap up to 65,536. But at the
next step he has to calculate the value of 65,536
two's multiplied together, and it is so great that we
could not possibly compute it, the mere expression of
the result requiring 19,729 places of figures. But go
one step more and we pass the bounds of all reason.
The sixth order of the powers of two becomes so
great, that we could not even express the number of

222 THE PKIXCIPLES OF SCIENCE.

figures required in writing it down, without using
about 19,729 figures for the purpose.

The successive orders of the powers of two have then
the following values :—

First order . • 2

Second order . • 4

Third order . 16

Fourth order . . 65,536

Fifth order, number expressed by 19,729 figures.
Sixth order, number expressed by
figures, to express the number
of which figures would require
about . . . . i9,729 %ures.

It may give us a powerful notion of infinity to remem
ber that at this sixth step, having long surpassed all
bounds of conception, we have made no approach to the
goal. Nay, were we to make a hundred such steps, we
should be as far away as ever from actual infinity.

It is well worth observing that our powers of ex
pression rapidly overcome the possible multitude of
finite objects which may exist in any assignable space.
Archimedes showed long ago, in one of the most won
derful writings of antiquity,* that the grains of sand
in the world could be numbered, or rather, that if
numbered, the result could readily be expressed in
arithmetical notation. Let us extend his problem, and
ascertain whether we could express the number of
atoms which could exist in the visible universe. The
most distant stars which can now be seen by telescopes
—those of the sixteenth magnitude — are supposed to
have a distance of about 33,900,000,000,000,000 miles.u
Sir W. Thomson, again, has shown reasons for supposing

i ' Liber de Arenas Numero.'

u Chambers' s 'Astronomy' (1861), p. 272.

COMBINATIONS AND PERMUTATIONS. 223

that there do not exist more than from 3Xio21 to io26
molecules in a cubic centimetre of a solid or liquid sub
stance/ Assuming these data to be true, for the sake
of argument, a simple calculation enables us to show that
the almost inconceivably vast sphere of our stellar system
if entirely filled with solid matter, would not contain
more than about 68 x io90 atoms, that is to say, a number
requiring for its expression 92 places of figures. Now,
this number would be immensely less than the fifth order
of the powers of two.

In the variety of logical relations, which may exist
between a certain number of logical terms, we also meet
a case of higher variations. Two terms, as it has been
shewn (p. 154), may form four distinct combinations,
but the possible selections from these series of com
binations will be sixteen in number, or, excluding cases
of contradiction, seven. Three terms may form eight
combinations, allowing 256 selections, or with exclu
sion of contradictory cases, 193. Four terms give sixteen
combinations, and no less than 65,536 possible selec
tions from those combinations, the nature of which I
naturally abstained from exhaustively examining. Five
terms give thirty-two combinations, and 4,294,967,296
possible selections ; and for six terms the corresponding
numbers are sixty-four and 18,446,744,073,709,551,616.
Considering that it is the most common thing in the
world to use an argument involving six objects or terms,
it may excite some surprise that the complete investiga
tion of the relations in which six such terms may stand
to each other, should involve an almost inconceivable
number of cases. Yet these numbers of possible logical
relations belong only to the second order of combina
tions.

x 'Nature/ vol. i. p. 553.

CHAPTER X.

THEORY OF PROBABILITY.

t THE subject upon which we now enter must not be
regarded as an isolated and curious branch of speculation.
It is the necessary basis of nearly all the judgments
and decisions we make in the prosecution of science, or
the conduct of ordinary affairs. As Butler truly said,
'Probability is the very guide of life.' Had the science of
numbers been developed for no other purposes, it must
have been developed for the calculation of probabilities.
All our inferences concerning the future are merely pro
bable, and a due appreciation of the degree of probability
depends entirely upon a due comprehension of the prin
ciples of the subject. I conceive that it is impossible
even to expound the principles and methods of induction
as applied to natural phenomena, in a sound manner, with
out resting them upon the theory of probability. Perfect
knowledge alone can give certainty, and in nature perfect
knowledge would be infinite knowledge, which is clearly
beyond our capacities. We have, therefore, to content our
selves with partial knowledge — knowledge mingled with
ignorance, producing doubt.

Almost the greatest difficulty in this subject consists in
acquiring a precise notion of the matter treated. What
is it that we number, and measure, and calculate in the
theory of probabilities \ Is it belief, or opinion, or doubt,
or knowledge, or chance, or necessity, or want of art \

THE THEORY OF PROBABILITY. 225

Does probability exist in the things which are probable,
or in the mind which regards them as such \ The
etymology of the name lends us no assistance : for,
curiously enough, probable is ultimately the same word
as provable, a good instance of one word becoming differ
entiated to two opposite meanings.

Chance cannot be the subject of the theory, because
there is really no such thing as chance,a regarded as pro
ducing and governing events. This name signifies^//?'?/ (7,
and the notion is continually used as a simile to express
uncertainty, because we can seldom predict how a die,
or a coin, or a leaf will fall, or when a bullet will hit
the mark. But every one knows, on a little reflection, that
it is in our knowledge the deficiency lies, not in the cer
tainty of nature's laws. There is no doubt in lightning
as to the point it shall strike ; in the greatest storm there
is nothing capricious ; not a grain of sand lies upon the
beach, but infinite knowledge would account for its lying
there ; and the course of every falling leaf is guided by
the same principles of mechanics as rule the motions of
the heavenly bodies.

Chance then exists not in nature, and cannot co-exist
with knowledge ; it is merely an expression for our
ignorance of the causes in action, and our cons;quent
inability to predict t'.e result, or to bring it about in
fallibly. In nature the happening of a physical event
has been pre-determined from the first fashioning of the
universe. Probability belongs wholly to the mind ; this
indeed is proved by the fact that different minds may
regard the very same event at the same time with totally
different degrees of probability. A steam-vessel, for in
stance, is missing and some persons believe that she has
sunk in mid-ocean ; others think differently. In the

a Dufau, ' De la Methode d'Observation,' chap. iii.
Q

226 THE PRINCIPLES OF SCIENCE.

event itself there can be no such uncei tainty ; the steam-
vessel either has sunk or has not sunk, and no subsequent
discussion of the probable nature of the event can alter
the fact. Yet the probability of the event will really
vary from day to day, and from mind to mind, according
as the slightest information is gained regarding the vessels
met at sea, the weather prevailing there, the signs of
wreck picked up, or the previous condition of the vessel.
Probability thus belongs to our mental condition, to the
light in which we regard events, the occurrence or non-
occurrence of which is certain in themselves. Many
writers accordingly have asserted that probability is con
cerned with degree or quantity of belief. De Morgan
says,b 'By degree of probability we really mean or ought
to mean degree of belief/ The late Professor Donkin
expressed the meaning of probability as 'quantity of
belief;' but I have never felt satisfied with such a defini
tion of probability. The nature of belief is not more
clear to my mind than the notion it is used to define.
But an all-sufficient objection is, that the theory does not
measure what the belief is, but what it ought to be. Few
minds think in close accordance with the theory, and
there are many cases of evidence in which the belief
existing is habitually different from what it ought to be.
Even if tlje state of belief in any mind could be measured
and expressed in figures, the results would be worthless.
The very value of the theory consists in correcting and
guiding our belief, and rendering our states of mind and
consequent actions harmonious with our knowledge of
exterior conditions.

This objection has been clearly perceived by some of
those who still used quantity of belief as a definition of
probability. Thus De Morgan adds — ' Belief is but another
name for imperfect knowledge.' Professor Donkin has

b 'Formal Logic,' p. 172.

THE THEORY OF PROBABILITY. 227

well said that the quantity of belief is ' always relative
to a particular state of knowledge or ignorance ; but it
must be observed that it is absolute in the sense of not
being relative to any individual mind ; since, the same
information being presupposed, all minds ought to dis
tribute their belief in the same way/6 Dr. Boole, too,
seemed to entertain a like view, when he described the
theory as engaged with ' the equal distribution of ignor
ance,' d but we may just as well say that it is engaged
with the equal distribution of knowledge.

I prefer to dispense altogether with this obscure word
belief, and to say that the theory of probability deals with
quantity of knowledge, an expression of which a precise
explanation and measure can presently be given. An
event is only probable when our knowledge of it is
diluted with ignorance, and exact calculation is needed
to discriminate how much we do and do not know. The
theory has been described by some as professing to evolve
knowledge out of ignorance; but as Professor Donkin has
admirably remarked, it is really ' a method of avoiding
the erection of belief upon ignorance/ e It defines rational
expectation by measuring the comparative amounts of
knowledge and ignorance, and teaches us to regulate our
action with regard to future events in a way which will,
in the long run, lead to the least amount of disappointment
and injury. It is, as Laplace as happily expressed it, good
sense reduced to calculation.

This theory appears to me the noblest creation of
human intellect, and it passes my conception how two
men possessing such high intelligence as Auguste Cornte
and J. S. Mill, could have been found depreciating it,
or even vainly attempting to question its validity. To

c ' Philosophical Magazine,' 4th Series, vol. i. p. 355.

d ' Transactions of the Royal Society of Edinburgh,' vol. xxi. part iv.

e 'Philosophical Magazine,' 4th Series, vol i. p. 355.

228 THE PRINCIPLES OF SCIENCE.

eulogise the theory is as needless as to eulogise reason
itself.

Fundamental Principles of the Tlieory.

The calculation of probabilities is really founded, as
I conceive, upon the principle of reasoning set forth in
preceding chapters. We must treat equals equally, and
what we know of one case may be affirmed of every
other case resembling it in the necessary circumstances.
The theory consists in putting similar cases upon a par,
and distributing equally among them whatever knowT-
ledge we may possess. Throw a penny into the air, arid
consider what we know with regard to its mode of falling.
We know that it will certainly fall upon a flat side, so
that either the head or tail will be uppermost, but as
to whether it will be head or tail, our knowledge is
equally divided. Whatever we know concerning head,
we know as much concerning tail, so that we have no

O '

reason for expecting one more than the other. The least
predominance of belief to either side would be irrational,
as it would consist in treating unequally things of which
our knowledge is equal.

The theory does not in the least require, as some
writers have erroneously supposed, that we should first
ascertain by experiment the equal facility of the events
we are considering. So far as we can examine and
measure the causes in operation, events are removed
out of the sphere of probability. The theory comes into
play where ignorance begins, and the knowledge we
possess requires to be distributed over many cases.
Nor does ' the theory show that the coin will fall as
often on one side as the other. It is almost impossible
that this should happen, because some inequality in the
form of the coin, or some uniform manner in throwing
it up, is almost sure to occasion a slight preponderance

THE THEORY OF PROBABILITY. 229

in one direction. But as we do not previously know in
which way a preponderance will exist, we have no more
reason for expecting head than tail. Our state of know
ledge will be changed, indeed, should we throw up the
coin many times in succession and register the result.
Every throw gives us some slight information as to the
probable tendency of the coin, and in subsequent calcula
tions we must take this into account. In other cases
experience might show that we had been entirely mis
taken ; we might expect that a die would fall as often
on each of the six sides as on each other one in the long
run ; trial might show that the die was a loaded one,
and fell much the most often on a particular face. The
theory would not have misled us : it treated correctly
the information we had, which is all that any theory
can do.

It may be asked, Why spend so much trouble in calcu
lating from imperfect data, when a very little trouble
would enable us to render a conclusion certain by actual
trial ? Why calculate the probability of a measurement
being correct, when we can try whether it is correct ?
But I shall fully point out in later parts of this work
that in measurement we never can attain perfect coin
cidence. Two measurements of the same base line in a
survey may show a difference of some inches, and there
may be no means of knowing which is the better result.
A third measurement would probably agree with neither.
To select any one of the measurements, would imply that
we knew it to be the most nearly correct one, which we
do not. In this state of ignorance, the only guide is the
theory of probability, which proves that in the long run
the mean of different quantities will come most nearly to
the truth. In all other scientific operations whatsoever, per
fect knowledge is impossible, and when we have exhausted
all our instrumental means in the attainment of truth,

230 THE PRINCIPLES OF SCIENCE.

there is a margin of error which can only be safely treated
by the principles of probability.

The method which we employ in the theory consists
in calculating the number of all the cases or events

O

concerning which our knowledge is equal. If we have
even the slightest reason for suspecting that one event
is more likely to occur than another, we should take this
knowledge into account. This being done, we must
determine the whole number of events which are, so far
as we know, equally likely. Thus, if we have no reason
for supposing that a penny will fall more often one way
than another, there are two cases, head and tail, equally
likely. But if from trial or otherwise we know, or think
we know, that of 100 throws 55 will give tail, then the
probability is measured by the ratio of 55 to 100.

The mathematical formulae of the theory are exactly the
same as those of the theory of combinations. In this
latter theory, we determine in how many ways events may
be joined together, and we now proceed to use this know
ledge in calculating the number of ways in which a certain
event may come about, and thus defining its probability.
If we throw three pennies into the air, what is the proba
bility that two of them will fall tail uppermost ? This
amounts to asking in how many possible ways can we
select two tails out of three, compared with the whole
number of ways in which the coins can be placed. Now,
the fourth line of the Arithmetical Triangle (p. 208) gives
us the answer.- The whole number of ways in which we
can select or leave three things is eight, and the possible
combinations of two things at a time is three ; hence the
probability of two tails is the ratio of three to eight-
From the numbers in the triangle we may draw all the
following probabilities :—

One combination gives o tail. Probability £.

Three combinations give i tail. Probability f .

THE THEORY OF PROBABILITY. 231

Three combinations give 2 tails. Probability f .

One combination gives 3 tails. Probability -J-.
We could apply the same considerations to the ima
ginary causes of the difference of stature, the combina
tions of which were shown in p. 213. There are alto
gether 128 ways in which seven causes can be combined
together. Now, twenty-one of these combinations give
an addition of two inches, so that the probability of a
person under the circumstances being five feet two inches
is i2—. The probability of five feet three inches is ™258 ;
of five feet one inch is —-g ; of five feet r*8, and so on.
Thus the eighth line of the Arithmetical Triangle gives all
the probabilities arising out of the combinations of seven
causes or things,

Rules for the Calculation of Probabilities.

I will now explain as simply as possible the rules
for calculating probabilities. The principal rule is as
follows : —

Calculate the number of events which may happen
independently of each other, and which are as far as
is known equally probable. Make this number the de
nominator of a fraction, and take for the numerator the
number of such events as imply or constitute the hap
pening of the event, whose probability is required.

Thus, if the letters of the word Roma be thrown down
casually in a row, what is the probability that they will
form a significant Latin word"? The possible arrange
ments of four letters are 4x3x2x1, or 24 in number
(p. 201), and if all the arrangements be examined, seven
of these will be found to have meaning, namely Roma,
ramo, oram, mora, maro, armo, and amor. Hence the
probability of a significant result is -/4-.f

*' AVallis ' Of Combinations,' p. n1/.

232 THE PRINCIPLES OF SCIENCE.

We must distinguish comparative from absolute pro
babilities. In drawing a card casually from a pack, there
is no reason to expect any one card more than any other.
Now, there are four kings and four queens in a pack, so that
there are just as many ways of drawing one as the other,
and the probabilities are equal. But there are thirteen
diamonds, so that the probability of a king is to that of
a diamond as four to thirteen. Thus the probabilities
of each are propoitional to their respective numbers of
ways of happening. Now, I can draw a king in four
ways, and not draw one in forty-eight, so that the pro
babilities are in this proportion, or, as is commonly said,
the odds against drawing a king are forty-eight to four.
The odds are seven to seventeen in favour, or seventeen
to seven against the letters R,o,m,a, accidentally forming
a significant word. The odds are five to three against
two tails appearing in three throws of a penny. Con
versely, when the odds of an event are given, and the
probability is required, take the number in favour of the
event for numerator, and the sum of the numbers for
denominator.

It is obvious that an event is certain when all the
combinations of causes which can take place produce
that event. Now, if we were to represent the pro
bability of any such event according to our rule, it would
give the ratio of some number to itself, or unity. An
event is certain not to happen when no possible combina
tion of causes gives the event, and the ratio by the same
rule becomes that of o to some number. Hence it follows
that in the theory of probability certainty is expressed
by i, and impossibility by o ; but no mystical meaning
should be attached to these symbols, as they merely
express the fact that all or no possible combinations give
the event.

By a compound event, we mean an event which may be

THE THEORY OF PROBABILITY. 233

distinguished into two or more simpler events. Thus
the firing of a gun may be distinguished into pulling the
trigger, the fall of the hammer, the explosion of the cap,
&c. In this example the simple events are not inde
pendent, because if the trigger is pulled, the other events
will under proper conditions necessarily follow, and their
probabilities are therefore the same as that of the first
event. Events are independent when the happening of
one does not render the other either more or less probable
than before. Thus the death of a person is neither more
nor less probable because the planet Mars happens to be
visible. When the component events are independent,
a simple rule can be given for calculating the probability
of the compound event, thus — Multiply together the fi ac
tions expressing the probabilities of the independent
component events.

The probability of throwing tail twice with a penny
is 7=r x -ET, or ^ ; the probability of throwing it three times
running is ^ x -^ x ^, or ^ ; a result agreeing with that
obtained in an apparently different manner (p. 230). In
fact when we multiply together the denominators, we get
the whole number of ways of happening of the compound
event, and when we multiply the numerators, we get the
number of ways favourable to the required event.

Probabilities may be added to or subtracted from each
other under the important condition that the events in
question are exclusive of each other, so that not more than
one of them can happen. It might be argued that as
the probability of throwing head at the first trial is ^, and
at the second trial also ^, the probability of throwing
it in the first two throws is \ + ^, or certainty. Not only
is this result evidently absurd, but a repetition of the
process would lead us to a probability of i-| or of any
greater number, results which could have no meaning
whatever. The probability we wish to calculate is that of

234 THE PRINCIPLES OF SCIENCE.

one head in two throws, but in our addition we have
involved the case in which two heads also appear. The
true result is ^ + ^ x \ or f , or the probability of head at
the first throw, added to the exclusive probability that if it
does not come at the first, it will come at the second.
Some of the greatest difficulties of the theory and the
subtlest errors arise from the confusion of exclusive and
unexclusive alternatives. I may remind the reader that
the possibility of unexclusive alternatives was a point
previously discussed (p. 81), and to the reasons then given
for considering alternation as logically unexclusive, may be
added the existence of these difficulties in the theory of
probability. The expression

Headfirst throw or head second throw
ought to be interpreted in our logical system as including
both cases at once, and so it is in practice.

Employment of the Logical Abecedarian in questions of

Probability.

When the probabilities of certain events are given, and
it is required to deduce the probabilities of compound
events, the Logical Abecedarium may give assistance, pro
vided that there are no special logical conditions and all
the combinations are possible. Thus, if there be three
events A, B, C, of which the probabilities are a, (3, 7, then
the negatives of those events, expressing the absence
of the events, will have the probabilities i — a, i — /3, 1—7.
We have only to insert these values for the letters of the
combinations and multiply, and we obtain the probability
of each combination. Thus the probability of ABC is
0/37 ; of A6c, a(i — /3)(i - 7).

We can now clearly distinguish between the probabilities
of exclusive and unexclusive events. Thus if A and B
are events which may happen together like rain and high

THE THEORY OF PROBABILITY. 235

tide, or an earthquake and a storm, the probability of
A or B happening is not the sum of their separate proba
bilities. For by the Laws of Thought we develop A I B
into AB | Kb \ aB, and substituting a and {3, the proba
bilities of A and B respectively we obtain a./3 + a.(i — /3) +
(i — a)./3 or a + /3 — a . /3. But if events are incompossible or
incapable of happening together, like a clear sky and rain,
or a new moon and a full moon, then the events are not
really A or B but A not-B, or B not-A or in symbols
Kb } ttB. Now if we take

fj. — probability of Kb

v = probability of aB,

then we may add simply, and probability of Kb \ aB = /m + v.

Let the reader observe that since the combination AB
cannot exist, the probability of Kb is not the product of
the probabilities of A and b.

But when certain combinations are logically impossible,
it is no longer allowable to substitute the probability of
each term for the term, because the multiplication of
probabilities presupposes the independence of the events.
A large part of the late Dr. Boole's Laws of Thought is
devoted to an attempt to overcome this difficulty and
produce a General Method in Probabilities, by which from
certain logical conditions and certain given probabilities it
wrould be possible to deduce the probability of any other
combinations of events under those conditions. Boole
pursued his task with wonderful ingenuity and power, but
after spending much study on his work, I am compelled to
adopt the conclusion that his method is fundamentally
erroneous. As pointed out by Mr. Wilbraham &, Boole
obtains his results by an arbitrary assumption, which is
only the most probable, and not the only possible assump-

K 'Philosophical Magazine,' 4th Series, vol. vii. p. 465; vol. viii.
p. 91.

23G THE PRINCIPLES OF SCIENCE.

tion. The answer obtained is therefore not the real
probability, which is usually indeterminate, but only, as it
were, the most probable probability. Certain problems
solved by Boole are free from logical conditions and
therefore may admit of valid answers. These as I have

V

shown h may also be solved by the simple combinations
of the Abecedarium, but the remaind er of the problems
do not admit of a determinate answer, at least by Boole's
method.

Comparison of the Theory with Experience.

The Laws of Probability rest upon the simplest principles
of reasoning, and cannot be really negatived by any
possible experience. It might happen that a person
should always throw a coin head uppermost, and appear
incapable of getting tail by chance. The theory. would
not be falsified, because it contemplates the possibility of
the most extreme runs of luck. Our actual experience
might be counter to ah1 that is probable ; the whole
course of events might seem to be in complete contra
diction to what we should expect, and yet a casual con
junction of events might be the real explanation. It is
just possible that some regular coincidences which we
attribute to fixed laws of nature, are due to the accidental
conjunction of phenomena in the cases to which our
attention is directed. All that we can learn from
finite experience is capable, according to the theory of
probabilities, of misleading us, and it is only infinite
experience that could assure us of any inductive truths.

At the same time, the probability that any extreme
runs of luck will occur is so excessively slight, that it
would be absurd seriously to expect their occurrence. It

'Memoirs of the Manchester Literary and Philosophical Society,'
3rd Series, vol. iv. p. 347.

THE THEORY OF PROBABILITY. 237

is almost impossible, for instance, that any whist player
should have played in any two games where the distri
bution of the cards was exactly the same, by pure accident
(p. 217). Such a thing as a person always losing at
a game of pure chance, is wholly unknown. Coincidences
of this kind are not impossible, as I have said, but they
are so unlikely that the lifetime of any person, or indeed
the whole duration of history does not give any appreciable
probability of their being encountered. Whenever we
make any extensive series of trials of chance results, as in
throwing a die or coin, the probability is great that the
results will agree nearly with the predictions yielded by
theory. Precise agreement must not be expected, for that,
as the theory could show, is highly improbable. Several
attempts have been made to test, in this way, the accord
ance of theory and experience. The celebrated naturalist,
Buffon, caused the first trial to be made by a young
child who threw a coin many times in succession, and he
obtained 1992 tails to 2048 heads. A pupil of Professor
De Morgan repeated the trial for his own satisfaction, and
obtained 2044 tails to 2048 heads. In both cases the
coincidence with theory is as close as could be expected,
and the details may be found in De Morgan's ' Formal
Logic,' p. 185.

Quetelet also tested the theory in a rather more com
plete manner, by placing 20 black and 20 white balls in an
urn and drawing a ball out time after time in an
indifferent manner, each ball being replaced before a
new drawing was made. He found, as might be expected,
that the greater the number of drawings made the more
nearly were the white and black balls equal in number.
At the termination of the experiment he had registered
2066 white and 2030 black balls, the ratio being i'O21.

i ' Letters on the Theoi'y of Probabilities,' translated by Downcs, 1849,
PP- 36.37-

238

THE PRINCIPLES OF SCIENCE.

I have made a series of experiments in a third manner,
which seemed to me even more interesting, and capable
of more extensive trial. Taking a handful of ten coins,
usually shillings, I threw them up time after time, and
registered the numbers of heads which appeared each
time. Now the probability of obtaining 10 , 9 , 8 , 7, &c.,
heads is proportional to the number of combinations of
10,9,8,7, &c., things out of 10 things. Consequently
the results ought to approximate to the numbers in the
eleventh line of the Arithmetical Triangle. I made
altogether 2048 throws, in two sets of 1024 throws each,
and the numbers obtained are given in the following
table : —

Character of Throw.

Theoretical
Numbers.

First
Series.

Second
Series.

Average.

Divergence.

10 Heads o Tail

I

3

i

2

+ i

9 .. i

10

12

^3

»7*

+ 7i

8 „ 2

45

57

73

65

+ 20

7 .. 3

1 20

129

123

126

+ 6

6 „ 4

210

181

190

1*5*

-*54

5 .. 5

252

257

232

244!

~ 'U

4 H 6

210

201

197

199

— ii

3 ,, 7

I 2O

III

119

i«5

- 5

2 „ 8

45

52

50

51

+ 6

i „ 9

10

2 1

15

18

+ 8

o „ 10

I

O

i

a

- i

Totals.

1024

1O24

1024

1024

— i

The whole number of single throws of coins amounted
to 10x2048 or 20,480 in all, one half of which or
10,240 should theoretically give head. The total number
of heads obtained was actually 10,353, or 5222 in the
first series, and 5131 in the second. The coincidence
with theory is pretty close, but considering the large
number of throws there is some reason to suspect a
tendency in favour of heads.

The special interest of this trial consists in the ex
hibition, in a practical form, of the results of Bernouilli's
theorem, and the law of error or divergence from the

THE THEORY OF PROBABILITY. 239

mean to be afterwards more fully considered. It illus
trates the connection between combinations and permu
tations, which is exhibited in the Arithmetical Triangle,
and which underlies many of the most important
theorems of science.

Probable Deductive Arguments.

With the aid of the theory of probabilities, we may
extend the sphere of deductive argument. Hitherto we
have treated propositions as certain, and on the hypo
thesis of certainty have deduced conclusions equally
certain. But the information on which we reason in
ordinary life is seldom or never certain, and almost all
reasoning is really a question of probability. We ought
therefore to be fully aware of the mode and degree in
which the forms of deductive reasoning are affected by
the theory of probability, and many persons might be
surprised at the results which must be admitted. Many
controversial writers appear to consider, as De Morgan
remarked k, that an inference from several equally pro
bable premises is itself as probable as any of them, but
the true result is very different. If a fact or argument
involves many propositions, and each of them is uncertain,
the conclusion will be of very little force.

The truth of a conclusion may be regarded as a com
pound event, depending upon the premises happening
to be true ; thus, to obtain the probability of the conclusion,
we must multiply together the fractions expressing the
probabilities of the premises. Thus, if the probability is
TJ that A is B, and also -^ that B is C, the conclusion that
A is C, on the ground of these premises, is ^ x -| or ^.
Similarly if there be any number of premises requisite to
k ' Encyclopaedia Metrop.' art. Probabilities, p. 396.

240 THE PRINCIPLES OF SCIENCE.

the establishment of a conclusion and their probabilities
be m, n, p, q, r, &c., the probability of the conclusion on

the ground of these premises is m x n x p x q x r x

This product has but a small value, unless each of the
quanties m, n, &c., be nearly unity.

But it is particularly to be noticed that the probability
thus calculated is not the whole probability of the con
clusion, but that only which it derives from the premises
in question. Whately's1 remarks on this subject might
mislead the reader into supposing that the calculation is
completed by multiplying together the probabilities of the
premises. But it has been fully explained by De Morgan111
that we must take into account the antecedent probability
of the conclusion ; A may be C for other reasons besides
its being B, and as he remarks, ' It is difficult, if not
impossible, to produce a chain of argument of which the
reasoner can rest the result on those arguments only.'
We must also bear in mind that the failure of one argu
ment does not, except under special circumstances, disprove
the truth of the conclusion it is intended to uphold, other
wise there are few truths which could survive the ill
considered arguments adduced in their favour. But as
a rope does not necessarily break because one strand in it
is weak, so a conclusion may depend upon an endless
number of considerations besides those immediately in
view. Even when we have no other information we must
not consider a statement as devoid of all probability. The
true expression of complete doubt is a ratio of equality
between the chances in favour of and against it, and this
ratio is expressed in the probability ^.

Now if A and C are wholly unknown things, we have
no reason to believe that A is C rather than A is not C.
The antecedent probability is then ^. If we also have the

'Elements of Logic,' Book III, sections, n and 18.
m ' Encyclopaedia Metrop.' art. Probabilities, p. 400.

THE THEORY OF PROBABILITY. 241

probabilities that A is B, J and that B is C, J, we have
no right to suppose that the probability of A being C
is reduced by the argument in its favour. If the conclu
sion is true on its own grounds, the failure of the argument
does not affect it ; thus its total probability is its ante
cedent probability, added to the probability that this
failing, the new argument in question establishes it.
There is a probability ^ that we shall not require the
special argument ; a probability ^ that we shall, and
a probability ^ that the argument does in that case
establish it. Thus the complete result is i + i x ^, or -f.
In general language, if a be the probability formed on
a particular argument, and c the antecedent probability,
then the general result is

i — ( i - a) ( i — c), or a + c — ac.

We may put it still more generally in this way : — Let
a, b, c, d, &c., be the probabilities of a conclusion grounded
on various arguments or considerations of any kind. It is
only when all the arguments fail that our conclusion
proves finally untrue ; the probabilities of each failing
are respectively i — a, i—b, i — c, &c. ; the probability
that they will all fail (i — a)(i — 6)(i — c)... ; therefore
the probability that the conclusion will not fail is
i — (i — a)(i — 6)(i — c)...&c. On this principle it follows
that every argument in favour of a fact, however flimsy
and slight, adds probability to it. When it is unknown
whether an overdue vessel has foundered or not, every
slight indication of a lost vessel will add some proba
bility to the belief of its loss, and the disproof of any
particular evidence will not disprove the event.

We must apply these principles of evidence with great
care, and observe that in a great proportion of cases the
adducing of a weak argument does tend to the disproof of
its conclusion. The assertion may have in itself great
inherent improbability as being opposed to other evidence

R

242 THE PRINCIPLES OF SCIENCE.

or to the supposed laws of nature, and every reasoner may
be assumed to be dealing plainly, and putting forward the
whole force of evidence which he possesses in its favour.
If he brings but one argument, and its probability a is
small, then in the formula i — (i —a)(i — c) both a and c
are small, and the whole expression has but little value.
The whole effect of an argument thus turns upon the
question whether other arguments remain so that we can
introduce other factors (i —6), (i — c), &c., into the above
expression. In a court of justice, in a publication having
an express purpose, and in many other cases, it is doubtless
right to assume that the whole evidence considered to
have any value as regards the conclusion asserted, is
put forward.

To assign the antecedent probability of any proposi
tion, may be a matter of great difficulty or impos
sibility, and one with which logic and the theory of pro
bability has little concern. From the general body of
science or evidence in our possession, we must in each
case make the best judgment we can. But in the absence
of all knowledge the probability should be considered = ^,
for if we make it less than this we incline to believe it
false rather than true. Thus before we possessed any
means of estimating the magnitudes of the fixed stars, the
statement that Sirius was greater than the sun had
a probability of exactly ^ ; for it was as likely that it
would be greater as that it would be smaller ; and so of
any other star. This indeed was the assumption which
Michell made in his admirable speculations.0 It might
seem indeed that as every proposition expresses an agree
ment, and the agreements or resemblances between phe
nomena are infinitely fewer than the differences (p. 52),
eveiy proposition should in the absence of other informa
tion be infinitely improbable, or c — o. But in our logical

0 ' Philosophical Transactions' (1767). Abridg. vol. xii. p. 435.

THE THEORY OF PROBABILITY. 243

system every term may be indifferently positive or nega
tive, so that we express under the form A is B or A = AB
as many differences as agreements. It is impossible
therefore that we should have any reason to disbelieve
rather than to believe it. We can hardly indeed invent
a proposition concerning the truth of which we are
absolutely ignorant, except when we are absolutely ignorant
of the terms used. If I ask the reader to assign the
odds that a ' Platythliptic Coefficient is positive'? he
will hardly see his way to doing so, unless he regard
them as even.

The assumption that complete doubt is properly ex
pressed by 4 has been called in question by Bishop Terrot,i
who proposes instead the indefinite symbol § ; and he
considers that 'the d priori probability derived from
absolute ignorance has no effect upon the force of a
subsequently admitted probability/ But a writer of far
greater power, the late Professor Donkin, has strongly
defended the commonly adopted expression of complete
doubt. If we grant that the probability may have any
value between o and i, and that every separate value
is equally likely, then n and i-n are equally likely,
and the average is always J. Or we may take p . dp
to express the probability that our estimate concerning
any proposition should lie between p and p + dp. The
complete probability of the proposition is then the in
tegral taken between the limits i and o, or again ^r.

«*

Difficulties of the Theory.

The doctrine of probability, though undoubtedly true,
requires very careful application. Not only is it a branch

v 'Philosophical Transactions,' vol. 146. part i. p, 273.

9 ' Transactions of the Edinburgh Philosophical Society,' vol. xxi. p. 375,

r 'Philosophical Magazine,' 4th Series, vol. i. p. 361.

B 2

244 THE PRINCIPLES OF SCIENCE.

of mathematics in which positive blunders are frequently
committed, but it is a matter of great difficulty in many
cases, to be sure that the formulae correctly represent the
data of the problem. These difficulties often arise from
the logical complexity of the conditions, which might be,
perhaps to some extent cleared up by constantly bearing
in mind the system of combinations as developed in the
Indirect Logical Method. In the study of probabilities,
mathematicians had unconsciously employed logical pro
cesses far in advance of those in possession of logicians,
and the Indirect Method is but the full statement of
these processes.

It is very curious how often the most acute and power
ful intellects have gone astray in the calculation of
probabilities. Seldom was Pascal mistaken, yet he in
augurated the science with a mistaken solution.8 Leibnitz
fell into the extraordinary blunder of thinking that the
number twelve was as probable a result in the throwing
of two dice as the number eleven.* In not a few cases the
false solution first obtained seems more plausible to the
present day than the correct one since demonstrated.
James Bernouilli candidly records two false solutions of
a problem which he at first thought self-evident ; u and he
adds an express warning against the risk of error, especially
when we attempt to reason on this subject without a rigid
adherence to the methodical rules and symbols. x Mont-
mort was not free from similar mistakes/ and as to
D'Alembert, great though his reputation was, and perhaps
is, he constantly fell into blunders which must diminish
the weight of his opinions.2 He could not perceive, for

K Montucla, ' Histoire des Hathdmatiques,' vol. iii. p. 386.
* Leibnitz ' Opera/ Dutens' Edition, vol. vi. part i. p. 217. Todhunter's
' History of the Theory of Probability,' p. 48.

11 Todlmnter, pp. 67-69. * Ibid. p. 63. y Ibid. p. 100.

''• Hii«l. pp. 258-59, 286.

THE THEORY OF PROBABILITY. 245

instance, that the probabilities would be the same when
coins are thrown successively as when thrown simul
taneously. a Some men of high ability, such as Ancillon,
Moses Mendelssohn, Garve,b Auguste Comte c and J. S.
Mill,d have so far misapprehended the theory, as to
question its value or even to dispute altogether its
vaKdity.

Many persons have a fallacious tendency to believe that
when a chance event has happened several times together
in an unusual conjunction, it is less likely to happen
again. D'Alembert seriously held that if head was thrown
three times running with a coin, tail would more probably
appear at the next trial.6 Bequelin adopted the same
opinion, and yet there is no reason for it whatever. If
the event be really casual, what has gone before cannot in
the slightest degree influence it.

As a matter of fact, the more often the most casual
event takes place the more likely it is to happen again;
because there is some slight empirical evidence of a
tendency, as will afterwards be pointed out. The source of
the fallacy is to be found entirely in the feelings of
surprise with which we witness an event happening by
apparent chance, in a manner which seems to proceed from
design.

Misapprehension may also arise from overlooking the
difference between permutations and combinations. To
throw ten heads in succession with a coin is no more
unlikely than to throw any other particular succession
of heads and tails, but it is much less likely than five
heads and five tails without regard to their order, be-

a Todhunter, p. 279. b Ibid. p. 453.

c 'Positive Philosophy/ translated by Martineau, vol. ii. p. 120.
d 'System of Logic,' bk. iii. chap. 18. 5th Ed. vol. ii. p. 61.
e Montucla, ' Histoire/ vol. iii. p. 405. Todhunter, p. 263.

246 THE PRINCIPLES OF SCIENCE.

cause there are no less than 252 different particular
throws which will give this result, when we abstract
the difference of order.

Difficulties arise in the application of the theory from
our habitual disregard of slight probabilities. We are
obliged practically to accept truths as certain which are
nearly so, because it ceases to be worth while to calculate
the difference. No punishment could be inflicted if
absolutely certain evidence of guilt were required, and as
Locke remarks, ' He that will not stir till he infallibly
knows the business he goes about will succeed, will
have but little else to do but to sit still and perish.'f
There is not a moment of our lives when we do not lie
under a slight danger of death, or some most terrible fate.
There is not a single action of eating, drinking, sitting
down, or standing up which has not proved fatal to some
person. Several philosophers have tried to assign the
limit of the probabilities which we regard as zero ; Buffon
named f^Voo* because it is the probability that a man of
56 years of age would die the next day, and is practically
disregarded. Pascal had remarked that a man would be
esteemed a fool for hesitating to accept death when three
dice gave sixes twenty times running, if his reward in
case of a different result was to be a crown ; but as the
chance of death in question is only in-660, or unity divided
by a number of 47 places of figures, we may be said
every day to incur greater risks for less motives. There
is far greater risk of death, for instance, in a game of
cricket.

Nothing is more requisite than to distinguish carefully
between the truth of a theory and the truthful application
of the theory to actual circumstances. As a general rule,
events in nature or art will present a complexity of

1 'Essay on the Human Understanding,' bk. IV. ch. 14. § i.

THE THEORY OF PROBABILITY. 247

relations exceeding our powers of treatment. The infinitely
intricate action of the mind often intervenes and renders
complete analysis hopeless. If, for instance, the probability
that a marksman shall hit the target in a single shot be
i in 10, we might seem to have no difficulty in calculating
the probability of any succession of hits ; thus the proba
bility of three successive hits would be one in a thousand.
But, in reality, the confidence and experience derived from
the first successful shot would render a second success
more probable. The events are not really independent,
and there would generally be a far greater preponderance
of runs of apparent luck, than a simple calculation
of probabilities could account for. In. many persons,
however, a remarkable series of successes will produce a
degree of excitement rendering continued success almost
impossible.

Attempts to apply the theory of probabilities to the
results of judicial proceedings have proved of little value,
simply because the conditions are far too intricate. As
Laplace said,g ' Tant de passions, d'mterets divers et de
circonstances compliquent les questions relatives a ces
objets, qu'elles sont presque toujours insolubles.' Men
acting on a jury, or giving evidence before a court, are
subject to so many complex influences that no mathema
tical formulae can be framed to express the real conditions.
Jurymen or even judges on the bench cannot be regarded
as acting independently, with a definite probability in
favour of each delivering a correct judgment. Each man
of the jury is more or less influenced by the opinion of the
others, and there are subtle effects of character and manner
and strength of mind which defy human analysis. Even
in physical science we shall in comparatively few cases be
able to apply the theory in a definite manner, because the

s Quoted by Todhunter, 'History of the Theory of Probability,' p. 410.

248 THE PRINCIPLES OF SCIENCE.

data required for the estimation of probabilities are too
complicated and difficult to obtain. But such failures in
no way diminish the truth and beauty of the theory itself;
for in reality there is no branch of science in which, as we
shall afterwards fully consider, our symbols can cope with
the complexity of Nature. As the late Professor Donkin
excellently said,—

' I do not see on what ground it can be doubted that
every definite state of belief concerning a proposed
hypothesis, is in itself capable of being represented
by a numerical expression, however difficult or im
practicable it may be to ascertain its actual value. It
would be very difficult to estimate in numbers the vis
viva of all the particles of a human body at any instant ;
but no one doubts that it is capable of numerical ex
pression/11

The difficulty, in short, is merely relative to our know
ledge and skill, and is not absolute or inherent in the
subject. We must distinguish between what is theo
retically conceivable and what is practicable with our
present mental resources. Provided that our aspirations
are pointed in a right direction, we must not allow them
to be damped by the consideration that they pass beyond
what can now be turned to immediate use. In spite of
its immense difficulties of application, and the aspersions
which have been mistakenly cast upon it, the theory of
probabilities, I repeat, is the noblest, as it will in course
of time prove, perhaps the most fruitful branch of mathe
matical science. It is the very guide of life, and hardly
can we take a step or make a decision of any kind without
correctly or incorrectly making an estimation of proba
bilities. In the next chapter we proceed to consider how
the whole cogency of inductive reasoning, as applied to

h <

Philosophical Magazine,' 4th Series, vol. i. p. 354.

THE THEORY OF PROBABILITY. 249

physical' science rests upon probability. The truth or
untruth of a natural law, when carefully investigated,
resolves itself into a high or low degree of probability,
and this is the case whether or not we are capable of pro
ducing precise numerical data.

CHAPTEE XI.

PHILOSOPHY OF INDUCTIVE INFERENCE.

WE have inquired into the nature of the process of
perfect induction, whereby we pass backwards from certain
observed combinations of qualities or events, to the logical
conditions governing such combinations. We have also
investigated the grounds of that theory of probability,
which must be our guide when we leave certainty behind
us, and dilute knowledge with ignorance. There is now
before us the difficult task of endeavouring to decide how,
by the aid of that theory, we can ascend from the facts to
the laws of nature ; and may then with more or less
success anticipate the future course of events. All our
knowledge of natural objects must be ultimately derived
from observation, and the difficult question arises — How
can we ever know anything which we have not directly
observed through one of our senses, the apertures of the
rnind "? The practical utility of reasoning is to assure
ourselves that, at a determinate time or place, or under
specified conditions, a certain phenomenon may be ob
served. When we can use our senses and perceive that
the phenomenon does occur, reasoning is superfluous. If
the senses cannot be used, because the event is in the
future, or out of reach, how can reasoning take their
place 1 Apparently, at least, we must infer the unknown
from the known, and the mind must itself create an
addition to the sum of knowledge. But I hold that it is
quite impossible to make any real additions to the con-

PHILOSOPHY OF INDUCTIVE INFERENCE. 251

tents of our knowledge, except through new impressions
upon the senses, or upon some seat of feeling. I shall
attempt to show that inference, whether inductive or
deductive, is never more than an unfolding of the contents
of our experience, and that it always proceeds upon the
assumption that the future and the unperceived will be
governed by the same conditions as the past and the
perceived, an assumption which will often prove to be
mistaken.

In inductive just as in deductive reasoning, the con
clusion never passes beyond the premises. Reasoning
adds no more to the implicit contents of our knowledge,
than the arrangement of the specimens in a museum adds
to the number of those specimens. This arrangement adds
to our knowledge in a certain sense : it allows us to per
ceive the similarities and peculiarities of the individual
specimens, and on the assumption that the museum is an
adequate representation of nature, it enables us to judge
of the prevailing forms of natural objects. Bacon's first
aphorism holds perfectly true, that man knows nothing
but what he has observed, provided that we include his
whole sources of experience, and the whole implicit con
tents of his knowledge. Inference but unfolds the hidden
meaning of our observations, and the theory of probability
shows how far we go beyond our data in assuming that
new specimens will resemble the old ones, or that the
future may be regarded as proceeding uniformly with the
past.

Various Classes of Inductive Truths.

It will be desirable, in the first place, to distinguish
between the several kinds of truths which we endeavour
to establish by induction. Although there is a certain
common and universal element in all our processes of

252 THE PRINCIPLES OF SCIENCE.

reasoning, yet a diversity arises in their application.
Similarity of conditions between the events from which
we argue, and those to which we argue, must always be
the ground of inference; but this similarity may have
regard either to time or place, or the simple logical
combination of events, or to any conceivable junction of
circumstances involving quality, time, and place. Having
met with many pieces of substance possessing ductility,
and a bright yellow colour, and having discovered, by
perfect induction, that they all possess in addition a high
specific gravity, and a freedom from the corrosive action
of acids, we are led to expect that every piece of substance,
possessing like ductility, and a similar yellow colour, will
have an equally high specific gravity, and a like freedom
from corrosion by acids. This is a case of the co-existence
of qualities ; for the character of the specimens examined
alters not with time or place.

In a second class of cases, time will enter as a prin
cipal ground of similarity. When we hear a clock
pendulum beat moment after moment, at equal in
tervals, and with a uniform sound, we confidently expect
that the stroke will continue to be repeated uniformly.
A comet having appeared several times at nearly equal
intervals, we infer that it will probably appear again
at the end of another like interval. A man who has
returned home evening after evening for many years,
and found his house standing, may, on like grounds,
expect that it will be standing the next evening, and on
many succeeding evenings. Even the continuous exist
ence of an object in an unaltered state, or the finding
again of that which we have hidden, is but a matter of
inference to be decided by experience.

A still larger and more complex class of cases involves
the relations of space, in addition to those of time and
quality. Having observed that every triangle drawn upon

PHILOSOPHY OF INDUCTIVE INFERENCE. 253

the diameter of a circle, with its apex upon the circum
ference, apparently contains a right angle, we may
ascertain that all triangles in similar circumstances will
contain right angles. This is a case of pure space reason
ing, apart from circumstances of time or quality, and it
seems to be governed by different principles of reasoning.
I shall endeavour to show, however, that geometrical
reasoning differs but in degree from that which applies
to other natural relations. If we observe that the com
ponents of a binary star have moved for a length of time
in elliptic curves, we have reason to believe that they will
continue so to move. Time and space relations are here
complicated together.

The Relation of Cause and Effect.

In a very large part of the scientific investigations
which must be considered, we deal with events which
follow from previous events, or with existences which
succeed existences . Science, indeed, might arise even were
material nature a fixed and changeless whole. Endow
mind with the power to travel about, and compare part
with part, and it could certainly draw inferences concern
ing the similarity of forms, the co-existence of qualities,
or the preponderance of a particular kind of matter in
a changeless world. A solid universe, in at least approxi
mate equilibrium, is not inconceivable, and then the rela
tion of cause and effect would evidently be no more than
the relation of before and after. As nature exists, how
ever, it is a progressive existence, ever moving and
changing as time, the great independent variable, pro
ceeds. Hence it arises that we must continually compare
what is happening now with what happened a moment
before, arid a moment before that moment, and so on,
until we reach indefinite periods of past time. A comet

254 THE PRINCIPLES OF SCIENCE.

is seen moving in the sky, or its constituent particles
illumine the heavens with their tails of fire. We cannot
explain the present movements of such a body without
supposing its prior existence, with a determinate amount
of energy and direction of motion ; nor can we validly
suppose that our task is concluded when we find that it
came wandering to our solar system through the un
measured vastness of surrounding space. Every event
must have a cause, and that cause again a cause, until
we are lost in the obscurity of the past, and are driven
to the belief in one First Cause, by whom the whole
course of nature was determined.

Fallacious Use of the Term Cause.

The words Cause and Causation have given rise to in
finite trouble and obscurity, and have in no slight degree
retarded the progress of science. From the time of
Aristotle, the work of philosophy has been often de
scribed as the discovery of the causes of things, and
Francis Bacon adopted the notion when he said a ' vere
scire esse per causas scire.' Even now it is not uncom
monly supposed that the knowledge of causes is some
thing different from other knowledge, and consists, as it
were, in getting possession of the keys of nature. A
single word may thus act as a spell, and throw the
clearest inteUect into confusion, as I have often thought
that Locke was thrown into confusion when endeavouring
to find a meaning for the word power.1' In Mr. Mill's
' System of Logic ' the term cause seems to have re
asserted its old noxious power. Not only does Mr. Mill
treat the Laws of Causation as almost co-extensive with

a 'Novum Organum/ bk. ii. Aphorism 2.

b 'Essay on the Human Understanding/ bk. ii. chap. xxi.

PHILOSOPHY OF INDUCTIVE INFERENCE. 255

science, but he so uses the expression as to imply that
when once we pass within the circle of causation we deal
with certainties.

The philosophical danger which attaches to the use of
this word may be thus described. A cause is defined as
the necessary or invariable antecedent of an event, so
that when the cause exists the effect will also exist or
soon follow. If then we know the cause of an event, we
know when it will certainly happen ; and as it is implied
that science, by a proper experimental method, may attain
to a knowledge of causes, it follows that experience may
give us a certain knowledge of future events. Now, no
thing is more unquestionable than that finite experience
can never give us certain knowledge of the future, so that
either a cause is not an invariable antecedent, or else we
can never gain certain knowledge as to causes. The first
horn of this dilemma is hardly to be accepted. Doubtless
there is in nature some invariably acting mechanism, such
that from certain fixed conditions an invariable result
always emerges. But we, with our finite minds and
short experience, can never penetrate the mystery of
those existences which embody the Will of the Creator,
and evolve it throughout time. We are in the position
of spectators who witness the productions of a compli
cated machine, but are not allowed to examine its inti
mate structure. We learn what does happen and what
does appear, but if we ask for the reason, the answer
would involve an infinite depth of mystery. The simplest
bit of matter, or the most trivial incident, such as the
stroke of two billiard balls, offers infinitely more to learn
than ever the human intellect can fathom. The word cause
covers just as much untold meaning as any of the words
substance, matter, thought, existence.

256 THE PRINCIPLES OF SCIENCE.

Confusion of Two Questions.

The subject is much complicated, too, by the confusion
of two distinct questions. An event having happened, we
may ask —

(j) Is there any cause for the event \
(2) Of what kind is that cause 1

No one would assert that the mind possesses any
faculty capable of inferring, prior to experience, that the
occurrence of a sudden noise with flame and smoke indi
cates the combustion of a black powder, formed by the
mixture of black, white, and yellow powders. The greatest
upholder of d priori doctrines will allow that the parti
cular aspect, shape, size, colour, texture, and other qualities
of a cause must be gathered from experience and through
the senses.

The question whether there is any cause at all for an
event, is of a totally different kind. If an explosion could
happen without any prior existing conditions, it must be
a new creation — a distinct addition to the universe. It
may be plausibly held that we can imagine neither the
creation nor annihilation of anything. As regards matter,
this has long been held true ; as regards force, it is now
almost universally assumed as an axiom that energy can
neither come into nor go out of existence without distinct
acts of Creative Will. That there exists any instinctive
belief to this effect, indeed, seems doubtful. We find
Lucretius, a philosopher of the utmost intellectual power
and cultivation, gravely assuming that his raining atoms
could turn aside from their straight paths in a self-deter
mining manner, and by this spontaneous origination of
energy determine the form of the universe.0 Sir George
Airy, too, seriously discussed the mathematical conditions

0 ' De Rerum Natura/ bk. ii. 11. 216-293.

PHILOSOPHY OF INDUCTIVE INFERENCE. 257

under which a perpetual motion, that is, a perpetual
source of self-created energy might exist.d The larger
part of the philosophic world has long held that in mental
acts there is free will — in short, self-causation. It is in
vain to attempt to reconcile this doctrine with that of an
intuitive belief in causation, as Sir W. Hamilton candidly
allowed.

It is quite obvious, moreover, that to assert the exist
ence of a cause for every event, cannot do more than
remove into the indefinite past the inconceivable fact and
mystery of creation. At any given moment matter and
energy were equal to what they are at present, or they
were not ; if equal, we may make the same inquiry con
cerning any other moment, however long prior, and we
are thus obliged to accept one horn of the dilemma — ex
istence from infinity, or creation at some moment. This
is but one of the many cases in which we are compelled
to believe in one or other of two alternatives, both incon
ceivable. My present purpose, however, is to point out
that we must not confuse this supremely difficult question
with that into which inductive science inquires on the
foundation of facts. By induction we gain no certain
knowledge ; but by observation, and the inverse use of
deductive reasoning, we estimate the probability that an
event which has occurred was preceded by conditions of
specified character, or that such conditions will be followed
by the event.

Definition of the Term Cause.

Clear definitions of the word cause have been given by
several philosophers. Hobbes has said, ' A cause is the
sum or aggregate of all such accidents both in the agents

d 'Cambridge Philosophical Transactions/ [1830] vol. iii. pp. 369^
372.

S

258 THE PRINCIPLES OF SCIENCE.

and the patients, as concur in the producing of the effect
propounded ; all which existing together, it cannot be
understood but that the effect existeth with them; or
that it can possibly exist if any of them be absent/
Dr. Brown, in his 'Essay on Causation/ gave a nearly
corresponding statement. 'A cause,' he sayse, 'may be
defined to be the object or event which immediately
precedes any change, and which existing again in similar
circumstances will be always immediately followed by a
similar change/ Of the kindred word poiver, he like
wise says : f ' Power is nothing more than that invariable-
ness of antecedence which is implied in the belief of
causation/

These definitions may be accepted with the qualifica
tion that our knowledge of causes in such a sense can
be probable only. The work of science consists in ascer
taining the combinations in which phenomena present
themselves. Concerning every event we shall have to
determine its probable conditions, or group of antecedents
from which it probably follows. An antecedent is any
thing which exists prior to an event ; a consequent is
anything which exists subsequently to an antecedent. It
will not usually happen that there is any probable con
nection between an antecedent and consequent. Thus
nitrogen is an antecedent to the lighting of a common
fire ; but it is so far from being a cause of the lighting,
that it renders the combustion less active. Daylight is
an antecedent to all fires lighted during the day, but it
probably has no appreciable effect one way or the other.
But in the case of any given event it is usually pos
sible to discover a certain number of antecedents which

0 'Observations on the Nature and Tendency of the Doctrine of
Mr. Hume, concerning the Relation of Cause and Effect.' Second ed.
P- 44- i Ibid. p. 97.

PHILOSOPHY OF INDUCTIVE INFERENCE. 259

seem to be always present, and with more or less pro
bability we conclude that when they exist the event will
follow.

Let it be observed that the utmost latitude is at present
enjoyed in the use of the term cause. Not only may a
cause be an existent thing endowed with powers, as
oxygen is the cause of combustion, gunpowder the cause
of explosion, but the very absence or removal of a thing
may also be a cause. It is quite correct to speak of the
dryness of the Egyptian atmosphere, or the absence of
moisture, as being the cause of the preservation of
mummies, and other remains of antiquity. The cause of
a mountain elevation, Ingleborough for instance, is the
excavation of the surrounding valleys by denudation. It
is not so usual to speak of the existence of a thing at one
moment as the cause of its existence at the next, but to
me it seems the commonest case of causation which can
occur. The cause of motion of a billiard ball may be the
stroke of another ball ; and recent philosophy leads us to
look upon all motions and changes, as but so many mani
festations of prior existing energy. In all probability
there is no creation of energy and no destruction, so that as
regards both mechanical and molecular changes, the cause
is really the manifestation of existing energy. In the
same way I see not why the prior existence of matter is
not also a cause as regards its subsequent existence. All
science tends to show us that the existence of the universe
in a particular state at one moment, is the condition of its
existence at the next moment, in an apparently different
state. When we analyse the meaning which we can
attribute to the word cause, it amounts to the existence of
suitable portions of matter endowed with suitable quan
tities of energy. If we may accept Home Tooke's asser
tion, cause has etymologically the meaning of thing before.
Though, indeed, the origin of the word is very obscure, its

S 2

2GO THE PRINCIPLES OF SCIENCE.

derivatives the Italian cosa, and the French chose, mean
simply thing. In the German equivalent iirsache, we have
plainly the original meaning of thing before, the sache
denoting 'interesting or important object,' the English
sake, and ur being the equivalent of the English ere,
before*1. We abandon, then, both etymology and philo
sophy, when we attribute to the laivs of causation any
meaning beyond that of the conditions in which an event
may be expected to happen, according to our observation
of the previous course of nature.

I have no objection to use the words cause and
causation, provided they are never allowed to lead us to
imagine that our knowledge of nature can attain to cer
tainty. I repeat that if a cause is an invariable and
necessary condition of an event, we can never know
certainly whether the cause exists or not. To us, then, a
cause is not to be distinguished from the group of positive
or negative conditions which, with more or less probability,
precede an event. In this sense, there is no particular
difference between knowledge of causes and our general
knowledge of the combinations, or succession of com
binations, in which the phenomena of nature are presented
to us, or found to occur in experimental inquiry.

Distinction of Inductive and Deductive Results.

We must carefully avoid confusing together inductive
investigations which terminate in the establishment of
general laws, and those which seem to lead directly to
the knowledge of future particular events. That process
only can be called induction which gives general laws,
and it is by the subsequent employment of deduction that
we can alone anticipate particular events. If the ob
servation of a number of cases shews that alloys of metals

11 Leslie, ' Inquiry into the Nature of Heat/ Note xvi. p. 521.

PHILOSOPHY OF INDUCTIVE INFERENCE, 261

fuse at lower temperatures than their constituent metals,
I may with more or less probability draw a general in
ference to that effect, and may thence deductively ascer
tain the probability that the next alloy examined will fuse
at a lower temperature than its constituents. It has been
asserted, indeed, by Mr. J. S. Mill1, and partially admitted
by Mr. Fowler k, that we can argue directly from case to
case, so that what is true of some alloys will be true of
the next. Doubtless, this is the usual result of our
reasoning, regard being had to degrees of probability ; but
these logicians fail entirely to give any explanation of the
process by which we get from case to case. To point, as
Mr. Mill has done, to the reasoning, if such it can be
called, of brute animals, is little better than to parody
philosophy1. It may well be allowed, indeed, that the
knowledge of future particular events is one main purpose
of our investigations, and if there were any process of
thought by which we could pass directly from event to
event without ascending into general truths, this method
would be sufficient, and certainly the most brief and
simple. It is true, also, that the laws, of mental asso
ciation lead the mind always to expect the like again in
apparently like circumstances, and even animals of very
low intelligence must have some trace of such powers of
association, serving to guide them more or less correctly,
in the absence of true reasoning faculties. But it is the
very purpose of logic, according to Mr. Mill, to ascertain
whether inferences have been correctly drawn, rather than
to discover themm. Even if we can, then, by habit,

i ' System of Logic,' bk. II. chap. iii. Mr. Bain has not adopted the
views of Mr. Mill, on this particular point, so far as I can ascertain. See
his 'Inductive Logic/ p. i.

k 'Inductive Logic,' pp. 13-14.

1 'System of Logic,' bk. II. chap. 3, § 3. Fifth ed. pp. 212-213.

"» Ibid., Introduction, § 4. Fifth ed. pp. 8-9.

262 THE PRINCIPLES OF SCIENCE.

association, or any rude process of inference, infer the
future directly from the past, it is the work of logic to
analyse the conditions on which the correctness of this
inference depends. Even Mr. Mill would admit that such
analysis involves the consideration of general truths11, and
in this, as in several other important points, we might
controvert Mr. Mill's own views by his own statements.

On the Grounds of Inductive Inference.

I hold that, in all cases of inductive inference, we must
invent hypotheses, until we fall upon some hypothesis
which yields deductive results in accordance with experi
ence. Such accordance renders the chosen hypothesis
more or less probable, and we may then deduce, with some
degree of likelihood, the nature of our future experience, on
the assumption that no arbitrary change takes place in
the conditions of nature. We can only argue from the
past to the future, on the general principle set forth in the
commencement of this work, that what is true of a thing
will be true of the like. So far then as one object or
event differs from another, all inference is impossible ;
particulars as particulars can no more make an inference
than grains of sand can make a rope. We must always
rise to something which is general or same in the cases,
and assuming that sameness to be extended to new cases
we learn their nature. Hearing a clock tick five thousand
times without exception or variation, we adopt the very
probable hypothesis that there is some invariably acting
machine which produces those uniform sounds, and which
will, in the absence of change, go on producing them.
Meeting twenty times with a bright yellow ductile sub
stance, and finding it to be always very heavy and in
corrodible, I infer that there was some natural condition,

n ' System of Logic,' bk. II. chap. iii. § 5. pp. 225, &c.

PHILOSOPHY OF INDUCTIVE INFERENCE. 263

which tended, in the creation of things, to associate these
properties together, and I expect to find them associated
in the next instance. But there always is the possibility
that some unknown change may take place between past
and future cases. The clock may run down, or be subject
to any one of a hundred accidents altering its condition.
There is no reason in the nature of things, so far as known
to us, why yellow colour, ductility, high specific gravity,
and incorrodibility, should always be associated together ;
and in other like cases, if not in this, men's expectations
have been deceived. Our inferences, therefore, always
retain more or less of a hypothetical character, and are so
far open to doubt. Only in proportion as our induction
approximates to the character of perfect induction, does
it approximate to certainty. The amount of uncertainty
corresponds to the probability that other objects than
those examined, may exist and falsify our inferences ; the
amount of probability corresponds to the amount of infor
mation yielded by our examination ; and the theory of
probability will be needed to prevent our over-estimating
or under-estimating the knowledge we possess.

Illustrations of the Inductive Process.

To illustrate the passage from the known to the ap
parently unknown, let us suppose that the phenomena
under investigation consist of numbers, and that the
following six numbers being exhibited to us, we are
required to infer the character of the next in the
series : —

5, 15, 35, 45, 65, 95.

The question first of all arises, How may we describe this
series of numbers ? What is uniformly true of them ?
The reader cannot fail to perceive at the first glance that
they all end in five, and the problem is, from the proper-

264 THE PRINCIPLES OF SCIENCE.

ties of these six numbers, to infer the properties of the
next number ending in five. If we proceed to test their
properties by the process of perfect induction, we soon
perceive that they have another common property, namely
that of being divisible by jive without remainder. May
we then assert that the next number ending in five is also
divisible by five, and, if so, upon what grounds 1 Or
extending the question, Is every number ending in five
divisible by five 1 Does it follow that because six num
bers obey a supposed law, therefore 376,685,975 or any
other number, however large, obeys the law r( I answer
certainly not. The law in question is undoubtedly true ;
but its truth is not proved by any finite number of exam
ples. All that these six numbers can do, is to suggest to
my mind the possible existence of such a law ; and I then
ascertain its truth, by proving deductively from the rules
of decimal numeration, that any number ending in five
must be made up of multiples of five, and must therefore
be itself a multiple.

To make this more plain, let the reader now examine
the numbers—

7, i7> 37> 47> 67, 97.

They all obviously end in 7 instead of 5, and though not
at equal intervals, the intervals are exactly the same as in
the previous case. After a little consideration, the reader
will perceive that these numbers all agree in being prime
numbers, or multiples of unity only. May we then infer
that the next, or any other number ending in 7, is a
prime number '{ Clearly not, for on trial we find that
27, 57, 117 are not primes. Six instances, then, treated
empirically, lead us to a true and universal law in one
case, and mislead us in another case. We ought, in fact
to have no confidence in any law until we have treated it
deductively, and have shown that from the conditions
supposed the results expected must ensue. From the

PHILOSOPHY OF INDUCTIVE INFERENCE. 265

principles of number, no one can show that numbers
ending in 7 should be primes.

From the history of the theory of numbers some good
examples of false induction can be adduced. Taking the
following series of prime numbers

41, 43, 47, 53, 61, 71, 83, 97, 113, 131, 151, &c.,
it will be found that they all agree in being values of
the general expression x2 + x + 4 1 , putting for x in succes
sion the values, o, i, 2, 3, 4, &c. We thus seem always
to obtain a prime number, and the induction is apparently
very strong, to the effect that this expression always will
give primes. Yet a few more trials will disprove this false
conclusion. Put x = 40, and we obtain 40 x 40 + 40 + 41,
or 41 x 4 1 . Now such a failure could never have hap
pened, had we shown any deductive reason why x2 + x + 4 1
should give primes.

There can be no doubt that what here happens with
forty instances, might happen with forty thousand or
forty million instances. An apparent law never once
failing up to a certain point may then suddenly break
down, so that inductive reasoning, as it has been described
by some writers, can give no sure knowledge of what is to
come. Mr. Babbage admirably pointed out, in his Ninth
Bridgewater Treatise, that a machine could be constructed
to give a perfectly regular series of numbers, through
a vast series of steps, and yet to break the law of progres
sion suddenly at any required point. No number of
particular cases as particulars enables us to pass by
inference to any new case. It is hardly needful to inquire
here what can be inferred from an infinite series of facts,
because they are never practically within our power ; but
we may unhesitatingly accept the conclusion, that no
finite number of instances can ever prove a general law,
or can give us sure knowledge of even one other instance.

General mathematical theorems have indeed been dis-

266 THE PRINCIPLES OF SCIENCE.

covered by the observation of particular cases, and may
again be so discovered. We have Newton's own state-

o

ment, to the effect that he was thus led to the all-impor
tant Binomial Theorem, the basis of the whole structure
of mathematical analysis. Speaking of a certain series of
terms, expressing the area of a circle or hyperbola, he says,
'I reflected that the denominators were in arithmetical
progression ; so that only the numerical co-efficients of
the numerators remained to be investigated. But these,
in the alternate areas, were the figures of the powers of
the number eleven, namely 1 1°, 1 11, 1 12, 1 13, 1 14 ; that is,
in the first i ; in the second i, i ; in the third i, 2, i ; in
the fourth i, 3, 3, i ; in the fifth i, 4, 6, 4, i.° I inquired,
therefore, in what manner all the remaining figures could
be found from the first two ; and I found that if the first
figure be called m, all the rest could be found by the
continual multiplication of the terms of the formula

ni—o m—i in— 2 m—^

X X X X &c. P

1234

It is pretty evident, from this most interesting statement,
that Newton having simply observed the succession of the
numbers, tried various formula? until he found one which
agreed with them all. He was so little satisfied with this
process, however, that he verified particular results of his
new theorem by comparison with the results of common
multiplication, and the rule for the extraction of the
square root. Newton, in fact, gave no demonstration of
his theorem ; and a number of the first mathematicians
of the last century, James Bernouilli, Maclaurin, Landen,
Euler, Lagrange, &c., occupied themselves with discovering
a conclusive method of deductive proof.

0 These are the figurate numbers considered in pages 206-216.

P ' Commercium Epistolicum. Epistola ad Oldenburgum,' Oct. 24,
1676. Horsley's '"Works of Newton', vol. iv. p. 541. See De Morgan
in 'Penny Cyclopaedia', art. Binomial TJworem, p. 412.

PHILOSOPHY OF INDUCTIVE INFERENCE. 267

Sir George Airy has also recorded a curious case, in
which he accidentally fell by trial on a new geometrical
property of the sphered Many of the most important and
now trivial propositions in geometry, were probably thus
discovered by the ancient Greek geometers ; and we have
pretty clear evidence of this in the Commentaries of
Proclus.r But discovery in such cases means nothing
more than suggestion, and it is always by pure deduction
that the general law is really established. As Proclus
puts it, we must pass from sense to consideration.

Given, for instance, the series of figures in the accom
panying diagram, a little examination and measurement
will show that the curv
ed lines approximate to
semicircles, and the rec
tilineal figures to right-
angled triangles. These
figures may seem to sug
gest to the mind the gen
eral law that angles in
scribed in semicircles are right angles ; but no number of
instances, and no possible accuracy of measurement would
really establish the truth of that general law. Availing
ourselves of the suggestion furnished by the figures, we
can only investigate deductively the consequences which
flow from the definition of a circle, until we discover
among them the property of containing right angles.
Many persons, after much labour, have thought that they
had discovered a method of trisecting angles by plane
geometrical construction, because a certain complex ar
rangement of lines and circles had appeared to trisect an
angle in every case tried by them, and they inferred, by a

<i 'Philosophical Transactions/ [1866] vol. 146, p. 334.
r Bk. ii. chap. iv.

268 THE PRINCIPLES OF SCIENCE.

supposed act of induction, that it would succeed in all
other cases. Professor de Morgan has recorded a proposed
mode of trisecting the angle which could not be dis
criminated by the senses from a true general solution,
except when it was applied to very obtuse angles.3 In
all such cases, it has always turned out either that the
angle was not trisected at all, or that only certain
particular angles could be thus trisected. They were
misled by some apparent or special coincidence, and only
deductive proof could establish the truth and generality
of the result. In this case, deductive proof shows that the
problem, as attempted, is impossible, and that angles
generally cannot be trisected by common geometrical
methods.

Geometrical Reasoning.

This view of the matter is strongly supported by the
further consideration of geometrical reasoning. No skill
and care could ever enable us to verify absolutely any one
geometrical proposition. Rousseau, in his Emile? tells us
that we should teach a child geometry by causing him to
measure and compare figures by superposition. While a
child was yet incapable of general reasoning, this would
doubtless be an instructive exercise ; but it never could
teach geometry, nor prove the truth of any one proposition.
All our figures are rude approximations, and they may
happen to seem unequal when they should be equal,
and equal when they should be unequal. Moreover,
figures may from chance be equal in case after case, and
yet there may be no general reason why they should be
so. The results of deductive geometrical reasoning are

s 'Budget of Paradoxes,' p. 257.

t i2mo. Amsterdam, 1762, vol. i. p. 401.

PHILOSOPHY OF INDUCTIVE INFERENCE. 269

absolutely certain, and are either exactly true or capable
of being carried to any required degree of approximation.
In a perfect triangle, the angles must be equal to one half-
revolution precisely ; even an infinitesimal divergence
would be impossible ; and I believe with equal confidence,
that however many are the angles of a figure, provided
there are no re-entrant angles, the sum of the angles will
be precisely and absolutely equal to twice as many right-
angles as the figure has sides, less by four right-angles.
In such cases, the deductive proof is absolute and com
plete ; empirical verification can at the most guard against
accidental oversights.

There is a second class of geometrical truths which can
only be proved by approximation ; but, as the mind sees
no reason why that approximation should not always go
on, we arrive at complete conviction. We thus learn that
the surface of a sphere is equal exactly to two-thirds of
the whole surface of the circumscribing cylinder, or to four
times the area of the generating circle. The area of a
parabola is exactly two-thirds of that of the circumscribing
parallelogram. The area of the cycloid is exactly three
times that of the generating circle. These are truths that
we could never ascertain, nor even verify by observation ;
for any finite amount of difference, vastly less than what
the senses can discern, would falsify them. There are
again geometrical relations which we cannot assign ex
actly, but can carry to any desirable degree of approxi
mation. Thus, the ratio of the circumference to the
diameter of a circle is that of 3' 14159265358979323846....
to i, and the approximation may be carried to any ex
tent by the expenditure of sufficient labour, as many as 607
places of figures having been calculated." Some years since,
I amused myself by trying how near I could get to this
ratio, by the careful use of compasses, and I did not come

" ' English Cyclopaedia/ art. Tabhs.

270 THE PRINCIPLES OF SCIENCE.

nearer than i part in 540. We might imagine measure
ments so accurately executed as to give us eight or ten
places correctly. But the power of the hands and senses
must soon stop, whereas the mental powers of deductive
reasoning can proceed to an unlimited degree of approxi
mation. Geometrical truths, then, are incapable of verifi
cation ; and, if so, they cannot even be learnt by observa
tion. How can I have learnt by observation a proposition
of which I cannot even prove the truth by observation,
when I am in possession of it \ All that observation or
empirical trial can do is to suggest propositions, of which
the truth may afterwards be proved deductively. By
drawing a number of right-angled triangles on paper,
with squares upon their sides, and cutting out and
weighing these squares very accurately, I might have
reason to suspect the existence of the relation of equality
proved in Euclid's 47th Proposition ; but no process of
weighing or measuring could ever prove it, nor could it
ever assure me that the like degree of approximation
would exist in untried cases.

Much has been said about the peculiar certainty of
mathematical reasoning, but it is only certainty of deduc
tive reasoning, and equal certainty attaches to all correct
logical deduction. If a triangle be right-angled, the
square on the hypothenuse will undoubtedly equal the
sum of the two squares on the other sides ; but I can
never be sure that a triangle is right-angled : so I can be
certain that nitric acid will not dissolve gold, provided I
know that the substances employed really correspond to
those on which I tried the experiment previously. Here
is like certainty of inference, and like doubt as to the
facts.

PHILOSOPHY OF INDUCTIVE INFERENCE. 271

Discrimination of Certainty and Probability in the
Inductive Process.

We can never recur too often to the truth that our
knowledge of the laws and future events of the external
world is only probable. The mind itself is quite capable
of possessing certain knowledge, and it is well to discri
minate carefully between what we can and cannot know
with certainty. In the first place, whatever feeling is
actually present to the mind is certainly known to that
mind. If I see blue sky, I may be quite sure that I
do experience the sensation of blueness. Whatever I do
feel, I do feel beyond all doubt. We are indeed very
likely to confuse what we really feel with what we are
inclined to associate with it, or infer inductively from
it; but the whole of our consciousness, as far as it is
the result of pure intuition arid free from inference, is
certain knowledge beyond all doubt.

In the second place, we may have certainty of inference ;
the first axiom of Euclid, the fundamental laws of thought,
and the rule of substitution (p. u), are certainly true;
and if my senses could inform me that A was indistin
guishable in colour from B, and B from C, then I should
be equally certain that A was indistinguishable from C.
In short, whatever truth there is in the premises, I can
certainly embody in their correct logical result. But
practically the certainty generally assumes a hypothetical
character. I never can be quite sure that two colours
are exactly alike, that two magnitudes are exactly equal,
or that two bodies whatsoever are identical even in their
apparent qualities. Almost all our judgments involve
quantitative relations, and, as will be shown in succeeding
chapters, we can never attain exactness and certainty
where continuous quantity enters. Judgments concerning

272 THE PRINCIPLES OF SCIENCE.

discontinuous quantity or numbers, however, allow of cer
tainty ; for I may establish beyond doubt, for instance, that
the difference of the squares of 17 and 13 is the product
of 17 + 13 and 17-13, and is therefore 30 x 4, or 120.

Inferences which we draw concerning natural objects
are never certain except in a hypothetical point of
view. It might seem, indeed, to be certain that iron is
magnetic, or that gold is incapable of solution in nitric
acid ; but, if we carefully investigate the meanings of
these statements, they will be found to involve no cer
tainty but that of consciousness and that of hypothetical
inference. For what do I mean by iron or gold ? If I
choose a remarkable piece of yellow substance, call it
gold, and then immerse it in a liquid which I call nitric
acid, and find that there is no change called solution,
then consciousness has certainly informed me that with
my meaning of the terms, ' Gold is insoluble in nitric
acid.' I may further be certain of something else ; for if
this gold and nitric acid remain what they were, I may be
sure there will be no solution on again trying the experi
ment. If I take other portions of gold and nitric acid,
and am sure that they really are identical in properties
with the former portions, I can be certain that there will
be no solution. But at this point my knowledge becomes
purely hypothetical ; for how can I be sure without trial
that the gold and acid are really identical in nature with
what I formerly called gold and nitric acid. How do
I know gold when I see it \ If I judge by the appa
rent qualities — colour, ductility, specific gravity, &c., I
may be misled, because there may always exist a sub
stance which to the colour, ductility, specific gravity, and
other specified qualities, joins others which we do not
expect. Similarly, if iron is magnetic, as shown by an
experiment with objects answering to those names, then
all iron is magnetic, meaning all pieces of matter identical

PHILOSOPHY OF INDUCTIVE INFERENCE. 273

with my assumed piece. But in trying to identify iron, I
am always open to mistake. Nor is this liability to mis
take a matter of speculation only v.

The history of chemistry shows that the most confident
inferences may have been falsified by the confusion of one
substance with another. Thus strontia was never discri
minated from baryta until Klaproth and Haiiy detected
differences between some of their properties x. Accordingly
chemists must often have inferred concerning strontia
what was only true of baryta, and vice versd. There is
now no doubt that the recently discovered substances,
coesium and rubidium were long mistaken for potassiumv.
Other elements have often been confused together, for
instance, tantalum and niobium ; sulphur and selenium ;
cerium, lanthanum, and didymium ; yttrium and erbium.

Even the best-established laws of physical science do
not exclude false inference. No law of nature has been
better established than that of universal gravitation, and
we believe with the utmost confidence that any body
capable of affecting the senses will attract other bodies,
and fall to the earth if not prevented. Euler remarks
that, although he had never made trial of the stones
which compose the church of Magdeburg, yet he had
not the least doubt that all of them were heavy, and
would fall if unsupported. But he adds, that it would
be extremely difficult to give any satisfactory explanation
of this confident belief2. The fact is, that the belief ought
not to amount to certainty until the experiment has been
tried, and in the meantime a slight amount of uncer-

v Professor Bowen has excellently stated this view. ' Treatise on
Logic.' Cambridge, U.S.A., 1866. P. 354.

x Whewell's 'History of the Inductive Sciences,' vol. iii. p. 174.

y Hoscoe's 'Spectrum Analysis/ ist edit. p. 99.

z Euler's ' Letters to a German Princess,' translated by Hunter.
2nd ed. vol. ii. pp. 17-18.

T

274 THE PRINCIPLES OF SCIENCE.

tainty enters, because we cannot be sure that the stones of
the Magdeburg church resemble other stones in all their
properties.

In like manner, not one of the inductive truths which
men have established, or think they have established, is
really safe from exception or reversal. Lavoisier, when
laying the foundations of chemistry, met with so many
instances tending to show the existence of oxygen in
all acids, that he adopted a general conclusion to that
effect, and devised the name oxygen accordingly. He
entertained no appreciable doubt that the acid existing
in sea salt also contained oxygen a ; yet subsequent ex
perience falsified his expectations.

This instance refers to a science in its infancy, speaking
relatively to the possible achievements of men. But all
sciences are and will ever remain in their infancy, relatively
to the extent and complexity of the universe which they
undertake to investigate. Euler expresses no more than
the truth when he says that it would be impossible to fix
on any one thing really existing, of which we could have
so perfect a knowledge as to put us beyond the reach of
mistake b.

Like remarks may be made concerning all other in
ductive inferences. We may be quite certain that a comet
will go on moving in a similar path if all circumstances
remain the same as before ; but if we leave out this exten
sive qualification, our predictions will always be subject
to the chance of falsification by some wholly unexpected
event, such as the division of Biela's comet, or the un
foreseen interference of some planetary or other gravitating
body.

Inductive inference might attain to certainty if our

a Lavoisier's 'Chemistry/ translated by Kerr. 3rd edit. pp. 114, 121,
I23- b Euler's 'Letters,' vol. ii. p. 21.

PHILOSOPHY OF INDUCTIVE INFERENCE. 275

knowledge of the agents existing throughout the universe
were complete, and if we were at the same time certain
that the same Power which created the universe would
allow it to proceed without arbitrary change. There is
always a possibility of causes being in existence without
our knowledge, and these may at any moment produce
an unexpected effect. Even when by the theory of pro
babilities \ve succeed in forming some notion of the com
parative confidence with which we should receive in
ductive results, it yet appears to me that we must make
an assumption. Events come out like balls from the vast
ballot-box of nature, and close observation will enable us
to form some notion, as we shall see in the next chapter,
of the contents of that ballot-box. But we must still
assume that between the time of an observation and that
to which our inferences relate, no change in the ballot-box
shall have been made.

T 2

CHAPTER XII.

THE INDUCTIVE OK INVERSE APPLICATION OF THE
THEORY OF PROBABILITIES.

WE have hitherto considered the theory of probability
only in its simple deductive employment, by which it
enables us to determine from given conditions the probable
character of events happening under those conditions.
But as deductive reasoning when inversely applied con
stitutes the process of induction, so the calculation of
probabilities may be inversely applied ; from the known
character of certain events we may argue backwards to
the probability of a certain law or condition governing
those events. Having satisfactorily accomplished this
work, we may indeed calculate forwards to the probable
character of future events happening under the same con
ditions ; but this part of the process is a direct use of
deductive reasoning (p. 260).

Now it is highly instructive to find that whether the
theory of probabilities be deductively or inductively ap
plied, the calculation is always performed according to
the principles and rules of deduction. The probability
that an event has a particular condition entirely depends
upon the probability that if the condition existed the
event would follow. If we take up a pack of common
playing cards, and observe that they are arranged in per
fect numerical order, we conclude beyond all reasonable
doubt that they have been thus intentionally arranged

THE INDUCTIVE OR INVERSE METHOD. 277

by some person acquainted with the usual order of
sequence. This conclusion is quite irresistible, and rightly
so ; for there are but two suppositions which we can make
as to the reason of the cards being in that particular
order :—

1. They have been intentionally arranged by some one
who would probably prefer the numerical order.

2. They have fallen into that order by chance, that is,
by some series of conditions which, being wholly unknown
in nature, cannot be known to lead by preference to the
particular order in question.

The latter supposition is by no means absurd, for any
one order is as likely as any other when there is no prepon
derating tendency. But we can readily calculate by the
doctrines of permutation the probability that fifty-two
objects would fall by chance into any one particular order.
Fifty-two objects can be arranged in—

52x5ix5Ox .... x4x3x2xi or 8066 x (io)64
possible orders, the number obtained requiring 68 places
of figures for its full expression. Hence it is excessively
unlikely, and, in fact, practically impossible, that any one
should ever meet with a pack of cards arranged in perfect
order by pure accident. If we do meet with a pack so
arranged, we inevitably adopt the other supposition, that
some person having reasons for preferring that special
order, has thus put them together.

We know that of the almost infinite number of possible
orders the numerical order is the most remarkable ; it is
useful as proving the perfect constitution of the pack, and
it is the intentional result of certain games. At any rate,
the probability that intention should produce that order is
incomparably greater than the probability that chance
should produce it; and as a certain pack exists in that
order, we rightly prefer the supposition which most
probably leads to the observed result.

278 THE PRINCIPLES OF SCIENCE.

By a similar mode of reasoning we every day arrive,
and validly arrive, at conclusions approximating to cer
tainty. Whenever we observe a perfect resemblance
between two objects, as, for instance, two printed pages,
two engravings, two coins, two foot-prints, we are warranted
in asserting that they proceed from the same type, the
same plate, the same pair of dies, or the same boot. And
why ? Because it is almost impossible that with different
types, plates, dies, or boots some minute distinction of
form should not be discovered. It is barely possible for
the hand of the most skilful artist to make two objects
alike, so that mechanical repetition is the only probable
explanation of exact similarity. We can often establish
with extreme probability that one document is copied
from another. Suppose that each document contains
10,000 words, and that the same word is incorrectly
spelt in each. There is then a probability of less than
i in 10,000 that the same mistake should be made in
each.

If we meet with a second error occurring in each docu-

O

ment, the probability is less than i in 10,000 x 9999, that
such two coincidences should occur by chance, and the
numbers grow with extreme rapidity for more numerous
coincidences. We cannot indeed make any precise calcu
lations without taking into account the character of the
errors committed, concerning the conditions of which we
have no accurate means of estimating probabilities.
Nevertheless, abundant evidence may thus be obtained
as to the derivation of documents from each other. In
the examination of many sets of logarithmic tables, six
remarkable errors were found to be present in all but
two, and it was proved that tables printed at Paris, Berlin,
Florence, Avignon, and even in China, besides thirteen
sets printed in England, between the years 1633 and
1822, were derived directly or indirectly from some

THE INDUCTIVE OR INVERSE METHOD. 279

common source a. With a certain amount of labour, it
is possible to establish beyond reasonable doubt the rela
tionship or genealogy of any number of copies of one
document, proceeding possibly from parent copies now
lost. Tischendorf has thus investigated the relations
between the manuscripts of the New Testament now
existing, and the same work has been performed by
German scholars for several classical writings.

Principle of the Inverse Method.

The inverse application of the rules of probability
entirely depends upon a proposition which may be thus
stated, nearly in the words of Laplace b. If an event can
be produced by any one of a certain number of different
causes, the probabilities of the existence of these causes as
inferred from the event, are proportional to the proba
bilities of the event as derived from these causes. In other
words, the most probable cause of an event which has
happened is that which would most probably lead to the
event supposing the cause to exist ; but all other possible
causes are also to be taken into account with probabilities
proportional to the probability that the event would have
happened if the cause existed. Suppose, to fix our ideas
clearly, that E is the event, and Cj C2 C3 are the three
only conceivable causes. If Cx exist, the probability is pl
that E would follow ; if C2 and C3 exist, the like pro
babilities are respectively p2 and p3. Then as p± is top.z, so
is the probability of d being the actual cause to the
probability of C2 being it ; and, similarly, as p.2 is to p3, so
is the probability of C2 being the actual cause to the
probability of C3 being it. By a very simple mathematical

a Lardner, 'Edinburgh Review/ July 1834, p. 277.
b ' Mcmoires par clivers Savans,' torn. vi. ; quoted by Todliunter in his
' History of Theory of Probability,' p. 458.

280 THE PRINCIPLES OF SCIENCE.

process we arrive at the conclusion that the actual pro
bability of C, being the cause is

Pi + P* + P3
and the similar probabilities of the existence of C2 and

1\ + P2+ 1>3 1\ + P, + P3

The sum of these three fractions amounts to unity, which
correctly expresses the certainty that one cause or other
must be in o*peration.

We may thus state the result in general language.
If it is certain that one or other of the supposed causes
exists, the probability that any one does exist is the
probability that if it exists the event happens, divided by
the sum of all the similar probabilities. There ir.ay seem
to be an intricacy in this subject which may prove dis
tasteful to some readers ; but this intricacy is essential
to the subject in hand. No one can possibly understand
the principles of inductive reasoning, unless he will take
the trouble to master the meaning of this rule, by which
we recede from an event to the probability of each of its
possible causes.

This rule or principle of the indirect method is that
which common sense leads us to adopt almost instinctively,
before we have any comprehension of the principle in its
general form. It is easy to see, too, that it is the rule
which will, out of a great multitude of cases, lead us most
often to the truth, since the most probable cause of an
event really means that cause which in the greatest
number of cases produces the event. But I have only
met with one attempt at a general demonstration of the
principle. Poisson imagines each possible cause of an
event to be represented by a distinct ballot-box, containing
black and white balls, in such ratio that the probability of
a white ball being drawn is equal to that of the event

THE INDUCTIVE OR INVERSE METHOD. 281

happening. He further supposes that each box, as is
possible, contains the same total number of balls, black
and white ; and then, mixing all the contents of the boxes
together, he shows that if a white ball be drawn from the
aggregate ballot-box thus formed, the probability that it
proceeded from any particular ballot-box is represented
by the number of white balls in that particular box,
divided by that total number of white balls in all the
boxes. This result corresponds to that given by the
principle in question c.

Thus, if there be three boxes, each containing ten balls
in all, and respectively containing seven, four, and three
wThite balls, then on mixing all the balls together we have
fourteen Avhite ones ; and if we draw a white ball, that is
if the event happens, the probability that it came out of

the first box is T7T ; which is exactly equal to — ^ — ^-,the

Tff + T& + T(>

fraction given by the rule of the Inverse Method.

Simple Applications of the Inverse Method.

In many cases of scientific induction wTe may apply the
principle of the inverse method in a simple manner. If
only two, or at the most a few hypotheses, may be made
as to the origin of certain phenomena, or the connection of
one phenomenon with another, we may sometimes easily
calculate the respective probabilities of these hypotheses.
It was thus that Professors Bunsen and Kirchhoff esta
blished, with a probability little short of certainty, that
iron exists in the sun. On comparing the spectra of sun
light and of the light proceeding from the incandescent
vapour of iron, it became apparent that at least sixty
bright lines in the spectrum of iron coincided with dark

c Poisson, ' Rechcrclies sur la Probability cles Jugcmcnts,' Paris, 1837,
pp. 82, 83.

282 THE PRINCIPLES OF SCIENCE.

lines in the sun's spectrum. Such coincidences could
never be observed with certainty, because, even if the lines
only closely approached, the instrumental imperfections of
the spectroscope would make them apparently coincident,
and if one line came within half a millemetre of another,
on the map of the spectra, they could not be pronounced
distinct. Now the average distance of the solar lines on
KirchhofFs map is 2 millemetres, and if we throw down
a line, as it were, by pure chance on such a map, the pro
bability is about one-half that the new line will fall within
^ millemetre on one side or the other of some one of the
solar lines. To put it in another way, we may suppose
that each solar line, either on account of its real breadth
or the defects of the instrument, possesses a breadth of
TJ millemetre, and that each line in the iron spectrum has
a like breadth. The probability then is just one-half that
the centre of each iron line will come by chance within
i millemetre of the centre of a solar line, so as to appear
to coincide with it. The probability of casual coincidence
of each iron line with a solar line is in like manner ^.
Coincidence in the case of each of the sixty iron lines is
a very unlikely event if it arises casually, for it would
have a probability of only (^)co or less than i in a trillion.
The odds, in short, are more than a million million millions
to unity against such casual coincidence fl. But on the
other hypothesis, that iron exists in the sun, it is highly
probable that such coincidences would be observed ; it is
immensely more probable that sixty coincidences would
be observed if iron existed in the sun, than that they
should arise from chance. Hence by our principle it is
immensely probable that iron does exist in the sun.

All the other interesting results given by the com
parison of spectra, rest upon the same principle of proba-

d Kirchhoff's ' Researches on the Solar Spectrum.' First part, trans
lated by Professor Roscoe, pp. 18, 19.

THE INDUCTIVE OK INVERSE METHOD. 283

bility. The almost complete coincidence between the
spectra of solar, lunar, and planetary light renders it prac
tically certain that the light is all of solar origin, and is
reflected from the surfaces of the moon and planets,
suffering only slight alteration from the atmospheres of
some of the planets. A fresh confirmation of the truth of
the Copernican theory is thus furnished.

A vast probability may be shown to exist that the heat,
light, and chemical effects of the sun are due to the same
rays, and are so many different manifestations of the same
undulations. For a photograph of the spectrum corre
sponds exactly with what the eye observes, allowance being
made for the great differences of chemical activity in dif
ferent parts of the spectrum ; and delicate experiments
with the thermopile also show that, where there is a dark
line, there also the heat of the rays is absent.

Sir J. Herschel proved the connexion between the di
rection of the oblique faces of symmetrical quartz crystals,
and the direction in which the same crystals rotate the
plane of the polarisation of light. For if it is found in a
second crystal that the relation is the same as in the first,
the probability of this happening by chance is ^ ; the
probability that in another crystal also the direction
would be the same is 5, and so on. The probability that
in n + i crystals there would be casual agreement of direc
tion is the wth power of ^, Thus, if in examining fourteen
crystals the same relation of the two phenomena is dis
covered in each, the probability that it proceeds from
uniform conditions is more than 8000 to i e. Now, since
the first observations on this subject were made in 1820,
no exceptions have been observed, so that the probability
of invariable connexion is incalculably great.

e 'Edinburgh Review,' No. 185, vol. xcii. July 1850, p. 32 ; Herschel's
'Essays,' p. 421; 'Transactions of the Cambridge Philosophical Society,'
vol. i. p. 43.

284 THE PRINCIPLES OF SCIENCE.

A good instance of this method is furnished by the
agreement of numerical statements with the truth. Thus,

o '

iii a manuscript of Diodorus Siculus, as Dr. Young states g,
the ceremony of an ancient Egyptian funeral is described
as requiring the presence of forty-two persons sitting in
judgment on the merits of the deceased, and in many
ancient papyrus rolls the same number of persons are
found delineated. The probability is but slight that Dio
dorus, if inventing his statements or writing without
proper information, would have chosen such a number as
forty-two, and though there are not the data for an exact
calculation, Dr. Young considers that the probability in
favour of the correctness of the manuscript and the
veracity of the writer on this ground alone, is at least
100 to i.

It is exceedingly probable that the ancient Egyptians
had exactly recorded the eclipses occurring during long
periods of time, for Diogenes Laertius mentions that 373
solar and 832 lunar eclipses had been observed, and the
ratio between these numbers exactly expresses that which
would hold true of the eclipses of any long period, of
say 1 200 or 1.300 years, as estimated on astronomical
grounds h.

It is evident that an agreement between small numbers,
or customary numbers, such as seven, one hundred, a
myriad, &c., is much more likely to happen from chance,
and therefore gives much less presumption of dependence.
If two ancient writers spoke of the sacrifice of oxen, they
would in all probability describe it as a hecatomb, and
there would be nothing remarkable in the coincidence.

On similar grounds, we must inevitably believe in the
human origin of the flint flakes so copiously discovered of
late years. For though the accidental stroke of one stone

g Young's 'Works,' vol. ii. pp. 18, 19.

11 'History of Astronomy,' Library of Useful Knowledge, p. 14.

THE INDUCTIVE OR INVERSE METHOD. 285

against another may often produce flakes, such as are
occasionally found on the sea-shore, yet when several
flakes are found in close company, and each one bears
evidence, not of a single blow only, but of several suc
cessive blows, all conducing to form a symmetrical knife-
like form, the probability of a natural and accidental
origin becomes incredibly small, and the contrary suppo
sition, that they are the work of intelligent beings,
approximately certain '.

An interesting calculation concerning the probable con
nexion of languages, in which several or many words are
similar in sound and meaning, was made by Dr. Young k.

Application of the TJieory of Probabilities in
Astronomy.

The science of astronomy, occupied with the simple
relations of distance, magnitude, and motion of the
heavenly bodies, admits more easily than almost any
other science of interesting conclusions founded on the
theory of probability. More than a century ago, in
1 767, Michell showed the extreme probability of bonds
connecting together systems of stars. He was struck
by the unexpected number of fixed stars which have
companions close to them. Such a conjunction might
happen casually by one star, although possibly at a
great distance from the other, happening to lie on the
same straight line passing near the earth. But the
probabilities are so greatly against such an optical union
happening often in the expanse of the heavens, that
Michell asserted the existence of a bond between most of

1 Evans' 'Ancient Stone Implements of Great Britain.' London,
1872 (Longmans).

k 'Philosophical Transactions,' 1819; Young's 'Works,' vol. ii. pp.
15-18.

286 THE PRINCIPLES OF SCIENCE.

the double stars. It has since been estimated by Struve,
that the odds are 95 70 to I against any two stars of not
less than the seventh magnitude falling within the appa
rent distance of four seconds of each other by chance, and
yet ninety-one such cases were known when the estimation
was made, and many more cases have since been discovered.
There were also four known triple stars, and yet the odds
against the appearance of any one such conjunction are
173,524 to i l. The conclusions of Michell have been en
tirely verified by the discovery that many double stars are
in connexion under the law of gravitation.

Michell also investigated the probability that the six
brightest stars in the Pleiades should have come
by accident into such striking proximity. Estimating
the number of stars of equal or greater brightness at
1500, he found the odds to be nearly 500,000 to i
against casual conjunction. Extending the same kind of
argument to other clusters, such as that of Prsesepe, the
nebula in the hilt of Perseus' sword, he saysm : ' We
may with the highest probability conclude, the odds
against the contrary opinion being many million millions
to one, that the stars are really collected together in
clusters in some places, where they form a kind of system,
while in others there are either few or none of them, to
whatever cause this may be owing, whether to their
mutual gravitation, or to some other law or appointment
of the Creator.'

The calculations of Michell have been called in question
by the late James D. Forbes n, and Mr. Todhunter vaguely

I Herschel, 'Outlines of Astronomy,' 1849, p. 565; but Todhunter,
in his 'History of the Theory of Probability,' p. 335, states that the
calculations do not agree with those published by Struve.

111 'Philosophical Transactions,' 1767, vol. Ivii. p. 431.

II 'Philosophical Magazine,' 3rd Series, vol. xxxvii. p. 401, December,
1850; also August, 1849.

THE INDUCTIVE OR INVERSE METHOD. 287

countenances his objections °, otherwise I should not have
thought them of much weight. Certainly Laplace accepts
Michell's views P, and if Michell be in error, it is in the
methods of calculation, not in the general validity of his
conclusions.

Similar calculations might no doubt be applied to the
peculiar drifting motions which have been detected by
Mr. .R. A. Proctor in some of the constellations (i. Against
a general tendency of stars to move in .one direction by
chance, the odds are very great. It is on a similar ground
that a considerable proper motion of the sun is found to
exist with immense probability, because on the average
the fixed stars show a tendency to move apparently from
one point of the heavens towards that diametrically op
posite. The sun's motion in the contrary direction would
explain this tendency, otherwise we must believe that
myriads of stars accidentally agree in their direction of
motion, or are urged by some common force from which the
sun is exempt. It may be said that the rotation of the
earth is proved in like manner, because it is immensely
more probable that one body would revolve than that
the sun, moon, planets, comets, and the whole of the stars
of the heavens should be whirled round the earth daily,
with a uniform motion superadded to their own peculiar
motions. This appears to be nearly the reason which led
Gilbert, one of the earliest English Copernicans, and in
every way an admirable physicist, to admit the rotation
of the earth, while Francis Bacon denied it r.

In contemplating the planetary system, we are struck
with the similarity in direction of nearly all its move-

0 ' History,' &c., p. 334.
P ' Essai Philosophique/ p. 57.

(i ' Proceedings of the Royal Society,' 20 January, 1870. ' Philosophical
Magazine,' 4th Series, vol. xxxix. p. 381.

r Hallam's ' Literature of Europe,' ist ed. vol. ii. p. 464.

288 THE PRINCIPLES OF SCIENCE.

ments. Newton remarked upon the regularity arid uni
formity of these motions, and contrasted them with the
eccentricity and irregularity of the cometary orbits s-
Could we, in fact, look down upon the system from the
northern side, we should see all the planets moving round
from west to east, the satellites moving round their
primaries and the sun, planets, and all the satellites
rotating in the same direction, with some exceptions on
the verge of the system. Now in the time of Laplace
eleven planets were known, and the directions of rotation
were known for the sun, six planets, the satellites of Jupiter,
Saturn's ring, and one of his satellites. Thus there were
altogether 43 motions all concurring, namely :—

Orbital motions of eleven planets . .11
Orbital motions of eighteen satellites . .18
Axial rotations . . . . . 14

43

The probability that 43 motions independent of each
other would coincide by chance is the 4 2nd power of ^, so
that the odds are about 4,400,000,000,000 to i in favour
of some common cause for the uniformity of direction. This
probability, as Laplace observes*, is higher than that of
many historical events which we undoubtingly believe.
In the present day, the probability is much increased by
the discovery of additional planets, and the rotation of
other satellites, and it is only slightly weakened by the
fact that some of the outlying satellites are exceptional in
direction, there being considerable evidence of an acci
dental disturbance in the more distant parts of the
system.

Hardly less remarkable than the uniformity of motion

s ' Principia,' bk. ii. General scholium.

' Essai Philosophique,' p. 55. Laplace appears to count the rings of
Saturn as giving two independent movements.

THE INDUCTIVE OB INVERSE METHOD. 289

is the near approximation of all the orbits of the planets
to a common plane. Daniel Bernouilli roughly estimated
the probability of such an agreement arising from accident
at r^c, the greatest inclination of any orbit to the sun's

(12) ' J

equator being i-i2th part of a quadrant. Laplace de
voted to this subject some of his most ingenious investi
gations. He found the .probability that the sum of the
inclinations of the planetary orbits would not exceed by
accident the actual amount ("914187 of a right angle for
the ten planets known in 1801) to be j$ ('Qi/j-iS/)10, or
about '00000011235. This probability may be combined
with that derived from the direction of motion, and it
then becomes immensely probable that the constitution of
the planetary system arose out of uniform conditions, or,
as we say, from some common cause11.

If the same kind of calculation be applied to the orbits
of comets the result is very different y. Of the orbits
which have been determined 48*9 per cent, only are direct
or in the same direction as the planetary motions z. Hence
it becomes apparent that comets do not properly belong
to the solar system, and it is probable that they are stray
portions of nebulous matter which have become accidently
attached to the system by the attractive powers of the
sun or Jupiter.

Statement of the General Inverse Problem.

In the instances described in the preceding sections,
we have been occupied in receding from the occurrence

« Lubbock, 'Essay on Probability/ p. 14. De Morgan, ' Encyc.
Metrop.' art. Probability, p. 412. Todhunter's ' History of the Theory of
Probability,' p. 543. Concerning the objections raised to these conclu
sions by the late Dr. Boole, see the ' Philosophical Magazine,' 4th Series,
vol. ii. p. 98. Boole's ' Laws of Thought,' pp. 364-375.

y Laplace, 'Essai Philosophique,' pp. 55, 56.

z Chambers's ' Astronomy,' 2nd ed. pp. 346-49.

U

290 THE PRINCIPLES OF SCIENCE.

of certain similar events to the probability that there
must have been a condition or cause for such events. We
have found that the theory of probability, although never
yielding a certain result, often enables us to establish an
hypothesis beyond the reach of reasonable doubt. There
is, however, another method of applying the theory,
which possesses for us even greater interest, because it
illustrates, in the most complete manner, the theory of
inference adopted in this work, which theory indeed it
suggested. The problem to be solved is as follows : —

An event having happened a certain number of times,
and failed a certain number of times, required the pro
bability that it will happen any given number of times
in the future under the same circumstances.

All the larger planets hitherto discovered move in one
direction round the sun ; what is the probability that, if a
new planet exterior to Neptune be discovered, it will move
in the same direction ? All known permanent gases, ex
cept chlorine, are colourless ; what is the probability that,
if some new permanent gas should be discovered, it will
be colourless 1 In the general solution of this problem, we
wish to infer the future happening of any event from the
number of times that it has already been observed to
happen. Now, it is very instructive to find that there is
no known process by which we can pass directly from the
data to the conclusion. It is always requisite to recede
from the data to the probability of some hypothesis, and
to make that hypothesis the ground of our inference
concerning future happenings. Mathematicians, in fact,
make every hypothesis which is applicable to the question
in hand ; they then calculate, by the inverse method, the
probability of every such hypothesis according to the
data, and the probability that if each hypothesis be true,
the required future event will happen. The total pro
bability that the event will happen, is the sum of the

THE INDUCTIVE OR INVERSE METHOD. 291

separate probabilities contributed by each distinct hypo
thesis.

To illustrate more precisely the method of solving the
problem, it is desirable to adopt some concrete mode of
representation, and the ballot-box, so often employed by
mathematicians, will best serve our purpose. Let the
happening of any event be represented by the drawing of
a white ball from a ballot-box, while the failure of an
event is represented by the drawing of a black ball. Now,
in the inductive problem we are supposed to be ignorant
of the contents of the ballot-box, and are required to
ground all our inferences on our experience of those con
tents as shown in successive drawings. Rude common
sense would guide us nearly to a true conclusion. Thus
if we had drawn twenty balls, one after another, replacing
the ball after each drawing, and the ball had in each case
proved to be white, we should believe that there was a
considerable preponderance of white balls in the urn, and
a probability in favour of drawing a white ball on the
next occasion. Though we had drawn white balls for
thousands of times without fail, it would still be possible
that some black balls lurked in the urn and would at last
appear, so that our inferences could never be certain. On
the other hand, if black balls came at intervals, I should
expect that after a certain number of trials the future
results would agree more or less closelv with the past

o «•'

ones.

The mathematical solution of the question consists in
nothing more than a close analysis of the mode in which
our common sense proceeds. If twenty white balls have
been drawn and no black ball, my common sense tells me
that any hypothesis which makes the black balls in the
urn considerable compared -with the white ones is im
probable ; a preponderance of white balls is a more pro
bable hypothesis, and as a deduction from this more

u 2

292 THE PRINCIPLES OF SCIENCE.

probable hypothesis, I expect a recurrence of white balls.
The mathematician merely reduces this process of thought
to exact numbers. Taking, for instance, the hypothesis
that there are 99 white and one black ball in the urn,
he can calculate the probability that 20 white balls
should be drawn in succession in those circumstances ; he
thus forms a definite estimate of the probability of this
hypothesis, and knowing at the same time the probability
of a white ball reappearing if such be the contents of the
urn, he combines these probabilities, and obtains an exact
estimate that a white ball will recur in consequence of
this hypothesis. But as this hypothesis is only one out
of many possible ones, since the ratio of white and black
balls may be 98 to 2, or 97 to 3, or 96 to 4, and so on,
he has to repeat the estimate for every such possible
hypothesis. To make the method of solving the problem
perfectly evident, I will describe in the next section a
very simple case of the problem, originally devised for the
purpose by Condorcet, which was also adopted by Lacroixa,
and has passed into the works of De Morgan, Lubbock,
and others.

Simple Illustration of the Inverse Problem.

Suppose it to be known that a ballot-box contains only
four black or white balls, the ratio of black and white balls
being unknown. Four drawings having been made with
replacement, and a white ball having appeared on each
occasion but one, it is required to determine the proba
bility that a white ball will appear next time. Now the
hypotheses which can be made as to the contents of the
urn are very limited in number, and are at most the
following five : —

a 'Traite dldmentaire du Calcul ties Probabilites,' 3rd ed. (1833),
p. 148.

THE INDUCTIVE OR INVERSE METHOD. 293

4 white and o black balls

The actual occurrence of black and white balls in the
drawings renders the first and last hypotheses out of the
question, so that we have only three left to consider.

If the box contains three white and one black, the
probability of drawing a white each time is f , and a black
3- ; so that the compound event observed, namely, three
white and one black, has the probability f x f x f x \, by
the rule already given (p. 233). But as it is indifferent
to us in what order the balls are drawn, and the black
ball might come first, second, third, or fourth, we must
multiply by four, to obtain the probability of three white
and one black in any order, thus getting ~.

Taking the next hypothesis of two white and two
black balls in the urn, we obtain for the same proba
bility the quantity ^x±x±x±x^,or Jf, and from the
third hypothesis of one white and three black we deduce
likewise ^ x ^ x ^ x f x 4, or f^. According, then, as we
adopt the first, second, or third hypothesis, the proba
bility that the result actually noticed would follow is f-f,
i|, and j,34> Now it is certain that one or other of these

O 4 ' O i

hypotheses must be the true one, and their absolute
probabilities are proportional to the probabilities that the
observed events would follow from them (see p. 279). All
we have to do, then, in order to obtain the absolute pro
bability of each hypothesis, is to alter these fractions in
a uniform ratio, so that their sum shall be unity, the
expression of certainty. Now since 27 + 16 + 3 =46,
this will be effected by dividing each fraction by 46 and

294 THE PRINCIPLES OF SCIENCE.

multiplying by 64. Thus the probabilities of the first,
second, and third hypotheses are respectively —

46' 46' 46

The inductive part of the problem is now completed, since
we have found that the urn most likely contains three
white and one black ball, and have assigned the exact
probability of each possible supposition. But we are now
in a position to resume deductive reasoning, and infer the
probability that the next drawing will yield, say a white
ball. For if the box contains three white and one black
ball, the probability of drawing a white one is certainly f ;
and as the probability of the box being so constituted is
f|, the compound probability that the box will be so filled
and will give a white ball at the next trial, is

27 3 81
_JL x 2 or — _

46 4 104

Again, the probability is ~ that the box contains two
white and two black, and under those conditions the
probability is | that a white ball will appear ; hence the
probability that a white ball will appear in consequence
of that condition, is

16 i 32

~r x or -1 ^

46 2 184

From the third supposition we get in like manner the

probability

i

jL x _ or -^ .
46 4 184

Now since one and not more than one hypothesis can be
true, we may add together these separate probabilities,
and we find that

81 i 32 3 116
184 ' ' 184 ' 784 184

is the complete probability that a white ball will be next
drawn under the conditions and data supposed.

TUB INDUCTIVE OR INVERSE METHOD. 295

General Solution of the Inverse Problem.

In the instance of the inverse method described in the
last section, a very few balls were supposed to be in the
ballot-box for the purpose of simplifying the calculation.
In order that our solution may apply to natural phe
nomena, we must render our hypothesis as little arbitrary
as possible. Having no a priori knowledge of the con
ditions of the phenomena in question, there is no limit
to the variety of hypotheses which might be suggested.
Mathematicians have therefore had recourse to the most
extensive suppositions which can be made, namely, that
the ballot-box contains an infinite number of balls ; they
have thus varied the proportion of white balls to black
balls continuously, from the smallest to the greatest
possible proportion, and estimated the aggregate proba
bility which results from this comprehensive supposition.

To explain their procedure, let us imagine that, instead
of an infinite number, the ballot-box contained a large
finite number of balls, say 1000. Then the number of
white balls might be i or 2 or 3 or 4, and so on, up
to 999. Supposing that three white and one black ball
have been drawn from the urn as before, there is a certain
very small probability that this would have occurred in
the case of a box containing one white and 999 black
balls ; there is also a small probability that from such a
box the next ball would be white. Compound these
probabilities, and we have the probability that the next
ball really will be white, in consequence of the ex
istence of that proportion of balls. If there be two
white and 998 black balls m the box, the probability
is greater, and will increase until the balls are supposed
to be in the proportion of those drawn. Now 999 different
hypotheses are possible, and the calculation is to be made
for each of these, arid their aggregate taken as the final

296 THE PRINCIPLES OF SCIENCE.

result. It is apparent that as the number of balls in the
box is increased, the absolute probability of any one hypo
thesis concerning the exact proportion of balls is decreased,
but the aggregate results of all the hypotheses will assume
the character of a wide average.

When we take the step of supposing the balls within
the urn to be infinite in number, the possible proportions
of white and black balls also become infinite, and the
probability of any one proportion actually existing is
infinitely small. Hence the final result that the next ball
drawn will be white is really the sum of an infinite
number of infinitely small quantities. It might seem,
indeed, utterly impossible to calculate out a problem
having an infinite number of hypotheses, but the wonderful
resources of the integral calculus enable this to be done
with far greater facility than if we supposed any large
finite number of balls, and then actually computed the
results. I will not attempt to describe the processes by
which Laplace finally accomplished the complete solution
of the problem. They are to be found described in several
English works, especially De Morgan's * Treatise on Proba
bilities/ in the ' Encyclopaedia Metropolitana,' and Mr. Tod-
hunter's ' History of the Theory of Probability.' The ab
breviating power of mathematical analysis \vas never more
strikingly shown. But I may add that though the integral
calculus is employed as a means of summing infinitely
numerous results, we in no way abandon the principles of
combinations already treated. We calculate the values of
infinitely numerous factorials, not, however, obtaining their
actual products, which would lead to an infinite number of
figures, but obtaining the final answer to the problem by
devices which can only be comprehended after study of the
integral calculus.

It must be allowed that the hypothesis adopted by
Laplace is in some degree arbitrary, so that there was some

THE INDUCTIVE OR INVERSE METHOD. 297

opening for the doubt which Boole has cast upon it1'.
But it may be replied, (i) that the supposition of an infinite
number of balls treated in the manner of Laplace is less
arbitrary and more comprehensive than any other that
could be suggested. (2) The result does not differ much
from that which would be obtained on the hypothesis of
any very large finite number of balls. (3) The supposition
leads to a series of simple formulae which can be applied
with ease in many cases, and which bear all the appearance
of truth so far as it can be independently judged by a
sound and practiced understanding.

Rules of the Inverse Method.

By the solution of the problem, as described in the last
section, we obtain the following series of simple rules.

i. To find the probability that an event which has not
hitherto been observed to fail ivill happen once more,
divide the number of times the event has been observed
increased by one, by the same number increased by two.

If there have been m occasions on which a certain event
might have been observed to happen, and it has happened
on all those occasions, then the probability that it will

happen on the next occasion of the same kind is wl—

m+2

For instance, we may say that there are nine places in
the planetary system where planets might exist obeying
Bode's law of distance, and in every place there is a
planet obeying the law more or less exactly, although
no reason is known for the coincidence. Hence the pro
bability that the next planet beyond Neptune will
conform to the law is fj.

2. To find the probability that an event which has not
hitherto failed will not fail for a certain number of new
occasions, divide the number of times the event has hap-

b 'Laws of Thought/ pp. 368-375.

298 THE PRINCIPLES OF SCIENCE.

pened increased by one, by the same number increased by
one and the number of times it is to happen.

An event having happened m times without fail, the

probability that it will happen n more times is

Thus the probability that three new planets would obey
Bode's law is ff, but it must be allowed that this, as well
as the previous result, would be much weakened by the
fact that Neptune can barely be said to obey the law.

3. An event having happened and failed a certain
number of times, to find the probability that it will happen
the next time, divide the number of times the event has
happened increased by one, by the whole number of times
the event has happened or failed increased by two.

Thus, if an event has happened m times and failed n times,
the probability that it will happen on the next occasion

TO+ I

8 m -\-n-\- 2*

Thus, if we assume that of the elements yet discovered
50 are metallic and 14 non-metallic, then the proba
bility that the next element discovered will be metallic

° (To"'

Again since of 37 metals which have been sufficiently
examined only four, namely, sodium, potassium, lan
thanum and lithium, are of less density than water, the
probability that the next metal examined or discovered

will be less dense than water is ^— - or ~

37 + 2 39.

We may state the results of the method in a more
general manner thus, — If under given circumstances cer
tain events A, B, 0, &c., have happened respectively m, n,
p, &c., times, and one or other of these events must
happen, then the probabilities of these events are propor
tional to m + i, ?i -f i, p+ i, &c. so that the probability

of A will be - But if new events

TOrf

THE INDUCTIVE OR INVERSE METHOD. 299

may happen in addition to those which have been ob
served, we must assign unity for the probability of such
new event. The proportional probabilities then become
i for a new event, m + i for A, n + i for B, and so on, and

the absolute probability of A is

+»+J+ &c.

It is very interesting to trace out the variations of
probability according to these rules under diverse circum
stances. Thus the first time a casual event happens it is
i to i, or as likely as not that it will happen again ; if it
does happen it is 2 to i that it will happen a third time ;
and on successive occasions of the like kind the odds
become 3, 4, 5, 6, &c., to i. The odds of course will be
discriminated from the probabilities which are successively
i|, ^, ^, &c. Thus on the first occasion on which a person
sees a shark, and notices that it is accompanied by a little
pilot fish, the odds are i to i , or the probability ^, that the
next shark will be so accompanied.

When an event has happened a very great number of
times, its happening once again approaches nearly to cer
tainty. Thus if we suppose the sun to have risen demon
stratively one thousand million times, the probability that it
will rise again, on the ground of this knowledge merely, is

1,000,000,000+1 ^ But then the bability that it
1,000,000,000+ i + 1
continue to rise for as long a period as we know it to have

1 I,OOO,OOO,OOO+I i ,1 i rni

risen is only — . or almost exactly A. I he

y 2,000,000,000+1

probability that it will continue so rising a thousand times
as long is only about 5-^3-. The lesson which we may
draw from these figures is quite that which we should
adopt on other grounds, namely that experience never
affords certain knowledge, and that it is exceedingly im
probable that events will always happen as we observe

c De Morgan's ' Essay on Probabilities,' Cabinet Cyclopaedia, p. 67.

.300 THE PRINCIPLES OF SCIENCE.

them. Inferences pushed far beyond their data soon lose
any considerable probability. De Morgan has saidd, ' No
finite experience whatsoever can justify us in saying that
the future shall coincide with the past in all time to come,
or that there is any probability for such a conclusion.' On
the other hand, we gain the assurance that experience
sufficiently extended and prolonged will give us the
knowledge of future events with an unlimited degree of
probability, provided indeed that those events are not
subject to arbitrary interference.

It must be clearly understood that these probabilities are
only such as arise from the mere happening of the events,
irrespective of any knowledge derived from other sources
concerning those events or the general laws of nature.
All our knowledge of nature is indeed founded in like
manner upon observation, and is therefore only probable.
The law of gravitation itself is only probably true. But
when a number of different facts, observed under the most
diverse circumstances, are found to be harmonized under a
supposed law of nature, the probability of the law approxi
mates closely to certainty. Each science rests upon so
many observed facts, and derives so much support from
analogies or direct connections with other sciences, that
there are comparatively few cases where our judgment of
the probability of an event depends entirely upon a few
antecedent events, disconnected from the general body of
physical science.

Events may often again exhibit a regularity of suc
cession or preponderance of character, which the simple
formula will not take into account. For instance, the
majority of the elements recently discovered are metals,
so that the probability of the next discovery being that of
a metal, is doubtless greater than we calculated (p. 298).
At the more distant parts of the planetary system, there

ll 'Treatise on Probability,' Cabinet Cyclopaedia, p. 128.

THE INDUCTIVE OR INVERSE METHOD. 301

are symptoms of disturbance which would prevent our
placing much reliance on any inference from the prevailing
order of the known planets to those undiscovered ones
which may possibly exist at great distances. These and
all like complications in no way invalidate the theoretic
truth of the formula, but render their sound application
much more difficult.

Erroneous objections have been raised to the theory of
probability, on the ground that we ought not to trust to
our a priori conceptions of what is likely to happen, but
should always endeavour to obtain precise experimental
data to guide use. This course, however, is perfectly in
accordance with the theory, which is our best and only
guide, whatever data we possess. We ought to be always
applying the inverse method of probabilities so as to take
into account all additional information. When we throw
up a coin for the first time, we are probably quite ignorant
whether it tends more to fall head or tail upwards, and
we must therefore assume the probability of each event
as TT. But if it shows head, for instance, in the first throw,
we now have very slight experimental evidence in favour
of a tendency to show head. The chance of two heads is
now slightly greater than £, which it appeared to be at
firstf, and as we go on throwing the coin time after time,
the probability of head appearing next time constantly
varies in a slight degree according to the character of our
previous experience. As Laplace remarks, we ought
always to have regard to such considerations in common
life. Events when closely scrutinized will hardly ever
prove to be quite independent, and the slightest pre
ponderance one way or the other is some evidence of
connexion, and in the absence of better evidence should
be taken into account.

e J. S. Mill, ' System of Logic,' 5th Edition, bk. iii. chap, xviii. § 3.
f Todhuntcr's 'History,' pp. 472, 598.

302 THE PRINCIPLES OF SCIENCE.

The grand object of seeking to estimate the probability
of future events from past experience, seems to have been
entertained by James Bernoulli! and De Moivre, at lea,st
such was the opinion of Condorcet ; and Bernouilli may be
said to have solved one case of the problem?'. The English
writers Bayes and Price are, however, undoubtedly the
first who put forward any distinct rules on the subject11.
Condorcet and several other eminent mathematicians ad
vanced the mathematical theory of the subject; but it was
reserved to the immortal Laplace to bring to the subject
the full power of his genius, and carry the solution of the
problem almost to perfection. It is instructive to observe
that a theory which arose from the consideration of the
most petty games of chance, the rules and the very names
of which are in many cases forgotten, gradually advanced,
until it embraced the most sublime problems of science,
and finally undertook to measure the value and certainty
of all our inductions.

Fortuitous Coincidences.

We should have studied the theory of probability to
very little purpose, if we thought that it would furnish
us with an infallible guide. The theory itself points out
the possibility, or rather the approximate certainty, that
we shall sometimes be deceived by extraordinary, but
fortuitous coincidences. There is no run of luck so ex
treme that it may not happen, and it may happen to us,
or in our time, as well as to other persons or in other
times. We may be forced by all correct calculation to
refer such coincidences to some necessary cause, and yet
we may be deceived. AH that the calculus of probability

s Todlnmter's 'History,' pp. 378, 79.

11 ' Philosophical Transactions' [1763], vol. liii. p. 370, and [1764],
vol. liv. p. 296. Todhunter, pp. 294-300.

THE INDUCTIVE OR INVERSE METHOD. 303

pretends to give, is the result in the long run, as it is
called, and this really means in an infinity of cases.
During any finite experience, however long, chances may
be against us. Nevertheless the theory is the best guide
we can have. If we always think and act according to
its well interpreted indications, we shall have the best
chance of escaping error ; and if all persons, throughout
all time to come, obey the theory in like manner, they
will undoubtedly thereby reap the greatest advantage.

No rule can be given for descriminating between
coincidences which are casual and those which are the
effect of law or common conditions. By a fortuitous or
casual coincidence, we mean an agreement between events,
which nevertheless arise from wholly independent and
different causes or conditions, and which will not always
so agree. It is a fortuitous coincidence, if a penny thrown
up repeatedly in various ways always falls on the same
side ; but it would not be fortuitous if there were any
similarity in the motions of the hand, and the height of
the throw, so as to cause or tend to cause a uniform
result. Now among the infinitely numerous events, ob
jects, or relations in the universe, it is quite likely that
we shall occasionally notice casual coincidences. There
are seven intervals in the octave, and there is nothing very
improbable in the colours of the spectrum happening to
be apparently divisible into the same or similar series of
seven intervals. It is hardly yet decided whether this
apparent coincidence, with which Newton was much
struck, is well founded or not1, but the question will
probably be decided in the negative.

It is certainly a casual coincidence which the ancients
noticed between the seven vowels, the seven strings of the
lyre, the seven Pleiades, and the seven chiefs at Thebesk.

i 'Nature/ vol. i. p. 286.

k Aristotle's ' Metaphysics,' xiii. 6. 3.

304 THE PRINCIPLES OF SCIENCE.

The accidents connected with the number seven have mis
led the human intellect throughout the historical period.
Pythagoras imagined a connection between the seven
planets, and the seven intervals of the monochord. The
alchemists were never tired of drawing inferences from
the coincidence in numbers of the seven planets and the
seven metals, not to speak of the seven days of the
week.

A singular circumstance was pointed out concerning
the dimensions of the earth, sun, and moon ; the sun's
diameter was almost exactly no times as great as the
earth's diameter, while in almost exactly the same ratio
the mean distance of the earth was greater than the sun's
diameter, and the mean distance of the moon from the
earth was greater than the moon's diameter1. The agree
ment was so close that it might have proved more than
casual, but its fortuitous character is sufficiently shown
by the fact, that the coincidence ceases to be remarkable
when we adopt the amended dimensions of the planetary
system.

A considerable number of the elements have atomic
weights, which are apparently exact multiples of that
of hydrogen. If this be not a law to be ultimately ex
tended to all the elements, as supposed by Prout, it is a
most remarkable coincidence. But, as I have observed,
we have no means of absolutely discriminating accidental
coincidences from those which imply a deep producing
cause. A coincidence must either be very strong in
itself, or it must be corroborated by some explanation or
connection with other laws of nature. Little attention
was ever given to the coincidence concerning the dimen
sions of the sun, earth, and moon, because it was not very
strong in itself, and had no apparent connexion with the

1 Chambers's 'Astronomy,' ist. ed. p. 23.

THE INDUCTIVE OR INVERSE METHOD. 305

principles of physical astronomy. Front's Law bears more
probability because it would bring the constitution of the
elements themselves in close connexion with the atomic
theory, representing them as built up out of a simpler
substance.

In historical and social matters, coincidences are fre
quently pointed out which are due to chance, although
there is always a strong popular tendency to regard them
as the work of design, or as having some hidden cause.
It has been pointed out that if to 1 794, the number of
the year in which Robespierre fell, we add the sum of its
digits, the result is 1815, the year in which Napoleon
fell; the repetition of the process gives 1830, the year
in which Charles the Tenth abdicated. Again, the French
Chamber of Deputies, in 1830, consisted of 402 members,
of whom 221 formed the party called, 'La queue de Robes
pierre/ while the remainder, 181 in number, were named
' Les honnetes gens.' If we give to each letter a numerical
value corresponding to its place in the alphabet, it will
be found that the sum of the values of the letters in each
name exactly indicates the number of the party m.

A number of such coincidences, often of a very curious
character, might be adduced, and the probability against
the occurrence of each may be enormously great. They
must be attributed to chance, because they cannot be
shown to have the slightest connexion with the general
laws of nature ; but persons are often found to be greatly
influenced by such coincidences, regarding them as evidence
of fatality, that is of a system of causation governing
human affairs independently of the ordinary laws of nature.
Let it be remembered that there are an infinite number of
opportunities in life for some strange coincidence to pre
sent itself, so that it is quite to be expected that remark
able conjunctions will sometimes happen.

m S. B. Gould's 'Curious Myths,' p. 222.
X

306 THE PRINCIPLES OF SCIENCE.

In all matters of judicial evidence, we must bear in
mind the necessary occurrence from time to time of un
accountable coincidences. The Roman jurists refused for
this reason to invalidate a testamentary deed, the wit
nesses of which had sealed it with the same seal. For
witnesses independently using their own seals miglit be
found to possess identical ones by accident". It is well
known that circumstantial evidence of apparently over
whelming completeness will sometimes lead to a mistaken
judgment, and as absolute certainty is never really attain
able, every court must act upon probabilities of a very
high amount, and in a certain small proportion of cases
they must almost of necessity condemn the innocent
victims of a remarkable conjuncture of circumstances0.
Popular judgments usually turn upon probabilities of
far less amount, as when the palace of Nicomedia, and
even the bedchamber of Diocletian, having been on fire
twice within fifteen days, the people entirely refused to
believe that it could be the result of accident. The
Romans believed that there was a fatality connected with
the name of Sextus.

' Semper sub Sextis perdita Roma fuit.'

The utmost precautions will not provide against all
contingencies. To avoid errors in important calculations,
it is usual to have them repeated by different computers,
but a case is on record in which three computers made
exactly the same calculations of the place of a star, a,nd
yet all did it wrong in precisely the same manner, for no
apparent reason P.

n Possunt autem omnes testes et uno annulo signare testamentum.
Quid enim si septem annul! una sculptura fuerint, secundum quod Pom-
ponio visum estl — 'Justinian/ ii. tit. x. 5.

0 See Wills on ' Circumstantial Evidence,' p. 148.

P ' Memoirs of the Koyal Astronomical Society,' vol. iv. p. 290, quoted
by Lardner, 'Edinburgh Review,' July 1834, p. 278.

THE INDUCTIVE OR INVERSE METHOD. 307

Summary of the Theory of Inductive Inference.

The theory of inductive inference adopted in this and
the previous chapter, was chiefly suggested by the study
of the Inverse Method of Probabilities, but it also bears
much resemblance to the so-called Deductive Method
described by Mr. J. S. Mill, in his well known ' System of
Logic Q/ Mr. Mill's views concerning the Deductive Method,
probably form the most original and valuable part of his
treatise, and I should have ascribed the doctrine entirely
to him, had I not found that the opinions put forward in
other parts of his work are entirely inconsistent with the
theory here upheld. As this subject is the most impor
tant and difficult one with which we have to deal, I will
try to remedy the imperfect manner in which I have
treated it, by giving a brief recapitulation of the views
adopted.

All inductive reasoning is but an inverse application
of deductive reasoning. Being in possession of certain
particular facts or events expressed in propositions, we
imagine some more general proposition expressing the
existence of a law or cause ; and, deducing the particular
results of that supposed general proposition, we observe
whether they agree with the facts in question. Hypo
thesis is thus always employed, consciously or unconsci
ously. The sole conditions to which we need conform in
framing any hypothesis is, that we both have and exercise
the power of inferring deductively from the hypothesis,
to the particular logical combinations or results, which are
to be compared with the known facts. Thus there are
but three steps in the process of induction : —

(i) Framing of some hypothesis as to the character of
the general law.

q Book iii. chap. u.
X 2

308 THE PRINCIPLES OF SCIENCE.

(2) Deducing consequences from that law.

(3) Observing whether the consequences agree with the
particular facts under consideration.

In very simple cases of inverse reasoning, hypothesis
may sometimes seem altogether needless. Thus, to take
numbers again as a convenient illustration, I have only
to look at the series,

i, 2, 4, 8, 16, 32, &c.,

to know at once that the general law is that of geo
metrical progression ; I need no successive trial of vari
ous hypotheses, because I am familiar with the series,
and have long since learnt from what general formula
it proceeds. In the same way a mathematician becomes
acquainted with the integrals of a number of common
formulae, so that we have no need to go through any pro
cess of discovery. But it is none the less true that when
ever previous reasoning does not furnish the knowledge,
hypotheses must be framed and tried. (See p. 142.)

There naturally arise two different cases, according as
the nature of the subject admits of certain or only pro
bable deductive reasoning. Certainty, indeed, is but a
singular case of probability, and the general principles of
procedure are always the same. Nevertheless, when
certainty of inference is possible the process is simplified.
Of several mutually inconsistent hypotheses, the results of
which can be certainly compared with fact, but one hypo
thesis can ultimately be entertained. Thus in the inverse
logical problem, two logically distinct conditions could not
yield the same series of possible combinations. Accord
ingly in the case of two terms we had to choose one of
seven different kinds of propositions, or in the case of
three terms, our choice lay among 192 possible distinct
hypotheses (pp. 1 54-1 64). Natural laws, however, are often
quantitative in character, and the possible hypotheses are
then infinite in varietv.

THE INDUCTIVE OE INVERSE METHOD. 309

When deduction is certain, comparison with fact is
needed only to assure ourselves that we have rightly
selected the hypothetical conditions. The law establishes
itself, and no number of particular verifications can add
to its probability. Having once deduced from the prin
ciples of algebra that the difference of the squares of two
numbers is equal to the product of their sum and dif
ference, no number of particular trials of its truth will
render it more certain. On the other hand, no finite
number of particular verifications of a supposed law will
render that law certain. In short, certainty belongs only
to the deductive process, and to the teachings of direct
intuition ; and as the conditions of nature are not given
by intuition, we can only be certain that we have got a
correct hypothesis when, out of a limited number con
ceivably possible, we select that one which alone agrees
with the facts to be explained.

In geometry and kindred branches of mathematics,
deductive reasoning is conspicuously certain, and it would
often seem as if the consideration of a single diagram
yields us certain knowledge of a general proposition.
But in reality all this certainty is of a purely hypothetical
character. Doubtless if we could ascertain that a sup
posed circle was a true and perfect circle, we could be
certain concerning a multitude of its geometrical pro
perties. But geometrical figures are physical objects, and
the senses can never assure us as to their exact forms.
The figures really treated in Euclid's 'Elements' are
imaginary, and we never can verify in practice the con
clusions which we draw with certainty in inference ;
questions of degree and probability enter.

Passing now to subjects in which deduction is only
probable, it ceases to be possible to adopt one hypothesis
to the exclusion of the others. We must entertain at the
same time all conceivable hypotheses, and regard each

310 THE PRINCIPLES OF SCIENCE.

with the degree of esteem proportionate to its proba
bility. We go through the same steps as before.

(1) We frame an hypothesis.

(2) We deduce the probability of various series of pos
sible consequences.

(3) We compare the consequences with the particular
facts, and observe the probability that such facts would
happen under the hypothesis.

The above processes must be performed for every con
ceivable hypothesis, and then the absolute probability of
each will be yielded by the principle of the inverse
method (p. 279). As in the case of certainty we accept
that hypothesis which certainly gives the required results,
so now we accept as most probable that hypothesis which
most probably gives the results ; but we are obliged to
entertain at the same time all other hypotheses with
degrees of probability proportionate to the probabilities
that they would give the results.

So far we have treated only of the process by which
we pass from special facts to general laws, that inverse
application of deduction which constitutes induction.
But the direct employment of deduction is often com
bined with the inverse. No sooner have we established
a general law, than the mind rapidly draws other particular
consequences from it. In geometry we may almost seem
to infer that because one equilateral triangle is equi
angular, therefore another is so. In reality it is not
because one is that another is, but because all are. The
geometrical conditions are perfectly general, and by what is
sometimes called parity of reasoning whatever is true of
one equilateral triangle, so far as it is equilateral, is true
of all equilateral triangles.

Similarly, in all other cases of inductive inference,
where we seem to pass from some particular instances to
a new instance, we go through the same process. We

THE INDUCTIVE OR INVERSE METHOD. 311

form an hypothesis as to the logical conditions under
which the given instances might occur; we calculate
inversely the probability of that hypothesis, and com
pounding this with the probability that a new instance
would proceed from the same conditions, we gain the
absolute probability of occurrence of the new instance in
virtue of this hypothesis. But as several, or many, or
even an infinite number of mutually inconsistent hypo
theses may be possible, we must repeat the calculation for
each such conceivable hypothesis, and then the complete
probability of the future instance will be the sum of the
separate probabilities. The complication of this process
is often very much reduced in practice, owing to the fact
that one hypothesis may be nearly certainly true, and
other hypotheses, though conceivable, may be so im
probable as to be neglected without appreciable error.
But when we possess no knowledge whatever of the con
ditions from which the events proceed, we may be unable
to form any probable hypotheses as to their mode of
origin. We have now to fall back upon the general
solution of the problem effected by Laplace, which consists
in admitting on an equal footing every conceivable ratio
of favourable and unfavourable chances for the production
of the event, and then accepting the aggregate result as
the best which can be obtained. This solution is only to
be accepted in the absence of all better means, but like
other results of the calculus of probabilities, it comes
to our aid where knowledge is at an end and ignorance
begins, and it prevents us from over-estimating the know
ledge we possess. The general results of the solution are
in accordance with common sense, namely, that the more
often an event has happened the more probable, as a
general rule, is its subsequent occurrence. With the
extension of experience this probability indefinitely in
creases, but at the same time the probability is slight

312 THE PRINCIPLES OF SCIENCE.

that events will long continue to happen as they have
previously happened.

We have now pursued the theory of inductive inference,
as far as can be done with regard to simple logical or
numerical relations. The laws of nature deal with time
and space, which are indefinitely, or rather infinitely, divi
sible. As we passed from pure logic to numerical logic,
so we must now pass from questions of discontinuous,
to questions of continuous quantity, encountering fresh
considerations of much difficulty. Before, therefore, we
consider how the great inductions and generalizations of
physical science illustrate the views of inductive reason
ing just explained, we must break off for a time, and
review the means which we possess of measuring and
comparing magnitudes of time, space, mass, force, mo
mentum, energy, and the various manifestations of energy
in motion, heat, electricity, chemical change, and the other
phenomena of nature.

BOOK III.

METHODS OF MEASUREMENT.

CHAPTER XIII.

THE EXACT MEASUREMENT OF PHENOMENA.

As physical science advances, it becomes more and
more accurately quantitative. Questions of simple logical
fact after a time resolve themselves into questions of
degree, time, distance, or weight. Forces hardly suspected
to exist by one generation, are clearly recognised by the
next, and precisely measured by the third generation.
But one condition of this rapid advance is the invention
of suitable instruments of measurement. We need what
Francis Bacon called Instantice citantes, or evocantes,
methods of rendering minute phenomena perceptible to
the senses ; and we also require Instantice radii or curri-
culi, that is measuring instruments3. Accordingly, the
introduction of a new instrument often forms an epoch in
the history of science. As Davy said, ' Nothing tends so
much to the advancement of knowledge as the application
of a new instrument. The native intellectual powers of
men in different times, are not so much the causes of the
different success of their labours, as the peculiar nature
of the means and artificial resources in their possession13'.

In the absence indeed of advanced theory and analyti-

a ' Novum Orgaiium,' bk. ii. Aphorisms 40, 45 and 46.
b 'Chemical Philosophy/ Works, vol. iv. p. 39. Quoted by Young,
"Works, vol. i. p. 576.

314 THE PRINCIPLES OF SCIENCE.

cal power, a very precise instrument would be useless.
Measuring apparatus and mathematical theory should ad-
vanceparipassu, and with just such precision as the theorist
can anticipate results, the experimentalist should be able
to compare them with experience. The laborious and scrupu
lously accurate observations of Flamsteed, were the proper
complement to the intense mathemetical powers of Newton.
Every branch of knowledge commences with quantita
tive notions of a very rude character. After we have far
progressed, it is often amusing to look back into the
infancy of the science, and contrast present with past
methods. At Greenwich Observatory in the present day,
the hundredth part of a second is not thought an in
considerable portion of time. The ancient Chaldseans
recorded an eclipse to the nearest hour, and even the
early Alexandrian astronomers thought it superfluous to
distinguish between the edge and centre of the sun.
By the introduction of the astrolabe, Ptolemy and the
later Alexandrian astronomers could determine the places
of the heavenly bodies within about ten minutes of arc.
But little progress then ensued for thirteen centuries,
until Tycho Brahe made the first -great step towards
accuracy, not only by employing better instruments,
but even more by ceasing to regard an instrument
as correct. Tycho, in fact, determined the errors of his
instruments, and corrected his observations. He also took
notice of the effects of atmospheric refraction, and suc
ceeded in attaining an accuracy often sixty times as great
as that of Ptolemy. Yet Tycho and Hevelius often erred
several minutes in the determination of a star's place, and
it was a great achievement of Rcemer and Flamsteed to
reduce this error to seconds. Bradley, the modern Hip-
parchus, carried on the improvement, his errors in right
ascension being under one second of time, and those of
declination under four seconds of arc according to Bessel.

THE EXACT MEASUREMENT OF PHENOMENA. 315

In the present day the average error of a single observa
tion is probably reduced to the half or quarter of what it
was in Bradley 's time; and further extreme accuracy is
attained by the multiplication of observations, and their
skilful combination according to the theory of error.

Some of the more important constants, for instance that
of nutation, have been determined within the tenth part
of a second of space0.

It would be a matter of great interest to trace out the
dependence of this vast progress upon the introduction of
new instruments. The astrolabe of Plotemy, the tele
scope of Galileo, the pendulum of Galileo and Huygens,
the micrometer of Horrocks, and the telescopic sights and
micrometer of Gascoygne and Picard, Koemer's transit in
strument, Newton's and Hadley's quadrant, Dollond's
achromatic lenses, Harrison's chronometer, and Eamsden's
dividing engine — such were some of the principal addi
tions to. astronomical apparatus. The result is, that we
now take note of quantities, 300,000 or 400,000 times as
small as in the time of the Chaldseans.

It would be interesting again to compare the scrupulous
accuracy of a modern trigonometrical survey with Erato
sthenes' rude but ingenious guess at the difference of lati
tude between Alexandria and Syene — or with Norwood's
measurement of a degree of latitude in 1635. ' Sometimes
I measured, sometimes I paced/ said Norwood ; ' and I
believe I am within a scantling of the truth/ Such was
the germ of those elaborate geodesical measurements
which have made the dimensions of the globe known to
us within a few hundred yards.

In other branches of science, the invention of an instru
ment has usually marked, if it has not made, an epoch.
The science of heat might be said to commence with the

c Baily, 'British Association Catalogue of Stars,' pp. "7, 23.

316 THE PRINCIPLES OF SCIENCE.

construction of the thermometer, and it has recently been
advanced by the introduction of the thermo-electric pile.
Chemistry has been created chiefly by the careful use of
the balance, which forms a unique instance of an instru
ment remaining substantially in the form in which it was
first applied to scientific purposes by Archimedes. The
balance never has been and probably never can be im
proved, except in details of construction. On the other
hand, the torsion balance, introduced by Coulomb towards
the end of last century, has rapidly become essential in
many branches of investigation. In the hands of Caven
dish and Baily, it gave a determination of the earth's
density ; applied in the galvanometer, it gave a delicate
measure of electrical forces, and was essential to the
introduction of the thermo-electric pile. This balance is
made by simply suspending any light rod by a thin wire
or thread attached to the middle point. And we owe to
it almost all the more delicate investigations in the theo
ries of heat, electricity, and magnetism.

Though we can now take note of the millionth of an
inch in space, and the millionth of a second in time, wre
must not overlook the fact that in other operations of
science we are yet in the position of the Chaldeeans. Not
many years have elapsed since the magnitudes of the
stars, meaning the amount of light they send to the
observer's eye, were guessed at in the rudest manner, and
the astronomer adjudged a star to this or that order of
magnitude by a rough comparison with other stars of the
same order. To the late Sir John Herschel we owe an
attempt to introduce an uniform method of measurement
and expression, bearing some relation to ' the real photo
metric magnitudes of the stars'*. Previous to the re-

d 'Outlines of Astronomy,' 4th ed. sect. 781, p. 522. 'Eesults of Ob
servations at the Cape of Good Hope,' &c., p. 371.

THE EXACT MEASUREMENT OF PHENOMENA. 317

searches of Bunsen and Roscoe on the chemical action of
light, we were absolutely devoid of any mode of measuring
the energy of light ; even now the methods are tedious,
and it is not clear that they give the energy of light so
much as one of its special effects. Many natural phe
nomena have hardly yet been made the subject of mea
surement at all, such as the intensity of sound, the phe
nomena of taste and smell, the magnitude of atoms, the
temperature of the electric spark or of the sun's photo
sphere.

To suppose, then, that quantitative science treats only of
exactly measurable quantities, is a gross if it be a common
mistake. Whenever we are treating of an event which
either happens altogether or does not happen at all, we are
engaged with a non-quantitative phenomenon, a matter of
fact, not of degree ; but whenever a thing may be greater or
less, or twice or thrice as great as another, whenever, in short,
ratio enters even in the rudest manner, there science will
have a quantitative character. There can be little doubt,
indeed, that every science as it progresses will become
gradually more and more quantitative. Numerical pre
cision is doubtless the very soul of science, as Herschel
said6, and as all natural objects exist in space, and involve
molecular movements, measurable in velocity and extent,
there is no apparent limit to the ultimate extension of
quantitative science. But the reader must not for a
moment suppose that, because we depend more and more
upon mathematical methods, we leave logical methods
behind us. Number, as I have endeavoured to show,
is logical in its origin, and quantity is but a development
of number, or is analogous thereto.

e 'Preliminary Discourse on the Study of Natural Philosophy/ p. 122.

318 THE PRINCIPLES OF SCIENCE.

Division of the Subject.

The general subject of quantitative investigation will
have to be divided into several parts. We shall, firstly,
consider the means at our disposal for measuring phe
nomena, and thus rendering them more or less amenable
to mathematical treatment. This task will involve an
analysis of the principles on which accurate methods of
measurement are founded, forming the subject of the
remainder of the present chapter. As measurement, how
ever, only yields ratios, we have in the next chapter
(XIV) to consider the establishment of unitary mag
nitudes, in terms of which our results may be expressed.
As every phenomenon is usually the sum of several dis
tinct quantities proceeding from different causes, we have
next to investigate in Chapter XV the methods by which
we may disentangle complicated effects, and refer each
part of the joint effect to its separate cause.

It yet remains for us in subsequent chapters to treat of
quantitative induction, properly so called. We must
follow out the inverse logical method, as it presents itself
in problems of a far higher degree of difficulty than those
which treat of objects related in a simple logical manner,
and incapable of merging into each other by addition and
subtraction.

Continuous Quantity.

The phenomena of nature are for the most part mani
fested in quantities which increase or decrease continu
ously. When we inquire into the precise meaning of
continuous quantity, we find that it can only be described
as that which is divisible without limit. We can divide
a millemetre into ten, or one hundred, or one thousand, or
ten thousand parts, and mentally at any rate we can carry

THE EXACT MEASUREMENT OF PHENOMENA. 319

on the process ad infinitum. Any finite space, then, must
be conceived as made up of an infinite number of parts,
each of which must consequently be infinitely small. We
cannot entertain some of the simplest geometrical notions
without allowing this. The conception of a square in
volves the conception of a side and diagonal, which, as
Euclid admirably proves in the ii7th proposition of his
tenth book, have no common measure f, meaning, as I
apprehend, no finite common measure. Incommensurable
quantities are, in fact, those which have for their only
common measure an infinitely small quantity. It is
somewhat startling to find, too, that in theory incommen
surable quantities will be infinitely more frequent than
commensurable. Let any two lines be drawn haphazard ;
it is infinitely unlikely that they will be commensurable,
so that the commensurable quantities, which we are sup
posed to deal with in practice, are but singular cases
among an infinitely greater number of incommensurable
cases.

Practically, however, we treat all quantities as made up
of the least quantities which our senses, assisted by the
best measuring instruments, can appreciate. So long as
microscopes were unin vented, it was sufficient to regard
an inch as made up of a thousand thousandths of an
inch ; now we must treat it as composed of a million
millionths. We might apparently avoid all mention of
infinitely small quantities, by never carrying our approxi
mations beyond quantities, which the senses can appreciate.
In geometry, as thus treated, we should never assert two
quantities to be equal, but only to be apparently equal.
Legendre really adopts this mode of treatment in the
twentieth proposition of the first book of his Geometry ;
and it is practically adopted throughout the physical
sciences, as we shall afterwards see. But though our

f See De Morgan, 'Study of Mathematics/ in TJ. K. S. Library, p. 81.

320 THE PRINCIPLES OF SCIENCE.

fingers, and senses, and instruments must stop somewhere,
there is no reason why the mind should not go on. We
can see that a proof which is only carried through a few
steps, in fact, might be carried on without limit, and it is
this consciousness of no stopping place, which renders
Euclid's proof of his 1 1 7th proposition so impressive. Try
how we will to circumvent the matter, we cannot really
avoid the consideration of the infinitely small and the
infinitely great. The same methods of approximation
which seem confined to the finite, mentally extend them
selves to the infinite £.

One result which immediately follows from these con
siderations is, that we cannot possibly adjust any two
quantities in absolute equality. The suspension of Ma
homet's coffin between two precisely equal magnets, is
theoretically conceivable but practically impossible. The
story of the ' Merchant of Venice,' turns upon the infinite
improbability, that an exact quantity of flesh could be
cut. Unstable equilibrium cannot exist in nature, for it
is that which is destroyed by an infinitely small displace
ment. It might be possible to balance an egg on its end
practically, because no egg has a surface of perfect curva
ture. Suppose the egg shell to be perfectly smooth, and
the feat would become impossible.

The Fallacious Indications of the Senses.

I may briefly remind the reader how little we can trust
to our unassisted senses in estimating the degree, quantity,
or magnitude of any phenomenon. The eye cannot cor
rectly estimate the comparative brightness of two lumi
nous bodies which differ much in brilliancy ; for we know
that the iris is constantly adjusting itself to the intensity

8 Lacroix, ' Essai sur 1'Enseignement ou maniere d'dtudier les Mathe-
matiques/ 2nd ed. Paris, 1816, pp. 292-294.

THE EXACT MEASUREMENT OF PHENOMENA. 321

of the light received, and thus admits more or less light
according to circumstances. The moon which shines with
almost dazzling brightness by night, is pale and nearly
imperceptible while the eye is yet affected by the vastly
more powerful light of day. Much has been recorded
concerning the comparative brightness of the zodiacal
light at different times h, but it would be difficult to prove
that these changes are not due to the varying darkness
at the time, or the different acuteness of the observer's
eye. For a like reason it is exceedingly difficult to esta
blish the existence of any change in the form or compara
tive brightness of nebula? ; the appearance of a nebula
greatly depends upon the keenness of sight of the ob
server, or the accidental condition of freshness or fatigue
of his eye; the same is true of lunar observations'; and
even the use of the best telescope fails to remedy this
difficulty. In judging of colours again, we must remember
that light of any given colour tends to dull the sensibility
of the eye for light of the same colour.

Nor is the eye when unassisted by instruments a much
better judge of magnitude. Our estimates of the size of
minute bright points, such as the fixed stars, are com
pletely falsified by the effects of irradiation. Tycho calcu
lated from the apparent size of the star-discs, that no
one of the principal fixed stars could be contained within
the area of the earth's orbit. Apart, however, from irradia
tion or other distinct causes of error, our visual estimates
of sizes and shapes are often astonishingly incorrect.
Artists almost invariably draw distant mountains or other
objects in ludicrous disproportion to nearer objects, as a
comparison of a sketch with a photograph at once shows.
The extraordinary apparent difference of size of the sun

11 'Cosmos,' Translated by Ottd, vol. i. pp. 131-134.
i 'Report of the British Association,' 1871, p. 84. Grant's 'History
of Physical Astronomy/ pp. 568-9.

Y

322 THE PRINCIPLES OF SCIENCE.

or moon, according as it is high in the heavens or near
the horizon, should be sufficient to make us cautious in
accepting the plainest indications of our senses, unassisted
by instrumental measurement. As to statements concern
ing the height of the aurora and the distance of meteors,
they are to be utterly distrusted. When Captain Parry
says that a ray of the aurora shot suddenly downwards
between him and the land which was only 3000 yards dis
tant, we must consider him subject to an error of sense1.

It is true that errors of observation are more usually
errors of judgment than of sense. That which is actually
seen must be truly seen so far ; and if we correctly
interpret the meaning of the phenomenon, there would
be no error at all. But the weakness of the bare senses
as measuring instruments, arises from the fact that they
import varying conditions of unknown amount, and we
cannot make the requisite corrections and allowances as in
the case of a solid and invariable instrument.

Bacon has excellently stated the insufficiency of the
senses for estimating the magnitudes of objects, or de
tecting the degrees in which phenomena present them
selves. ' Things escape the senses/ he saysm, 'because the
object is not sufficient in quantity to strike the sense : as
all minute bodies ; because the percussion of the object is
too great to be endured by the senses : as the form of the
sun when looking directly at it in mid- day ; because the
time is not proportionate to actuate the sense : as the
motion of a bullet in the air, or the quick circular motion
of a firebrand, which are too fast, or the hour-hand of
a common clock, which is too slow ; from the distance
of the object as to place: as the size of the celestial
bodies, and the size and nature of all distant bodies ;

1 Loomis, ' Ou the Aurora Borealis.' Smithsonian Transactions, quot
ing Parry's Third Voyage, p. 61.
m ' Novnm Organum.'

THE EXACT MEASUREMENT OF PHENOMENA. 323

from prepossession by another object : as one powerful
smell renders other smells in the same room imper
ceptible ; from the interruption of interposing bodies :
as the internal parts of animals ; and because the object
is unfit to make an impression upon the sense : as the
air or the invisible and untangible spirit which is in
cluded in every living body/

Complexity of Quantitative. Questions.

One remark which we may well make in entering
upon quantitative questions, has regard to the great variety
and extent of phenomena presented to our notice. So
long as we deal only with a simply logical question, that
question is merely, Does a certain event happen \ or, Does
a certain object exist ? No sooner do we regard the event
or object as capable of more or less, than one question
branches out into many. We must now ask, How much
is it compared with its cause or necessary condition ?
Does it change when the amount of the cause changes \
If so, does it change in the same or opposite direction ? Is
the change in simple proportion to that of the cause \ If
not, what more complex law of connection holds true 1
This law determined satisfactorily in one series of cir
cumstances may be varied under new conditions, and the
most complex relations of several quantities may ultimately
be established.

In every question of physical science there is thus a
series of steps of progress, the first one or two of which
are usually made with ease, while the succeeding ones
demand more and more careful measurement. We cannot
lay down any single invariable series of questions which
must be asked from nature. The exact character of the
questions will vary according to the nature of the case,
but they will usually be of a very evident kind, and we
may readily illustrate them by actual examples. Suppose,

Y 2

324 THE PRINCIPLES OF SCIENCE.

for instance, that we are investigating the solution of some
salt in water. The first is a purely logical question : Is
there solution, or is there not ? Assuming the answer to
be in the affirmative, we next inquire, Does the solubility
vary with the temperature, or not I In all probability
some variation will be found to exist, and we shall have
at the same time an answer to the further question,
Does the quantity dissolved increase, or does it dimmish
with the temperature ? In by far the greatest number of
cases salts and substances of all kinds dissolve more freely
the higher the temperature of the water, but there are a
few salts, such as calcium sulphate, which follow the
opposite rule. A considerable number of salts resemble
sodium sulphate in becoming more soluble up to a certain
temperature, and then varying in the opposite direction.
We next require to assign the amount of variation as
compared with that of the temperature, assuming at first
that the increase of solubility is proportional to the in
crease of temperature. Common salt is an instance of
very slight variation, and potassium nitrate of very con
siderable increase with temperature. Very accurate ob
servations will probably show, however, that the simple
law of proportionate variation is only approximately true,
and some more complicated law involving the second,
third, or higher powers of the temperature may ultimately
be established. All these investigations have to be
carried out for each salt separately, since no distinct prin
ciples by which we may infer from one substance to
another have yet been detected. There is still an in
definite field for further research open ; for the solubility
of salts would probably vary with the pressure under
which the medium is placed ; the presence of other salts
already dissolved may have effects yet unknown. The
researches already effected as regards the solvent power of
water must be repeated as regards alcohol, ether, carbon

THE EXACT MEASUREMENT OF PHENOMENA. 325

bisulphide, and other media, so that unless general laws
can be detected, this one phenomenon of solution can
never be exhaustively treated. The same kind of questions
recur as regards the solution or absorption of gases in
liquids, the pressure as well as the temperature having
then a most decided effect, and Professor Roscoe's re
searches on the subject present an excellent example of
the successive determination of various complicated laws11.

There is hardly a single branch of scientific research
in which similar complications are not ultimately en
countered. In the case of gravity, indeed, we arrive at
the final law, that the force is invariably the same for all
kinds of matter, and depends only on the distance of
action. But in other subjects the laws, if simple in their
ultimate nature, are disguised and complicated in their
apparent results. Thus the effect of heat in expanding
solids, or the reverse effect of forcible extension or com
pression upon the temperature of a body, will vary
from one substance to another, will vary as the tem
perature is already higher or lower, and will probably
follow a highly complex law, which in some cases gives
negative or exceptional results. In crystalline substances
the same researches have to be repeated in each distinct
axial direction.

In the sciences of pure observation again, such as those
of astronomy, meteorology, and terrestrial magnetism, we
meet with many interesting series of quantitative deter
minations. The so-called fixed stars, as Giordano Bruno
divined, are not really fixed, and may be more truly
described as vast wandering orbs, each pursuing its own
path through space. We must then determine separately
for each star the following questions : —

1 . Does it move 1

2. In what direction I

» Watt's 'Dictionary of Chemistry,' vol. ii. p. 790.

326 THE PRINCIPLES OF SCIENCE.

3. At what velocity ?

4. Is this velocity variable or uniform ?

5. If variable, according to what law ?

6. Is the direction uniform ?

7. If not, what is the form of the apparent path ?

The successive answers to such questions in the case of
certain binary stars, have afforded a proof that the
motions are due to a central force coinciding in law with
gravity, and doubtless identical with it. In other cases
the motions are usually so small that it is exceedingly
difficult to distinguish them with certainty. A coincidence
of motions in some constellations has been pointed out
by Mr. Proctor, and the parallactic effect due to the sun's
proper motion has been surely detected ; but the time is
yet far off when any general results as regards stellar
motions can be established.

The variation in the brightness of stars opens an un
limited field for curious observation. There is not a star
in the heavens concerning which we might not have to
determine—

1 . Does it vary in brightness 1

2. Is the brightness increasing or decreasing ?

3. Is the variation uniform, that is, simply proportional
to time "?

4. If not, according to what law does it vary ?

In a majority of cases the change will probably be
found to have a periodic character, in which case several
other questions will arise, such as —

5. What is the length of the period ?

6. Are there minor periods within the principal
period ?

7. What is the form or law of variation within the
period ?

8. Is there any change in the amount of variation ?

THE EXACT MEASUREMENT OF PHENOMENA. 327

9. If so, is it a secular, i. e. a continually growing-
change, or does it give evidence of a greater period I

Already the periodic changes of a certain number of
stars have been determined with accuracy, and the lengths
of the periods vary from less than three days up to in
tervals of time at least 250 times as great. Periods
within periods have also been detected0.

There is, perhaps, no subject in which more complicated
quantitative conditions have to be determined than ter
restrial magnetism. Since the time when the declination
of the compass was first noticed, as some suppose by
Columbus, we have had successive discoveries from time
to time of the progressive change of declination from
century to century ; of the periodic character of this
change ; of the difference of the declination in various
parts of the earth's surface ; of the varying laws of the
change of declination ; of the dip or inclination of the
needle, and the corresponding laws of its periodic changes ;
the horizontal and perpendicular intensities have also
been the subject of exact measurement, and have been
found to vary by place and time, like the directions of the
needle ; daily and yearly periodic changes have also been
detected, and all the elements are found to be subject to
occasional storms or abnormal perturbations, in which the
eleven year period, now known to be common to many
planetary relations, is apparent. The complete solution
of these motions of the compass needle involves nothing
less than a determination of its position and oscillations in
every part of the world at any epoch, the like deter
mination for another epoch, and so on, time after time,
until the periods of all changes are ascertained, and the
character of the variations determined. This one subject
offers to men of science an almost inexhaustible field for

o Humboldt's 'Cosmos,' translated by Otte, vol. iii. p. 228.

328 THE PRINCIPLES OF SCIENCE.

interesting quantitative research P, in which we shall
doubtless at some future time discover the operation of
causes now most mysterious and unaccountable.

The Methods of Accurate Measurement.

In studying the modes by which physicists have ac
complished very exact measurements, we find that they
are very various, but that they may perhaps be reduced
under the following three classes : —

1. The increase or decrease of the quantity to be
measured in some determinate ratio, so as to bring it
within the scope of our senses, and to equate it with the
standard unit, or some determinate multiple or sub-mul
tiple of this unit.

2. The discovery of some natural conjunction of events
which will enable us to compare directly the multiples of
the quantity with those of the unit, or a quantity related
in a definite ratio to that unit.

3. Indirect measurement, which gives us not the quan
tity itself, but some other quantity connected with it by
known mathematical relations.

Conditions of Accurate Measurement.

Several conditions are requisite in order that a mea
surement may be made with great accuracy, and that
the result may be closely accordant when several inde
pendent measurements are made.

In the first place the magnitude must be exactly defined
by sharp terminations, or precise marks of inconsiderable
thickness. When a boundary is vague and graduated,
like the penumbra in a lunar eclipse, it is impossible to
say where the end really is, and different people wih1 come

P Gauss, 'General Theory of Terrestrial Magnetism'; Taylor's
' Scientific Memoirs,' vol. ii. p, 228.

THE EXACT MEASUREMENT OF PHENOMENA. 329

to different results. We may sometimes overcome this
difficulty to a certain extent, by observations repeated in
a special manner, as we shall afterwards see ; but when
possible, we should choose opportunities for measure
ment when precise definition is easy. The moment of
occultation of a star by the moon can be observed with
great accuracy, because the star disappears with perfect
suddenness ; but there are many other astronomical con
junctions, eclipses, transits, &c., which occupy a certain
length of time in happening, and thus open the way to
differences of opinion. It would be impossible to observe
with precision the movements of a body possessing no
definite points of reference. The spots on the sun, for
instance, furnish the only direct criterion of its rotation,
and the possibility that these spots have a tendency to
move in one direction throws a doubt upon all deter
minations of the sun's axial movement.

The colours of the complete spectrum shade with perfect
continuity into each other, so that their separation is
entirely an arbitrary matter. Exact determinations of
refractive indices would have been impossible, had we not
the fixed dark lines of the solar spectrum as precise points
for measurement, or, what comes to the same thing, various
kinds of homogeneous light, such as that of sodium, pos
sessing a nearly uniform length of vibration.

In the second place, we cannot measure accurately
unless we have the means either of multiplying or dividing
a quantity without considerable error, so that we may
correctly equate one magnitude with the multiple or sub-
multiple of the other. In some cases we operate upon the
quantity to be measured, and bring it into accurate coin
cidence with the actual standard, as when in photometry
we vary the distance of our luminous body, until its
illuminating power at a certain point is equal to that of a
standard lamp. In other cases we repeat the unit until it

330 THE PRINCIPLES OF SCIENCE.

equals the object, as in surveying land, or determining a
weight by the balance. The requisites of accuracy now
are : — (T) That we can repeat unit after unit of exactly
equal magnitude ; (2) That these can be joined together
so that the aggregate shall really be the sum of the
parts. The same conditions apply to subdivision, which
may be regarded as a multiplication of subordinate units.
In order to measure to the thousandth of an inch, we
must be able to add thousandth after thousandth without
error in the magnitude of these spaces, or in their con
junction.

The condenser electrometer, as remarked by Thomson
and Tait <i, is a good example of an instrument unfitted to
give any sure measure of electro-motive force, because the
friction between the parts of the condenser often produces
more electricity than the original quantity which was to
be measured.

Measuring Instruments.

To consider the mechanical construction of scientific
instruments, is no part of my purpose in this book. I
wish to point out merely the general purpose of such
instruments, and the methods adopted to carry out that
purpose with great precision. In the first place we must
distinguish between the instrument which effects a com
parison between two quantities, and the standard mag
nitude which often forms one of the quantities compared.
The astronomer's clock, for instance, is no standard of the
efflux of time ; it serves but to subdivide, with approxi
mate accuracy, the interval of successive passages of a
star across the meridian, which it may effect perhaps to
the tenth part of a second, or -oV-uinr part of the whole.

i 'Elements of Natural Philosophy,' sect. 326, p. 108.

THE EXACT MEASUREMENT OF PHENOMENA. 331

The moving globe itself is the real standard clock, and the
transit instrument the finger of the clock, while the stars
are the hour, minute, and second marks, none the less
useful or accurate because they are disposed at unequal
intervals. The photometer is a simple instrument, by
which we compare the relative intensity of rays of light
falling upon a given spot. The galvanometer shows the
comparative intensity of electric currents passing through
a wire. The calorimeter guages the quantity of heat
passing from a given object. But no such instruments
furnish the standard unit in terms of which our results are
to be expressed. In one peculiar case alone does the same
instrument combine the unit of measurement and the
means of comparison. A theodolite, mural circle, sextant,
or other instrument for the measurement of angular mag
nitudes has no need of an additional physical unit ; for
the very circle itself, or complete revolution, is the natural
unit to which all greater or lesser amounts of angular
magnitude are referred.

The result of every measurement is to make known the
purely numerical ratio existing between the magnitude
to be measured, and a certain other magnitude, which
should, when possible, be a fixed unit or standard magni
tude, or at least an intermediate unit of which the value
can be ascertained in terms of the ultimate standard. But
though a ratio is the required result, an equation is the
mode in which the ratio is determined and expressed. In
every measurement we equate some multiple or submul-
tiple of one quantity, with some multiple or submultiple
of another, and equality is always the fact which we
ascertain by the senses. By the eye, the ear, or the touch,
we judge whether there is a discrepancy or not between
two lights, two sounds, two intervals of time, two bars of
metal. Often indeed we substitute one sense for the other,
as when the efflux of time is judged by the marks upon

332 THE PRINCIPLES OF SCIENCE.

a moving slip of paper, so that equal intervals of time are
represented by equal lengths. There is, perhaps, a ten
dency to reduce all comparisons to the comparison of space
magnitudes, but in any case one of the senses must be the
ultimate judge of coincidence or non-coincidence.

Since the equation to be established may exist between
any multiples or submultiples of the quantities compared,
there naturally arise several different modes of comparison
adapted to different cases. Let p be the magnitude to
be measured, and q that in terms of which it is to be
expressed. Then we wish to find such numbers x and y,

that the equation p = ~q may be true. Now this same

equation may be presented in four slightly different forms,
namely : —

First Form. Second Form. Third Form. Fourth Form.
x y PI

P = - q p~ = q py = qx - - -

y x x y

Each of these modes of expressing the same equation cor
responds to one mode of effecting a measurement.

When the standard quantity is greater than that to be
measured, we often adopt the first mode, and subdivide
the unit until we get a magnitude equal to that measured.
The angles observed in surveying, in astronomy, or in
goniometry are usually smaller than a whole revolution,
and the measuring circle is divided by the use of the
microscope and screw, until we obtain an angle undistin-
guishable from that observed. The dimensions of minute
objects are determined by subdividing the inch or centi
metre, the screw micrometer being the most accurate
means of subdivision. Ordinary temperatures are esti
mated by division, of the standard interval between the
freezing and boiling points of water, as marked on a
thermometer tube.

In a still greater number of cases, perhaps, we multiply

THE EXACT MEASUREMENT OF PHENOMENA. 333

the standard unit until we get a magnitude equal to that
to be measured. Ordinary measurement by a foot rule,
a surveyor's chain, or the excessively careful measurements
of the base line of a trigonometrical survey by standard
bars form a sufficient instance of this case.

In the second case, where p- — q, we multiply or divide

y
a magnitude until we get what is equal to the unit, or to

some magnitude easily comparable with it. As a general
rule the quantities which we desire to measure in
physical science are too small rather than too great for
easy determination, and the problem consists in multiply
ing them without introducing error. Thus the expansion
of a metallic bar when heated from o° C to 100° may be
multiplied by a train of levers or cog wheels. In the
common thermometer the expansion of the mercury is
rendered very apparent, and easily measurable by the
fineness of the tube, and many other cases might be
quoted. There are some phenomena, on the contrary,
which are too great or rapid to come within the easy
range of our senses, and our task is then the opposite
one of diminution. Galileo found it difficult to measure
the velocity of a falling body, owing to the very consider
able velocity acquired in a single second. He adopted
the elegant device, therefore, of lessening the rapidity
by letting the body roll down an inclined plane, which
enables us to reduce the accelerating force in any required
ratio. The same purpose is effected in the well known
experiments performed on Attwood's machine, and the
measurement of gravity by the pendulum really depends
on the same principle applied in a far more advantageous
manner. Sir Charles Wheatstone has invented a beauti
ful method of galvanometry for strong currents, which
consists in drawing off from the main current a certain
determinate portion, which is equated by the galvano-

334 THE PRINCIPLES OF SCIENCE.

meter to a standard current r. In short, he measures not
the current itself but a known fraction of it.

In many electrical and other experiments, we wish to
measure the movements of a needle or other body, which
are not only very slight in themselves, but the manifes
tations of exceedingly small forces. We cannot even
approach a delicately balanced needle without disturbing
it. Under these circumstances the only mode of proceed
ing with accuracy, is to attach a very small mirror to the
moving body, and employ a ray of light reflected from
the mirror as an index of its movements. The ray may
be considered quite incapable of affecting the body, and
yet by allowing the ray to pass to a sufficient distance,
the motions of the mirror may be increased to almost any
extent. A ray of light is in fact a perfectly weightless
finger or index of indefinite length, with the additional
advantage that the angular deviation is by the law of
reflection double that of the mirror. This method, was
introduced by Gauss, and is now of great importance ;
but in Wollaston's reflecting goniometer a ray of light
had previously been employed as an index finger. Lavoi
sier and Laplace had also used a telescope in connection
with the pyrometer.

It is a great advantage in some instruments that they
can be readily made to manifest a phenomenon in a greater
or less degree, by a very slight change in the construction.
Thus either by enlarging the bulb or contracting the tube
of the thermometer, we can make it give more conspicuous
indications of change of temperature. The barometer, on
the other hand, always gives the variations of pressure
on one scale. The torsion balance is especially remark
able for the extreme delicacy which may be attained
by increasing the length and lightness of the rod, and the

r De la Rive's 'Electricity/ vol. ii. pp. 897, 98.

THE EXACT MEASUREMENT OF PHENOMENA. 335

length and thinness of the supporting thread. Forces so
minute as the attraction of gravitation between two balls,
or the magnetic and diamagnetic attraction of common
liquids and gases, may thus be made apparent, and even
measured. The common chemical balance, too, is capable
theoretically of indefinite sensibility8.

The third mode of measurement, which may be called
the Method of Eepetition, is of such great importance and
interest that we must consider it in a separate section. It
consists in multiplying both magnitudes to be compared
until some multiple of the first is found to coincide very
nearly with some multiple of the second. If the multipli
cation can be effected to an indefinite extent, without the
introduction of countervailing errors, the accuracy with
which the required ratio can be determined is unlimited,
and we thus account for the extraordinary precision with
which intervals of time in astronomy are compared to
gether.

The fourth mode of measurement in which we equate
submultiples of two magnitudes is comparatively seldom
employed, because it does not conduce to accuracy. In
the photometer, perhaps, we may be said to use it ; we
compare the intensity of two sources of light, by placing
them both at such distances from a given surface, that the
light falling on the surface is tolerable to the eye, and
equally intense from each source. Since the intensity of
rays diminishes, as the inverse squares of the distances,
the relative intensities of the luminous bodies are propor
tional to the squares of their distances. The equality of
intensity of two rays of similarly coloured light, may be
most accurately ascertained in the mode suggested by
Arago, namely, by causing the rays to pass in opposite
directions through two nearly flat lenses pressed together.

s Watt's 'Dictionary of Chemistry/ art. Balance, vol. i. p. 487.

33G THE PRINCIPLES OF SCIENCE.

There is an exact equation between the intensities of the
beams when Newton's rings disappear, the ring created
by one ray being exactly the complement of that created
by the other1.

The Method of Repetition.

The ratio of two quantities can be determined with
unlimited accuracy, if we can multiply both the object
of measurement and the standard unit without error, and
then observe what multiple of the one coincides or nearly
coincides with some multiple of the other. Although per
fect coincidence can never be really attained, the error
thus arising may be indefinitely reduced. For if the
equation py = qx be uncertain to the amount e, so

1C P

that py = qx±e, then we have p = q-±-} and

J J

as we are supposed to be able to make y as great as we
like without increasing the error e, it follows that we
can approximate as closely as we like to the required
ratio x + y.

This method of repetition is naturally employed when
ever quantities can be repeated, or repeat themselves
without error of juxtaposition, which is especially the
case with the motions of the earth and heavenly bodies.
In determining the length of the sidereal day, we really
determine the ratio between the earth's revolution round
the sun, and its rotation on its own axis. We might
ascertain the ratio by observing the successive passages
of a star across the zenith, and comparing the interval by
a good clock with that between two passages of the sun,
the difference being due to the angular movement of the
earth round the sun. In such observations w7e should
have an error of a considerable part of a second at ?ach

1 Humboldt's 'Cosmos/ (Bolin), vol. iii. p. 129.

THE EXACT MEASUREMENT OF PHENOMENA. 337

observation, in addition to the irregularities of the clock.
But the revolutions of the earth repeat themselves day
after day, and year after year, without the slightest in
terval between the end of one period and the beginning
of another. The operation of multiplication is perfectly
performed for us by nature. If, then, we can find an obser
vation of the passage of a star across the meridian a hun
dred years ago. that is of the interval of time between

*/ CD *

the passage of the sun and the star, the instrumental
errors in measuring this interval by a clock and telescope
may be greater than in the present day, but will be
divided by about 36,524 days, and rendered excessively
small. It is thus that astronomers have been able to
ascertain the ratio of the mean solar to the sideral day
to the 8th place of decimals (i '002 7 3 791 to i), or to the
hundred millionth part, probably the most accurate result
of measurement in the whole range of science.

The antiquity of this mode of comparison is almost as
great as that of astronomy itself. Hipparchus made the
first clear • application of it, when he compared his own
observations with those of Aristarchus, made 145 years
previously u, and thus ascertained the length of the year.
This calculation may in fact be regarded as the earliest
attempt at an exact determination of the • constants of
nature. The method is the main resource of astronomers ;
Tycho, for instance, detected the slow diminution of the
obliquity of the earth's axis, by the comparison of ob
servations at long intervals. Living astronomers use the
method as much as earlier ones ; but so superior in ac
curacy are all observations taken during the last hundred
years to aU previous ones, that it is often found pre
ferable to take a shorter interval, rather than incur the
risk of greater instrumental errors in the earlier observa
tions.

u Montucla, 'Histoire des Mathematiques,' vol. i. p. 258,

Z

338 THE PRINCIPLES OF SCIENCE.

It is obvious that many of the slower changes of the
heavenly bodies must require the lapse of large intervals
of time to render their amount perceptible. Hipparchus
could not possibly have discovered many of the smaller
inequalities of the heavenly motions, because there were
no previous observations of sufficient age or exactness to
exhibit them. And just as the observations of Hipparchus
formed the starting-point for subsequent comparisons, so
a large part of the labour of present astronomers is di
rected to recording the present state of the heavens so
exactly, that future generations of astronomers may detect
many changes, which cannot possibly become known in
the present age.

The principle of repetition was very ingeniously em
ployed in an instrument first proposed by Mayer in 1767,
and carried into practice in the Repeating Circle of Borda v.
The exact measurement of angles is indispensable, not
only in astronomy but also in trigonometrical surveys, and
the highest skill in the mechanical execution of the gradu
ated circle and telescope will not prevent terminal errors
of considerable amount. If instead of one telescope, the
circle be provided with two similar telescopes, these may
be alternately directed to two distant points, say the
marks in a trigonometrical survey, so that the circle shall
be turned through any multiple of the angle subtended
by those marks, before the amount of the angular revolu
tion is read off upon the graduated circle. Theoretically
speaking, all error arising from -imperfect graduation might
thus be indefinitely reduced, being divided by the number
of repetitions. In practice, however, the advantage of
the invention is not found to be great, probably because
a certain error is introduced at each observation in the
changing or fixing of the telescopes. It is moreover in-

v Young, 'Works,' vol. ii. p. 546.

THE EXACT MEASUREMENT OF PHENOMENA. 339

applicable to moving objects like the heavenly bodies, so
that its use is confined to important trigonometrical
surveys.

The pendulum is the most perfect of all instruments,
chiefly because it naturally admits of almost indefinite
repetition. Since the force of gravity never ceases, one
swing of the pendulum is no sooner ended than the other
is begun, so that the juxtaposition of successive units is
absolutely perfect. Provided that the oscillations be equal,
then one thousand oscillations will occupy exactly one
thousand times as great an interval of time as one oscil
lation. Not only is the subdivision of time entirely de
pendent on this fact, but in the accurate measurement of
gravity, and many important determination it is of
the greatest service. In the deepest mine, we could not
observe the rapidity of fall of a body for more than a
quarter of a minute, and the measurement of its velocity
would be difficult, and subject to uncertain errors from
resistance of air, &c. In the pendulum, we have a body
which can be kept rising and falling for many hours, in
a medium entirely under our command or if desirable in
a vacuum. Moreover, the comparative force of gravity at
different points, at the top and bottom of a mine for
instance, can be determined with wonderful precision, by
comparing the oscillations of two exactly similar pendu
lums, with the aid of electric clock signals. To ascertain
the comparative lengths of vibration of two pendulums, it
is only requisite to swing them one in front of the other,
to record by a clock the moment when they coincide in
swing, so that one hides the other, and then count the
number of vibrations until they again come to similar
coincidence. If one pendulum makes m vibrations and
the other n, we at once have our equation pn = qm ;
which gives the length of vibration of either pendulum in
terms of the other. This method of coincidence, embody-

z 2

340 THE PRINCIPLES OF SCIENCE.

ing the principle of repetition in perfection, was employed
with wonderful skill by Sir George Airy, in his experi
ments on the Density of the Earth at the Harton Colliery ;
the pendulums above and below being compared with
clocks, which again were compared with each other by
electric signals. So exceedingly accurate was this method
of observation, as carried out by Sir George Airy, that he
was able to measure a total difference in the vibrations at
the top and bottom of the shaft, amounting to only 2-24
seconds in the twenty-four hours, with an error of less
than one hundredth part of a second, or one part in
8,640,000 of the whole day x.

The principle of repetition has been elegantly applied
in observing the motion of waves in water. If the canal
in which the experiments are made be short, say twenty
feet long, the waves will pass through it so rapidly that
an observation of one length, as practised by Walker, will
be subject to much terminal error, even when the observer
is very skilful. But it is a result of the undulatory theory
that a wave is quite unaltered, and loses no time by com
plete reflection, so that it may be allowed to travel back
wards and forwards in the same canal, and its motion, say
through sixty lengths, or 1200 feet, may be observed with
the same accuracy as in a canal 1200 feet long, with the
advantage of greater uniformity in the condition of the
canal and water y. It is always desirable, if possible, to
bring an experiment into a small compass, so as to be well
under command, and yet we may often by repetition
enjoy at the same time the advantage of extensive obser
vation.

One reason of the great accuracy of weighing with a
good balance is the fact, that weights placed in the same

x 'Philosophical Transactions,' (1856) vol. 146, Part i. p. 297.
y Airy, ' On Tides and Waves,' Encyclopaedia Metropolitana, p. 345.
Scott Russell, ' British Association Report/ 1837, p. 432.

THE EXACT MEASUREMENT OF PHENOMENA. 341

scale are naturally added together without the slightest
error. There is no difficulty in the precise juxtaposition
of two grammes, but the juxtaposition of two metre mea
sures can only be effected with tolerable accuracy, by the
use of microscopes and many precautions. Hence, the
extreme trouble and cost attaching to the exact measure
ment of a base line for a survey, the risk of error entering
at every juxtaposition of the measuring bars, and indefatig
able attention to all the requisite precautions being neces
sary throughout the operation z.

Measurements by Natural Coincidence.

In certain cases a peculiar conjunction of circumstances
enables us to dispense more or less with instrumental
aids, and to obtain the most exact numerical results in
the simplest manner. The mere fact, for instance, that
no human being has ever seen a different face of the moon
from that familiar to us, conclusively proves that the
period of rotation of the moon on its own axis is equal
to that of its revolution round the earth. Not only have
we the repetition of these movements during 1000 or
2000 years at least, but we have observations made for
us at very remote periods, free from instrumental error,
no instrument being needed. We learn that the seventh
satellite of Saturn is subject to a similar law, because its
light undergoes a variation in each revolution, owing to
the existence of some dark tract of land ; now this failure
of light always occurs while it is in the same position
relative to Saturn, clearly proving the equality of the
axial and revolutional periods, as Huyghens perceived a.

z Herschel's, ' Familiar Lectures on Scientific Subjects/ p. 184.
a ' Hugenii Cosmo theoros,' pp. 117-18. Laplace's ' Systbme,' trans
lated, vol. i. p. 67.

312 THE PRINCIPLES OF SCIENCE.

A like peculiarity in the motions of Jupiter's fourth satel
lite was similarly detected by Maraldi in 1713.

Remarkable conjunctions of the planets may sometimes
allow us to compare their periods of revolution, through
long intervals of time, with great accuracy. Laplace in
explaining the long inequality in the motions of Jupiter
and Saturn, was much assisted by a conjunction of these
planets, observed by Ibyn Jounis at Cairo, towards the
close of the eleventh century. Laplace calculated that
such a conjunction must have happened on the 3ist of
October, A. D. 1087 ; and the discordance between the dis
tances of the planets as recorded, and as assigned by
theory, was less than one-fifth of the apparent diameter
of the sun. This difference being less than the probable
error of the early record, his theory was confirmed as far
as facts were available1*.

The ancient astronomers often shewed the highest in
genuity in turning any opportunities of measurement
which occurred to good account. Eratosthenes, as early
as -2 50 B.C., happening to hear that the sun at Syene, in
Upper Egypt, was visible at the summer solstice at the
bottom of a well, proving that it was in the zenith, pro
posed to determine the dimensions of the earth, by mea
suring the length of the shadow of a rod at Alexandria on
the same day of the year. He thus learnt in a rude
manner the difference of latitude between Alexandria and
Syene, and finding it to be about one fiftieth part of the
whole circumference, he ascertained the dimensions of the
earth within about one sixth part of the truth. The use
of wells in astronomical observation appears to have been
occasionally practised in comparatively recent times, as
by Flamsteed in 1679°. Hipparchus employed the moon
as an instrument of measurement in several sagacious

b Grant's, 'History of Physical Astronomy,' p. 129.
c Baily's, ' Account of Flamsteed,' p. lix.

THE EXACT MEASUREMENT OF PHENOMENA. 343

modes. When the moon is exactly half full, the moon,
sun, and earth, are at the angles of a right-angled triangle.
He proposed therefore at such a time to measure the
moon's elongation from the sun, which would give him
the two other angles of the triangle, and enable him to
judge of the comparative distances of the moon and sun
from the earth. His result, though very rude, was far
more accurate than any notions previously entertained,
and enabled him to form some estimate of the comparative
magnitudes of the bodies. Eclipses of the moon were also
very . useful in ascertaining the longitudes of the stars,
which were invisible when the sun was above the horizon.
For the moon when eclipsed must be 180° distant from
the sun ; hence it was only requisite to measure the
distance of a fixed star in longitude from the eclipsed
moon to obtain with ease its angular distance from the
sun.

In later times the eclipses of Jupiter have usefully
served to give a measure of an angle ; for at the middle
moment of the eclipse the satellite must be exactly in the
same straight line with the planet and sun, so that we
can learn from the known laws of movement of the
satellite the longitude of Jupiter as seen from the sun.
If at the same time we measure the elongation or ap
parent angular distance of Jupiter from the sun, as seen
from the earth, we have all the angles of the triangle
between Jupiter, the sun, and the earth, and can cal
culate the comparative magnitudes of the sides of the
triangle by simple trigonometry.

The transits of Venus over the sun's face are other
natural events which seem to give most accurate measure
ments of the sun's parallax, or apparent difference of
position as seen from distant points of the earth's surface.
The sun forms a kind of background on which the place
of the planet is marked, and serves as a measuring instru-

344 THE PRINCIPLES OF SCIENCE.

ment free from all the errors of construction, which affect
human instruments. The rotation of the earth, too, by
variously affecting the apparent velocity of ingress or
egress of Venus, as seen from different places, discloses
the amount of the parallax. It has been sufficiently
shown that by rightly choosing the moments of obser
vation, the planetary bodies may often be made to reveal
their relative distance, to measure their own position, to
record their own movements with a high degree of ac
curacy. With the improvement of astronomical instru
ments, such conjunctions become less necessary to the
progress of the science, but it will always remain ad
vantageous to choose those moments for observation
when instrumental errors enter with the least effect.

In other sciences, exact quantitative laws can occasion
ally be obtained without instrumental measurement, as
when we learn the exactly equal velocity of sounds of
different pitch, by observing that a peal of bells or a
musical performance is heard harmoniously at any dis
tance to which the sound penetrates ; this could not be
the case, as Newton remarked, if one sound overtook
the other. One of the most important principles of the
atomic theory, was proved by implication, before the use
of the balance was introduced into chemistry. Wenzel
observed, before 1777, that when two neutral substances
decompose each other, the resulting salts are also neutral.
In mixing sodium sulphate and barium nitrate, we
obtain insoluble barium sulphate and neutral sodium
nitrate. This result could not follow unless the nitric
acid, requisite to saturate one atom of sodium, were
exactly equal to that required by one atom of barium,
so that an exchange could take place without leaving
either acid or base in excess d.

A very important principle of mechanics may also be

d Daubeny, ' Atomic Theory/ p. 30.

THE EXACT MEASUREMENT OF PHENOMENA. 345

established by a simple acoustical observation. When
a rod or tongue of metal fixed at one end is set in
vibration, the pitch of the sound may be observed to
be exactly the same, whether the vibrations be small or
great ; hence the oscillations are isochronous, or equally
rapid, independently of their magnitude. On the ground
of theory, it can be shown that such a result only
happens when the flexure is proportional to the deflecting
force. Thus the simple observation that the pitch of
the sound of a harmonium, for instance, does not change
with its loudness, establishes an exact law of nature °.

A closely similar instance is found in the proof that the
intensity of light or heat rays varies inversely as the
square of the distance increases. For the apparent mag
nitude certainly varies according to this law; hence, if the
intensity of light varied according to any other law, the
brightness of an object would be different at different
distances, which is not observed to be the case. Mellorii
applied the same kind of reasoning, in a somewhat dif
ferent form, to the radiation of heat-rays f.

Modes of Indirect Measurement.

Some of the most conspicuously beautiful experiments
in the whole range of science, have been devised for the
purpose of indirectly measuring quantities, which in their
extreme greatness or smalmess surpass the powers of
sense. All that we need to do, is to discover some
other conveniently measurable phenomenon, which is re
lated in a known ratio or according to a known law,
however complicated, with that to be measured. Having

e Jamin, ' Cours de Physique,' vol. i. p. 152.

f Balfour Stewart's, 'Elementary Treatise on Heat/ ist edit. pp. 164,

165-

346 THE PRINCIPLES OF SCIENCE.

once obtained experimental data, there is no further
difficulty beyond that of arithmetic or algebraic calcu
lation.

Gold is reduced by the gold-beater to leaves so thin,
that the most powerful microscope would not detect any
measurable thickness. If we laid several hundred leaves
upon each other to multiply the thickness, we should
still have no more than Yiroth of an inch at the most to
measure, and the errors arising in the superposition and
measurement would be considerable. But we can readily
obtain an exact result through the connected amount of
weight. Faraday weighed 2000 leaves of gold, each
3§ inch square, and found them equal to 384 grains.
From the known specific gravity of gold, it was easy to
calculate that the average thickness of the leaves was

,821ooo Of an incn g-

We must ascribe to Newton the honour of leading the
way in methods of minute measurement. He did not
call waves of light by their right name, and did not
understand their nature ; yet he measured "their length,
though it did not exceed the 2,ooo,oooth part of a metre
or the one fifty thousandth, part of an inch. He pressed
together two lenses of very large but known radii. It
was not difficult to calculate the interval between the
lenses at any point, by measuring the distance from the
central point of contact. Now, with homogeneous rays the
successive rings of light and darkness mark the points at
which the interval between the lenses is equal to one
half, or any multiple of half a vibration of the light, so
that the length of the vibration became known. In a
similar manner many phenomena of interference of rays
of light admit of the measurement of the wave lengths,
The fringes of interference arise from rays of light which
cross each other at a small angle, and an excessively

g Faraday, 'Chemical Researches,' p. 393.

THE EXACT MEASUREMENT OF PHENOMENA. 347

minute difference in the lengths of the waves makes a
very perceptible difference in the position of the point
at which two rays will interfere and produce darkness.

M. Fizeau has recently employed Newton's rings in an
inverse manner, to measure small amounts of motion. By
merely counting the number of rings of sodium mono
chromatic light passing a certain point where two glass
plates are in close proximity, he is able to ascertain with
the greatest accuracy and ease the change of distance
between these glasses, produced, for instance, by the ex
pansion of a metallic bar, connected with one of the glass
plates11.

Nothing excites more admiration than the mode in
which scientific observers can occasionally measure quan
tities, which seem bevond the bounds of human obser-

tj

vation. We know the average depth of the Pacific
Ocean to be 14,190 feet, not by actual sounding, which
would be impracticable in sufficient detail, but by noticing
the rate of transmission of earthquake waves from the
South American to the opposite coasts, the rate of move
ment being connected by theory with tbe depth of the
water1. In. the same way the average depth of the
Atlantic Ocean is inferred to be no less than 22,157 feet,
from the velocity of the ordinary tidal waves. A tidal
wave again gives beautiful evidence of an effect of the
law of gravity, which we could never in any other way
detect. Newton estimated that the moon's force in mov
ing: the ocean is onlv — —part of the whole force of

2,871, 400 L

gravity, which even the pendulum, used with the utmost
skill, would fail to render apparent. Yet the immense
extent of the ocean allows the accumulation of the effect
into a very palpable amount ; and from the comparative

h 'Proceedings of the Royal Society/ 3oth November, 1866.
i Herschel, ' Physical Geography/ § 40.

348 THE PRINCIPLES OF SCIENCE.

heights of the lunar and solar tides, Newton roughly
estimated the comparative forces of the moon's and sun's
gravity at the earth k.

A few years ago it might have seemed impossible that
we should ever measure the velocity with which a star
approaches or recedes from the earth, since the apparent
position of the star is thereby unaltered. But the spec
troscope now enables us to detect and even measure such
motion with considerable accuracy, by the alteration which
it causes in the apparent rapidity of vibration, and conse
quently in the refrangibility of rays of light of definite
colour. And while our estimates of the lateral move
ments of stars depend upon our very uncertain know
ledge of their distance, the spectroscope gives the motion
in another direction in absolute quantity, irrespective of
all other quantities known or unknown, excepting the
motion of the earth itself1.

The rapidity of vibration for each musical tone, hav
ing been accurately determined by comparison with the
Syren (p. 12), we can use sounds as indirect indications of
rapid vibrations. It is now known that the contraction of
a muscle arises from the periodical contractions of each
separate fibre, and from a faint sound or susurrus which
accompanies the action of a muscle, it is inferred that

each contraction lasts for about — "— of a second. Minute

300

quantities of radiant heat are now always measured indi
rectly by the electricity which they produce when falling
upon a thermopile. The extreme delicacy of the method
seems to be due to the power of multiplication at several
points in the apparatus. The number of elements or junc
tions of different rnetals in the thermopile can be increased

'Principia,' bk. iii. Prop. 37, 'Corollaries,' 2 and 3. Motte's trans
lation, vol. ii. p. 310.

1 Roscoe's, ' Spectrum, Analysis,' ist ed. p. 296.

THE EXACT MEASUREMENT OF PHENOMENA. 349

so that the tension of the electric current derived from
the same intensity of radiation is multiplied ; the effect of
the current upon the magnetic needle can be multiplied
within certain bounds, by passing the current many times
round it in a coil ; the excursions of the needle can be
increased by rendering it astatic and increasing the deli
cacy of its suspension ; lastly, the angular divergence can
be observed, with any required accuracy, by the use of an
attached mirror and distant scale viewed through a tele
scope (p. 234). Such is the delicacy of this method of
measuring heat, that Dr. Joule succeeded in making a

thermopile which would indicate a difference of - - part

8800

of a degree centigrade m.

A striking case of indirect measurement is furnished by
the revolving mirror of Wheatstone and Foucault, whereby
a minute interval of time is estimated in the form of an
angular deviation. Wheatstone viewed an electric spark
in a mirror rotating so rapidly, that if the duration of the

spark had been more than - - of a second, the point of

72,000

light would have appeared elongated to an angular extent
of one-half degree. In the spark, as drawn directly from a
Leyden jar, no elongation was apparent, so that the dura
tion of the spark was immeasurably small ; but when the
discharge took place through a bad conductor, the elonga
tion of the spark denoted a sensible duration11. In the
hands of Foucault the rotating mirror gave a measure
of the time occupied by light in passing through a few
metres of space.

Comparative Use of Measuring Instruments.
In almost every case a measuring instrument serves,

m 'Philosophical Transactions' (1859), v°l- cxlix. p. 94.
n Watts' ' Dictionary of Chemistry/ vol. ii. p. 393.

350 THE PRINCIPLES OF SCIENCE.

and should serve only as a means of comparison between
two or more magnitudes. As a general rule, we should
not even attempt to make the divisions of the measuring
scale exact multiples or submultiples of the unit, but,
regarding them as arbitrary marks, should determine their
values by comparison with the standard itself. Thus the
perpendicular wires in the field of a transit telescope, are
fixed at nearly equal but arbitrary distances, and those
distances are afterwards determined, as first suggested by
Malvasia, by watching the passage of star after star across
them, and noting the intervals of time by the clock.
Owing to the perfectly regular motion of the earth, these
time intervals give an exact determination of the angular
intervals. In the same way, the angular value of each
turn of the screw micrometer attached to a telescope, can
be easily and accurately ascertained.

When a thermopile is used to observe radiant heat, it
would be almost impossible to calculate on a priori
grounds what is the value of each division of the galvano
meter circle, and still more difficult to construct a galva
nometer, so that each division should have a given value.
But this is quite unnecessary, because by placing the ther
mopile before a body of known dimensions, at a known
distance, with a known temperature, and radiating power,
we measure a known amount of radiant heat, and in
versely measure the value of the indications of the ther
mopile. In a similar way Mr. Joule ascertained the actual
temperature produced by the compression of bars of metal.
For having inserted a simple thermopile composed of a
single junction of copper and iron wire, and noted the
deflections of the galvanometer, he had only to dip the
bars into water of different temperatures, until he pro
duced a like deflection, in order to ascertain the temperature
developed by pressure0.

0 'Philosophical Transactions' (1859), vol. cxlix. p. 119, &<?.

THE EXACT MEASUREMENT OF PHENOMENA. 351

In many instances we are indeed obliged to accept a
very carefully constructed instrument as a standard, as in
the case of a standard barometer. But it is then best to
treat all inferior instruments comparatively only, and
determine the values of their scales by comparison with
the assumed standard.

Systematic Performance of Measurements.

When a large number of accurate measurements have
to be effected, it is usually desirable to make a certain
number of determinations with scrupulous care, and after
wards use them as points of reference for the remaining
determinations. In the trigonometrical survey of a coun
try, the principal triangulation fixes the relative positions
and distances of a few points with rigid accuracy. A
minor triangulation refers every prominent hill or village
to one of the principal points, and then the details are
filled in by reference to the secondary points. The survey
of the heavens is effected in a like manner. The ancient
astronomers compared the right ascensions of a few prin
cipal stars with the moon, and thus ascertained their posi
tions with regard to the sun ; the minor stars were after
wards referred to the principal stars. Tycho followed
the same method, except that he used the more slowly
moving planet Venus instead of the moon. Flamsteed
was in the habit of using about seven stars, favourably
situated at points all round the heavens. The distances
of the other stars from these standard points, were deter
mined in his early observations by the use of the quadrant P.
Even since the introduction of the transit telescope and
mural circle, tables of standard stars are formed at
Greenwich, the positions being determined with every

P Baily's 'Account of Flamsteed/ pp. 378-380.

352 THE PRINCIPLES OF SCIENCE.

possible accuracy, so that they can be employed for pur
poses of reference by all astronomers.

In ascertaining the specific gravities of substances, all
gases are referred to atmospheric air at a given tempera
ture and pressure ; all liquids and solids are referred to
water. We require to compare the densities of water
and air with great care, and the comparative densities of
any two substances whatever can then be with ease
ascertained.

In comparing a very great with a very small magni
tude, it is usually desirable to break up the process into
several steps, using intermediate terms of comparison.
We should never think of measuring the distance from
London to Edinburgh by laying down measuring rods
throughout the whole distance. A base of several miles
in length is selected on level ground, and compared on
the one hand with the standard yard, and on the other
with the distance of London and Edinburgh, or any other
two points, by trigonometrical survey. It would be ex
ceedingly difficult to compare the light of a star with
that of the sun, which would be about thirty thousand
million times greater; but Sir J. Herscheli effected the
comparison by using the full moon as an intermediate
unit. Wollaston ascertained that the sun gave 801,072
times as much light as the full moon, and Herschel
determined that the light of the latter exceeded that of
a Centauri 27,408 times, so that we find the ratio be
tween the light of the sun and star to be that of about
22,000,000,000 to i.

The Pendulum.

By far the most perfect and beautiful of all instru
ments of measurement is the pendulum. Consisting

<i Herschel's 'Astronomy/ § 817, 4th. ed. p. 553.

THE EXACT MEASUREMENT OF PHENOMENA. 353

merely of a heavy body suspended freely at an invari
able distance from a fixed point, it is the most simple
in construction ; and yet all the highest problems of phy
sical measurement depend upon its careful use. Its
excessive value arises from two circumstances, which
render it at once most accurate and indispensable.

(1) The method of repetition is eminently applicable
to it, as already described (p. 339.)

(2) Unlike any other instrument, it connects together
three different variable quantities, those of space, time,
and force.

In most works on natural philosophy it is shown, that
when the oscillations of the pendulum are infinitely small,
the square of the time occupied by an oscillation is directly
proportional to the length of the pendulum, and indirectly
proportional to the force affecting it, of whatever kind.
The whole theory of the pendulum is contained in the
formula, first given by Huyghens in his Horologium Oscil-
latorium,

time of OScillation= 3-14159. . x / length of pendulum

V force.

The quantity 3-14159 is the constant ratio of the circum
ference and radius of a circle, and is of course known with
accuracy. Hence, any two of the three quantities con
cerned being given, the third may be found ; or any two
being maintained invariable, the third will be invariable.
Thus a pendulum of invariable length suspended at the
same place, where the force of gravity may be considered
uniform, furnishes a theoretically perfect measure of time.
The same invariable pendulum being made to vibrate at
different points of the earth's surface, and the time of
vibration being astronomically determined, the force of
gravity becomes accurately known. Finally, with a known
force of gravity, and time of vibration ascertained by refer
ence to the stars, the length is determinate.

A a

354 THE PRINCIPLES OF SCIENCE.

In the first use all astronomical observations depend
upon it. In the second employment it has been almost
equally indispensable. The primary principle that gravity
is equal in all matter was proved by Newton's and Gauss'
pendulum experiments. The torsion pendulum of Michell,
Cavendish, and Baily, depending upon exactly the same
principles as the ordinary pendulum, gave the density of
the earth, one of the foremost natural constants. Kater
arid Sabine, by pendulum observations in different parts
of the earth, ascertained the variation of gravity, whence
comes a determination of the earth's ellipticity. The laws
of electric and magnetic attraction have also been deter
mined by the method of vibrations, which is in constant
use in the measurement of the horizontal force of terres
trial magnetism.

We must not confuse with the ordinary use of the
pendulum its application by Newton, to show the absence
of internal friction against space1', or to ascertain the laws
of motion and elasticity8. In such cases the extent of
vibration is the quantity measured, and the principles of
the instrument are different.

Attainable Accuracy of Measurement.

It is a matter of some interest to compare the degrees
of accuracy, which can be attained in the measurement of
different kinds of magnitude. Few measurements of any
kind are exact to more than six significant figures*, but it
is seldom that such a point of accuracy can be hoped for.
Time is the magnitude which seems to be capable of the
most exact discrimination, owing to the properties of the

r ' Principia,' bk. ii. Sect. 6. Prop. 31. Motte's Translation, vol. ii
p. 107.

p Ibid. bk. i. Law iii. Corollary 6. Motte's Translation, vol. i. p. 33.
4 Thomson and Tail's 'Natural Philosophy/ vol. i. p. 333.

THE EXACT MEASUREMENT OF PHENOMENA. 355

pendulum, and the principle of repetition already described
(PP- 339: 353)- As regards short intervals of time, it has
already been stated that Sir George Airy was able to
estimate a difference of 2^ seconds per day, between two
pendulums with an uncertainty of less than *oi of a second,
or one part in 8,640,000, an exactness, as he truly remarks,
' almost beyond conception11'. The ratio between the mean
solar and the sidereal day, too, is known to about one part
in one hundred millions, or to the eighth place of decimals

(P- 337)-

Determinations of weight seem to come next in exact
ness, owing to the fact that repetition without error is
applicable to them (p. 340). An ordinary good balance
should show about one part in 500,000 of the loadx. The

finest balance employed by M. Stas, turned with - - of a

33

milligramne, when loaded with 25 grammes in each pan,
that is, with one part in 825,000 of the loady. But balances
have certainly been constructed to show one part in a
million z, and Eamsden is commonly said to have con
structed a balance for the Boyal Society, to indicate one
part in seven millions, though this is hardly credible.
Professor Clerk Maxwell takes it for granted that one
part in five millions can be detected, but we ought to
discriminate between what a balance can do when first
constructed, and when in continuous use.

Determinations of lengths, unless performed with extra
ordinary care, are open to much error in the junction of
the measuring bars. Even in measuring the base line of
a trigonometrical survey, the accuracy generally attained
is only that of about one part in 60,000, or an inch in the

u 'Philosophical Transactions/ (1856), vol. cxlvi pp. 330-1.
x Thomson and Tait, 'Natural Philosophy/ vol. i. p. 333.
y 'First Annual Report of the Mint/ p. 106.
z Jevons, in Watts' ' Dictionary of Chemistry/ vol. i. p. 483.

A a 2

356 THE PRINCIPLES OF SCIENCE.

milea ; but it is said that in four measurements of a
base line carried out very recently at Cape Comorin, the
greatest error was 0*077 inch in i'68 mile, or one part in
1,382,400, an almost incredible degree of accuracy b.. Sir J.
Whitworth has shown that touch is even a more delicate
mode of measuring lengths than sight, and by means of a
splendidly executed screw, and a small cube of iron placed
between two flat-ended iron bars, so as to be suspended
when touching them, he can detect a change of dimension
in a bar, amounting to no more than one-millionth of an
inchc.

a Thomson and Tait, ' Natural Philosophy,' vol. i. p. 333.
b 'Athenaeum/ February 28, 1870, p. 295.

c British Association, Glasgow, 1856. 'Address of the President of
the Mechanical Section.'

CHAPTER XIV.

UNITS AND STANDARDS OF MEASUREMENT.

INSTRUMENTS of measurement are, as we have seen,
only means of comparison between one magnitude and
another, and as a general rule we must assume some
one arbitrary magnitude, in terms of which all results
of measurement are to be expressed. Mere ratios be
tween any series of objects will never tell us their
absolute magnitudes ; we must have at least one ratio
for each, and we must have one absolute quantity. The
number of ratios n are expressible in n equations, which
will contain at least n + i quantities, so that if we
employ them to make known n magnitudes, we must
have one magnitude known. Hence, whether we are
measuring time, space, density, weight, mass, energy, or
any other physical quantity, we must refer to some con
crete standard, some actual object, which if once lost and
irrecoverable, all our measures lose their absolute mean
ing. This concrete standard is in all, except two, cases
absolutely arbitrary in point of theory, and its selection
a question of practical convenience.

Of the two cases in which a natural standard unit is
ready made for us, one case is that of number itself.
Abstract number needs no special unit ; for any object
by existing or being thought of as separate from other
objects (p. 176), furnishes us with a unit, and is the only
standard required.

358 ' THE PRINCIPLES OF SCIENCE.

Angular magnitude is the second case in which we
have a natural and almost necessary unit of reference,
namely, the whole revolution or perigon, as it has been
called by Mr. Sandemana.

It is a necessary result of the uniform properties of
space, that all complete revolutions are equal to each
other, so that we need not select any one, and can always
refer anew to space itself. Whether we take the whole
perigon, its half, or its quarter, is really immaterial ;
Euclid took the right angle, because the Greek geome
ters had never generalized their notions of angular
magnitude sufficiently to conceive clearly angles of all
magnitude, or of unlimited quantity of revolution. But
Euclid defines a right angle as half that made by a line
with its own continuation, not called by him an angle, and
which is of course equal to half a revolution. In mathe
matical analysis, again, a different fraction of the perigon
is taken, namely, such a fraction that the arc or portion
of the circumference included within it is equal to the
radius of the circle-. This angle, called by De Morgan the
arcual unit, is equal to about 57°, 17', 44-"'8, or decimally
57°'2957795i3 , and is such that the half revolu
tion contains 3*141 59265 .... such unitsb. Though this
standard angle is naturally employed in mathematical
analysis, and any other unit would introduce needless
complexity, we must not look upon it as a distinct unit,
since its amount is connected with that of the half peri
gon, by a natural constant 3*14159 usually signified

by the letter TT.

When we pass to other species of quantity, the choice
of unit is found to be entirely arbitrary. There is abso-

a ' Pelicotetics, or the Science of Quantity ; an Elementary Treatise on
Algebra, and its groundwork Arithmetic.' By Archibald Sandeman, M.A.
Cambridge, (Deighton, Bell, and Co.) 1868, p. 304.

b De Morgan's ' Trigonometry and Double Algebra,' p. 5.

UNITS AND STANDARDS OF MEASUREMENT. 359

lutely no mode of defining a length, but by selecting some
physical object exhibiting that length between certain
obvious points — as, for instance, the extremities of a bar,
or marks made upon its surface.

Standard Unit of Time.

Time is the great independent variable of all change,
that which itself flows on uninterruptedly, and brings the
variety which we call life and motion. When we reflect
upon its intimate nature, Time, like every other element of
existence, proves to be an inscrutable mystery. We can
only say with St. Augustin, to one who asks us what is
time, ' I know when you do not ask me/ The mind of
man will ask what can never be answered, but one result
of a true and rigorous logical philosophy must be to
convince us, that scientific explanation can only take place
between phenomena which have something in common,
and that when we get down to primary notions, like those
of time and space, the mind must meet a point of mystery
beyond which it cannot penetrate. A definition of time
must not be looked for; if we say with Hobbesc, that it
is 'the phantasm of before and after in motion,' or with
Aristotle that it is 'the number of motion according to
former and latter ; ' we obviously gain nothing, because
the notion of time is involved in the expressions before
and after, former and latter. Time is undoubtedly one
of those primary notions which can only be defined physi
cally, or by observation of phenomena which proceed in
time.

If we have not advanced a step beyond Augustin's acute
reflections on this subject01, it is curious to observe the

c ' English Works of Thos. Hobbes,' Edit, by Molesworth, vol. i. p. 95.
d ' Confessions,' bk. xi. chapters 20-28.

360 THE PRINCIPLES OF SCIENCE.

wonderful advances which have been made in the practical
measurement of its efflux. The rude sun-dial or the
rising of a conspicuous star, gave points of reference, while
the flow of water from the clepsydra, the burning of a
candle, or, in the monastic ages, even the continuous
equable chanting of psalms, gave the means of roughly
subdividing periods, and marking the hours of the day and
night6. The sun and stars still furnish the standard of
time, but means of accurate subdivision have become
requisite, and this has been furnished by the pendulum
and the chronoscope. By the pendulum we can accurately
divide the day into seconds of time. By the chronograph
we can subdivide the second into a hundred, a thousand,
or even a million parts. Wheatstone measured the dura
tion of an electric spark, and found it to be no more than

— -^ part of a second, while more recently Captain Noble

has been able to appreciate intervals of time, not exceed
ing the millionth part of a second.

When we come to inquire precisely what phenomenon
it is that we thus so minutely measure, we meet insur
mountable difficulties. Newton distinguished time accord
ing as it was absolute or apparent time, in the following
words : —

' Absolute, true, and mathematical time of itself and
from its own nature, flows equably without regard to any
thing external, and by another name is called duration ;
relative, apparent and common time, is some sensible and
external measure of duration by the means of motion f'.
Though we are perhaps obliged to assume the existence
of a uniformly increasing quantity which we call time,

e Sir G. C. Lewis gives many curious particulars concerning the mea
surement of time, 'Astronomy of the Ancients,' pp. 241, &c.

f ' Principia/ bk. i. ' Scholium to Definitions.' Translated by Motte,
vol. i. p. ix. See also, p. u.

UNITS AND STANDARDS OF MEASUREMENT. 3G1

yet we cannot feel or know abstract and absolute time.
Duration must be made manifest to us by the recurrence

»/

of some phenomenon. The succession and change of our
own thoughts is no. doubt the first and simplest measure
of time, but a very rude one, because in some persons and
circumstances the thoughts evidently flow with much
greater rapidity than in other persons and circumstances.
In the absence of all other phenomena, the interval be
tween one thought and another, would necessarily become
the unit of time. The earth, as I have already said, is
the real clock of the astronomer, and is practically assumed
as invariable in its movements. But on what ground is
it so assumed ? According to the first law of motion, every
body perseveres in its state of rest or of uniform motion
in a right line, unless it is compelled to change that state
by forces impressed thereon. Rotatory motion is subject
to a like condition, namely, that it perseveres uniformly
unless disturbed by extrinsic forces. Now uniform mo
tion means motion through equal spaces in equal times,
so that if we have a body entirely free from all resistance
or perturbation, and can measure equal spaces of its path,
we have a perfect measure of time. But let it be remem
bered at the same time, that this law has never been
absolutely proved by experience ; for we cannot point to
any body, and say that it is wholly unresisted or undis
turbed ; and even if we had such a body, we should need
some entirely independent standard of time to ascertain
wrhether its motion was really uniform. As it is in moving
bodies that we find the best standard of time, we cannot
theoretically speaking use them to prove the uniformity
of their own movements, which would amount to a petitio
principii. Our experience amounts to this, that when
we examine and compare the movements of bodies which
seem to us nearly free from disturbance, we find them
give nearly harmonious measures of time. If any one

3G2 THE PRINCIPLES OF SCIENCE.

body which seems to us to move uniformly is not doing
so, but is subject to fits and starts unknown to us, because
we have no absolute standard of time, then all other
bodies must be subject to exactly the same arbitrary fits
and starts, otherwise there would be a discrepancy be
tween them disclosing the irregularities. Just as in com
paring together a number of chronometers, we should
soon detect bad ones by their irregular going, as measured
by the others, so in nature we detect disturbed movement
by its discrepancy from that of other bodies, which we
believe to be undisturbed, and which agree very nearly
among themselves. But inasmuch as the measure of motion

O

involves time, and the measure of time involves motion,
there must be ultimately an assumption. We may define
equal times, as times during which a moving body under
the influence of no force describes equal spaces e, but all
we can say in its support is, that it leads us into no
known difficulties, and that to the best of our experience,
one freely moving body gives exactly the same results as
any other.

When we inquire where the freely moving body is, no
satisfactory answer can be given. Practically the rotating
globe is sufficiently accurate, and Thomson and Tait say :
' Equal times are times during which the earth turns
through equal angles11'. No long time has passed since
astronomers thought it impossible to detect any inequality
in its movement. Poisson was supposed to have proved
that a change in the length of the sidereal day, amounting
to one ten-millionth part in 2500 years, was incompatible
with an ancient eclipse recorded by the Chaldeans, and
similar calculations were made by Laplace. But it is now
known that these calculations were somewhat in error,

8 Rankine, 'Philosophical Magazine/ Feb. 1867, vol. xxxiii. p. 91.
b ' Treatise on Natural Philosophy,' vol. i. p. 179.

UNITS AND STANDARDS OF MEASUREMENT. 363

and that the dissipation of energy arising out of the fric
tion of tidal waves, and the radiation of the heat into
space, has slightly decreased the rapidity of the earth's
rotatory motion. The sidereal day is now longer by one
part in 2,700,000, than it was in 720 B.C. Even before
this discovery, it was certain that the invariable rotation
depended upon the perfect maintenance of the earth's
internal heat, which is requisite in order that the earth's
dimensions shall be unaltered. Now the earth being far
superior in temperature to empty space, must cool more
or less rapidly, so that it cannot furnish an absolute
measure of time. Similar objections could be raised to
all other rotating bodies within our cognizance.

The moon's motion round the earth, and the earth's
motion round the sun, form the next best measure of
time. They are subject, indeed, to all kinds of disturb
ance from other planets, but it is believed that these
must in the course of time run through their rhythmical
courses, and leave the mean distances unaffected, and con
sequently, by the third Law of Kepler, the periodic times
unchanged. But there is more reason than not to believe
that the earth encounters a certain slight resistance in
passing through space, like that which is so apparent in
Encke's comet. There may also be a certain dissipation
of energy in the electrical relations of the earth to the

<_/»/

sun, possibly identical with that which is manifested in
the retardation of comets1. It is probably an untrue
assumption then, that the earth's orbit remains quite
invariable, and if so our last hope of getting a really
uniform measure of time disappears, and we are reduced
to accepting such as are sufficient for all practical pur
poses.

i ' Proceedings of the Manchester Philosophical Society,' 28th Nov.
1871, vol. xi. p. 33.

364 THE PRINCIPLES OF SCIENCE.

It is just possible that in the course of time, some other
body may be found to furnish a better standard of time
than the earth in its annual motion. The greatly superior
mass of Jupiter and its satellites, and their greater
distance from the sun, may render the electrical dissipa
tion of energy less considerable even than in the case of
the earth. But the choice of the best measure will always
be an open one, "and whatever moving body we assume,
may ultimately be shown to be subject to disturbing
forces.

The pendulum, although so admirable an instrument
for subdivision of time, entirely fails as a standard ; for
though the same pendulum affected by the same force of
gravity would perform equal vibrations in equal times,
yet the slightest change in the form or weight of the
pendulum, the slightest corrosion of any part, or the most
minute displacement of the point of suspension, would
falsify the results, and there enter many other diffi
cult questions of temperature, resistance, length of vibra
tion, &c.

Thomson and Tait are of opinion k that the ultimate
standard of chronometry must be founded on the physical
properties of some body of more constant character than
the earth ; for instance, a carefully arranged metallic
spring, hermetically sealed in an exhausted glass vessel.
Although their suggestion is no doubt theoretically cor
rect, it is hard to see how we can be sure that the dimen
sions and elasticity of a piece of wrought metal will
remain perfectly unchanged for the few millions of years
contemplated by them. A nearly perfect gas, like hydrogen,
is perhaps the only kind of substance in the unchanged
elasticity of which we could have confidence. Moreover,
it is difficult to perceive how the undulations of such a

k 'The Elements of Natural Philosophy/ pai't i. p. 119.

UNITS AND STANDARDS OF MEASUREMENT. 3G5

spring could be observed with the requisite accuracy. We
thus appear to be devoid of any hope of establishing a
sure standard of the efflux of time.

The Unit of Space and the Bar Standard.

Next in importance after the measurement of time is
that of space. Time comes first in theory, because pheno
mena, our internal thoughts for instance, may change in
time without regard to space magnitude. As to the phe
nomena of outward nature, they tend more and more to re
solve themselves into the motion of molecules, and motion
cannot be conceived or measured without reference both
to time and space.

Turning now to space measurements, we find it almost
equally difficult to fix and define once and for ever, a unit
magnitude. There are three different modes in which
it has been proposed to attempt the perpetuation of a
standard length.

1 i ) By constructing an actual specimen of the standard
yard or metre, in the form of a bar.

(2) By assuming the globe itself to be the ultimate
standard of magnitude, the practical unit being a sub-
multiple of some dimension of the globe.

(3) By adopting the length of a simple pendulum,
beating seconds as a standard of reference.

At first sight it might seem that there was no great
difficulty in this matter, and that any one of these methods
might serve well enough ; but the more minutely we
inquire into the details, the more hopeless appears to be
the attempt to establish an invariable standard. We must
in the first place point out a principle not of an obvious
character, namely, that the standard length must be defined
by one single object1. To make two bars of exactly the

1 See Harris' 'Essay upon Money and Coins/ part ii. [1758] p. 127.

3GG THE PRINCIPLES OF SCIENCE.

same length, or even two bars bearing a perfectly defined
ratio to each other, is beyond the power of human art. If
two copies of the standard metre be made and declared
equally correct, future investigators will certainly discover
some discrepancy between them, proving of course that they
cannot both be the standard, and giving cause for dispute
as to what magnitude should then be taken as correct.

If one invariable bar could be constructed and main
tained as the absolute standard, no such inconvenience
could arise. Each successive generation as it acquired
higher powers of measurement, would detect errors in
the copies of the standard, but the standard itself would
be unimpeached, and would, as it were, become by degrees
more and more accurately known. Unfortunately to con
struct and preserve a metre or yard is also a task which
is either impossible, or what comes nearly to the same
thing, cannot be shown to be possible. Passing over the
practical difficulty of defining the ends of the standard
length with complete accuracy, whether by dots or lines
on the surface, or by the terminal points of the bar, we
have no means of proving that substances remain of in
variable dimensions. Just as we cannot tell whether the
rotation of the earth is uniform, except by comparing it
with other moving bodies, believed to be more uniform
in motion, so we cannot detect the change of length in a
bar, except by comparing it with some other bar sup
posed to be invariable. But how are we to know which
is the invariable bar 1 It is certain that many rigid
and apparently invariable substances do change in di
mensions. The bulb of a thermometer certainly contracts
by age, besides undergoing rapid changes -of dimensions
when warmed or cooled through 100° Cent.m Can we

111 Watts' 'Dictionary of Chemistry/ vol. v. pp. 766, ^67. Dr. Joule
has recently confirmed the statements concerning the contraction of a
thermometer-bulb.

UNITS AND STANDARDS OF MEASUREMENT. 367

be sure that even the most solid metallic bars do not
slightly contract by age, or undergo variations in their
structure by change of temperature. M. Fizeau was in
duced to try whether a quartz crystal, subjected to several
hundred alternations of temperature, would be modified in
its physical properties, and he was unable to detect any
change in the coefficients of expansion11. It does not
follow however, that, because no apparent change was
discovered in a quartz crystal, newly-constructed bars of
metal would undergo no change.

The only principle, as it seems to me, upon which the
perpetuation of a standard of length can be ultimately
rested, is that, if a variation of length occurs, it will in
all probability be of different amount in different sub
stances. If then a great number of standard metres were
constructed of all kinds of different metals, alloys; hard
rocks, such as granite, serpentine, slate, quartz, limestone ;
artificial substances, such as porcelain, glass, &c., &c., care
ful comparison would show from time to time the com
parative variations of length of these different substances.
The most variable substances would be the most divergent,
and the true standard would be furnished by the mean
length of those which agreed most closely with each other,
just as uniform motion is that of those bodies which agree
most closely in indicating the efflux of time.

The Terrestrial Standard.

The second method assumes that the globe itself is a
body of invariable dimensions. The founders of the me
trical system selected the ten-millionth part of the dis
tance from the equator to the pole as the definition of the
metre, and the late Sir John Herschel proposed0 that

11 'Philosophical Magazine/ (1868), 4th Scries, vol. xxxvi. p. 32.
0 ' Familiar Lectures on Scientific Subjects,' (1866) p. 191.

3G8 THE PRINCIPLES OF SCIENCE.

the English inch, which is now almost exactly the
5OO,5oo,oooth part of the polar axis of the earth, should
be made exactly equal to the 5OO,ooo,oooth part, and be
adopted as our standard. The first imperfection in such
a method is that the earth is certainly not invariable in
size; for we know that it is superior in temperature to sur
rounding space, and must be slowly cooling and contract
ing. There is much reason to believe that all earthquakes,
volcanoes, mountain elevations, and changes of sea level,
are evidences of this contraction as asserted by Mr. Mallet P.
But such is the vast bulk of the earth and the duration
of its past existence, that this contraction is perhaps less
rapid in proportion than that of any bar or other material
standard which we can construct.

The second and chief difficulty of this method arises
from the vast size of the earth, which prevents us from
making any comparison with the ultimate standard, ex
cept by a trigonometrical survey of a most elaborate and
costly kind. The French physicists, who first proposed
the method, attempted to obviate this inconvenience by
carrying out the survey once for all, and then constructing
a standard metre, which should be exactly the one ten
millionth part of the distance from the pole to the
equator. But since all measuring operations are merely
approximate, as so often stated in previous pages, it was
impossible that this operation could be perfectly achieved.
Accordingly it was shown by Colonel Puissant in 1838,
that the supposed French metre was erroneous to the con
siderable extent of one part in 5527, the quadrant of the
earth's circumference measuring 10,001,789 instead of
10,000,000 of such metres. It then became necessary
either to alter the length of the assumed metre, or
otherwise to abandon its supposed relation to the earth's
dimensions.

P 'Proceedings of the Koyal Society,' 20th June, 1872, vol. xx. p. 438.

UNITS AND STANDARDS OF MEASUREMENT. 369

The French Government and the present International
Metrical Commission have for obvious reasons decided in
favour of the latter course, and have thus reverted to the
first method of defining the metre by a given bar. As
from time to time the ratio between this assumed
standard metre and the dimensions of the earth becomes
more and more accurately known, we have the better
means of restoring that metre by actual reference to the
globe if required. But until lost, destroyed, or for some
clear reason discredited, the bar metre and not the globe
is the standard. Any of the more accurate measurements
of the English trigonometrical survey might in like
manner be employed to restore our standard yard, in terms
of which the results are recorded ^

The Pendulum Standard.

The third method of defining a standard length, by
reference to the seconds' pendulum, was first proposed by
Huyghens, and was at one time adopted by the English
Government. From the principle of the pendulum (p. 353)
it clearly appears that if the time of oscillation and the
force actuating the pendulum be the same, the length
must be the same. We do not get rid of theoretical
difficulties, for we must practically assume the attraction
of gravity at some point of the earth's surface, say
London, to be unchanged from time to time, and the
sidereal day to be invariable, neither assumption being
absolutely correct so far as we can judge. The pendulum,
in short, is only an indirect means of making one physi
cal quantity of space depend upon two other physical
quantities of time and force.

The practical difficulties are, however, of a far more

q Thomson and Tait's ' Elements of Natural Philosophy/ Part i.
p. 119.

Bb

370 THE PRINCIPLES OF SCIENCE.

serious character than the theoretical ones. The length
of a pendulum is not the ordinary length of the instru
ment, which might be greatly varied, without affecting
the duration of a vibration, but the distance from the
centre of suspension to the centre of oscillation. There is
no direct means of determining this centre, which depends
upon the average momentum of all the particles of the
pendulum as regards the centre of suspension. Huyghens
discovered that the centres of suspension and oscillation
are interchangeable, and Captain Kater pointed out that
if a pendulum vibrates with exactly the same rapidity
when suspended from two different points, the distance
between these points is the true length of the equivalent
simple pendulum1". But the practical difficulties in em
ploying Rater's reversible pendulum are considerable, and
questions regarding the disturbance of the air, the force
of gravity or even the interference of electrical attractions
have to be entertained. It has been shown that all the
experiments made under the authority of government for
establishing the ratio between the standard yard and the
seconds' pendulum, were vitiated by an error in the correc
tions for the resisting, adherent or buoyant power of the
air in which the pendulum swung. Even if such correc
tions were rendered unnecessary by operating in a vacuum,
other difficult questions remain3. Gauss' mode of com
paring the vibrations of a wire pendulum when suspended
at two different lengths is open to equal or greater practi
cal difficulties. Thus it is found that the pendulum
standard cannot compete in accuracy arid certainty with
the simple bar standard, and the method would only be
useful as an accessory mode of restoring the bar standards
if at any time again destroyed.

r Rater's ' Treatise on Mechanics/ Cabinet Cyclopaedia, p. 154.
8 Grant's ' History of Physical Astronomy,' p. 156.

UNITS AND STANDARDS OF MEASUREMENT. 371

Unit of Density.

Before we can measure and define the phenomena of
nature, we require a third independent unit, which shall
enable us to define the quantity of matter which occupies
any given space. All the motions and changes of nature,
as we shall see, are probably so many manifestations of
energy ; but energy requires some substratum or material
machinery of molecules, in and by which it may be
exerted. Very little observation shows that, as regards
force, there may be two modes of variation of matter.
The force required to set a body in motion, varies in
simple proportion to the bulk or cubic dimensions of the
matter, but also according to its quality. Two cubic
inches of iron of uniform quality, will require twice as
much force to produce a certain velocity in a given time
as one cubic inch ; but one cubic inch of gold will require
more force than one cubic inch of iron. There is then
some new measurable quality in matter apart from its
bulk, which we may call density, and which is, strictly
speaking, indicated by its capacity to resist and absorb
the action of force. For the unit of density we may
assume that of any substance which is uniform in quality,
and can readily be referred to from time to time. Pure
water at any definite temperature, for instance that of
snow melting under an inappreciable pressure, furnishes
a natural and invariable standard of density, and by
testing equal bulks of various substances compared with a
like bulk of ice-cold water, as regards the velocity pro
duced in a unit of time by the same force, we should
ascertain the densities of those substances as expressed in
that of water.

Practically the force of gravity is used to measure
density ; for a simple and beautiful experiment with the

B b 2

372 THE PRINCIPLES OF SCIENCE.

pendulum, performed by Newton and Gauss, shows that
all kinds of matter equally gravitate, that is, the attractive
power of a substance is exactly proportional to its density.
Two portions of matter then which are in equilibrium in
the balance, may be assumed to possess equal inertia, and
their densities will therefore be inversely as their cubic
dimensions.

Unit of Mass.

Multiplying the number of units of density of a portion
of matter, by the number of units of space occupied by it,
we arrive at the quantity of matter, or, as it is usually
called, the units of mass, as indicated by the inertia and
gravity it possesses. To proceed in the most simple and
logical manner, the unit of mass ought to be that of a
cubic unit of matter of the standard density. The
founders of the French metrical system took as their unit
of mass, the cubic centimetre of water, at the temperature
of maximum density (about 4° Centigrade). They called
this unit of mass the gramme, and constructed standard
specimens of the kilogram, which might be readily re
ferred to by all who required to employ accurate weights.
Unfortunately, however, the determination of the bulk of
a given weight of water at a certain temperature is an
operation involving many practical and theoretical dif
ficulties, and it can not be performed in the present day
with a greater exactness than that of about one part in
5000, the results of careful observers being sometimes
found to differ as much as one part in 1000*.

Weights, on the other hand, can be compared with
each other to at least one part in a million. Hence if
different specimens of the kilogram be prepared by direct

* Clerk Maxwell's 'Theory of Heat/ p. 79.

UNITS AND STANDARDS OF MEASUREMENT. 373

weighing against water, they will not agree very closely
with each other ; and, as a matter of fact, the two principal
standard kilograms neither agree with each other, nor
with their true definition u. The so-called Kilogram des
Archives weighs 15432-34874 grains according to Prof.
W H. Miller, while the kilogram deposited at the
Ministry of the Interior in Paris, as the standard for
commercial purposes, weighs 15432*344 grains x.

Now since a standard weight constructed of platinum,
or platinum and iridium, can be preserved in all proba
bility free from any appreciable alteration, and since it
can be very accurately compared with other weights, we
shall ultimately attain the greatest exactness in our
recorded measurements of weight and mass, by assuming
some single standard kilogram as a provisional standard,
leaving the determination of its actual mass in units of
space and density for future investigation. This is what
is practically done at the present day, and thus a unit of
mass takes the place of the unit of density, both in the
French and the present English systems. The English
pound is defined by a certain lump of platinum, carefully
preserved at Westminster, and is an entirely arbitrary
mass, made to agree as nearly as possible with old English
pounds. The gallon, the old English unit of cubic mea
surement, is defined by the condition that it shall con
tain exactly ten pounds weight of water at 62° Fahr.; and
although it is stated that it has the capacity of about
277^274 cubic inches, this ratio between the cubic and
linear system of measurement is not legally enacted, but
is left open to investigation from time to time. While
the French metric system as originally designed was
theoretically perfect, it does not seem to differ practically
in this point from the English system.

" Thomson and Tait's 'Treatise on Natural Philosophy,' vol. i.
p. 325. * Ibid.

374 THE PRINCIPLES OF SCIENCE.

Subsidiary Units.

Having once established the standard units of time,
space, and density or mass, we might employ them for the
expression of all quantities of such nature. But it is often
found convenient in particular branches of science, to use
multiples or submultiples of the original units, for the ex
pression of quantities, in a clear and simple manner. We
use the mile rather than the yard when treating of the
magnitude of the globe, and the mean distance of the earth
and sun is not too large a unit when we have to describe

O

the distances of the stars. On the other hand, when we are
occupied with microscopic objects, the inch, the line or the
millimetre, become the most convenient terms of expression.
It is allowable for a scientific man to introduce a new
unit in any branch of knowledge, provided that it assists
precise expression, and is carefully brought into relation
with the primary units. Thus Prof. A. W. Williamson
has proposed as a convenient unit in chemical science, an
absolute volume equal to about 11*2 litres, representing
the bulk of one gramme of hydrogen gas at standard
temperature and pressure, or the equivalent weight of any
other gas, such as 16 grammes of oxygen, 14 grammes
of nitrogen, &c. ; in short, the bulk of that quantity of
any one of those gases which weighs as many grammes
as there are units in the number expressing its atomic
weight y. Professor Hofmann has also proposed a new con
crete unit for chemists, called a crith, to be defined by the
weight of one cubic decimetre or litre of hydrogen gas
at o° C. and o°'76mm., weighing about 0*0896 grammes2.
Both these units if adopted must be regarded as purely
subordinate units, ultimately defined by reference to the
primary units, and not involving any new assumption.

y ' Chemistry for Students,' by A. W. Williamson. Clarendon Press
Series, 2nd ed. Preface p. vi. " z ' Introd. to Chemistry,' p. 131.

UNITS AND STANDARDS OF MEASUREMENT. 375

Derived Units.

The standard units of time, space, and mass having been
once fixed, it becomes obvious that many kinds of magnitude
are naturally measured by units immediately derived from
one or more of the three principal ones. From the standard
metre of linear magnitude follows in the most obvious
manner the centaire or square metre, the unit of super
ficial magnitude, and the litre or cube of the tenth part
of a metre, the standard of capacity or volume. Velocity
of motion, again, is expressed by the ratio of the space-
passed over, when the motion is uniform, to the time
occupied ; hence the unit velocity will be that of a
body which passes over a unit of space in a unit of time,
say one metre per second. Momentum is measured by
the mass moving, regard being paid both to the amount
of matter and the velocity at which it is moving. Hence
the unit of momentum wih1 be that of a unit volume of
matter of the unit density moving with the unit velocity,
or in the French system, a cubic centimetre of water of
the maximum density moving one metre per second.

An accelerating force is measured by the ratio of the
momentum generated to the time occupied, the force
being supposed to act uniformly. The unit of force will
therefore be that which generates a unit of momentum
in a unit of time, or which causes, in the French system,
one cubic centimetre of water at maximum density to
acquire in one second a velocity of one metre per second.
The force of gravity is the most familiar kind of force,
and as when acting unimpeded upon any substance it
produces in a second a velocity of 9'8o868 .... metres
per second in Paris, it follows that the absolute unit
of force is about the tenth part of the force of gravity.
If we employ British weights and measures, the absolute
unit of force is represented by the gravity of about half

376 THE PRINCIPLES OF SCIENCE.

an ounce, since the force of gravity of any portion of
matter acting upon that matter during one second, pro
duces a final velocity of 3 2 '1889 feet per second or about
32 units of velocity. Although from its perpetual presence
and approximate uniformity we find in gravity the most
convenient force for reference, and thus habitually employ
it to estimate quantities of matter or mass, we must re
member that it is only one of many instances of force.
Strictly speaking, we should express weight in terms of
force, but practically we express all forces in terms of
weight.

We still require the unit of energy, a more com
plex notion. The momentum of a body expresses the
quantity of motion which belongs or would belong to the
aggregate of the particles, but when we consider how this
motion is related to the action of a force producing or
removing it, we find that the effect of a force is pro
portional to the mass multiplied by the square of the
velocity and it is most convenient to take half this product
as the expression required. But it is shown in books
upon Dynamics that it will be exactly the same thing if
we define energy by a force acting through a certain space.
The natural unit of energy will then be that which over
comes a unit of force acting through a unit of space ; when
we lift one kilogram through one metre, against gravity,
we therefore accomplish 9*80868 . . . . units of work,
that is, we turn so many units of potential energy ex
isting in the muscles, into potential energy of gravitation.
In lifting one pound through one foot there is in like
manner a conversion of 32*1889 units of energy. Accord
ingly the unit of energy will be that required to lift a
kilogram through about one tenth part of a metre against
gravity, or, in the English system, to lift one pound through
the thirty-second part of a foot.

Every person is at perfect liberty to measure and record

UNITS AND STANDARDS OF MEASUREMENT. 377

quantities in terms of any unit which he likes to adopt.
He may use the yard for linear measurement and the
litre for cubic measurement, only there will then be a
complicated relation between his different results. The
system of derived units which we have been briefly con
sidering, is that which gives the most simple and natural
relation between quantitative expressions of different
kinds, and therefore conduces to ease of comprehension
and saving of laborious calculation.

Provisionally Independent Units.

Ultimately, as we can hardly doubt, all phenomena
will be recognised as so many manifestations of energy ;
and, being expressed in terms of the unit of energy, will
be referable to the primary units of space, time, and
mass. To effect this reduction, however, in any particular
case, we must not only be able to compare different
quantities of the phenomenon, but to trace the whole
series of steps by which it is connected with the primary
notions. We can readily observe that the intensity of
one source of light is greater than that of another ; and,
knowing that the intensity of light decreases as the
square of the distance, we can easily determine their
comparative brilliance. Hence we can express the inten
sity of light falling upon any surface, if we have a unit
in which to make the expression. Light is undoubtedly
one form of energy, and the unit ought therefore to be
the unit of energy. But at present it is quite impossible
to say how much energy there is in any particular
amount of light. The question then arises, — Are we to
defer the measurement of light until we can fully and
accurately assign its relation to other forms of energy ?
If we answer Yes, it is equivalent to saying that the
science of light must stand still perhaps for a generation ;

378 THE PRINCIPLES OF SCIENCE.

and not only this science but almost every other. The
true course evidently is to select, as the provisional unit
of light, some light of convenient intensity, which can be
reproduced from time to time in exactly the same in
tensity, and which is denned by physical circumstances.
All the phenomena of light may be experimentally investi
gated relatively to this unit, for instance that obtained
after much labour by Bunsen and Koscoea. In after
years it will become a matter of inquiry what is the
energy exerted in such unit of light ; but it may be long
before the relation is exactly determined.

A provisionally independent unit, then, means one which
is assumed and physically denned in a safe and repro
ducible manner, in order that particular quantities may
be compared inter se more accurately than they can yet
be referred to the primary units. In reality almost all
our measurements are made by such independent units.
Even the unit of mass is practically an independent one,
as we have seen (p. 373).

Similarly the unit of heat ought to be simply the
unit of energy, already described. But a weight can
be measured to the one-millionth part, and temperature
to less than the thousandth part of a degree Fahrenheit,
and to less therefore than the five-hundredth thousandth
part of the absolute temperature, whereas the mechanical
equivalent of heat is probably not known to the thousandth
part. Hence the need of a provisional unit of heat, which
is often taken as that requisite to raise a unit weight of
water (say one gramme) through one degree Centigrade
of temperature, that is from o° to i°. This quantity of
heat is capable of approximate expression in terms of
time, space, and mass ; for by the natural constant,
determined by Dr. Joule, and called the mechanical

a ' Philosophical Transactions ' (1859), vol. cxlix. p. 884, &c.

UNITS AND STANDARDS OF MEASUREMENT. 379

equivalent of heat, we know that the assumed unit of
heat is equal to the energy of 423'55 gramme -metres,
or that energy which will raise the mass of 423*55
grammes through one metre against 9*80868 absolute
units of force. Heat may also be expressed in terms of
the quantity of ice at o° Cent., which it is capable of con
verting into water under an inappreciable pressure.

The science of electricity has lately become so much a
matter of quantity, that it is necessary to have some
means of accurate expression. When we know exactly
the mechanical equivalent of electricity, we can express
quantities of electricity in terms of energy, but in the
meantime we need some easy available unit. The British
Association accordingly have selected as the unit of
electrical force that which can just overcome the resistance
offered by a piece of pure silver wire i metre in length,
and i millemetre in diameter. This unit must be re
garded as merely a convenient provision for working
purposes, to be employed for the easy expression of
quantities not yet brought into precise relation with the
ultimate standards of time, space, and mass. There may
also be other provisionally independent units employed
in electrical science, such as the voltametric unit of cur
rent strength, namely, that current which by decomposing
water produces one cubic centimetre of detonating gas at
o° Cent, and 760 mm. of pressure in one minute. The unit
of electrical quantity, again, is that quantity which when
concentrated in a point and acting on an equal quantity
also concentrated in a point at a unit of distance, exerts
a repulsion equal to the unit of force. There must also
be a unit of electro-magnetic force. All these electrical
units must, however, be definitely related to each other,
and to the fundamental units, and it is a matter for
continual investigation to determine such relations more
and more accurately.

380 THE PRINCIPLES OF SCIENCE.

Natural Constants and Numbers.

Having acquired accurate measuring instruments, and
decided upon the units in which the results shall be
estimated and expressed, there remains the question,
What use shall he made of our powers of measurement ?
Our principal object must be to discover general quanti
tative laws of nature ; but a very large amount of pre
liminary labour is employed in the accurate determination
of the dimensions of existing objects, and the numerical
relations between diverse forces and phenomena. Step
by step every part of the material universe is surveyed
and brought into known relations with other parts. Each
manifestation of energy is correlated with each other kind
of manifestation. Professor Tyndall has described the care
with which such operations are conducted b.

'Those who are unacquainted with the details of
scientific investigation, have no idea of the amount of
labour expended on the determination of those numbers
on which important calculations or inferences depend.
They have no idea of the patience shown by a Berzelius
in determining atomic weights ; by a Eegnault in deter
mining coefficients of expansion ; or by a Joule in deter
mining the mechanical equivalent of heat. There is a
morality brought to bear upon such matters which, in
point of severity, is probably without a parallel in any
other domain of intellectual action/

Every new natural constant which is recorded brings
many fresh inferences within our power. For if n be the
number of such constants known, then ^ (n2— n) is the
number of ratios which are within our powers of cal
culation, and this increases with the square of n. We
thus gradually piece together a map of nature, in which
the lines of inference from one phenomenon to another

b Tyndall's ' Sound/ ist ed. p. 26.

UNITS AND STANDARDS OF MEASUREMENT. 381

rapidly grow in complexity, and the powers of scientific
prediction are correspondingly augmented.

The late Mr. Babbagec proposed the formation of a
complete collection of all the constant numbers of nature ;
but such a collection would be almost coextensive with
the whole mass of scientific literature. Almost all numbers
occurring in works on Chemistry, Mineralogy, Physics,
Astronomy, &c. are natural constants, and it would be
impracticable to give in any one work more than a
selection of the more important numbers.

Our present object will be to classify these constant
numbers roughly, according to their comparative gener
ality and importance, under the following heads : —

(1) Mathematical constants.

(2) Physical constants.

(3) Astronomical constants.

(4) Terrestrial numbers.

(5) Organic numbers.

(6) Social numbers.

Mathematical Constants.

At the head of the list of natural constants must come
those which express the necessary relations of numbers
to each other. The ordinary Multiplication Table is the
most familiar and the most important of such series of
constants, and is, theoretically speaking, infinite in extent.
Next we must place the Arithmetical Triangle, the sig
nificance of which has already been pointed out (p. 206.)
Tables of logarithms also contain vast series of natural
constants, arising out of the relations of pure numbers.
At the base of all logarithmic theory is the mysterious
natural constant commonly denoted by E, e, or e, being

equal to the infinite series i + - + 1 — - — I \- . ,

i 1.2 1.2.3 1-2.3.4

c British Association, Cambridge, 1833. Report, pp. 484-490.

382 THE PRINCIPLES OF SCIENCE.

and thus consisting of the sum of the ratios between the
numbers of permutations and combinations of o, i , 2, 3, 4,
&c. tilings.

Tables of prime numbers and of the factors of composite
numbers must not be forgotten.

Another vast and in fact infinite series of numerical
constants contains those connected with the measure
ment of angles, and embodied in trigonometrical tables,
whether as natural or logarithmic sines, cosines, and
tangents. It should never be forgotten that though
these numbers find their chief employment in connexion
with trigonometry, or the measurement of the sides of a
right-angled triangle, yet the numbers themselves arise
out of simple numerical relations bearing no special rela
tion to space.

Foremost among trigonometrical constants is the well
known number v, usually employed as expressing the
ratio of the circumference and the diameter of a circle ;
from TT follows the value of the arcual or natural unit
of angular value as expressed in ordinary degrees (see

P- 358).

Among other mathematical constants not uncommonly
used may be mentioned tables of factorials (p. 202), tables
of Bernouilli's numbers, tables of the error function11,
which latter are indispensable not only in the theory of
probability but also in several other branches of science.

It should also be clearly understood that the mathe
matical constants and tables of reference already in our
possession, although very extensive, are only an infinitely
small part of what might be formed. With the progress
of science the tabulation of new functions will be con
tinually demanded, and it is worthy of consideration
whether public money should not be constantly available

d See J. W. L. Glaisher, ' Philosophical Magazine/ 4th Series, vol. xlii.
p. 421.

UNITS AND STANDARDS OF MEASUREMENT. 383

to reward the enormous labour which must be undertaken
in these calculations. Such labours once successfully com
pleted must benefit the whole human race as long as it
shall exist. A valuable account of all the chief mathe
matical tables yet published will be found in De Morgan's
article on Tables, in the ' English Cyclopaedia,' Division
of Arts and Sciences, vol. vii. p. 976.

Physical Constants.

The second class of constants contains those which refer
to the actual constitution of matter. For the most part
they depend upon the peculiarities of the chemical sub
stance in question, but we may begin with those which
are of the most general character. In a first sub-class
we may place the velocity of light or heat undulations,
the numbers expressing the relation between the lengths
of the undulations, and the rapidity of the undulations,
these numbers depending only on the properties of the
ethereal medium, and being probably the same in all parts
of the universe. The theory of heat gives rise to several
numbers of the highest importance, especially Joule's
mechanical equivalent of heat, the absolute zero of tempe
rature, the mean temperature of empty space, &c.

Taking into account the diverse properties of the
elements we must have tables of the atomic weights,
the specific heats, the specific gravities, the refractive
powers, not only of the elements, but their almost in
finitely numerous compounds. The properties of hardness,
elasticity, viscosity, expansion by heat, conducting powers
for heat and electricity, must also be determined in
immense detail. There are, however, certain of these
numbers which stand out prominently because they serve
as intermediate units or terms of comparison. Such are,
for instance, the absolute coefficients of expansion of air,

384 THE PRINCIPLES OF SCIENCE.

water, and mercury, the temperature of the maximum
density of water (39°*ioi Fahr. or 4°*o Cent.), the latent
heats of water and steam, the boiling-point of water
under standard pressure, the melting and boiling-points
of mercury, and so on.

Astronomical Constants.

The third great class consists of numbers possessing far
less generality because they refer, not to the universal
properties of matter, but to the special forms and dis
tances in which matter has been disposed in the part of
the universe open to our examination. We have, first of
all, to define the magnitude and form of the earth, its mean
density, the constant of aberration of light expressing the
relation between the earth's mean velocity in space and
the velocity of light. From the earth, as our observatory,
we then proceed to lay down the mean distances of the
sun, and of the planets from the same centre ; all the
elements of the planetary orbits, the magnitudes, densities,
masses, periods of axial rotation of the several planets
are by degrees determined with growing accuracy. The
same labours must be gone through for the satellites.
Catalogues of comets with the elements of their orbits,
as far as ascertainable, must not be omitted.

From the earth's orbit as a new base of observations,
we next proceed to survey the heavens and lay down the
apparent positions, magnitudes, motions, distances, periods
of variation, &c. of the stars. All catalogues of stars from
those of Hipparchus and Tycho, are full of numbers ex
pressing rudely the conformation of the visible universe.
But there is obviously no limit to the labours of astrono
mers ; not only are minions of distant stars awaiting their
first measurements, but those already registered require
endless scrutiny as regards their movements in the three

UNITS AND STANDARDS OF MEASUREMENT. 385

dimensions of space, their periods of revolution, their
changes of brilliancy and colours. It is obvious that
though astronomical numbers are conventionally called
constant, they are in all cases probably subject to more
or less rapid variation.

Terrestrial Numbers.

Our knowledge of the globe we inhabit involves many
numerical determinations, which have little or no con
nexion with astronomical theory. The extreme heights
of the principal mountains, the mean elevation of con
tinents, the mean or extreme depths of the oceans, the
specific gravities of rocks, the temperature of mines, all
the host of numbers expressing the meteorological or
magnetic conditions of every part of the surface must
fall into this class. Many of such numbers are hardly
to be called constant, being subject to periodic or even
secular changes, but they are no more variable in fact
than many which in astronomical science are set down
as constant. In many cases quantities which seem most
variable may go through rhythmical changes resulting
in a nearly uniform average, and it is only in the long
progress of physical investigation that we can hope to
discriminate successfully between those elemental num
bers which are absolutely fixed and those which vary.
In the latter case the law of variation becomes the
constant relation which is the object of our search.

Organic Numbers.

All the forms and properties of brute nature having
been sufficiently defined by the previous classes of numbers,
the organic world, both vegetable and animal, remains
outstanding, and offers a higher series of phenomena for

c c

386 THE PRINCIPLES OF SCIENCE.

our investigation. All exact knowledge relating to the
forms and sizes of living things, their numbers, the
quantities of various compounds which they consume,
contain, or excrete, their muscular or nervous energy, &c.
must be placed apart in a class by themselves. All such
numbers are doubtless more or less subject to variation,
and but in a minor degree capable of exact determination.
Man, so far as he is an animal, and as regards his physical
form, must also be treated in this class.

Social Numbers.

Little or no allusion has hitherto been made in this
work to the fact that man in his economical, sanitary,
intellectual, aesthetic, or moral relations may become the
subject of exact sciences, the highest and most useful
of all sciences. Every one who is in any degree engaged
in statistical inquiry or study must so far acknowledge
the possibility of natural laws governing such statistical
facts. Hence we must certainly allot a distinct place to
all numerical information relating to the numbers, ages,
physical and sanitary condition, mortality, of all different
peoples, in short, to vital statistics. Economic statistics,
comprehending the quantities of commodities produced,
existing, exchanged, and consumed, constitute another
most extensive body of science. In the progress of reason
exact investigation may possibly subdue regions of pheno
mena which at present defy all analysis and scientific
treatment. That scientific method can ever exhaust the
phenomena of the human mind is on the other hand in
credible.

CHAPTER XV.

ANALYSIS OF QUANTITATIVE PHENOMENA.

IN the two preceding chapters we have been engaged
in considering how a phenomenon may be accurately
measured and expressed. So delicate and complex an
operation is a measurement which pretends to any con
siderable degree of exactness, that no small part of the
skill and patience of physicists is usually spent upon this
operation. Much of this difficulty arises from the fact that
it is scarcely ever possible to measure one simple pheno
menon at a time. The ultimate object must be to discover
the mathematical equation or law connecting a quantitative
cause with its quantitative effect ; this purpose usually
involves, as we shall see, the varying of one condition at
a time, the other conditions being maintained constant.
The labours of the experimentalist would be comparatively
light if he could carry out this rule of varying one circum
stance at a time. He would then obtain a series of cor
responding values of the variable quantities concerned,
from which he might by proper hypothetical treatment
obtain the required law of connexion. But in reality
it is seldom possible to carry out this direction except
in an approximate manner. Before then we proceed to
the consideration of the actual process of quantitative
induction, it is necessary to review the several devices
by which the complication of effects can be disentangled.
Every phenomenon measured will usually be the sum
difference or product of two or more different effects,
and these must be in some way analysed and separately

C C 2

388 THE PRINCIPLES OF SCIENCE.

measured before we possess the materials for a true
inductive treatment.

Illustrations of the Complication of Effects.

It is easy to bring forward a multitude of instances to
show that a phenomenon is seldom to be observed simple
and alone. A more or less elaborate process of analysis
is almost always necessary. Thus if an experimentalist
wishes to observe and measure the expansion of a liquid
by heat, he places it in a thermometer tube and registers
the rise of the column of liquid in the narrow tube. But
he cannot heat the liquid without also heating the glass,
so that the change observed is really the difference between
the expansions of the liquid and the glass. More minute
investigation will show the necessity perhaps of allowing
for further effects, namely the compression of the liquid
or the expansion of the bulb due to the increased pressure
of the column as it becomes lengthened.

In a great many cases an observed effect will be ap
parently at least the simple sum of two separate and
independent effects. The heat evolved in the combustion
of oil is partly due to the carbon and partly to the
hydrogen. A measurement of the heat yielded by the two
jointly, cannot inform us how much proceeds from the
one and how much from the other. If by some separate
determination we can ascertain how much the hydrogen
yields, then by mere subtraction we learn what is due
to the carbon ; and vice versa. The heat conveyed by a
liquid, may be partly conveyed by true conduction, partly
by convection. The light dispersed in the interior of a
liquid consists both of what is reflected by floating
particles and what is due to true fluorescence51; and -we
must find some mode of determining one portion before
we can learn the other.

a Stokes, 'Philosophical Transactions' (1852), vol. cxlii. p. 529.

ANALYSIS OF QUANTITATIVE PHENOMENA. 389

The apparent motion of the spots on the sun, is the
algebraic sum of the sun's axial rotation, and of the
proper motion of the spots upon the sun's face ; hence
the difficulty of ascertaining by direct observations the
period of the sun's rotation.

We cannot obtain the weight of a portion of liquid
in a chemical balance without weighing it with the
containing vessel. Hence to have the real weight of
the liquid operated upon in an experiment, we must
have a separate weighing of the vessel, with or without
the adhering film of liquid according to circumstances.
This is likewise the mode in which a cart and its load
are weighed together, the tare or weight of the cart
previously ascertained being deducted. The variation
in the height of the barometer is a joint effect, partly
due to the real variation of the atmospheric pressure,
partly due to the expansion of the mercurial column by
heat. The effects may be discriminated, if, instead of
one barometer tube we have two tubes placed closely
side by side, so as to have exactly the same temperature.
If one of them be closed at the bottom so as to be
unaffected by the atmospheric pressure, it will show
the changes due to temperature only, and, by subtracting
these changes from those shown in the other tube, we
get the real oscillations of atmospheric pressure. But
this correction, as it is called, of the barometric reading,
is better effected by calculation from the readings of
an ordinary thermometer.

In a great many other cases a quantitative effect will be
the difference of two causes acting in opposite directions.
The late Sir John Herschel invented an instrument like a

large thermometer which he called the Actinometer b, and

,

M. Pouillet constructed a somewhat similar instrument
b 'Admiralty Manual of Scientific Enquiry,' edited by Sir John
Herschel, 2nd ed. p. 299.

390 THE PRINCIPLES OF SCIENCE.

called the Pyrheliometer, for ascertaining the heating
power of the sun's rays. In both instruments the heat
of the sun was absorbed by a reservoir containing water,
and the rise of temperature of the water was exactly
observed, either by its own expansion or by the readings
of a delicate thermometer immersed in it. The details
of the construction and use of these instruments are im
material to our immediate purpose. Now in exposing the
actinometer to the sun, we do not obtain the full effect
of the heat absorbed, because the receiving surface is at
the same time radiating heat into empty space. The
observed increment of temperature is in short the dif
ference between what is received from the sun and lost by
radiation. But the latter quantity is capable of ready
determination ; we have only to shade the instrument
from the direct rays of the sun, while leaving it exposed
to the rest of the open sky, and we can observe how
much it cools in a certain time. The total effect of the
sun's rays will obviously be the apparent effect plus the
cooling effect in an equal time. By alternate exposure
in sun and shade during equal intervals the desired result
may be obtained with considerable accuracy c.

Two quantitative effects were beautifully distinguished
in an experiment of John Canton, devised in 1761 for the
purpose of demonstrating the compressibility of water d.
He constructed a thermometer with a large bulb full of
water and a short capillary tube, the part of which above
the water was freed from air. Under these circumstances
the water was relieved from the pressure of the atmo
sphere, but the glass bulb in bearing that pressure was
somewhat contracted. He next placed the instrument
under the receiver of an airpump, and on exhausting the
air, observed the water sink in the tube. Having thus

c Pouillet, 'Taylor's Scientific Memoirs/ vol. iv. p. 45.
fl Jamin, ' Cours de Physique,' vol. i. p. 158.

ANALYSIS OF QUANTITATIVE PHENOMENA. 391

obtained a measure of the effect of atmospheric pressure
on the bulb, he opened the top of the thermometer tube
and admitted the air. The level of the water now sank
still more, partly from the pressure on the bulb being
now compensated, and partly from the compression of the
water by the atmospheric pressure. It is obvious that
the amount of the latter effect was approximately the
difference of the two observed depressions.

Not uncommonly indeed the actual phenomenon which
we wish to measure is considerably less than various
disturbing effects which enter into the question. Thus
the compressibility of mercury is considerably less than
the expansion of the vessels in which it is measured
under pressure, so that the attention of the experi
mentalist has chiefly to be concentrated on the change
of magnitude of the vessels. Many astronomical phe
nomena, such as the parallax or proper motions of the
fixed stars, are far less than the instrumental imper
fections, and the other phenomena of precession, nutation,
aberration, &c. Even Flamsteed imagined he had dis
covered the parallax of the pole star6, and time after
time astronomers mistook various other phenomena for
that minute motion which they were so desirous to
discover.

Methods of Eliminating Error.

In any particular experiment it is the object of the
experimentalist to measure a single effect only, and he
endeavours to obtain that effect free from any interfering
effects. If this cannot be, as it seldom or never can
really be, he makes the effect as considerable as possible
compared with the other effects, which he reduces to a
minimum, and treats as noxious errors. Those quantities,

«• Baily's ' Account of the Rev. John Flamsteed,' p. 58.

392 THE PRINCIPLES OF SCIENCE.

which are called errors in one case, may really be most
important and interesting phenomena in another inves
tigation. When we speak of eliminating error we really
mean disentangling the complicated phenomena of nature.
The physicist rightly wishes to treat one thing at a time,
but as this object can seldom be rigorously carried into
practice, he has to seek some mode of counteracting the
tendency to error.

The general principle of the subject is that a single
observation can render known only a single quantity.
Hence if several different quantitative effects are known
to enter into any investigation, we must have at least
as many distinct results of observation as there are
quantities to be determined. Every complete experiment
will therefore consist in general of several operations.
Guided if possible by previous knowledge of the causes
in action, we must arrange these determinations, so that
by a simple mathematical process we may distinguish the
separate quantities. There appear to be five principal
methods in which we may accomplish this object ; these
methods are specified below and illustrated in the suc
ceeding sections.

(1) The Method of Avoidance. The physicist may seek
for some special mode of experiment or opportunity of
observation, in which the error is lion- existent or inap
preciable.

(2) The Differential Method. He may find opportunities
of observation when all interfering phenomena remain
constant, and only the subject of observation is at one time
present and another time absent ; the difference between
two exact observations then gives its amount.

(3) The Method of Correction. He may endeavour to
estimate the amount of the interfering force by the best
available mode, and then make a corresponding correction
in the results of observation.

ANALYSIS OF QUANTITATIVE PHENOMENA. 393

(4) The Method of Compensation. He may invent some
mode of neutralizing the interfering force by balancing
against it an exactly equal and opposite force of unknown
amount.

(5) The Method of Reversal. He may so conduct the
experiment that the interfering force may act in opposite
directions, in alternate observations, the mean result being
free from interference.

1. Method of Avoidance of Error.

Astronomers always seek opportunities of observation
when errors will have the smallest effect. In spite of
elaborate observations and long continued theoretical
investigation, it is not found possible to assign any
satisfactory law to the refractive power of the atmo
sphere. Although the apparent change of place of a
heavenly body thus produced, may be more or less
accurately calculated, yet the error depends upon the
temperature and pressure of the atmosphere, and, when
a ray is highly inclined to the perpendicular, the un
certainty in the refraction becomes very considerable.
Hence astronomers always make their observations, if
possible, when the object is at the highest point of its
daily course, i.e. on the meridian. In some kinds of
investigation, as, for instance, in the determination of the
latitude of an observatory, the astronomer is at liberty
to select one or more stars out of the countless number
visible. There is an evident advantage in such a case,
in selecting a star which passes close to the zenith,
so that it may be observed almost entirely free from
atmospheric refraction, as was done by Hooke. It
was ingeniously suggested by Wallis that the parallax
of the fixed stars might perhaps be detected by ob
servations of the greatest azimuth east and west of some

394 THE PRINCIPLES OF SCIENCE.

circumpolar star, since the refractive power of the atmo
sphere which affects only the altitude would thus be
entirely avoided f.

Astronomers also endeavour to render their clocks as
accurate as possible, by removing the source of variation.
The pendulum is perfectly isochronous so long as its
length remains invariable, and the vibrations are exactly
of equal length. They render it nearly invariable in
length, that is in the distance between the centres of
suspension and oscillation, by a compensatory arrangement
for the change of temperature. But as this compensation
may not be perfectly accomplished, some astronomers
place their chief controlling clocks in a cellar, or other
apartment, where the changes of temperature may be as
slight as possible. At the Paris Observatory a clock has
been placed in the caves beneath the building, where
there is no appreciable difference between the summer
and winter temperature.

To avoid the effect of unequal oscillations Huyghens
made his beautiful investigations, which resulted in the
discovery that a pendulum, of which the centre of oscil
lation moved upon a cycloidal path, would be perfectly
isochronous, whatever the variation in the length of
oscillations. But though a pendulum may be rendered in
some degree cycloidal by the use of a steel suspension
spring, it is found that the mechanical arrangements
requisite to produce a truly cycloidal motion introduce
more error than they avoid. Hence astronomers seek to
reduce the error to the smallest amount by maintaining
their clock pendulums in uniform movement s ; and in
fact while a clock is in good order and has the same
weights, there need be little change in the length of
oscillation.

f Grant, ' History of Physical Astronomy,' p. 548.

B Montucla, ' Histoire des Mathematiques/ vol. ii. p. 420.

ANALYSIS OF QUANTITATIVE PHENOMENA. 395

When a pendulum cannot be made to swing uniformly,
as in experiments upon the force of gravity, it becomes
requisite to resort to the third method, and a correction
is introduced, calculated on theoretical grounds from
the amount of the observed change in the length of
vibration.

It has been mentioned that the apparent expansion
of a liquid by heat, when contained in a thermometer tube
or other vessel, is the difference between the real ex
pansion of the liquid and that of the containing vessel. The
effects can be accurately distinguished provided that we
can learn the real expansion by heat of any one convenient
liquid ; for by observing the apparent expansion of the
same liquid in any required vessel we can by difference
learn the amount of expansion of the vessel due to any
given change of temperature. When we once know the
change of dimensions of the vessel, we can of course-
determine the absolute expansion of any other liquid
tested in it. Thus it became an all-important object in
scientific research to measure with accuracy the absolute
dilatation by heat of some one liquid, and mercury owing
to several circumstances was by far the most suitable.
Dulong and Petit devised a beautiful mode of effecting
this by simply avoiding altogether the effect of the
change of size of the vessel. Two upright tubes full of
mercury were connected by a fine tube at the bottom,
and were maintained at two different temperatures. As
mercury was free to flow from one tube to the other
by the connecting tube, the two columns necessarily
exerted equal pressures by the principles of hydrostatics.
Hence it was only necessary to measure very accurately
by a cathetometer the difference of level of the surfaces
of the two columns of mercury, to learn the difference of
length of columns of equal hydrostatic pressure, which at
once gives the difference of density of the mercury, and

396 THE PRINCIPLES OF SCIENCE.

the dilatation by heat. The changes of dimension in the
containing tubes now became a matter of entire indifference,
and the length of a column of mercury at different tem
peratures was measured as easily as if it had formed a
solid bar. The experiment was carried out by Reguault
with many improvements of detail, and the absolute
dilatation of mercury, at temperatures between o° Cent,
and 350°, was determined almost as accurately as was
needful11.

The presence of a large and uncertain amount of error
may often render a method of experiment valueless.
Foucault's beautiful mode of demonstrating the rotation
of the earth by the motion of a pendulum was thus
frustrated. The slightest lateral disturbance of the
pendulum gave it an elliptical path with a progressive
motion of the axis of the ellipse, and this motion of an
•unknown amount disguised and overpowered that due
to the rotation of the earth1. Faraday's laborious ex
periments on the relation of gravity and electricity were
much obstructed, too, by the fact that it is almost im
possible to move a large weight of iron or even lead
without generating currents of electricity, either by friction
or induction. To distinguish the electricity directly due
to the action of gravity from the greater quantities
indirectly produced would have been a problem of ex
cessive difficulty. Baily in his experiments on the density
of -the earth was aware of the existence of inexplicable
disturbances which have since been referred to the action
of electricity with much probability k. The skill and
ingenuity of the experimentalist are often exhausted
in devising a form of apparatus in which such causes
of error shall be reduced to a minimum.

11 Jamin, ' Cours de Physique,' vol. ii. pp. 15-28.

i 'Philosophical Magazine/ 1851, 4th Series, vol. ii. passim.

k Hcarn, ' Philosophical Transactions,' 1847, vol. cxxxvii. pp. 217-221.

ANALYSIS OF QUANTITATIVE PHENOMENA. 397

In some rudimentary experiments we may wish merely
to establish the existence of a quantitative effect without
precisely measuring its amount ; if there exist causes of
error of which we can neither render the amount known
or inappreciable, the best way will be to make them all
negative so that the quantitative effects will be less than
the truth rather than greater. Mr. Grove, for instance,
in proving that the magnetization or demagnetization of
a piece of iron raises its temperature, took care to maintain
the electro-magnet by which the iron was acted upon at
a lower temperature, so that it would cool rather than
warm the iron by radiation or conduction l.

Rumford's celebrated experiment to prove that heat was
generated out of mechanical force in the boring of a
cannon was subject to the difficulty that heat might be
brought to the cannon by conduction from neighbouring
bodies. It was an ingenious device of Davy to produce
friction by a piece of clock-work resting upon a block
of ice in an exhausted receiver ; as the machine rose in
temperature above 32°, it was certain that no heat was
received by conduction from the support111. In many
other experiments ice may be employed to prevent the
access of heat by conduction, and this device, first put in
practice by Murray11, is beautifully employed in Bunsen's
calorimeter.

To obtain the true temperature of the air, though
apparently so easy, is really a very difficult matter,
because the thermometer employed is sure to be affected
either by the sun's rays, the radiation from neighbouring
objects, or the escape of heat into space. These sources

1 ' The Correlation of Physical Forces,' 3rd ed. p. 159.

m 'Collected works of Sir H. Davy,' vol. ii. pp. 12-14. 'Elements of
Chemical Philosophy,' p. 94.

n ' Nicholson's Journal,' vol. i. p. 241; quoted in 'Treatise on Heat,'
Useful Knowledge Society, p. 24.

398 THE PRINCIPLES OF SCIENCE.

of error are too fluctuating to allow of correction, so that
the only accurate mode of procedure is that devised by
Dr. Joule, of surrounding the thermometer with a copper
cylinder ingeniously adjusted to the temperature of the
air, as described by him, so that the effect of radiation
shall be nullified °.

When the avoidance of error cannot be carried into
effect, it will yet be desirable to reduce the absolute
amount of the interfering error as much as possible before
employing the succeeding methods to correct the result.
As a general rule we can determine a quantity with less
inaccuracy as it is smaller, so that if the error itself be
small the error in determining that error will be of a still
lower order of magnitude. But in some cases the absolute
amount of an error is of no consequence, as in the index
error of a divided circle, or the difference between a
chronometer and astronomical time. Even the rate at
which a clock gains or loses is a matter of little im
portance provided it remains constant, so that a sure
calculation of its amount can be made.

2. Differential Method.

When we cannot avoid the entrance of error, we can
often resort with great success to the second mode of
measuring phenomena under such circumstances that the
error shall remain nearly or quite the same in all the
observations, and neutralize itself as regards the purposes
in view. This mode is available whenever we want a
difference between quantities and not the absolute
quantity of either. The determination of the parallax
of the fixed stars is exceedingly difficult, because the
amount of paraUax is far less than most of the corrections

0 Clerk Maxwell, 'Theory of Heat,' p. 228. 'Proceedings of the
Manchester Philosophical Society,' Nov. 26, 1867, vol. vii. p. 35.

ANALYSIS OF QUANTITATIVE PHENOMENA. 399

for atmospheric refraction, nutation, aberration, pre
cession, instrumental irregularities, &c., and can with
difficulty be detected among these phenomena of various
magnitude. But, as Galileo long ago suggested, all
such difficulties would be avoided by the differential
observation of stars, which though apparently close
together are really far separated on the line of sight.
Two such stars in close apparent proximity will be sub
ject to almost exactly equal errors, so that all we
need do is to observe the apparent change of place of
the nearer star as referred to the more distant one.
A good telescope furnished with an accurate micrometer
is alone needed for the application of the method.
Huyghens appears to have been the first observer who
actually tried to employ the method practically P, but
it was not until 1835 *na^ the improvement of telescopes
and micrometers enabled Struve to detect in this way
the parallax of the star a Lyrse.

It is one of the many advantages of the observation
of transits of Venus for the determination of the solar
parallax that the refraction of the atmosphere affects
in an exactly equal degree the planet and the portion
of the sun's face over which it is passing. Thus the
observations are strictly of a differential nature.

By the process of substitutive weighing it is possible
to ascertain the equality or inequality of two weights
with almost perfect freedom from error. If two weights
A and B be placed in the scales of the best balance
we cannot be sure that the equilibrium of the beam
indicates exact equality, because the arms of the beam
may be unequal or unbalanced. But if we take B out
and put another weight C in, and equilibrium still
exists, it is apparent that the same causes of erroneous

P History of ' Physical Astronomy,' p. 549. Herschel's ' Outlines of
Astronomy/ 4th ed. p. 550.

400 THE PRINCIPLES OF SCIENCE.

weighing exist in both cases, supposing that the balance
has not been disarranged, and that B must be exactly
equal to C, since it has exactly the same effect under
the same circumstances. In like manner it is a general
rule that, if by any uniform mechanical process we get
a copy of an object, it is unlikely that this copy will
be precisely the same in magnitude and form as the
original, but two copies will equally diverge from the
original, and will therefore almost exactly resemble each
other.

Leslie's Differential Thermometer <i was well adapted
to the experiments for which it was invented. Having
two equal bulbs any alteration in the temperature of the
air will act equally by conduction on each and produce
no change in the indications of the instrument. Only
that radiant heat which is purposely thrown upon one
of the bulbs will produce any effect. This thermometer
in short carries out the principle of the differential method
in a mechanical manner.

3. Method of Correction.

Whenever the result of an experiment is affected by an
interfering cause to an amount either invariable or exactly
calculable, it is sufficient simply to add or subtract this
calculated amount. We are said to correct observations
when we thus eliminate what is due to extraneous causes,
although of course we are only separating the correct
effects of several agents. Thus the variation in the height
of the barometrical column is partly due to the change
of temperature, and since the coefficient of absolute
dilatation of mercury has been exactly determined, as
already described (p. 395), we have only to make cal-

(i Leslie's 'Inquiry into the Nature of Heat/ p. 10.

ANALYSIS OF QUANTITATIVE PHENOMENA. 401

Culations of a simple character, or, what is better still,
tabulate a series of such calculations for general use,
and the correction for temperature can be made with
all desired accuracy. The height of the mercury in the
barometer is also affected by capillary attraction, which
depresses it by a constant amount depending on the
diameter of the tube. The requisite corrections can be
estimated with accuracy sufficient for most purposes, more
especially as we can check the correctness of the reading
of a barometer by comparison with a perfect standard
barometer, and introduce if need be an index error
including both the error in the affixing of the scale
and the effect due to capillarity. But in constructing
the standard barometer itself we must take greater pre
cautions ; the capillary depression depends somewhat
upon the quality of the glass, the absence of air, and
the perfect cleanliness of the mercury, so that we cannot
with confidence assign the exact amount of the effect.
Hence a standard barometer is constructed with a wide
tube, sometimes even an inch in diameter, so that the
capillary effect may be rendered of little account1'.
Gay Lussac made barometers in the form of a siphon so
that the capillary forces acting equally at the upper and
lower surfaces should balance and destroy each other,
but the method fails in practice because the lower surface,
being open to the air, becomes sullied and subject to a
different force of capillarity.

In a great many mechanical experiments friction is
an interfering condition, and drains away a portion of
the energy intended to be operated upon in a definite
manner. We should of course reduce the friction in the
first place to the lowest possible amount, but as it cannot
be altogether prevented, and is not calculable with cer
tainty from any general laws, we must determine it

r "Watts' ' Dictionary of Chemistry,' vol. i. pp. 513-15-
Dd

402 THE PRINCIPLES OF SCIENCE.

separately for each apparatus by suitable experiments.
Thus Smeaton, in his admirable but now almost forgotten
researches concerning water-wheels, eliminated friction in
the most simple manner by determining by trial what
weight, acting by a cord and roller upon his model water-
wheel, would make it turn without water as rapidly as
the water made it turn. In short, he ascertained what
weight concurring with the water would exactly com
pensate for the friction8. In Dr. Joule's experiments to
determine the mechanical equivalent of heat by the con
densation of air, a considerable amount of heat was pro
duced by friction of the condensing pump, and a small
portion by stirring the water employed to measure the
heat. This heat of friction was ascertained by simply
repeating the experiment in an exactly similar manner
except that no condensation was effected, and observing
the change of temperature then produced1.

We may describe as test experiments any in which we
perform operations not intended to give the quantity of
the principal phenomenon, but some quantity which would
otherwise remain as an error in the result. Thus in
astronomical observations almost every source of error
may be avoided by increasing the number of observations
and distributing them in such a manner as to produce
in the final mean as much error in one way as in the
other. But there is one source of error, first discovered
by Maskelyne, which cannot be avoided, because it affects
all observations in the same direction and to the same
average amount, namely the Personal Error of the ob
server or the inclination to record the passage of a star
across the wires of the telescope a little too soon or a
little too late. This personal error was first described in
the 'Edinburgh Journal of Science/ vol. i. p. 178. The

8 ' Philosophical Transactions,' vol. li. p. i oo.

t 'Philosophical Magazine,' 3rd Series, vol. xxvi. p. 372.

ANALYSIS OF QUANTITATIVE PHENOMENA. 403

difference between the judgment of observers at the
Greenwich Observatory usually varies from ~0 to J of
a second, or even a little more, and remains pretty con
stant for the same observers u. In some observers it has
amounted to seven or eight-tenths of a second x. Do
Morgan appears to have entertained the opinion that
this source of error was essentially incapable of elimina
tion or correction y. But it seems clear that this personal
error might be determined absolutely with any desirable
degree of accuracy by test experiments, consisting in
making an artificial star move at a considerable distance
and recording by electricity the exact moment of its
passage over the wire. This method has in fact been
successfully employed in Leyden, Paris, and Neuchatel z.

Newton employed the pendulum for making experi
ments on the impact of balls. Two balls were hung in
contact, and one of them, being drawn aside through a
measured arc, was then allowed to strike the other, the
arcs of vibration giving sufficient data for calculating the
distribution of energy at the moment of impact. The
resistance of the air was an interfering; cause which he

o

estimated very simply by causing one of the balls to
make several complete vibrations and then marking the
reduction in the length of the arcs, a proper fraction
of which reduction was added to each of the other ob
served arcs of vibration a.

In the modern use of the pendulum, to measure
terrestrial gravity, it is not found convenient to annul

u 'Greenwich Observations for 1866,' p. xlix.

x 'Penny Cyclopaedia,' art. Transit, vol. xxv. pp. 129, 130.

y Ibid. art. Observation, p. 390.

z 'Nature,' vol. i. pp. 85, 337. See references to the Memoirs de
scribing the method,

a 'Principia,' Book I. Law III. Corollary VI. Scholium. Motte's
translation, vol. i. p. 33.

D d 2

THE PRINCIPLES OF SCIENCE.

the resistance of the air by operating in a vacuum.
Consequently this resistance has to be ascertained
by appropriate and tedious series of experiments,
which should be made if possible upon each pendulum
employed.

The exact definition of the standard of length is one
of the most important, as it is one of the most difficult
questions in physical science, and the different practice of
different nations introduces wholly needless confusion.
Were all standards constructed so as to give the true
length at a fixed uniform temperature, for instance the
freezing-point, then any two standards could be compared
without the interference of temperature by bringing them
both to exactly the same fixed temperature. Unfortu
nately the French metre is defined by a bar of platinum
at o°C, while our yard is defined by a bronze bar at 62°F.
It is quite impossible, then, to make a comparison of the
yard and metre without the introduction of a correction,
either for the expansion of platinum or bronze, or both.
Bars of metal differ too so much in their rates of expansion
according to their molecular condition that it is dangerous
to infer from one bar to another.

When we come to use instruments with great accuracy
there are many minute sources of error which must be
guarded against. If a thermometer has been graduated
when perpendicular, it will read somewhat differently
when laid down, as the pressure of a column of mercury
is removed from the bulb. The reading may also be
somewhat altered if it has recently been raised to a
higher temperature than usual, if it be placed under a
vacuous receiver, or if the tube be unequally heated as «
compared with the bulb. For these minute causes of
error we may have to introduce troublesome corrections,
unless we adopt the simple mode of using the thermometer
in circumstances of position, &c. exactly similar to those

ANALYSIS OF QUANTITATIVE PHENOMENA. 405

in which it was graduated c. There is no end to the
number of minute corrections which may ultimately be
required. A very large number of experiments on gases,
standard weights and measures, &c. depend upon the
height of the barometer ; but when experiments in dif
ferent parts of the world are compared together we ought
to take into account the varying force of gravity, which
even between London and Paris makes a difference of
•008 inch of mercury.

The measurement of quantities of heat is a matter of
great difficulty, because there is no known substance
impervious to heat, and the problem is therefore as
difficult as to measure liquids in porous vessels. To
determine the latent heat of steam we must condense a
certain amount of the steam in a known weight of water,
and then observe the rise of temperature of the water.
But while we are carrying out the experiment, part of the
heat will have escaped by radiation or conduction from
the condensing vessel or calorimeter. We may indeed
reduce the loss of heat by using vessels with double sides
and bright surfaces, surrounded with swan's-down wool or
other non-conducting materials ; and we may also avoid
raising the temperature of the water much above that of
the surrounding air. Yet we cannot by any such means
render the loss of heat inconsiderable. Eumford ingeni
ously proposed to reduce the loss to zero by commencing
the experiment when the temperature of the calorimeter
is as much below that of the air as it is at the end of the
experiment above it. Thus the vessel will first gain and
then lose by radiation and conduction, and these opposite
errors will approximately balance each other. But Reg-
nault has shown that the loss and gain do not proceed by
exactly the same laws, so that in very accurate inves
tigations Kumford's method is not sufficient. There

c Balfour Stewart, ' Elementary Treatise on Heat/ p. 16.

406 THE PRINCIPLES OF SCIENCE.

remains the method of correction which was beautifully
carried out by Regnault in his determination of the latent
heat of steam, He employed two calorimeters, made in
exactly the same way and alternately used to condense a
certain amount of steam, so that while one was measuring
the latent heat, the other calorimeter was engaged in
determining the corrections to be applied, whether on
account of radiation and conduction from the vessel or
on account of heat reaching the vessel by means of the
connecting pipes d.

4. Method of Compensation.

There are many cases in which a cause of error cannot
conveniently be rendered null, and is yet beyond the
reach of the third method, that of calculating the requisite
correction from independent observations. The magnitude
of an error may be subject to continual variations, on
account of change of weather, or other fickle circumstances
beyond our control. It may either be impracticable to
observe the variation of those circumstances in sufficient
detail, or, if observed, the calculation of the amount of
error may be subject to doubt. In these cases, and only
in these cases, it will be desirable to invent some artificial
mode of counterpoising the variable error against an equal
error subject to exactly the same variation.

We cannot weigh any object with great accuracy unless
we make a correction for the weight of the air displaced
by the object, and add this to the apparent weight. In
very accurate investigations relating to standard weights,
it is usual to note the barometer and thermometer at the
time of making a weighing, and, from the measured bulks
of the objects compared, to calculate the weight of air

d Graham's ' Chemical Reports and Memoirs/ Cavendish Society, pp.
247, 268, &c.

ANALYSIS OF QUANTITATIVE PHENOMENA. 407

displaced ; the third method in fact is adopted. To make
all the calculations in the frequent weighings requisite in
chemical analysis would be exceedingly laborious, hence
the correction is usually neglected. But when the chemist
wishes to weigh a quantity of gas contained in a glass
globe for the purpose of determining its specific gravity,
the correction becomes of much importance. Hence
chemists avoid at once the error, and the labour of cor
recting it, by attaching to the opposite scale of the balance
a sealed glass globe of exactly equal capacity to that
containing the gas to be weighed, noting only the dif
ference of weight when the globe is full and empty. The
correction, being exactly the same for both globes, may be
entirely neglected e.

A device of nearly the same kind is employed in the
construction of galvanometers which measure the force of
an electric current by the deflection of a suspended
magnetic needle. The resistance of the needle is partly
due to the directive influence of the earth's magnetism,
and partly to the torsion of the thread. But the former
force may often be inconveniently great as well as
troublesome to determine for different inclinations. Hence
it is customary to connect together two exactly equal
needles, with their poles pointing in opposite directions,
one needle being within and another without the coil of
wire. As regards the earth's magnetism, the needles are
now astatic or indifferent, the tendency of one needle
being exactly balanced by that of the other.

An elegant instance of the elimination of a disturbing
force by compensation is found in Faraday's researches
upon the magnetism of gases. To observe the magnetic
attraction or repulsion of a gas seems impossible unless we
enclose the gas in an envelope, probably best made of

c Rcgnault's ' Cours Elementaire dc Ohimie,' 1851, vol. i. p. 141.

408 THE PRINCIPLES OF SCIENCE.

glass. But any such envelope is sure to be more or less
affected by the magnet, so that it becomes difficult to
distinguish between three forces which enter the problem ;
namely, the magnetism of the gas in question, that of the
envelope, and that of the surrounding atmospheric air.
Faraday avoided all difficulties by employing two exactly
equal and similar glass tubes connected together, and so
suspended from the arm of a torsion balance that the
tubes were in similar parts of the magnetic field. One
tube being filled with nitrogen and the other with oxygen,
it was found that the oxygen seemed to be attracted and
the nitrogen repelled. The suspending thread of the
balance was then turned until the force of torsion restored
the tubes to their original places, where the magnetism of
the tubes as well as that of the surrounding air, being
exactly the same and in the opposite direction upon the
two tubes, could not produce any interference. The force
thus required to restore the tubes was measured by the
amount of torsion of the thread, and it indicated correctly
the comparative attractive powers of oxygen and nitrogen.
The oxygen was then withdrawn from one of the tubes,
and a second experiment made, so as to compare a vacuum
with nitrogen. No force was now required to maintain
the tubes in their places, so that nitrogen was found to
be, approximately speaking, indifferent to the magnet,
that is, neither magnetic nor diamagnetic, while oxygen
was proved to be positively magnetic f. It required the
highest experimental skill on the part of Faraday and
Tyndall, to distinguish between what is apparent and real
in magnetic attraction and repulsion.

Experience alone can absolutely decide when a com
pensating arrangement is conducive to accuracy. As a
general rule mechanical compensation is the last resource,

f Tyndall's 'Faraday/ pp. 114-15.

ANALYSIS OF QUANTITATIVE PHENOMENA. 409

and in the more accurate observations it is likely to
introduce more uncertainty than it removes. A multitude
of instruments involving mechanical compensation have
been devised, but they are usually of an unscientific
character £, because the errors compensated can be more
accurately determined and allowed for. But there are
exceptions to this rule, and it seems to be proved that in
the delicate and tiresome operation of measuring a base
line, invariable bars, compensated for expansion by heat,
give a very accurate result, the observation of their vary
ing temperature and the calculation of the corrections
being an uncertain and tedious work h.

O

We thus see that the choice of one or other mode of
eliminating a simple error depends entirely upon circum
stances and the object in view ; but we may safely lay
down the following conclusions. First of all, seek to avoid
the source of error altogether if it can be conveniently
done ; if not, make the experiment so that the error may
be as small, but more especially as constant, as possible.
If the means are at hand for determining its amount
by calculation from other experiments and principles
of science, allow the error to exist and make a correction
in the result. If this cannot be accurately done or in
volves too much labour for the purposes in view, then
throw in a counteracting error which shall as nearly as
possible be of equal amount in all circumstances with
that to be eliminated. There yet remains, however, one
important method, that of Reversal, which will form an
appropriate transition to the succeeding chapters on the
Method of Mean Results and the Law of Error.

s See, for instance, the Compensated Sympiesometer, ' Philosophical
Magazine,' 4th Series, vol. xxxix. p. 371.

h Grant, ' History of Physical Astronomy,' pp. 146, 147.

THE PRINCIPLES OF SCIENCE.

5. Method of Reversal.

The fifth method of eliminating error is most potent
and satisfactory whenever it can be applied, but it re
quires that we shall be able to reverse the apparatus and
mode of procedure, so as to make the interfering cause
act alternately in opposite directions. If we can get two
experimental results, one of which is as much too great as
the other is too small, the error is equal to half the dif
ference, and the true result is the mean of the two
apparent results. It is an unavoidable defect of the
chemical balance, for instance, that the points of sus
pension of the pans cannot be fixed at exactly equal
distances from the centre of suspension of the beam.
Hence two weights which seem to balance each other
will never be quite equal in reality. The difference is
detected by reversing the weights, and it may be esti
mated by adding sufficient small weights to the deficient
side to restore equilibrium, and then taking as the true
weight the geometric mean of the two apparent weights
of the same object. If the difference is small the arith
metic mean, that is half the sum, may be substituted for
the geometric mean, from which it will not appreciably
differ.

This method of reversal is most extensively employed
in practical astronomy. The apparent elevation of a
heavenly body is observed by a telescope moving upon
a divided circle, upon which the inclination of the
telescope is read off. Now this reading will be erroneous
if the circle and the telescope have not accurately the
same centre. But if we read off at the same time both
ends of the telescope, the one reading will be about as
much too small as the other is too great, and the mean
will be nearly free from error. In practice the observa-

ANALYSIS OF QUANTITATIVE PHENOMENA. 411

tion is differently conducted, but the principle is the
same ; the telescope is fixed to the circle, which moves
with it, and" the angle through which it moves is read
off at three, six, or more points, disposed of at equal
intervals round the circle. The older astronomers, down
even to the time of Flamsteed, were accustomed to use
portions only of a divided circle, generally quadrants, and
Homer made a vast improvement when he introduced
the complete circle.

The transit circle, employed to determine the meridian
passage of heavenly bodies, is so constructed that the
telescope and the axis bearing it, in fact the whole moving
part of the instrument, can be taken out of the bearing
sockets and turned over, so that what was formerly the
western pivot becomes the eastern one, and vice, versd.
It is impossible that the instrument could have been
so perfectly constructed, mounted, and adjusted that the
telescope should point exactly to the meridian, but the
effect of the reversal is that it will point as much to
the west in one position as it does to the east in the
other, and the mean result of observations in the two
positions must be free from such cause of error.

The accuracy with which the inclination of the compass
needle can be determined depends almost entirely on the
method of reversal. The dip needle consists of a bar
of magnetized steel, suspended like the beam of a delicate
balance on a slender axis passing through the centre of
gravity of the bar, so that it is at liberty to rest in that
exact degree of inclination in the magnetic meridian
which the magnetism of the earth induces. The in
clination is read off upon a vertical divided circle, but
to avoid any error in the centring of the needle and
circle, both ends are read, and the mean of the results
is taken. The whole instrument is now turned carefully
round through 180°, which gives two new readings, in

412 THE PRINCIPLES OF SCIENCE.

which any error due to the wrong position of the zero
of the division will be reversed. As the axis of the
needle may not be exactly horizontal, it is now reversed
in the same manner as the transit instrument, the end of
the axis which formerly pointed east being made to point
west, and a new set of readings is taken.

Finally, error may arise from the axis not passing
accurately through the centre of gravity of the bar, and
this error can only be detected and eliminated on re
versing the magnetic poles of the bar by the application
of a strong magnet. The error is thus made to act in
opposite directions. To ensure all possible accuracy each
reversal ought to be combined with each other reversal,
so that the needle will be observed in eight different
positions by sixteen different readings, the mean of the
whole of which will give the required inclination free
from all eliminable errors k.

There are certain cases of experiment in which a
disturbing cause can with much ease be made to act in
opposite directions, in alternate observations, so that the
mean of the results will be free from disturbance. Thus
in direct experiments upon the velocity of sound in
passing through the air between stations two or three
miles apart, the wind is a cause of error. It will be well,
in the first place, to choose a time for the experiment
when the air is very nearly at rest, and the disturbance
slight, but if at the same moment signal sounds be made
at each station and observed at the other, two sounds will
be passing in opposite directions through the same body
of air and the wind will accelerate one sound almost
exactly as much as it retards the other1. Again, in
trigonometrical surveys the apparent height of a point

k Quetelet, ' Sur la Physique du Globe,' p. 174. Jamin, ' Cours de
Physique,' vol. i. p. 504.

1 Herschel, On Sound, ' Encyclopaedia Metropolitana,' p. 748.

ANALYSIS OF QUANTITATIVE PHENOMENA. 413

will be affected by atmospheric refraction and the
curvature of the earth. But if in the case of two points
the apparent elevation of each as seen from the other be
observed, the corrections will be the same in amount, but
reversed in direction, and the mean between the two
apparent differences of altitude will give the true dif
ference of level m.

In the next two chapters we really pursue the Method
of Reversal into more complicated applications.

111 Hutton, 'Philosophical Transactions/ abridgment, vol. xiv. p. 422.

CHAPTER XVI.

THE METHOD OF MEANS.

ALL results of the measurement of continuous quantity
can only be approximately true. Were this assertion
doubted, it could readily be proved by direct experience.
For if any person, using an instrument of the greatest
precision, makes and registers successive observations in
an unbiassed manner, it will almost invariably be found
that the results differ from each other. When we operate
with sufficient care we cannot perform so simple an
experiment as weighing an object in a good balance
without getting discrepant numbers. Only the rough
and careless experimenter will think that his observations
agree, but in reality he will be found to overlook the
differences. The most elaborate researches, such as those
undertaken in connexion with standard weights and
measures, always render it apparent that complete coinci
dence is out of the question, and that the more accurate
our modes of observation are rendered, the more numerous
are the sources of minute error which become apparent.
We may look upon the existence of error in all measure
ments as the normal state of things. It is absolutely
impossible to eliminate separately the multitude of small
disturbing influences, except by balancing them off against
each other. And even in drawing a mean it is to be
expected that we shall come near the truth rather than
exactly to it. In the measurement of continuous quantity,

THE METHOD OF MEANS. 415

absolute coincidence, if it even occurs or seems to occur,
must be purely casual, and is no indication of precision. It
is one of the most embarrassing things we can meet when
experimental results agree too closely. Such coincidences
should raise our suspicion that the apparatus in use is in
some way restricted in its operation, so as not really to
give the true result at alla, or that the actual results have
not been faithfully recorded by the assistant in charge of
the apparatus.

If then we cannot get twice over exactly the same
result, the question arises, How can we ever attain the
truth or select the result which may be supposed to
approach most nearly to it "? The quantity of a certain
phenomenon is expressed in several numbers which differ
from each other ; no more than one of them at the most
can be true, and it is more probable that they are all
false. It may be suggested, perhaps, that the observer
should select the one observation which he judged to be
the best made, and there will often doubtless be a feeling
that one or more results were satisfactory, and the others
less trustworthy. This seems to have been the course
adopted by some of the early astronomers. Flamsteed
when lie had made several observations of a star probably
chose in an arbitrary manner that which seemed to him
nearest to the truth1'.

When Horrocks selects for his estimate of the sun's
semidiameter a mean between the results of Kepler and
Tycho he professes not to do it from any regard to the
idle adage, ' Medio tutissimus ibis,' but because he
thought it from his own observations to be correct c. But
this method will not apply at all when the observer has

lv Thomson and Tait, 'Treatise on Natural Philosophy,' vol. i. p. 309.
b Baily's 'Account of Flamsteed,' p. 376.

c 'The Transit of Venus across the Sun,' by Horrocks, London, 1859,
p. 146.

416 THE PRINCIPLES OF SCIENCE.

made a number of measurements which are equally good
in his opinion, and it is quite apparent that in using an
instrument or apparatus of considerable complication the
observer will not necessarily be able to judge whether
slight causes have affected its operation or not.

In this question, as indeed throughout inductive logic,
we deal only with probabilities. There is no infallible
mode of arriving at the absolute truth, which lies beyond
the reach of human intellect, and can only be the distant
object of our long continued and painful approximations.
Nevertheless there is a mode pointed out alike by common
sense and the highest mathematical reasoning, which is
more likely than any other, as a general rule, to bring us
near the truth. The apia-rov /merpoi', or the aurea mediocritas,
was highly esteemed in the ancient philosophy of Greece
and Rome ; but it is not probable that any of the ancients
should have been able clearly to analyse and express the
reasons why they advocated the mean as the safest course.
But in the last two centuries this apparently simple
question of the mean has been found to afford a field for
the exercise of the utmost mathematical skill. Roger
Cotes, the editor of the ' Principia/ appears to have had
some insight into the value of the mean ; but profound
mathematicians such as De Moivre, Daniel Bernoulli i, La
place, Lagrange, Gauss, Quetelet, De Morgan, Airy, Leslie,
Ellis and others have hardly exhausted the subject.

Several uses of the Mean Result.

The elimination of errors of unknown sources, is almost
always accomplished by the simple arithmetical process
of taking the mean, or, as it is often called, the average
of several discrepant numbers. To take an average is to
add the several quantities together, and divide by the
number of quantities thus added, which gives a quotient

THE METHOD OF MEANS. 417

lying among, or in the middle of, the several quantities.
Before however inquiring fully into the grounds of this
procedure, it is essential to observe that this one arith
metical process is really applied in at least three different
cases, for different purposes, and upon different principles,
and we must take great care not to confuse one applica
tion of the process with another. A mean result, then,
may have any one of the following significations.

(1) It may give a merely representative number,
expressing the general magnitude of a series of quantities,
and serving as a convenient mode of comparing them
with other series of quantities. Such a number is properly
called The fictitious mean or The average result.

(2) It- may give a result approximately free from
disturbing quantities, which are known to affect some
results in one direction, and other results equally in the
opposite direction. We may say that in this case we get

-a Precise mean result.

(3) It may give a result more or less free from unknown
and uncertain errors ; this we may call the Probable
mean result.

Of these three uses of the mean the first is entirely dif
ferent in nature from the two last, since it does not yield
an approximation to any natural quantity, but furnishes
us with an arithmetic result comparing the aggregate of
certain quantities with their number. The third use of
the mean rests entirely upon the theory of probability,
and will be more fully considered in a later part of this
, chapter. The second use is closely connected, or even
identical with, the Method of Reversal already described
(p. 410), but it will be convenient to enter somewhat fully
on all the three employments of the same arithmetical
process.

E e

418 THE PRINCIPLES OF SCIENCE.

The significations of the terms Mean
and Average.

Much confusion exists in the popular, or even the
scientific employment of the terms mean and average, and
they are commonly taken as synonymous. It is desirable
to ascertain carefully what significations we ought to
attach to them. The English word mean is exactly equi
valent to medium, being derived perhaps, through the
French moyen, from the latin medius, which again is un
doubtedly kindred with the Greek /zeo-o?. Etymologists
believe, too, that this Greek word is connected with the
preposition /xera, the German mitte, and the true English
mid or middle ; so that after all the mean is a technical
term identical in its root with the more popular equivalent
middle.

If we inquire what is a mean in a mathematical point
of view, the true answer is that there are several or many
kinds of means. The old arithmeticians recognised at
least ten kinds, which are stated by Boethius, and even
an eleventh was added by Jordanusd.

The arithmetic mean is the one by far the most
commonly denoted by the term, and that which we may
understand it to signify in the absence of any qualification.
It is the sum of any series of quantities divided by their
number, and may be represented by the formula J (a + ~b).
But there is also the geometric mean, which is the square
root of the product,, *Ja^xb, or that quantity the logar
ithm of which is the arithmetic mean of the logarithms
of the quantities. There is also the harmonic mean,
which is the reciprocal of the arithmetic mean of the
reciprocals of the quantities. Thus if a and I be the

d De Morgan, Supplement to the ' Penny Cyclopaedia,' art. Old Appel
lations of Numbers.

THE METHOD OF MEANS.

quantities, as before, their reciprocals are — and — , the

a b

mean of which is i(~ + y)3 and the reciprocal again is

— r. Other kinds of means might no doubt be invented
for particular purposes, and we might apply the term, as

De Morgan pointed out e, to any quantity a function of
which is equal to a function of two or more other
quantities, and is such, that the interchange of these latter
quantities among themselves will make no alteration in
the value of the function. Symbolically, if ^ (y, y, y . . . .)
= <p ( #„ x2, 03, ... .), then y is a kind of mean of the
quantities xt, x2, &c.

The geometric mean is necessarily adopted in certain
cases. Thus when we estimate the work done against
a force which varies inversely as the square of the
distance from a fixed point, the mean force is the geo
metric mean between the forces at the beginning and end
of the path f. When in an imperfect balance, we reverse
the weights to eliminate error, the true weight will be the
geometric mean of the two apparent weights of the one
body (see p. 410).

In almost all the calculations of statistics and commerce
the geometric mean ought, strictly speaking, to be used.
Thus if a commodity rises in price 100 per cent, and
another remains unaltered, the mean rise of price is not
50 per cent, because the ratio 150 : 200 is not the same
as 100 : 150. The mean ratio is as unity to x/i -00x2-00
or i to i '4 1. The difference between the three kinds of
mean in such a case, as I have elsewhere shown *, is very
considerable, being as follows—

e 'Penny Cyclopaedia,' art. Mean.

1 Thomson and Tait, ' Treatise on Natural Philosophy/ vol. i. p. 366.

g ' Journal of the Statistical Society,' June 865, vol. xxviii. p. 296.

E e 2

420 THE PRINCIPLES OF SCIENCE.

Arithmetic mean 50 per cent.

Geometric „ 41 „

Harmonic „ 33

In all calculations concerning the average rate of
progress of a community, or any of its operations, the
geometric mean should be employed. For if a quantity
increases 100 per cent, in 100 years, it would not on the
average increase 10 per cent, in each ten years, as the
10 per cent, would at the end of each decade be calculated
upon larger and larger quantities, and give at the end of
100 years much more than 100 per cent., in fact as much
as 159 per cent. The true mean rate in each decade
would be i£/2~ or about I'oy, that is, the increase
would be about 7 per cent, in each ten years. But
when the quantities differ but little, the arithmetic and
geometric means are approximately the same. Thus the
arithmetic mean of rooo and rooi is rooos, and the
geometric mean is about 1*0004998, the difference being
of an order inappreciable in almost all scientific or prac
tical processes. Even in the comparison of standard weights
by Gauss' method of transposition the arithmetic mean may
usually be substituted for the geometric mean which is
the true result.

Regarding the mean in the absence of express qualifica
tion to the contrary as the common arithmetic mean, we
must still distinguish between its two uses where it
defines with more or less accuracy and probability a
really existing quantity, and where it acts as a mere
representative of other quantities. If I make many
experiments to determine the specific gravity of a homo
geneous piece of gold there is a certain definite ratio
which I wish to approximate to, and the mean of my
separate results will, in the absence of any reasons to the
contrary, be the most probable approximate result. When
we determine on the other hand the mean density of the

THE METHOD OF MEANS. 421

earth, it is exceedingly unlikely that there is any part of
the earth exactly of that density, and, as the crust is only
about half the mean density, there must be other parts of
greater density. I may also determine the mean specific
gravity of a body composed of iron and gold, so that
there will certainly be no portion possessing the mean
density.

The very different signification of the word ' mean ' in

*/ O

these two uses has been fully explained by M. Quetelet1',
and the importance of the distinction has moreover been
pointed out by Sir John Herschel in reviewing his work \
It is much to be desired that scientific men would mark
the difference by using the word mean only in the former
sense when it denotes approximation to a definite exist
ing quantity ; and average, when the mean is only a
fictitious quantity, used for the convenience of thought
and expression. The etymology of this word ' average ' is
somewhat obscure ; but according to De Morgan k it comes
from averia, ' havings or possessions/ especially applied to
farm stock. By the accidents of language averagium
came to mean the labour of farm horses to which the lord
was entitled, and it probably acquired in this manner the
notion of distributing a whole into parts, a sense in which
it was very early applied to maritime averages or contri
butions of the other owners of cargo to those whose goods
have been thrown overboard or used for the safety of the
vessel.

h ' Letters on the Theory of Probabilities,' trans! . by Downes, Part ii.
i Herschel's 'Essays,' &c. pp. 404, 405.

k < On the Theory of Errors of Observations,' ' Cambridge Philosophical
Transactions/ vol. x. Part ii. 416.

122 THE PRINCIPLES OF SCIENCE.

On the Fictitious Mean or Average Result.

Although the average when employed in its proper
sense of a fictitious mean, represents no really existing
quantity, it is yet of the highest scientific importance, as
enabling us to conceive in a single result a multitude
of complex details. It enables us to make a hypothetical
simplification of a problem, and avoid complexity without
committing error. Thus the aggregate weight of a body is
the sum of the weights of the indefinitely small particles,
each acting at a different place, so that the simplest
mechanical problem concerning a body really resolves itself,
strictly speaking, into an infinite number of distinct pro
blems. We owe to Archimedes the first introduction of
the beautiful idea that one point might be discovered in
a gravitating body such that the weight of all the par
ticles might be regarded as concentrated in that point,
and yet the behaviour of the whole body would be exactly
represented by the behaviour of this heavy point. This
Centre of Gravity may be within the body, as in the
case of a sphere, or it may be in empty space, as in
the case of a ring. Any two bodies, whether connected
or separate, may be conceived as having a centre of
gravity ; that of the sun and earth, for instance, lying
within the sun and only 267 miles from its centre.

Although we most commonly use the notion of a centre
or average point with regard to gravity, the same notion
is applicable to many other cases. Terrestrial gravity
is only one case of approximately parallel forces, so that
the centre of gravity is but a special case of the more
general Centre of Parallel Forces. Wherever a number
of forces of whatever amount act in parallel lines, it
is possible to discover a point at which the algebraic
sum of the forces may be imagined to act with exactly
the same effect. Water in a cistern presses against the

THE METHOD OF MEANS. 423

side with a pressure varying according to the depth,
but always in a direction perpendicular to the side.
We may then conceive the whole pressure as exerted
on one point, which will be one-third from the bottom
of the cistern, and may be called the Centre of Pressure.
The Centre of Oscillation of a pendulum, discovered by
Huyghens, is that point at which the whole weight of
the pendulum may be considered as concentrated, without
altering the time of oscillation (see p. 370). Similarly
when one body strikes another the Centre of Percussion
is that point in the striking body at which all its mass
might be concentrated without altering the effect of the
stroke. Mathematicians have also described the Centre
of Gyration, the Centre of Conversion, the Centre of
Friction, &c.

We ought however carefully to distinguish between
those circumstances in which an invariable centre can
be assigned, and those in which it cannot. In perfect
strictness, there is no such thing as a true invariable
centre of gravity. As a general rule a body is capable
of possessing an invariable centre only for perfectly
parallel forces, and gravity never does act in absolutely
parallel lines. Thus, as usual, we find that our concep
tions are only hypothetically correct, and only approxi
mately applicable to real circumstances. There are indeed
certain geometrical forms, called Centrobaric}, such that
bodies of that shape would attract each other exactly
as if the mass were concentrated at the centre of gravity,
whether the forces act in a parallel manner or not.
Newton shewed that uniform spheres of matter have
this property, and this truth proved of the greatest im
portance in simplifying his calculations. But it is after
all a purely hypothetical truth, because we. can nowhere
meet with, nor can we construct, a perfectly spherical

1 Thomson and Tait, 'Treatise on Natural Philosophy/ vol. i. p. 394.

424 THE PRINCIPLES OF SCIENCE.

and homogeneous body. The slightest irregularity or pro
trusion from the surface will destroy the rigorous cor
rectness of the assumption. The spheroid, on the other
hand, has no invariable centre at which its mass may
always be regarded as concentrated. The point at which
its resultant attraction acts will move about according
to the distance and position of the other attracting body,
and it will only coincide with the centre as regards an in
finitely distant body whose attractive forces may be con
sidered as acting in parallel lines.

Physicists speak familiarly of the pole of a magnet,
and the term may be used with convenience. But, if
we attach any real and definite meaning to it, the pole
is not the end of the magnet, nor is it any one fixed
point within, but the variable point from which the
resultant of all the forces exerted by the particles in
the whole bar upon exterior magnetic particles may be
considered as acting. The pole is, in short, a Centre of
Magnetic Forces ; but as those forces are really never
parallel, this centre will vary in position according to
the relative place of the object attracted. Only when
we regard the magnet as attracting a very distant, or,
strictly speaking, infinitely distant particle, does the
centre become a fixed point, situated in short magnets
approximately at one sixth of the whole length from
each end of the bar. We have in the above instances
of centres or poles of force sufficient examples of the mode
in which the Fictitious Mean or Average is employed in
physical science.

The Precise Mean Result.

We now turn to that mode of employing the mean
result which is analogous to the method of reversal, but
which is brought into practice in a most extensive manner
throughout many branches of physical science. We find

THE METHOD OF MEANS. 425

the simplest possible case in the determination of the
latitude of a place by observations of the Pole-star.
Tycho Brahe suggested that if the elevation of any cir-
cumpolar star were observed at its higher and lower
passages across the meridian, half the sum of the elevations
would be the latitude of the place, which is equal to the
height of the pole. Such a star is as much above the
pole at its highest point, as it is below at its lowest, so
that the mean must necessarily give the height of the
pole itself free from doubt, except as regards incidental
errors of observation. The Pole-star is usually selected
for the purpose of such observations because it describes
the smallest circle, and is thus on the whole least affected
by atmospheric refraction.

Whenever several causes are in action, each of which
at one time increases and at another time decreases the
joint effect by equal quantities, we may apply this method
and disentangle the effects. Thus the solar and lunar
tides roll on in almost complete independence of each
other. When the moon is new or full the solar tide coin
cides, or nearly so, with that caused by the moon, and the
joint effect is the sum of the separate effects. When the
moon is in quadrature, or half full, the two tides are
acting in opposition, one raising and the other depressing
the water, so that we observe only the difference of the
effects. We have in fact —

Spring tide = lunar tide + solar tide
Neap tide = lunar tide — solar tide.
We have only then to add together the heights of the
maximum spring tide and the minimum neap tide, and
half the sum is the true height of the lunar tide. Half
the difference of the spring and neap tides on the other
hand gives the solar tide.

Effects of very small amount may with great approach
to certainty be detected among much greater fluctuations,

426 THE PRINCIPLES OF SCIENCE.

provided that we have a series of observations sufficiently
numerous and long continued to enable us to balance all
the larger effects against each other. For this purpose
the observations should be continued over at least one
complete cycle, in which the effects run through all their
variations, and return exactly to the same relative position
as at the commencement. If casual or irregular disturbing
causes exist, we should probably require many such cycles
of results to render their effect inappreciable. We obtain
the desired result by taking the mean of all the observa
tions in which a cause acts positively, and the mean of all
in which it acts negatively. Half the difference of these
means will be the desired quantity, provided indeed that
no other effect happens to vary in the same period.

Since the moon causes so considerable a movement of
the ocean, it is evident that its attraction must have some
effect upon the atmosphere. The laws of these tides were
investigated by Laplace, but as it would be impracticable
by theory to calculate their amount, we can only determine
them by observation, as Laplace predicted that they would
one day be determined m. But the oscillations of the
barometer thus caused are far smaller than the oscillations
due to several other causes. Storms, hurricanes, or changes
of weather produce movements of the barometer some
times as much as a thousand times as great as the tide in
question. There are also regular daily, yearly, or other
fluctuations, all greater than the desired quantity. To
detect and measure the atmospheric tide it was desirable
that observations should be made in a place as free as
possible from irregular disturbances. On this account
several long series of observations were made at St.
Helena, where the barometer is far more regular in its
movements than in a continental climate. The effect of
the moon's attraction was then detected by taking the

m ' Essai Philosophiqne sur les Probabilites/ pp. 49, 50.

TUB METHOD OF MEANS. 427

mean of all the readings when the moon was on the me
ridian and the similar mean when she was on the horizon.
The difference of these means was found to be only
•00365, yet it was possible to discover even the variation
of this tide according as the moon was nearer to or further
from the earth, though this difference was only "00056
inch n. It is quite evident that such minute effects could
never be discovered in a purely empirical manner. Having
no information but the series of observations before us,
we could have no clue as to the mode of grouping them
which would give so small a difference. In applying this
method of means in an extensive manner we must gener
ally then have d priori knowledge as to the periods at
which a cause will act in one direction or the other.

We are sometimes able to eliminate fluctuations and
take a mean result by purely mechanical arrangements.
The daily variations of temperature, for instance, become
imperceptible one or two feet below the surface of the
earth, so that a thermometer placed with its bulb at that
depth would give very nearly the true daily mean tem
perature. At a depth of twenty feet even the yearly
fluctuations would become nearly effaced, and the thermo
meter would stand a little above the true mean tempera
ture of the locality. In registering the rise and fall of the
tide by a tide-guage, it is desirable to avoid the oscilla
tions arising from surface waves, which is very readily
accomplished by placing the float which marks the level
of the water in a cistern communicating by a small hole
with the sea. Only a general rise or fall of the level is
then perceptible, just as in the marine barometer the
narrow tube prevents any casual fluctuations and allows
only a continued change of pressure to manifest itself.

11 Grant, ' History of Physical Astronomy,' p. 163.

428 THE PRINCIPLES OF SCIENCE.

Determination of the Zero point by the Method
of Means,

There are a number of important observations in which
one of the chief difficulties consists in defining exactly the
zero point from which we are to measure. We can point
a telescope with great precision to a star and can measure
the angle through which the telescope is raised or lowered
to a second of arc ; but all this precision will be useless
unless we can know exactly where the centre point of
the heavens is from which we measure, or, what comes to
the same thing, the horizontal line 90° distant from it.
Since the true horizon has reference to the figure of the
earth at the place of observation, we can only determine
it by the direction of gravity, as marked either by the
plumb-line or the surface of a liquid. The question re
solves itself then into the most accurate mode of observing
the direction of gravity, and as the plumb-line has long
been found hopelessly inaccurate, astronomers generally
employ the surface of mercury in repose as the criterion
of horizontality. They ingeniously observe the direction
of the surface by making a star the index. From the
Laws of Reflection it follows that the angle between the
direct ray from a star and that reflected from a surface
of mercury will be exactly double the angle between the
surface and the direct ray from the star. Hence the
horizontal or zero point is the mean between the apparent
place of any star or other very distant object and its
reflection in mercury.

A plumb-line is perpendicular, or a liquid surface is hori
zontal only in an approximate sense ; for any irregularity
of the surface of the earth, a mountain, or even a house
must cause some deviation by its attracting power. To
detect such deviation might seem very difficult, because
every other plumb-line or liquid surface would be equally

THE METHOD OF MEANS. 429

affected by the very principles of gravity. Nevertheless
it can be detected ; for if we place one plumb-line to the
north of a mountain, and another to the south, they will
be about equally deflected in opposite directions, and if
by observations on the same star we can measure the angle
between the plumb-lines, half the inclination will be the
deviation of either, after allowance has been made for the
inclination due to the difference of latitude of the two
places of observation. By this mode of observation ap
plied to the mountain Schehallien the deviation of the
plumb-line was accurately measured by Maskelyne, and
thus a comparison instituted between the attractive forces
of the mountain and the whole globe, which led to a very
probable estimate of the earth's average density.

In some cases it is actually better to determine the zero
point by the average of equally diverging quantities than
by direct observations. Thus in delicate weighings by a
chemical balance it is requisite to ascertain exactly the
point at which the beam comes to rest, and when standard
weights are being compared the position of the beam is
ascertained by a carefully divided scale viewed through a
microscope. But when the beam is just coming to rest,
friction, small impediments or other accidental causes
may readily obstruct it, because it is near the point at
which the force of stability becomes infinitely small.
Hence it is found better to let the beam vibrate and
observe the terminal points of the vibrations. The mean
between two extreme points will nearly indicate the posi
tion of rest. Friction and the resistance of air tend to
reduce the vibrations, so that this mean will be erroneous
by half the amount of this effect during a half vibration.
But by taking several observations we may determine
this retardation and allow for it. Thus if a, b, c be the
terminal points of three excursions of the beam from the
zero of the scale, then ^ (a + &) will be about as much

430 THE PRINCIPLES OF SCIENCE.

erroneous in one direction as ^ (b + c) in the other, so
that the mean of these two means, or what is the same,
^ (a + 2 b + c), will be exceedingly near to the point of
rest0. A still closer approximation may be made by
taking four readings and reducing them by the formula
-5- (a + 2 b + 2 c + d).

The accuracy of Baily's experiments, directed to deter
mine the density of the earth, entirely depended upon this
mode of observing oscillations. The balls whose gravi
tation was measured were so delicately suspended by a
torsion balance that they never came to rest. The ex
treme points of the oscillations were observed both when
the heavy leaden attracting ball was on one side and on the
other. The difference of the mean points when the leaden
ball was on the right hand and that when it was on the
left hand gave double the amount of the deflection.

A most beautiful instance of the mode of avoiding the
use of a zero point is to be found in Mr. E. J. Stone's
observations on the radiated heat of the fixed stars. The
great difficulty in these observations arose from the com
paratively great amounts of heat which were sent into the
telescope from the atmosphere, and which were sufficient
almost entirely to disguise the feeble heat rays of a star.
But Mr. Stone fixed at the focus of his telescope a double
thermo-electric pile of which the two parts were reversed
in order. Now any disturbance of temperature which
acted upon both piles uniformly produced no effect
upon the galvanometer needle, and when the rays of the
star were made to fall alternately upon one pile and
the other, the total amount of the deflection represented
double the heating power of the star. Thus Mr. Stone
was able to detect with much certainty a heating effect
of the star Arcturus, which even when concentrated
by the telescope amounted only to ro^h of a degree

0 Gauss, Taylor's ' Scientific Memoirs,' vol. ii. p. 43, &c.

THE METHOD OF MEAN'S. 431

Fahrenheit, and which represents a heating effect of the
direct ray of only about o°'OOOOOi37 Fahrenheit, equiva
lent to the heat which would be received from a three-
inch cubic vessel full of boiling water at the distance of
400 yards P. It is probable that Mr. Stone's arrangement
of the pile might be usefully employed in other delicate
thermometric experiments subject to considerable disturb
ing influences.

Determination of Maximum Points.

We employ the method of means in a certain number
of observations directed to determine the moment at which
a phenomenon reaches its highest point in quantity. In
noting the place of a fixed star at a given time there is
no difficulty in ascertaining the point to be observed, for a
star in a good telescope presents an exceedingly small disc.
In observing a nebulous body which from a bright centre
fades gradually away on all sides, it will not be possible
to select with certainty the middle point. In many such
cases the best method is not to select arbitrarily the sup
posed middle point, but points of equal brightness on
either side, and then take the mean of the observations of
these two points for the centre. As a general rule, a
variable quantity in reaching its maximum increases at a
less and less rate, and after passing the highest point be
gins to decrease by insensible degrees. The maximum may
indeed be defined as that point at which the increase or
decrease is insensibly small. Hence it will usually be the
most indefinite point in the whole course, and if we can
accurately measure the phenomenon we shall best deter
mine the place of the maximum by determining points on
either side at which the ordinates are equal. There is

P ' Proceedings of the Royal Society,' vol. xviii, p. 159 (Jan. 13, 1870).
'Philosophical Magazine' (4th Series), vol. xxxix. p. 376.

THE PRINCIPLES OF SCIENCE.

moreover this advantage in the method that several points
may be determined with the corresponding ones on the
other side, and the mean of the whole taken as the true
place of the maximum. But this method entirely depends
upon the existence of symmetry in the curve, so that of
two equal ordinates one shall be as far on one side of the
maximum as the other is on the other side. The method
fails when other laws of variation prevail.

In tidal observations great difficulty is encountered in
fixing the moment of high water, because the rate at
which the water is then rising or falling is almost imper
ceptible. Dr. Whewell proposed, therefore, to note the
time at which the water passes a fixed point somewhat
below the maximum both in rising and falling, and take
the mean time as that of high water. But this mode of
proceeding unfortunately does riot give a correct result,
because the tide follows different laws in rising and in
falling. There is a difficulty again in selecting the highest
spring tide, another object of much importance in tidology.
Laplace discovered that the tide of the second day pre
ceding the conjunction of the sun and moon is nearly
equal to that of the fifth day following ; and, believing
that the increase and decrease of the tides proceeded in a
nearly symmetrical manner, he decided that the highest
tide would occur about thirty-six hours after the con
junction, that is half-way between the second day before
and the fifth day after <i.

This method is also employed in determining the time
of passage of the middle or densest point of a stream of
meteors. The earth takes two or three days in passing
completely through the November stream ; but astro
nomers need for their calculations to have some definite
point fixed within a few minutes if possible. When near
to the middle they observe the numbers of meteors which

i Airy ' On Tides and Waves,' Encycl. Metrop. pp. 364

THE METHOD OF MEANS. 433

come within the sphere of vision in each half hour or
quarter hour, and then, assuming that the law of varia
tion is symmetrical, they select a moment for the passage
of the whole body equidistant between times of equal
frequency.

The eclipses of Jupiter's satellites are not only of great
interest as regards the motions of the satellites themselves,
but used to be, and perhaps still are, of importance in de
termining longitudes, because they are events occurring
at fixed moments of absolute time, and visible in all parts
of the planetary system at the same time, allowance being
made for the interval occupied by the light in travelling.
But as is excellently explained by Sir John Herschelr, the
moment of the event is wanting in definiteness, partly
because the long cone of Jupiter's shadow is surrounded
by a penumbra, and partly because the satellite has itself
a sensible disc, and takes a certain time in entering the
shadow. Different observers using different telescopes
would usually select different moments for that of the
eclipse. But it is evident that the increase of light in
the emersion will proceed according to a law exactly the
reverse of that observed in the immersion, so that if an
observer notes the time of both events with the same tele
scope, he will be as much too soon in one observation as
he is too late in the other, and the mean moment of the
two observations will represent with considerable accuracy
the time when the satellite is in the middle of the shadow.
The personal error of judgment of the observer is thus
eliminated, provided that he takes care to act at the
emersion as he did at the immersion.

r ' Outlines of Astronomy,' 4th edition, § 538.

Ff

CHAPTER XVII.

THE LAW OF ERROR.

To bring error itself under law might seem beyond
human power. He who errs surely diverges from law,
and it might well be deemed hopeless to suppose that out
of error we can draw truth. One of the most remarkable
achievements of the human intellect is the establishment
of a general theory which not only enables us among dis
crepant results to approximate to the truth, but to assign
the degree of probability which fairly attaches to this con
clusion. It would be a gross misapprehension indeed to
suppose that this law is necessarily the best guide under
all circumstances. Every measuring instrument and every
form of experiment may have its own special law of error ;
there may in one instrument be a tendency in one direc
tion and in another in the opposite direction. Every pro
cess has its peculiar liabilities to mistake and disturbance,
and we are never relieved from the necessity of vigilantly
providing against such special difficulties. The general
Law of Error is the best guide only when we have ex
hausted all other means of approximation, and still find
discrepancies, which are due to entirely unknown causes.
We must treat such residual differences in some way or
other, since they will occur in all accurate experiments,
and as their peculiar nature and origin is assumed to be
unknown, there is no reason why we should treat them
differently in different cases. Accordingly the ultimate
Law of Error must be a uniform and general one.

THE LAW OF ERROR. 435

It is perfectly recognised by mathematicians that in
each special case a special Law of Error may apply, and
should be discovered and adopted if possible. ' Nothing
can be more unlikely than that the errors committed in all
classes of observations should follow the same lawa/ and
the special Laws of Error which will apply to certain in
struments, as for instance the repeating circle, have been
investigated by M. Bravaisb. He concludes that every
partial and distinct cause of error gives rise to a curve of
possibility of errors, which may have any form whatever, —
a curve which we may either be able or unable to discover,
and which in the first case may be determined by con
siderations a priori, on the peculiar nature of this cause,
or which may be determined d posteriori by observation.
Whenever it is practicable and worth the labour, we ought
to investigate these special conditions of error ; never
theless, when there are a great number of different sources
of minute error, the general resultant will always tend to
obey that general law which we are about to consider.

Establishment of the Law of Error.

Mathematicians agree far better as to the nature of the
ultimate Law of Error than they do as to the manner in
which it can be deduced and proved. They agree that
among a number of discrepant results of observation, that
mean quantity is probably the most nearly approximate
to the truth which makes the sum of the squares of the
errors as small as possible. But there are at least three
different ways in which this principle has been arrived at
respectively by Gauss, by Laplace, by Quetelet and by
Sir John Herschel. Gauss proceeds much upon assump-

a 'Philosophical Magazine,' 3rd Series, vol. xxxvii. p. 324.
b ' Letters on the Theory of Probabilities,' by Quetelet, transl. by 0. G.
Dowries, Notes to Letter XXVI. pp. 286-295.

F f 2

436 THE PRINCIPLES OF SCIENCE.

tion ; Herschel rests upon geometrical considerations ; while
Laplace and Quetelet regard the Law of Error as a de
velopment of the doctrine of combinations ; that of Gauss
may be first noticed.

The Law of Error expresses the comparative probability
of errors of various magnitude, and partly from ex
perience, partly from a priori considerations, we may
readily lay down certain conditions to which the law will
certainly conform. It may fairly be assumed as a first
principle to guide us in the selection of the law, that large
errors will be far less frequent and probable than small
ones. We know that very large errors are almost im
possible, so that the probability must rapidly decrease as
the amount of the error increases. A second principle is
that positive and negative errors shall be equally pro
bable, which may certainly be assumed, because we are
supposed to be devoid of any knowledge as to the causes
of the residual errors. It follows that the probability of
the error must be a function of an even power of the
magnitude, that is of the square, or the fourth po\ver, or
the sixth power, otherwise the probability of the same
amount of error would vary accordingly as the error was
positive or negative. The even powers &2, x4, x6, &c., are
always intrinsically positive, whether x be positive or
negative. There is no d priori reason why one rather
than another of these even powers should be selected.
Gauss himself allows that the fourth or sixth powers would
fulfil the conditions as well as the second0, but in the
absence of any theoretical reasons we should prefer the
second power, because it leads to formulae of great com
parative simplicity. Did the Law of Error necessitate the
use of the higher powers of the error, the complexity of

c ' Me'thode des Moindres Carres.' ' Me'moires sur la Combinaison des
Observations, par Ch. Fr. Gauss. Traduit en Franyais par J. Bertram!,
Paris, 1855, pp. 6, 133, &c.

THE LAW OF ERROR. 437

the necessary calculations would much reduce the utility
of the theory.

By a process of reasoning, which it would be undesirable
to attempt to follow in detail in this place, it is shown
that, under these conditions, the most probable result of
any series of recorded observations is that which makes
the sum of the squares of the errors the least possible.
Let a, b, c, &c., be the results of observation, and x the
quantity selected as the most probable, that is the most
free from unknown errors : then we must determine x so

that (a — x)z + (b — x)z + (c - x)2 + shall be the least

possible quantity. Thus we arrive at the celebrated
Method of Least Squares, as it is usually called, which
appears to have been first distinctly put in practice by
Gauss in 1795, while Legendre first published in 1806 an
account of the process in his work, entitled, ' Nouvelles
Methodes pour la determination des Orbites des Cometes.'
It is worthy of notice, however, that Roger Cotes had
long previously recommended a method of equivalent
nature in his tract, 'Estimatio Erroris in Mixta Mathesid.'

Herschel's Geometrical Proof.

A second method of demonstrating the Principle of
Least Squares was proposed by Sir John Herschel, and
although only applicable to geometrical notions, it is re
markable as showing that from whatever point of view
we regard the subject, the same principle will be detected.
After assuming that some general law must exist, and
that it is subject to the general principles of proba
bility, he supposes that a ball is dropped from a high
point with the intention that it shall strike a given mark
on a horizontal plane. In the absence of any known
causes of deviation it will either strike that mark, or, as

d De Morgan, 'Penny Cyclopaedia,' art. Least Squares.

438 THE PRINCIPLES OF SCIENCE.

is infinitely more probable, diverge from it by an amount
which we must regard as error of unknown origin. Now,
to quote the words of Sir J. Herschel6, ' the probability of
that error is the unknown function of its square, i. e. of
the sum of the squares of its deviations in any two rect
angular directions. Now, the probability of any deviation
depending solely on its magnitude, and not on its direc
tion, it follows that the probability of each of these rect
angular deviations must be the same function of its square.
And since the observed oblique deviation is equivalent to
the two rectangular ones, supposed concurrent, and which
are essentially independent of one another, and is, there
fore, a compound event of which they are the simple in
dependent constituents, therefore its probability will be
the product of their separate probabilities. Thus the
form of our unknown function comes to be determined
from this condition, viz., that the product of such functions
of two independent elements is equal to the same function
of their sum. But it is shown in every work on algebra
that this property is the peculiar characteristic of, and
belongs only to, the exponential or antilogarithmic function.
This, then, is the function of the square of the error, which
expresses the probability of committing that error. That
probability decreases, therefore, in geometrical progression,
as the square of the error increases in arithmetical/

Laplace's and Quetelet's Proof of the Law
of Error.

However much presumption the modes of determining
the Law of Error, already described, may give in favour
of the law usually adopted, it is difficult to feel that the

e ' Edinburgh Eeview,' July 1 850, vol. xcii. p. 1 7. Reprinted ' Essays/
P- 399- This method of demonstration is discussed by Boole, ' Trans
actions of Royal Society of Edinburgh/ vol. xxi. pp. 627-630.

THE LA W OF ERROR.

arguments are satisfactory and conclusive. The law
adopted is chosen rather on the grounds of convenience
and plausibility, than because it can be seen to be the
true and necessary law. We can however approach the
subject from an entirely different point of view, and yet
get to the same result.

Let us assume that a particular observation is subject
to four chances of error, each of which will increase the
result one inch if it occurs. Each of these errors is to be
regarded as an event independent of the rest and we can
therefore assign, by the theory of probability, the com
parative probability and frequency of each conjunction of
errors. From the Arithmetical Triangle (pp. 208, 213) we
learn that the ways of happening are as follows :—

No error at all . . . i way.

Error of i inch . ... 4 ways.

Error of 2 inches . . .6 ways.

Error of 3 inches . . .4 ways.

Error of 4 inches . . i way.

We may infer that the error of two inches is the most
likely to occur, and will occur in the long run in six cases
out of sixteen. Errors of one and three inches will be
equally likely, but will occur less frequently ; while no
error at all, or one of four inches will be a comparatively
rare occurrence. If we now suppose the errors to act as
often in one direction as the other, the effect will be to
alter the average error by the amount of two inches, and
we shall have the following results :—

Negative error of 2 inches . . . i way.
Negative error of i inch . . .4 ways.

No error at all 6 ways.

Positive error of i inch . . .4 ways.
Positive error of 2 inches . . i way.

We may now imagine the number of causes of error

440

THE PRINCIPLES OF SCIENCE.

increased and the amount of each error decreased, and the
arithmetical triangle will always give us the proportional
frequency of the resulting errors. Thus if there be five
positive causes of error and five negative causes, the fol
lowing table shows the comparative numbers of aggregate
errors of various amount which will be the result : —

Direction of Error.

Positive Error.

Negative Error.

Amount of Error.

5. 4. 3. 2, i

O

i, 2, 3, 4, 5

Number of such Errors.

I, IO, 45, I2O, 210

252

210, 120, 45, 10, i

It is plain that from such numbers I can ascertain the
probability of any particular amount of error under the
conditions supposed. Thus the probability of a positive

error of exactly one inch is

1024'

in which fraction the

numerator is the exact number of combinations giving
one inch positive error, and the denominator the whole
number of possible errors of all magnitudes. I can also,
by adding together the appropriate numbers, get the pro
bability of an error not exceeding a certain amount. Thus
the probability of an error of three inches or less, positive
or negative, is a fraction whose numerator is the sum of
45 + 120 + 210+252 + 210+120 + 45, and the denomi
nator, as before, giving the result .

We may see at once that, according to these principles,
the probability of small errors is far greater than of large
ones: thus the odds are 1002 to 22, or more than 45 to i,
that the error will not exceed three inches ; and the odds
are 1022 to 2 against the occurrence of the greatest pos
sible error of five inches. The existence of no error at all
is the most likely event ; but a small error, such as that of
one inch positive, is little less likely.

THE LAW OF ERROR. 441

If any case should arise in which the observer knows
the number and magnitude of the independent errors
which may occur, he ought certainly to calculate from the
Arithmetical Triangle the special Law of Error which would
apply. But the general law, of which we are in search,
is to be used in the dark, when we have no knowledge
whatever of the sources of error. To assume any special
number of causes of error is then an arbitrary proceeding,
and mathematicians have chosen the least arbitrary course
of imagining the existence of an infinite number of in
finitely small errors, just as, in the inverse method of
probabilities, an infinite number of infinitely improbable
hypotheses were submitted to calculation (p. 296).

The reasons in favour of this choice are of several
different kinds.

1. It cannot be denied that there may exist infinitely
numerous causes of error in any act of observation.

2. The resulting law on the hypothesis of a large finite,
or even a moderate finite number of causes of error, does
not appreciably differ from that given by the hypothesis
of infinity.

3. We gain by the hypothesis of infinity a general law
capable of ready calculation, and applicable by uniform
rules to all problems.

4. This law, when tested by comparison with extensive
series of observations, is strikingly verified, as will be
shown in a later section.

When we imagine the existence of any large number of
causes of error, for instance one hundred, the numbers of
combinations become impracticably large, as may be seen
to be the case from a glance at the Arithmetical Triangle
(p. 208), which proceeds only up to the seventeenth line.
M. Quetelet, by suitable abbreviating processes, succeeded
in calculating out a table of probability of errors on the

442

THE PRINCIPLES OF SCIENCE.

hypothesis of one thousand distinct causes f ; but mathe
maticians have generally proceeded on the hypothesis of
infinity, and then, by some of the beautiful devices of
analysis, have substituted a general law of easy treatment.
In mathematical works upon the subject, it is shown that
the standard Law of Error is expressed in the formula

y= Ye-<*\

in which x is the amount of the error, Y the maximum
ordinate of the curve of error, and c a number constant
for each series of observations, and expressing the general
amount of the tendency to error, but varying between
one series of observations and another, while <? is the

peculiar constant, 2*71828 the base of the Naperian

logarithms. To show the close correspondence of this
general law with the special law which might be derived
from the supposition of any moderate number of causes
of error, I have in the accompanying figure drawn a

curved line representing accurately the variation of y
when x in the above formula is taken equal to o, -, i, -, 2,

2 2

&c., positive or negative, the arbitrary quantities Y and c

f ' Letters on the Theory of Probabilities,' Letter XV. and Appendix,
note pp. 256-266.

THE LA W OF ERROR. 443

being both assumed equal to unity, in order to simplify
the calculations. In the same figure are inserted eleven
dots, whose heights above the base line are proportional
to the numbers in the eleventh line of the Arithmetical
Triangle, thus representing the comparative probabilities
of errors of various amounts arising from ten equal causes
of error. It is apparent that the correspondence of the
general and the special Law of Error is almost as close as
can be exhibited in the figure, and the assumption of a
greater number of equal causes of error would render the
correspondence far more close.

It may be explained that the ordinates, for instance
NM, nm, n'mf, represent values of y in the equation ex
pressing the Law of Error. The occurrence of any one
definite amount of error is infinitely improbable, because
an infinite number of such ordinates might be drawn.
But the probability of an error occurring between certain
definite limits is finite, and is represented by a portion
of the area of the curve. Thus the probability that an
error, positive or negative, not exceeding unity will occur,
is represented by the area MmrmW, in short, by the area
standing upon the line nn'. Since every observation
must either have some definite error or none at all, it
follows that the whole area of the curve should be con
sidered as the unit expressing certainty, and the proba
bility of an error falling between particular limits will
then be expressed by the ratio which the area of the
curve between those limits bears to the whole area of
the curve.

Derivation of the Law of Error from Simple
Logical Principles.

It is worthy of notice that this Law of Error, abstruse
though the subject may seem, is really founded upon the
simplest principles. It arises entirely out of the difference

444 . THE PRINCIPLES OF SCIENCE.

between permutations and combinations, a subject upon
which I may seem to have dwelt with unnecessary pro
lixity in previous pages (pp. 200-216). The order in
which we add quantities together does not affect the
amount of the sum, so that if there be three positive
and five negative causes of error in operation, it does not
matter in which order they are considered a.s acting.
They may be indifferently intermixed in any arrange
ment, and yet the result will be the same. The reader
should not fail to notice how laws or principles which
appeared to be absurdly simple and evident when first
noticed, reappear in the most complicated and mysterious
processes of scientific method. The fundamental Laws
of Identity and Difference gave rise to the Logical Abe-
cedarium, which, after abstracting the character of the
differences, led to the Arithmetical Triangle (p. 214).
The Law of Error is defined by an infinitely high line
of that triangle, and the law proves that the mean is the
most probable result, and that divergencies from the
mean become much less probable as they increase in
amount. Now the comparative greatness of the numbers
towards the middle of each line of the Arithmetical
Triangle is entirely due to the indifference of order in
space or time, which was first prominently pointed out
as a condition of logical relations, and the symbols in
dicating them (pp. 40-42), and which was afterwards
shown to attach equally to numerical symbols, the deri
vatives of logical ferms (pp. 180, 181).

Verification of the Law of Error.

The theory of error which we have been considering
rests entirely upon an assumption, namely that when
known sources of disturbances are allowed for, there yet
remain an indefinite, possibly an infinite number of other

THE LA W OF ERROR.

445

minute sources of error, which will as often produce
excess as deficiency. Granting this assumption, the Law
of Error must be as it is usually taken to be, and there is
no more need to verify empirically than to test the truth
of one of Euclid's propositions mechanically, after we have
proved it theoretically. Nevertheless, it is an interesting
occupation to verify even the propositions of geometry in
an approximate manner, and it is still more instructive to
inquire whether a large number of observations will be
found to justify our assumption of the Law of Error.

Encke has given an excellent instance of the cor
respondence of theory with experience, in the case of
certain observations of the difference of Right Ascension
of the sun and two stars, namely a Aquilas and a Canis
minoris. The observations were 470 in number, and were
made by Bradley and reduced by Bessel, who found the
probable error of the final result to be only about one-
fourth part of a second (o/f'26^j). He then compared
the number of errors of each magnitude from — th part of

10

a second upwards, as actually given by the observations,
with what should occur according to the Law of Error.
The results were as follow #: —

Magnitude of the errors in parts

Number of errors of each magnitude
according to

of a second

Observation.

Theory.

O'O to O'l

94

95

'I '2

88

89

•2 -3

78

78

•3 -4

58

64

•4 -5

51

50

•5 '6

36

36

•6 -7

26

24

•7 -8

H

:5

•8 -9

10

9

•9 i-o

7

5

above , I'D

8

5

« Encke, ' On the Method of Least Squares/ Taylor's ' Scientific Me
moirs,' vol. ii. pp. 338, 339.

446

THE PRINCIPLES OF SCIENCE.

The reader will remark that the correspondence is
remarkably close, except as regards larger errors, which
are excessive in practice. It is one objection, indeed, to
the theory of error, that, being expressed in a continuous
mathematical function, it contemplates the possible exist
ence of errors of every magnitude, such indeed as could
not practically occur ; yet in this case the theory seems to
under-estimate the number of large errors.

Another excellent comparison of the law with observa
tion has been made by Quetelet, who has investigated the
errors of 487 determinations in time of the Right Ascen-

o

sion of the Pole-star, made at Greenwich during the four
years 1836-39. These observations, although carefully
corrected for all known causes of error, as well as for
nutation, precession, &c., are yet of course found to differ,
and being classified as regards intervals of one-half second
of time, and then proportionately increased in number, so
that their sum may be one thousand, give the following
results as compared with what theory would lead us to
expect h :—

Magnitude of

Number of errors

Magnitude of

Number of errors

error in tenths
of a second.

by

Observation.

by

Theory.

error in tenths
of a second.

by t

Observation.

by

Theory.

O'O

1 68

163

—

—

—

+ 0-5 148

H7

-o'S

15°

152

+ ro 129

112

— I'O

126

121

+ 1-5 78

72

-i'5

74

82

+ 2-0 33

40

— 2'O

43

46

+ 2-5

10

'9

-2'5

25

22

+ 3-0

2

10

_3-0

12

IO

— 1 —

—

-3'5

2

4

In this instance the correspondence is also satisfactory,
but the divergence between theory and fact is in the
opposite direction to that discovered in the former com-

11 Quetelet, ' Letters on tlie Theory of Probabilities,' translated by
Dowries, Letter XIX. p. 88. See also Galton's ' Hereditary Genius,'
P- 379-

THE LA W OF ERROR. 447

parison, the larger errors being less frequent than theory
would indicate.

We may also regard the experiments enumerated in
the chapter on Probabilities (p. 238), as forming an em
pirical verification of the theory of error.

Remarks on the General Law of Error.

The mere fact that the Law of Error allows of the
possible existence of errors of every assignable amount
shows that it is only approximately true. We may fairly
say that in measuring a mile it would be impossible to
commit an error of a hundred miles, and the length of life
would never allow of our committing an error of one
million miles. Nevertheless the general Law of Error
would assign a probability for an error of that amount or
more, but so small a probability as to ba utterly incon
siderable, and almost inconceivable. All that can, or in fact
need, be said in defence of the law is, that it may be made
to represent the errors in any special case to a very close
approximation, and that the probability of large and prac
tically impossible errors, as given by the law, will be so
small as to be entirely inconsiderable. And as we are
dealing with error itself, and our results pretend to no
thing more than approximation and probability, an in
definitely small error in our process of approximation is
of no importance whatever.

The Probable Mean Result as defined ~by the Law
of Error.

One immediate result of the Law of Error, as thus stated,
is that the mean result is the most probable one ; and
when there is only a single variable this mean is found by
the familiar arithmetical process. An unfortunate error

448 THE PRINCIPLES OF SCIENCE.

has crept into several works which allude to this subject.
Mr. Mill, in treating of the ' Elimination of Chance/ re
marks in a note l that ' the mean is spoken of as if it were
exactly the same thing with the average. But the mean,
for purposes of inductive inquiry, is not the average, or
arithmetical mean, though in a familiar illustration of the
theory the difference may be disregarded.' He goes on to
say that, according to mathematical principles, the most
probable result is that for which the sums of the squares
of the deviations is the least possible. In Bowen's ' Treatise
on Logic' (p. 439), we find the Method of Least Squares
mentioned as ' a mode of finding the most probable result
in those cases in which the arithmetical mean is not an
applicable expedient for determining the probability.' It
seems probable that these and other writers were misled
by Dr. Whe well's remarks on the subject ; for he saysk
that ' The Method of Least Squares is in fact a Method of
Means, but with some peculiar characters. . . . The
method proceeds upon this supposition ; that all errors
are not equally probable, but that small errors are more
probable than large ones.' He adds that this method
* removes much that is arbitrary in the Method of Means.'
It is strange to find a mathematician like Dr. Whewell
making such remarks, when there is no doubt whatever
that the Method of Means is only an application of the
Method of Least Squares. They are, in fact, the same
method, except that the latter method may be applied to
cases where two or more quantities have to be determined
at the same time. Many authorities might be quoted to
this effect, but it will be sufficient to mention Lubbock
and Drink water, who say ', ' If only one quantity has to be

» 'System of Logic,' bk. iii. chap. 17, § 3. 5th ed. vol. ii. p. 56.
k ' Philosophy of the Inductive Sciences/ 2nd ed. vol. ii. pp. 408, 409.
1 ' Essay on Probability,' by J. W. Lubbock and J. E. Drink\vat er,
Useful Knowledge Society, 1833, P- 41-

THE LA W OF ERROR.

determined, this method evidently resolves itself into
taking the mean of all the values given by observation.'
Encke, again, distinctly says m, that the expression for the
probability of an error 'not only contains in itself the
principle of the arithmetical mean, but depends so imme
diately upon it, that for all those magnitudes for which
the arithmetical mean holds good in the simple cases in
which it is principally applied, no other law of proba
bility can be assumed than that which is expressed by
this formula/

It can be shown, too, in a moment that the mean is the
result which gives the least sum of squares of errors.
For if a, b, c, &c., be the results of observation and x the
selected mean result, the sum of squares of the errors is
(a— x)2 + (b — x)2 + (c — x)2 + &c., which is at a minimum
when its differential coefficient 2 (a — x + b — x + c— x -\-
&c.) = o. From this equation we immediately obtain, de
noting by n the number of separate results, a, b, c, &c.,
x = (a + b + c + . . . ) -, or the ordinary arithmetic mean.

Weighted Observations.

It is to be distinctly understood that when we take the
mean of certain numerical results as the most probable
number aimed at, we regard all the different results as
equally good and probable in themselves. The theory
of error expresses no preference for any one number over
any other. If, then, an observer has reason to suppose
that some results are not so trustworthy as others, he
must take account of this difference in drawing the mean.
By the method of weighting observations this difference of
value is easily allowed for. Astronomers are in the habit

m Taylor's ' Scientific Memoirs,' vol. ii. p. 333.
G g

450 THE PRINCIPLES OF SCIENCE.

of recording with an observation the value or degree of
confidence with which they regard it, freely estimated
according to the impression of success or failure in accuracy
immediately after the observation. This value is usually
expressed in a decimal scale, so that 10 denotes the
highest degree of satisfaction with the result, and I the
least degree. Before taking the mean of the observations
each number is multiplied by its weight or value, and the
sum of the products is divided by the sum of the weights.
Thus if a, b, c, &c., be the observed numbers, arid w, w , iv",
&c., the weights, then the most probable mean is

cw" + . . . . ,„, . „ , . .,, , , .

71 — TT-T —' -Lhis formula, it will be observed, is
+w + . . . .

identical in form with that for finding the centre of
gravity of particles of different weights arranged in a
straight line. When we regard w, w', ID" , &c., as all equal,
it becomes identical with the formula for the ordinary
mean. This method of weighting observations, now of
much importance in astronomical and other very exactly
quantitative investigations, appears to have been first pro
posed by Roger Cotes, the editor of the ' Principia/ as
pointed out «by De Morgan11.

The practice of giving weights would open the way to
much error and abuse, if the weights were assigned when
the mean was being drawn, and when the divergence of
some results from the others would be likely to become
the guide. As a general rule the weights must be as
signed at the moment of observation, and afterwards
rigidly maintained, and they must be assigned not from
regard to the apparent intrinsic accuracy of the result,
but the extrinsic circumstances which seem to render it
valuable. . An observed result, in short, must be discre
dited, not because it is divergent, but because there were
other reasons to suppose that it would be divergent.

n ' Penny Cyclopsedia,' art. Least Squares,

THE LA W OF Eli ROB. 451

The Probable Error of Mean Results.

When we draw any conclusion from the numerical
results of observations we ought not to consider it suf
ficient, in cases of importance, to content ourselves with
finding the simple mean and treating it as true. We
ought also to ascertain what is the degree of confidence
we may place in this mean, and our confidence should be
measured by the degree of concurrence of the observations
from which it is derived. In some cases the mean may
be so close to the correct result that we may consider it
as approximately certain and accurate. In other cases it
may really be worth little or nothing. The Law of Error
enables us to give exact expression to the degree of con
fidence proper in any case ; for it shows how to calculate
the probability of a divergence of any amount from the
mean, and we can thence ascertain the probability that
the mean in question is within a certain distance from the
true number. The probable error is taken by mathema
ticians to mean the limits within which it is as likely as
not that the truth will fall. Thus if 5 -4 5 be the mean of
all the determinations of the density of the earth, and -20
be approximately the probable error, the meaning is that
the probability of the real density of the earth falling be
tween 5*25 and 5-65 is ^. Any other limits might have
been selected at will. We might readily calculate the limits
within which it was one hundred or one thousand to one
that the truth would fall ; but there is a general conven
tion to take the even odds, one to one, as the quantity of
probability of which the limits are to be estimated.

Many books on the subject of probability give rules for
making the calculations, but as, in the gradual progress of
science, all persons ought to be more familiar with these
processes, I propose to repeat the rules here and illustrate
their use. The calculations, when made in strict accordance
with the directions, involve none but arithmetic operation?.

452 THE PRINCIPLES OF SCIENCE.

Rules for finding the probable error of a mean result :—

1 . Draw the mean of all the observed results.

2. Find the excess or defect, that is, the error of each
result from the mean.

3. Square each of these reputed errors.

4. Add together all these squares of the errors.

5. Take the square root of this sum.

6. Divide the square root by the number of results.

7. Multiply the quotient by 0*67449 (or approxi
mately by 0*674, or even o-67), a natural constant
number derived from the Law of Error in a manner
which is described in mathematical works upon the
subject.

Suppose, for instance, that five measurements of the
height of a hill, by the barometer or otherwise, have given
the numbers of feet as 293, 301, 306, 307, 313 ; we want
to know the probable error of the mean, namely 304. Now
the differences between this mean and the above numbers,
paying no regard to direction, are n, 3, 2, 3, 9; their
squares are 121, 9, 4, 9, 81, and the sum of the squares
consequently 224. Taking the square root of this sum by
the common arithmetic process, or by logarithms, we ob
tain 14*966, and dividing by five, the number of observa
tions, we have 2*99, which has only to be multiplied by
•67 to yield us 2*019. This number is so close to 2, that
we may call the probable error equal to two. Thus the
probability is one-half, or the odds are even, that the true
height of the mountain lies between 302 and 306 feet.
We have thus an exact measure of the degree of credibility
of our mean result, which mean indicates the most likely
point for the truth to fall upon.

The reader should observe that as the object in these
calculations is only to gain a notion of the degree of con
fidence with which we view the mean, there is no real
use in carrying the calculations to any great degree of

THE LAW OF ERROR. 453

precision ; and whenever the neglecting of decimal frac
tions, or even the slight alteration of a number will much
abbreviate the computations, it may be fearlessly done, ex
cept in cases of high importance and precision. It has been
stated that the voyages of the Great Britain steamship to
Melbourne from Liverpool, up to May, 1871, have been
thirteen in number, with the following durations in days :
62, 63, 59, 60, 58, 61, 57, 57, 57, 57, 56, 63, 55. The
mean duration of the voyages is 58*85 days, which is the
most probable length of any similar future voyage ; but
to calculate the probable error, we may take the mean to
be 59 days. The sum of the squares of the errors is only
88, and the probable error thence calculated 0*49 day, or,
say half a day. It is as likely as not, then, that any par
ticular voyage will be not less than 58 J days, nor more
than 59^ days.

The experiments of Benzenberg to detect the revolution
of the earth, by the deviation of a ball from the exact
perpendicular line in falling down a deep pit, have been
cited by Encke0 as an interesting illustration of the Law
of Error. The mean deviation was 5*086 lines, and its
probable error was calculated by Encke to be not more
than '950 line, that is, the odds were even that the true
result lay between 4- 136 and 6'O36. As the deviation
should, according to astronomical theory be 4*6 lines,
which lies well within the limits, we may consider that
the experiments are consistent with the Copernican system
of the universe.

It will of course be understood that the probable error
has regard only to the differences of the results from
which the mean is drawn, and takes no account of con
stant errors. The true result accordingly will often fall
far beyond the limits of probable error.

0 Taylor's 'Scientific Memoirs,' vol. ii. pp. 330, 347, &c.

454 THE PRINCIPLES OF SCIENCE.

The Rejection of the Mean Result.

We ought always to bear in mind that the mean of any
series of observations is the best, that is, the most probable
approximation to the truth, only in the entire absence of
any knowledge to the contrary. The selection of the
mean rests entirely upon the probability that wholly un
known causes of error will in the long run fall as often in
one direction as the opposite, so that in drawing the mean
they will balance each other. If we have any presumption
to the contrary, any reason to suppose that there exists a
tendency to error in one direction rather than the other,
then to choose the mean would be to ignore that tendency.
Thus we may certainly approximate to the length of the
circumference of a circle, by measuring the perimeters of
inscribed and circumscribed polygons of an equal and large
number of sides. The correct length of the circular line
undoubtedly lies between the lengths of the two perimeters,
but it does not follow that the mean is the best approxi
mation. It may in fact be shown upon mathematical
principles that the circumference of the circle is very
nearly equal to the perimeter of the inscribed polygon,
together with one-third part of the difference between
the inscribed and circumscribed polygons of the same
number of sides. Having this knowledge we ought of
course to act upon it, instead of upon vague grounds of
probability.

We may often perceive that a series of measurements
tends towards an extreme limit rather than towards a
mean. Thus in endeavouring to obtain a correct estimate
of the apparent diameter of the brightest fixed stars, we
should find a continuous diminution in estimates as the
powers of observation increased. Kepler assigned to
Sirius an apparent diameter of 240 seconds ; Tycho Brahe
made it 126; Gassendi 10 seconds; Galileo, Hevelius,

THE LAW OF ERROR. 455

and J. Cassini, 5 or 6 seconds. Halley, Michell, and
subsequently Sir W. Herschel came to the conclusion
that the brightest stars in the heavens could not have
real discs of a second, and were probably much less in
diameter. It would of course be absurd to take the mean
of quantities which differ more than 240 times ; and as
the tendency has always been to smaller estimates, there
is a considerable indication in favour of the smallest?.

In the case of many experiments and measurements we
shall know on which side there is a tendency to error.
Thus the readings of a thermometer always tend to rise as
the age of the instrument increases, and no drawing of
means will correct this result. "Barometers, on the other
hand, are always likely to read too low instead of too high,
owing to the imperfection of the vacuum, or the action of
capillary attraction. If the mercury be perfectly pure and
no considerable error be due to the measuring apparatus,
the best barometer will be that which gives the highest
result.

When we have reasonable grounds for supposing that
certain experimental results are liable to grave errors, we
should exclude them in drawing a mean. If we want to
find the most probable approximation to the velocity of
sound in air, it would be absurd to go back to the old
experiments which made the velocity from 1200 to 1474
feet per second ; for we know that the old observers did
not guard against errors arising from wind and other
causes. Old chemical experiments are absolutely valueless
as regards quantitative results. The old chemists found
the atmosphere to differ in composition nearly ten per
cent, in different places, whereas modern accurate experi
menters find very slight variations. Any method of
measurement which we know to avoid a source of error
is far to be preferred to others which trust to probabilities

r Quetelet, 'Letters/ &c. p. 116.

456 THE PRINCIPLES OF SCIENCE.

for the elimination of the error. As Flamsteed says % ' OIK
good instrument is of as much worth as a hundred in
different ones.' But an instrument is good or bad only ir,
a comparative sense, and no instrument gives invariable
and truthful results. Hence we must always ultimately
fall back upon general probabilities for the selection of the
final mean, when our other precautions are exhausted.

Very difficult questions sometimes arise when one or
more results of a method of experiment diverge widely
from the mean of the rest. Are we or are we not to ex
clude them in adopting the supposed true mean result of
the method. The drawing of a mean result rests, as I
have frequently explained, upon the assumption that every
error acting in one direction will probably be balanced by
other errors acting in an opposite direction. If then we
know or can possibly discover any causes of error not
agreeing with this assumption, we shall be justified in
excluding results which seem to be affected by this cause.

In reducing large series of astronomical observations, it is
not uncommon to meet with numbers differing from others
by a whole degree or half a degree, or some considerable in
tegral quantity. These are errors which could hardly arise
in the act of observation or in instrumental irregularity ;
but they might readily be accounted for by misreading
of figures or mistaking of division marks. It would be
absurd to trust to chance that such mistakes would
balance each other in the long run, and it is therefore
better to correct arbitrarily the supposed mistake, or
better still, if new observations can be made, to strike
out the divergent numbers altogether. When results
come sometimes too great or too small in a regular
manner, we should suspect that some part of the instru
ment slips through a definite space, or that a definite
cause of error enters at times, and not at others. We
<i Baily, ' Account of Flamstced/ p. 56.

THE LAW OF ERROR. 457

should then make it a point of prime importance to dis
cover the exact nature and amount of such an error, and
either prevent its occurrence for the future or else intro
duce a corresponding correction. In many researches the
whole difficulty will consist in this detection and avoidance
of sources of error. Thus Professor Koscoe found that the
presence of phosphorus caused serious and almost una
voidable errors in the determination of the atomic weight
of vanadium1". Sir John Herschel, in reducing his obser
vations of double stars at the Cape of Good Hope, was
perplexed by an unaccountable difference of the angles of
position as measured by the Seven-feet Equatorial and
the Twenty-feet Eeflector Telescopes, and after a careful
investigation was obliged to be contented with introducing
a correction experimentally determined8.

Even the most patient and exhaustive investigations
will sometimes fail to disclose any reason why some results
diverge in an unusual and unexpected manner from others.
The question again recurs — Are we arbitrarily to exclude
them "? The answer should be in the negative as a general
rule. The mere fact of divergence ought not to be taken
as conclusive against a result, and the exertion of arbitrary
choice would open the way to the most fatal influence of
bias, and what is commonly known as the 'cooking' of
figures. It would amount in most cases to judging fact
by theory instead of theory by fact. The apparently
divergent number may even prove in time to be the true
one. It may be an exception of that peculiarly valuable
kind which upsets our false theories, a real exception, ex
ploding apparent coincidences, arid opening the way to a
wholly new view of the subject. To establish this position
for the divergent fact will of course require additional re
search ; but in the meantime we should give it a fair

r Bakerian Lecture, ' Philosophical Transactions' (1868), vol. clviii. p. 6.
8 ' Results of Observations at the Cape of Good Hope/ p. 283.

458 THE PRINCIPLES OF SCIENCE.

weight in our mean conclusions, and should bear in mind
the discrepancy as one demanding attention. To neglect
a divergent result is to neglect the possible clue to a great
discovery.

Method of Least Squares.

When two or more unknown quantities are so involved
that they cannot be separately determined by the single
Method of Means, we can yet obtain their most probable
amounts by the Method of Least Squares, without more
difficulty than arises from the length of the arithmetical
computations. If the result of each observation gives an
equation between two unknown quantities of the form

ax + l>y = c

then, if the observations were free from error, we should
only need two observations giving two equations ; but,
for the attainment of greater accuracy, we may take a
series of observations, and then reduce the equations so
as to give only a pair with average coefficients. This re
daction is effected by, firstly, multiplying the coefficients
of each equation by the first coefficient, and adding to
gether all the similar coefficients thus resulting for the
coefficients of a new equation ; and secondly, by repeating
this process, and multiplying the coefficients of each equa
tion by the coefficient of the second term. Thus meaning
by (sum of a2) the sum of all quantities of the same kind,
and having the same place in the equations as a2, we
may briefly describe the two resulting mean equations
as follows :—

(sum of a"'} . x + (sum of «?>) . y = (sum of ac),
(sum of ctb] . x + (sum of &2) . y — (sum of be].

When there are three or more unknown quantities the
process is exactly the same in nature, and we only need
additional mean equations to be obtained by multiply
ing by the third, fourth, &c., coefficients. As the numbers

THE LAW OF ERROR. 459

are in any case only approximate, it is usually quite un
necessary to make the computations with any great
degree of accuracy, and places of decimals may therefore
oe freely cut oif to save arithmetical work. The mean
equations having been computed, their solution by the
ordinary methods of algebra gives the most probable
values of the unknown quantities.

Works upon the Theory of Probability and the Law

of Error.

Regarding the Theory of Probability and the Law of
Error as constituting, perhaps, the most important subjects
of study for any one who desires to obtain a complete
comprehension of logical and scientific method as actually
applied in physical investigations, I will briefly indicate
the works in one or other of which the reader will best
pursue the study.

The best popular, and at the same time profound
English work on the subject is De Morgan's ' Essay on
Probabilities and on their Application to Life Contin
gencies and Insurance Offices/ published in the ' Cabinet
Cyclopedia/ and to be obtained from Messrs. Longman.
No mathematical knowledge beyond that of common
arithmetic is required in reading this work. Quetelet's
' Letters/ already often referred to, also form a most inter
esting and excellent popular introduction to the subject,
and the mathematical notes are also of value. Sir George
Airv's brief treatise ' On the Algebraical and Numerical

•/ O

Theory of Errors of Observation and the Combination of
Observations/ contains a complete explanation of the Law of
Error and its practical applications. De Morgan's treatise
' On the Theory of Probabilities' in the ' Encyclopaedia Me-
tropolitana/ presents an abstract of the more abstruse in
vestigations of Laplace, together with a multitude of pro-

460 THE PRINCIPLES OF SCIENCE.

found and original remarks concerning the theory generally.
In Lubbock and Drinkwater's work on ' Probability,' in the
Library of Useful Knowledge, we have a very concise but
good statement of a number of important problems. The
Rev. W. A. Whitworth has given, in an interesting little
work entitled ' Choice and Chance,' a number of good illus
trations of the calculations both in the theories of Com
binations and Probabilities. In Mr. Todhunter's admirable
History we have an exhaustive critical account of almost all
writings upon the subject of probability down to the cul
mination of the theory in Laplace's works. In spite of the
existence of these and some other good English works, there
seems to be a want of an easy and yet pretty complete
introduction to the study of the theory of probabilities.

Among French works the ' Traite Elementaire du Cal-
cul des Probabilites/ by S. F. Lacroix, of which several
editions have been published, and which is not difficult
to obtain, forms probably the best elementary treatise.
Poisson's ' Eecherches sur la Probabilite des Jugements,'
(Paris, 1837), commences with an admirable investigation
of the grounds and methods of the theory. While La
place's great ' Theorie Analytique des Probabilites ' is of
course the ' Principia' of the subject, his 'Essai Philo-
sophique sur les Probabilites ' is a popular discourse, and
is one of the most profound and interesting essays ever
published. It should be familiar to every student of
logical method, and has lost little or none of its import
ance by lapse of time.

Detection of Constant Errors.

The Method of Means is absolutely incapable of elimi
nating any error which is always the same, and which
always lies in one direction. We sometimes require to be
aroused from a false feeling of security, and to be urged

THE LAW OF ERROR. 461

to take suitable precautions against such occult errors.
' It is to the observer/ says Gauss*, 'that belongs the task
of carefully removing the causes of constant errors,' and
this is quite true when the error is absolutely constant.
When we have made a number of determinations with a
certain apparatus or method of measurement, there is a
great advantage in altering the arrangement, or even
devising some entirely different method of getting esti
mates of the same quantity. The reason obviously con
sists in the improbability that exactly the same constant
error will affect two or more different methods of experi
ment. If a discrepancy is found to exist, we shall at
least be aware of the existence of error, and can take
measures for finding in which way it lies. If we can try
a considerable number of methods, the probability becomes
considerable that errors constant in one method will be
balanced or nearly so by errors of an opposite effect in the
others. Suppose that there be three different methods
each affected by an error of equal amount. The pro
bability that this error wiU in all fall in the same direction
is only ^ ; and with four methods similarly f . If each
method be affected, as is always the case by several inde
pendent sources of error, the probability becomes very
great that in the mean result of all the methods some of
the errors will partially compensate the others. In this case,
as in all others, when human foresight and vigilance has
exhausted itself, we must trust the theory of probability.
In the determination of a zero point, of the magnitude
of the fundamental standards of time and space, in the
personal equation of an astronomical observer, we have
instances of such fixed errors ; but as a general rule a
change of procedure is likely to reverse the character of
the error, and many instances may be given of the value
of this precaution.

t Gauss, translated by Bertram!, p. 25.

4 02 THE PRINCIPLES OF SCIENCE.

If we measure over and over again the same angular
magnitude by the same divided circle, maintained in
exactly the same position, it is evident that the same
mark in the circle will be the criterion in each case, and
any error in the position of that mark will equally affect
all our results. But if in each measurement we use a
different part of the circle, a new mark will come into use,
and as the error of each mark can hardly be in the same
direction, the average result will be nearly free from
errors of division. It will be still better to use more
than one divided circle.

Even when we have no clear perception of the points of
our apparatus at which fixed error is likely to enter, we
may with advantage vary the construction of our appa
ratus with the hope that we shall accidentally detect
some latent imperfection. Baily's purpose in repeating
the experiments of Michell and Cavendish on the density
of the earth, was not merely to follow the same course
and verify the previous numbers, but to try whether
variations in the size and substance of the attracting
balls, the mode of suspension, the temperature of the sur
rounding air, &c., would yield different results. He per
formed no less than 62 distinct series, comprising 2153
experiments, and he carefully classified and discussed the
results so as to disclose the utmost differences. Again, in
experimenting upon the resistance of the air to the mo
tion of a pendulum, Baily employed no less than 80
pendulums of various forms and materials, in order to
ascertain exactly upon what conditions the resistance de
pends. Regnault, in his exact researches upon the dilata
tion of gases made arbitrary changes in the magnitude of
parts of his apparatus. He thinks that if, in spite of such
modification the results are unchanged, the errors are
probably of inconsiderable amount"; but in reality it is

11 Jamin, ' Cours cle Physique,' vol. ii. p. 60.

THE LA W OF ERROR. 463

always possible, and usually likely, that we overlook
sources of error which a future generation will detect.
Thus the pendulum experiments of Baily and Sabine
were directed to ascertain the nature and amount of a
correction for air resistance, which had been entirely mis
understood in the experiments upon which was founded
the definition of the standard yard, by means of the
seconds pendulum in the Act of 5th George IV. c. 74.
It has already been mentioned that a considerable error
was discovered in the determination of the standard
metre as the ten-millionth part of the distance from the
pole to the equator (p. 368).

We shall return in the second volume to the further
consideration of the methods by which we may as far as
possible secure ourselves against permanent and undetected
sources of error. In the meantime, having completed the
consideration of the special methods requisite for treating
quantitative phenomena, we must return to our principal
subject, and endeavour to trace out the course by which
the physicist, from observation and experiment, collects
the materials of natural knowledge, and then proceeds
by hypothesis and inverse calculation to educe from them
the laws of nature.

END OF THE FIRST VOLUME.

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