Full text of "The philosophy of the inductive sciences, founded upon their history"

GIFT OF
ASTRONOMICAL SOCIETY OF THE

PAHTFTfi

BR AR Y

THE

PHILOSOPHY

OF THE

INDUCTIVE SCIENCES,

FOUNDED UPON THEIR HISTORY.

BY WILLIAM WEEWELL, D.D,,

MASTER OF TRINITY COLLEGE, CAMBRIDGE.

A NEW EDITION,

WITH CORRECTIONS AND ADDITIONS, AND
AN APPENDIX, CONTAINING

PHILOSOPHICAL ESSAYS PREVIOUSLY PUBLISHED.

IN TWO VOLUMES.

Aau7ra8ta e^ciTe? $ia8a&gt;crovcriv aXXr/Xois.

VOLUME THE FIEST,

LONDON:
JOHN W. PARKER, WEST STRAND.

M.DCCC.XLVII.

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W4?
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Attron. a-o

ASTRONOMY

, 00-

REV. ADAM SEDGWICK, M.A.,

SENIOR FELLOW OF TRINITY COLLEGE,

WOODWARDIAN PROFESSOR OF GEOLOGY IN THE UNIVERSITY OF
CAMBRIDGE, AND PREBENDARY OF NORWICH.

MY DEAR SEDGWICK,

WHEN I showed you the last sheet of my History of the In
ductive Sciences in its transit through the press, you told me that
I ought to add a paragraph or two at the end, by way of Moral
to the story ; and I replied that the Moral would be as long as
the story itself. The present work, the Moral which you then
desired, I have, with some effort, reduced within a somewhat
smaller compass than I then spoke of ; and I cannot dedicate it
to any one with so much pleasure as to you.

It has always been my wish that, as far and as long as men
might know anything of me by my writings, they should hear of me
along with the friends with whom I have lived, whom I have loved,
and by whose conversation I have been animated to hope that I
too might add something to the literature of our country. There
is no one whose name has, on such grounds, a better claim than
yours to stand in the front of a work, which has been the subject
of my labours for no small portion of our long period of friend
ship. But there is another reason which gives a peculiar pro
priety to this dedication of my Philosophy to you. I have little
doubt that if your life had not been absorbed in struggling
with many of the most difficult problems of a difficult science,
you would have been my fellow-labourer or master in the work
which I have here undertaken. The same spirit which dictated
your vigorous protest against some of the errours which I also
attempt to expose, would have led you, if your thoughts had been

701543

iv DEDICATION.

more free, to take a leading share in that Reform of Philosophy,
which all who are alive to such errours, must see to be now in
dispensable. To you I may most justly inscribe a work which
contains a criticism of the fallacies of the ultra-Lockian school.

I will mention one other reason which enters into the satisfac
tion with which I place your name at the head of my Philosophy.
By doing so, I may consider myself as dedicating it to the College
to which we both belong, to which we both owe so much of all
that we are, and in which we have lived together so long and so
happily; and that, be it remembered, the College of Bacon and of
Newton. That College, I know, holds a strong place in your affec
tions, as in mine ; and among many reasons, not least on this
account ; we believe that sound and enduring philosophy ever
finds there a congenial soil and a fostering shelter. If the doc
trines which the present work contains be really true and valu
able, my unhesitating trust is, that they will spread gradually
from these precincts to every part of the land.

That this office of being the fosterer and diffuser of truth may
ever belong to our common Nursing Mother, and that you, my
dear Sedgwick, may long witness and contribute to these bene
ficial influences, is the hearty wish of

Yours affectionately,

W. WHEWELL.
Trinity College, May 1. 1840.

PREFACE

TO THE

SECOND EDITION.

IN the Preface to the first edition of this work, it was
stated that the work was intended as an application of
the plan of Bacon s Novum Organon to the present con
dition of Physical Science. Such an undertaking, it was
there said, plainly belongs to the present generation.
Bacon only divined how sciences might be constructed ;
we can trace, in their history, how their construction
has taken place. However sagacious were his conjec
tures, it may be expected that they will be further illus
trated by facts which we know to have really occurred.
However large were his anticipations, the actual progress
of science since his time may aid in giving comprehen
siveness to our views. And with respect to the methods
by which science is to be promoted, the structure and
operation of the Organ by which truth is to be collected
from nature, we know that, though Bacon s general
maxims still guide and animate philosophical enquirers
yet that his views, in their detail, have all turned out
inapplicable : the technical parts of his method failed in
his hands, and are forgotten among the cultivators of
science. It cannot be an unfit task, at the present day,
to endeavour to extract from the actual past progress
of science, the elements of a more effectual and sub-

VI PREFACE TO

stantial Method of Discovery. The advances which
have, during the last three centuries, been made in the
physical sciences; in Astronomy, in Physics, in Che
mistry, in Natural History, in Physiology ; these are
allowed by all to be real, to be great, to be striking :
may it not be, then, that these steps of progress have
in them something alike? that in each advancing move
ment there is some common process, some common prin
ciple? that the organ by which discoveries have been
made has had something uniform in its structure and
working ? If this be so, and if we can, by attending to
the past history of science, discover something of this
common element and common process in all discoveries,
we shall have a Philosophy of Science, such as our times
may naturally hope for : we shall have the New Organ
of Bacon, renovated according to our advanced intellec
tual position and office.

It was with the view to such a continuation and
extension of Bacon s design, that I undertook that sur
vey of the History of Science which I have given in
another work ; and that analysis of the advance of each
science which the present work contains. Of the doc
trines promulgated by Bacon, none has more completely
remained with us, as a stable and valuable truth, than
his declaration that true knowledge is to be obtained
from Facts by Induction : and in order to denote that I
start at once from the point to which Bacon thus led us,
I have, both in the History and in the Philosophy, termed
the sciences with which I have to do, the Inductive Sci
ences. By treating of the Physical Sciences only, while
I speak of the Inductive Sciences in the description of

THE SECOND EDITION. Vll

my design, I do not, (as I have already elsewhere said"*)
intend to deny the character of Inductive Sciences to
many other branches of knowledge, as for instance, Eth
nology, Glossology, Political Economy, and Psychology.
But I think it will be allowed that by taking, as I have
done, the Physical Sciences alone, in which the truths
established are universally assented to, and regarded with
comparative calmness, we are better able to discuss the
formal conditions and general processes of scientific
discovery, than we could do if we entangled ourselves
among subjects where the interest is keener and the
truth more controverted. Perhaps a more exact descrip
tion of the present work would be, The Philosophy of
the Inductive Sciences, founded upon the History of the
principal Physical Sciences.

I am well aware how much additional interest and
attractiveness are given to speculations concerning the
progress of human knowledge, when we include in them,
as examples of such knowledge, views on subjects of
politics, morals, beauty in art and literature, and the like.
Prominent instances of the effect of this mode of treating
such subjects have recently appeared. But I still think
that the real value and import of Inductive Philosophy,
even in its application to such subjects, are best brought
into view by making the progress of political, and moral
and caUesthetical-\ truth a subject of consideration apart
from physical science.

It can hardly happen that a work which treats of
Methods of Scientific Discovery shall not seem to fail in

* Hist. Ind. Sci. Second Edition. Note to the Introduction,
t Sec Vol. ii. On the Language of Science, Aphorism, xvn.

Vlll PREFACE TO

the positive results which it offers. For an Art of Dis
covery is not possible. At each step of the progress of
science, are needed invention, sagacity, genius ; elements
which no Art can give. We may hope in vain, as Bacon
hoped, for an organ which shall enable all men to construct
scientific truths, as a pair of compasses enables all men
to construct exact circles *. The practical results of the
Philosophy of Science must, we are persuaded, be rather
classification and analysis than precept and method. I
think however that the methods of discovery which
I have to recommend, though gathered from a wider
survey of scientific history, as to subject and as to
time, than, (so far as I am aware,) has been elsewhere
attempted, are quite as definite and practical as any
others which have been proposed ; with the great addi
tional advantage of being the methods by which all great
discoveries in physical science really have been made.
This may be said, for instance, of the Method of Grada
tion, and the Method of Natural Classification, spoken
of Book xin. Chap. vm. ; and in a narrower sense, of
the Method of Curves, the Method of Means, the Method
of Least Squares, and the Method of Residues, spoken
of in Chap. vn. of the same Book. Also the Remarks
on the Use of Plypotheses and on the Tests of Hypotheses
(Book xi. Chap, v.) point out features which mark the
usual course of discovery.

But undoubtedly one of the principal lessons which
results from the views here given is that different
sciences may be expected to advance by different modes
of procedure, according to their present condition ; and

* Noe. Org. Lib. i. A ph. 01.

THE SECOND EDITION. IX

that, in many of these sciences, an Induction per
formed by any of the methods just referred to, is not
the step which we may expect to see next made.
Several of the sciences may not be in a condition which
fits them for such a Colligation of Facts, (to use the
phraseology to which the succeeding analysis has led
me. See B. xi. C. i). The Facts may, at the present
time, require to be more fully observed, or the Idea by
which they are to be colligated may require to be more
fully unfolded.

But in this point also, our speculations are far from
being barren of practical results. The Philosophy of
each Science, as given in the present work, affords us
means of discerning whether that which is needed for
the further progress of the Science has its place in the
Observations, or in the Ideas, or in the union of the two.
If Observations be wanted, the Methods of Observation
given in Book xm. Chap. n. may be referred to; if
those who are to make the next discoveries need, for
that purpose, a developement of their Ideas, the modes
in which such a developement has usually taken place
are treated of in Chapters in. and iv. of that Book.

Perhaps one of the most prominent points of this
work is the attempt to show the place which discussions
concerning Ideas have had in the progress of science.
The metaphysical aspect of each of the physical sciences
is very far from being, as some have tried to teach, an
aspect which it passes through previously to the most
decided progress of the science. On the contrary, the
metaphysical is a necessary part of the inductive move
ment. This, which is evidently so by the nature of the

X PREFACE TO

case, is proved by a copious collection of historical evi
dences in the first ten Books of the present work. Those
Books contain an account of the principal philosophical
controversies which have taken place in all the physical
sciences, from Mathematics to Physiology; and these
controversies, which must be called metaphysical if any
thing be so called, have been conducted by the greatest
discoverers in each science, and have been an essential
part of the discoveries made. Physical discoverers have
differed from barren speculators, not by having no meta
physics in their heads, but by having good metaphysics
while their adversaries had bad ; and by binding their
metaphysics to their physics, instead of keeping the two
asunder. I trust that the ten Books of which I have
spoken are of some value, even as a series of analyses of
a number of remarkable controversies ; but I cannot con
ceive how any one, after reading these Books, can fail
to see that there is in progressive science a metaphysical
as well as a physical element ; ideas, as well as facts,
thoughts, as well as things : in short, that the Funda
mental Antithesis, for which I contend, is there most
abundantly and strikingly exemplified.

On the subject of this doctrine of a Fundamental
Analysis, which our knowledge always involves, I will
venture here to add a remark, which looks beyond the
domain of the physical sciences. This doctrine is suited
to throw light upon Moral and Political Philosophy, no
less than upon Physical. In Morality, in Legislation, in
National Polity, we have still to do with the opposition
and combination of two Elements ; of Facts and Ideas ;
of History, and an Ideal Standard of Action ; of actual

THE SECOND EDITION. XI

character and position, and of the aims which are placed
above the Actual. Each of these is in conflict with the
other ; each modifies and moulds the other. We can never
escape the control of the first ; we must ever cease to
strive to extend the sway of the second. In these cases,
indeed, the Ideal Element assumes a new form. It in
cludes the Idea of Duty. The opposition, the action
and re-action, the harmony at which we must ever
aim, and can never reach, are between what is and what
ought to be ; between the past or present Fact, and
the Supreme Idea. The Idea can never be independ
ent of the Fact, but the Fact must ever be drawn
towards the Idea. The History of Human Societies,
and of each Individual, is by the moral philosopher,
regarded in reference to this Antithesis ; and thus both
Public and Private Morality becomes an actual progress
towards an Ideal Form ; or ceases to be a moral reality.

I have made very slight alterations in the first
edition, except that the First Book is remodelled with
a view of bringing out more clearly the basis of the
work ; this doctrine of the Fundamental Antithesis of
Philosophy. This doctrine, and its relation to the rest
of the work, have become more clear in the years
which have elapsed since the first edition.

A separate Essay, in which this doctrine was ex
plained, and a few other Essays previously published in
various forms, and containing discussions of special
points belonging to the scheme of philosophy here de
livered, have attracted some notice, both in this and in
other countries. I have therefore added them as an
Appendix to the present edition.

Xll PREPACK TO

I have added a few Notes, in answer to arguments
brought against particular parts of this work. I have
written these in what I have elsewhere called an im
personal manner; wishing to avoid controversy, so far
as justice to philosophical Truth will allow me to do so.

I have not given any detailed reply to the criticisms
of this work which occur in Mr. Mill s System of Logic.
The consideration of these criticisms would be interest
ing to me, and I think would still further establish the
doctrines which I have here delivered. But such a dis
cussion would involve me in a critique of Mr. Mill s
work ; which if I were to offer to the world, I should
think it more suitable to publish separately.

More than one of my critics has expressed an opinion
that when I published this work, I had not given due at
tention to the Cours de Philosophic Positive of M. Comte.
I had, and have, an opinion of the value of M. Comte s
speculations very different from that entertained by my
monitors. I had in the former edition discussed, and,
as I conceive, confuted, some of M. Comte s leading
doctrines*. In order further to show that I had not
lightly passed over those portions of M. Comte s work
which had then appeared, I now publish f an additional
portion of a critique of the work which, though I had
written, I excluded from the former edition. This is
printed exactly as it existed in manuscript at the
period of that publication. To return to the subject and
to take it up in all its extent, would be an undertaking
out of the range of a new edition of my published
work.

* B. xr. c. vii. B. xni. c. iv. t 13. xn. c. xvi.

THE SECOND EDITION. Xlll

Bacon delivered his philosophy in Aphorisms ; a
series of Sentences which profess to exhibit rather the
results of thought than the process of thinking. A
mere Aphoristic Philosophy unsupported by reasoning,
is not suited to the present time. No writer upon
such subjects can expect to be either understood or
assented to, beyond the limits of a narrow school, who
is not prepared with good arguments as well as magis
terial decisions upon the controverted points of philo
sophy. But it may be satisfactory to some readers to
see the Philosophy, to which in the present work we are
led, presented in the Aphoristic form. I have therefore
placed a Series of Aphorisms at the end of the work.
In the former edition these, by being placed at the begin
ning of the work, might mislead the reader ; seeming
to some, perhaps, to be put forwards as the grounds, not
as the results, of our philosophy. I have also prefixed
an analysis of the work, in the form of a Table of Con
tents to each volume.

In that part of the second volume which treats of
the Language of Science, I have made a few alterations
and additions, tending to bring my recommendations
into harmony with the present use of the best scientific
works.

CONTENTS

THE FIRST VOLUME.

PREFACE

PART I.

OF IDEAS

PAGK
V

BOOK I.
OF IDEAS IN GENERAL.

CHAP. I. INTRODUCTION . . . Y

CHAP. II. OF THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY

Sect. 1. Thoughts and Things.

2. Necessary and Experiential Truths

3. Deduction and Induction ....

4. Theories and Facts

5. Ideas and Sensations . .

6. Reflexion and Sensation . .
7- Subjective and 01/jective .

8. Matter and Form . . . " . .

9. Man the Interpreter of Nature

10. The Fundamental Antithesis is inseparable .

11. /Successive Generalization * .

CHAP. III. OF TECHNICAL TERMS . . .

Art. 1. Examples.

2. Use of Terms.

CHAP. IV. OF NECESSARY TRUTHS * . .

Art. 1. The two Elements of Knowledge,
Shewn by necessary Truths.
Examples of necessary Truths in numbers.
The opposite cannot be distinctly conceived.
Other Examples.
Universal Truths.

CHAP. V. OF EXPERIENCE . . ^ .

Art. 1. Experience cannot prove necessary Truths,
2. Except when aided by Ideas.

2.
3.
4.
5.

1
16

19
21
23
24

27
29
33
37
38
46

XVI CONTENTS OF

PAGE

CHAP. VI. OF THE GROUNDS OF NECESSARY TRUTHS . . 66

Art. 1. These Grounds are Fundamental Ideas.

2. These are to be reviewed.

3. Definitions and Axioms.

4. Syllogism,

5. Produces no new Truths.

6. Axioms needed.

7. Axioms depend on Ideas :

8. So do Definitions.

9. Idea not completely expressed.

CHAP. VII. THE FUNDAMENTAL IDEAS ARE NOT DERIVED FROM

EXPERIENCE 74

Art. I. No connexion observed.

2. Faculties implied in observation.

3. We are to examine our Faculties.

CHAP. VIII. OF THE PHILOSOPHY OF THE SCIENCES . . . 78
Sciences arranged according to Ideas.

BOOK II.

THE PHILOSOPHY OF THE PURE SCIENCES.
CHAP. I. OF THE PURE SCIENCES . . , .82

Art. 1. Geometry, Arithmetic, Algebra,

2. Are not Inductive Sciences :

3. Are Mathematical Sciences.

4. Mixed Mathematics.

5. Space, Time, Number.

CHAP. II. OF THE IDEA OF SPACE . . 84

Art. 1. Space is an Idea,

2. Not derived from Experience,

3. As Geometrical Truth shews.

4. Space is a Form of Experience.

5. The phrase not essential.

CHAP. III. OF SOME PECULIARITIES OF THE IDEA OF SPACE . 88

Art. 1. Space is not an Abstract Notion.

2. Space is infinite.

3. Space is real.

4. Space is a Form of Intuition.

5. Figure.

6. Three Dimensions.

THE FIRST VOLUME. XV11

PAOK

CHAP. IV. OF THE DEFINITIONS AND AXIOMS WHICH RELATE TO

SPACE ... . .91

Art. 1. Geometry.

2. Definitions.

3. Axioms.

4. Not Hypotheses.

5. Axioms necessary.
6*. Straight lines.

7. Planes.

8. Elementary Geometry.

CHAP. V. OF SOME OBJECTIONS WHICH HAVE BEEN MADE TO THE

DOCTRINES STATED IN THE PREVIOUS CHAPTER . 101
Art. 1. How is Geometry hypothetical?

2. What was Stewart s view ?

3. " Legitimate filiations " of Definitions.

4. Is a Definition a complete explanation ?

5. Are some Axioms Definitions ?

6. Axiom concerning Circles.

7. Can Axioms become truisms ?

8. Use of such.

CHAP. VI. OF THE PERCEPTION OF SPACE . ., . . Ill

Art. 1. Which Senses apprehend Space?

2. Perception of solid figure.

3. Is an interpretation.

4. May be analysed.

5. Outline.

6. Reversed convexity.

7. Do we perceive Space by Touch ?

8. Brown s Opinion.

9. The Muscular Sense.

10. Bell s Opinion.

1 1 . Perception includes Activity.

12. Perception of the Skiey Dome.

13. Reid s Idomenians.

14. Motion of the Eye.

15. Searching Motion.

16. Sensible Spot.

17. Expressions implying Motion.

CHAP. VII. OF THE IDEA OF TIME . . . . 125

Art. 1. Time an Idea not derived from Experience.

2. Time is a Form of Experience.
VOL. I. W. P. h

XVlii CONTENTS OF

PAGE

Art. 3. Number.

4. Is Time derived from Motion ?

CHAP. VIII. OF SOME PECULIARITIES IN THE IDEA OF TIME . . 128

Art. 1. Time is not an Abstract Notion.

2. Time is infinite.

3. Time is a Form of Intuition.

4. Time is of one Dimension,

5. And no more.

6. Rhythm.

7- Alternation.

8. Arithmetic.

CHAP. IX. OF THE AXIOMS WHICH RELATE TO NUMBER .

Art. 1. Grounds of Arithmetic.

2. Intuition.

3. Arithmetical Axioms,

4. Are Conditions of Numerical Reasoning

5. In all Arithmetical Operations.

6. Higher Numbers.

CHAP. X. OF THE PERCEPTION OF TIME AND NUMBER . . 135
Art. 1. Memory.

2. Sense of Successiveness

3. Implies Activity.

4. Number also does so.

5. And apprehension of Rhythm.

Note to Chapter X . .139

CHAP. XI. OF MATHEMATICAL REASONING
Art. 1. Discursive Reasoning.

2. Technical Terms of Reasoning.

3. Geometrical Analysis and Synthesis.

CHAP. XII. OF THE FOUNDATIONS OF THE HIGHER MATHEMATICS 145

Art. 1. The Idea of a Limit.

2. The use of General Symbols.

3. Connexion of Symbols and Analysis.

CHAP. XIII. THE DOCTRINE OF MOTION . 150

Art. 1. Pure Mechanism.
2. Formal Astronomy.

CHAP. XIV. OF THE APPLICATION OF MATHEMATICS TO THE

INDUCTIVE SCIENCES ... .153

Art. 1. The Ideas of Space and Number are clear from the
first.

THE FIRST VOLUME XIX

PAC;K

Art. 2. Their application in Astronomy.

3. Conic Sections, &c.

4. Arabian Numerals.

5. Newton s Lemmas.

6. Tides.

7- Mechanics.
. Optics.
9. Conclusion.

BOOK III.

THE PHILOSOPHY OF THE MECHANICAL SCIENCES.
CHAP. I. OF THE MECHANICAL SCIENCES . ... , 164
CHAP. II. OF THE IDEA OF CAUSE f ,. . . . . 165

Art. 1. Not derived from Observation.

2. As appears by its use.

3. Cause cannot be observed.

4. Is Cause only constant succession ?

5. Other reasons.

CHAP. III. MODERN OPINIONS RESPECTING THE IDEA OF CAUSE . 701

Art. 1. Hume s Doctrine.

2. Stewart and Brown.

3. Kant.

4. Relation of Kant and Brown.

5. Axioms flow from the Idea.

6. The Idea implies activity in the Mind.

CHAP. IV. OF THE AXIOMS WHICH RELATE TO THE IDEA OF CAUSE 177

Art. 1. Causes are Abstract Conceptions.

2. First Axiom.

3. Second Axiom.

4. Limitation of the Second Axiom.

5. Third Axiom.

6. Extent of the Third Axiom.

CHAP. V. OF THE ORIGIN OF OUR CONCEPTIONS OF FORCE AND

MATTER . 1 05

Art. 1. Force.

2. Matter.

3. Solidity.

4. Inertia.

5. Application.

XX CONTENTS OF

PAGE

CHAP. VI. OF THE ESTABLISHMENT OP THE PRINCIPLES OF

STATICS .....

Art. 1. Object of the Chapter.

2. Statics and Dynamics.

3. Equilibrium.

4. Measure of Statical Forces.

5. The Center of Gravity.

6. Oblique Forces.

7- Force acts at any point of its Direction.

8. The Parallelogram of Forces

9. Is a necessary Truth.

10. Center of Gravity descends.

11. Stevinus s Proof.

12. Principle of Virtual Velocities.

13. Fluids press equally.

14. Foundation of this Axiom.

CHAP. VII. OF THE ESTABLISHMENT OF THE PRINCIPLES OF

DYNAMICS . . . . . 215

Art. 1. History.

2. The First Law of Motion.

3. Gravity is a Uniform Force.

4. The Second Law of Motion.

5. The Third Law of Motion.

6. Action and Reaction in Moving Bodies.
7- D Alembert s Principle.

8. Connexion of Statics and Dynamics.

9. Mechanical Principles grow more evident.
10. Controversy of the Measure of Force.

CHAP. VIII. OF THE PARADOX OF UNIVERSAL PROPOSITIONS

OBTAINED FROM EXPERIENCE . . 245

Art. 1. Experience cannot establish necessary Truths ;

2. But can interpret Axioms

3. Gives us the Matter of Truths.

4. Exemplifies Truths.

5. Cannot shake Axioms.

6. Is this applicable in other cases ?

CHAP. IX. OF THE ESTABLISHMENT OF THE LAW OF UNIVERSAL

GRAVITATION ...... 254

Art. 1 . General course of the History.
2. Particulars as to the Law.

THE FIRST VOLUME. XXI

PACK

Art. 3. As to the Gravity of Matter.

4. Universality of the Law.

5. Is Gravity an essential quality ?

6. Newton s Rule of Philosophizing.
7- Hypotheses respecting Gravity.
8. Do Bodies act at a distance ?

CHAP. X. OF THE GENERAL DIFFUSION OF CLEAR MECHANICAL

IDEAS . . . . . . . . 262

Art. 1. Nature of the Process

2. Among the Ancients.

3. Kepler, c.

4. Lord Monboddo, &c.

5. Schelling, c.

6. Common usage.

7. Effect of Phrases.

8. Contempt of Predecessors.

9. Less detail hereafter.

10. Mechanico-Chemical Sciences.

11. Secondary Mechanical Sciences.

Additional Note to Chapter IV. On the Axioms which relate to

the Idea of Cause . . ... . . . 274

Additional Note to Chapter VI. Sect. 5. On the Center of Gravity 275

BOOK IV.

THE PHILOSOPHY OF THE SECONDARY MECHANICAL
SCIENCES.

CHAP. I. OF THE IDEA OF A MEDIUM AS COMMONLY EMPLOYED . 277
Art. 1. Of Primary and Secondary Qualities.

2. The Idea of Externality.

3. Sensation by a Medium.

4. Process of Perception of Secondary Qualities.

CHAP. II. ON PECULIARITIES IN THE PERCEPTIONS OF THE DIF
FERENT SENSES . . . . . . 28(&gt;

Art. 1. Difference of Senses.
Sect. I. Prerogatives of Sight.
Art. 2. Position.
3. Distance.

XX11 CONTENTS OF

PAGE

Sect. II. Prerogatives of Hearing.

Art. 4. Musical Intervals.

5. Chords.

6. Rhythm.

Sect. III. The Paradoxes of Vision.

Art. 7- First Paradox.

8. Second Paradox.

9. The same for near Objects.
10. Objections answered.

Sect. IV. The Perception of Visible Figures.
Art. 11. Brown s Opinion.

CHAP. III. SUCCESSIVE ATTEMPTS AT THE SCIENTIFIC APPLICA
TION OF THE IDEA OF A MEDIUM . . 307

Art. 1. Introduction.

2. Sound.

3. Light.

4. Heat.

CHAP. IV. OF THE MEASURE OF SECONDARY QUALITIES . 319
Sect. I. Scales of Qualities in General.

Art. 1. Intensity.

2. Quantity and Quality.

Sect. II. The Musical Scale.

Art. 3. Musical Relations.

4. Musical Standard.

Sect. III. Scales of Colour.
Art. 5. The Prismatic Scale.

6. Newton s Scale.

7. Scales of Impure Colours.

8. Chromatometer.

Sect., IV. Scales of Light.
Art. 9. Photometer.

10. Cyanometer.

Sect. V. Scales of Heat.
Art. 11. Thermometers.

12. Their progress.

13. Fixed Points.

14. Concordance of Thermometers.

15. Natural Measure.

THE FIRST VOLUME. XX111

PAGE

Art. 16. Law of Cooling.

17- Theory of Exchanges.

18. Air Thermometer.

19. Theory of Heat.

20. Other Instruments.

Sect. VI. Scales of other Quantities.

Art. 21. Tastes and Smells.

22. Quality of Sounds.

23. Articulate Sounds.

24. Transition.

BOOK Y.

OF THE PHILOSOPHY OF THE MECHANICO-CHEMICAL
SCIENCES.

CHAP. I. ATTEMPTS AT THE SCIENTIFIC APPLICATION OF THE IDEA

OF POLARITY . . . .... 345

Art. 1. Introduction of the Idea.

2. Magnetism.

3. Electricity.

4. Voltaic Electricity.

5. Light.

6. Crystallization.

7- Chemical Affinity.
8. General Remarks.
9* Like repels like.

CHAP. II. OF THE CONNEXION OF POLARITIES . . . 357

Art. 1. Different Polar Phenomena from one Cause.

2. Connexion of Magnetic and Electric Polarity.

3. Ampere s Theory.

4. Faraday s views.

5. Connexion of Electrical and Chemical Polarity.

6. Davy s and Faraday s views

7- Depend upon Ideas as well as Experiments.

8. Faraday s Anticipations.

9. Connexion of Chemical and Crystalline Polarities.

10. Connexion of Crystalline and Optical Polarities.

11. Connexion of Polarities in general.

12. Schelling s Speculations.

13. Hegel s vague notions.

14. Ideas must guide Experiment.

XXIV CONTENTS OF

PAGE

BOOK VI.
THE PHILOSOPHY OF CHEMISTRY.

CHAP. I. ATTEMPTS TO CONCEIVE ELEMENTARY COMPOSITION . 376

Art. 1. Fundamental Ideas of Chemistry.

2. Elements.

3. Do Compounds resemble their Elements?

4. The Three Principles.

5. A Modern Errour.

6. Are Compounds determined by the Figure of Ele

ments ?

7. Crystalline Form depends on Figure of Elements.

8. Are Compounds determined by Mechanical Attrac

tion of Elements ?

9. Newton s followers.

10. Imperfection of their Hypotheses.

CHAP. II. ESTABLISHMENT AND DEVELOPMENT OF THE IDEA OF

CHEMICAL AFFINITY . . . . . . 388

Art. 1. Early Chemists.

2. Chemical Affinity.

3. Affinity or Attraction ?

4. Affinity preferable.

5. Analysis is possible.

6. Affinity is Elective.
7- Controversy on this.

8. Affinity is Definite.

9. Are these Principles necessarily true ?

10. Composition determines Properties.

11. Comparison on this subject.

12. Composition determines Crystalline Form.

CHAP. III. OF THE IDEA OF SUBSTANCE .... 404
Art. 1. Indestructibility of Substance.

2. The Idea of Substance.

3. Locke s Denial of Substance.

4. Is all Substance heavy ?

CHAP. IV. APPLICATION OF THE IDEA OF SUBSTANCE IN CHE
MISTRY ........ 412

Art. 1. A Body is Equal to its Elements.

2. Lavoisier.

3. Are there Imponderable Elements ?

THE FIRST VOLUME. XXV

PA OR
Art. 4. Faraday s views.

5. Composition of Water.

6. Heat in Chemistry.

CHAP. V. THE ATOMIC THEORY . . . . . .421

Art. 1. The Theory on Chemical Grounds.

2. Hypothesis of Atoms.

3. Its Chemical Difficulties.

4. Grounds of the Atomic Doctrine.

5. Ancient Atomists.

6. Francis Bacon.

7- Modern Atomists.

8. Arguments for and against.

9. Boscovich s Theory.

10. Molecular Hypothesis.

11. Poisson s Inference.

12. Wollaston s Argument.

13. Properties are Permanent.

BOOK VII.

THE PHILOSOPHY OF MORPHOLOGY, INCLUDING
CRYSTALLOGRAPHY.

CHAP. I. EXPLICATION OF THE IDEA OF SYMMETRY . ^ 439

Art. 1. Symmetry what.

2. Kinds of Symmetry.

3. Examples in Nature.

4. Vegetables and Animals.

5. Symmetry a Fundamental Idea.

6. Result of Symmetry.

CHAP. II. APPLICATION OF THE IDEA OF SYMMETRY TO CRYSTALS 447
Art. 1. " Fundamental Forms."

2. Their use.

3. " Systems of Crystallization."

4. Cleavage.

5. Other Properties.

CHAP. III. SPECULATIONS FOUNDED UPON THE SYMMETRY OF

CRYSTALS . . . . 4 fc . 452

Art. 1. Integrant Molecules

2. Difficulties of the Theory.

3. Merit of the Theory.

4. Wollaston s Hypothesis.

XXV111 CONTENTS OF

PAGE

CHAP. IV. OF THE IDEA OF NATURAL AFFINITY 535

Art. 1. The Idea of Affinity

2. Is not to be made out by Arbitrary Rules.

3. Functions of Living things are many,

4. But all lead to the same arrangement.

5. This is Cuvier s principle :

6. And Decandolle s.

7- Is this applicable to Inorganic Bodies ?
8. Yes ; by the agreement of Physical and Chemical
Arrangement.

BOOK IX.
THE PHILOSOPHY OF BIOLOGY.

CHAP. I. ANALOGY OF BIOLOGY WITH OTHER SCIENCES . 543

Art. I . Biology involves the Idea of Life.

2. This Idea to be historically traced.

3. The Idea at first expressed by means of other Ideas,

4. Mystical, Mechanical, Chemical, and Vital Fluid

Hypotheses.

CHAP. II. SUCCESSIVE BIOLOGICAL HYPOTHESES . . . 548

Sect. I. The Mystical School
Sect. II. The latrochemical School.
Sect. III. The latromathematical School.
Sect. IV. The Vital Fluid School.
Sect. V. The Psychical School

CHAP. III. ATTEMPTS TO ANALYSE THE IDEA OF LIFE . . 571

Art. 1. Definitions of Life,

2. By Stahl, Humboldt, Kant.

3. Definition of Organization by Kant.

4. Life is a System of Functions.

5. Bichat. Sum of Functions.

6. Use of Definition.
7- Cuvier s view.

8. Classifications of Functions.

9. Vital, Natural, and Animal Functions.

10. Bichat. Organic and Animal Life.

11. Use of this Classification.

THE FIRST VOLUME. XXIX

PAGE

CHAP. IV. ATTEMPTS TO FORM IDEAS OP SEPARATE VITAL

FORCES, AND FIRST, OF ASSIMILATION AND SECRE
TION 580

Sect. I. Course of Biological Research.

Art. 1. Observation and New Conceptions.

Sect. II. Attempts to form a distinct Conception of Assimila
tion and Secretion.

Art. 2. The Ancients.

3. Buffon. Interior Mould.

4. Defect of this view.

5. Cuvier. Life a Vortex.

6. Defect of this view.

7. Schelling. Matter and Form.

8. Life a constant Form of circulating Matter, &c.

Sect. III. Attempts to conceive the Forces of Assimilation and

Secretion.

Art. 9. Assimilation is a Vital Force.

10. The name "Assimilation."

11. Several processes involved in Assimilation.

12. Absorption. Endosmose.

13. Absorption involves a Vital Force.

14. Secretion. Glands.

15. Motions of Vital Fluids.

Sect. IV. Attempts to conceive the Process of Generation.

Art. 16. Reproduction figuratively used for Generation.

17. Nutrition different from

18. Generation.

19. Generations successively included.

20. Pre-existence of Germs.

21. Difficulty of this view.

22. Communication of Vital Forces.

23. Close similarity of Nutrition and Generation.

24. The Identity of the two Processes exemplified.

CHAP. V. ATTEMPTS TO FORM IDEAS OF SEPARATE VITAL FORCES,

continued. VOLUNTARY MOTION . . . (jOO

Art. 1. Voluntary Motion one of the animal Functions.

2. Progressive knowledge of it.

3. Nervous Fluid not electric.

4. Irritability. Glisson.

5. Haller.

XXX CONTENTS OF

PAGE

Art. 6. Contractility.

7- Organic Sensibility and Contractility not separable.

8. Improperly described by Bichat.

9. Brown.

10. Contractility a peculiar Power.

11. Cuvier s view.

12. Elementary contractile Action.

13. Strength of Muscular Fibre.

14. Sensations become Perceptions

15. By means of Ideas ;

16. And lead to Muscular Actions.

17. Volition comes between Perception and Action.

18. Transition to Psychology.

19. A center is introduced.

20. The central consciousness may be obscure.

21. Reflex Muscular Action.

22. Instinct.

23. Difficulty of conceiving Instinct.

24. Instinct opposed to Insight.

CHAP. YI. OF THE IDEA OF FINAL CAUSES . . . 618

Art. 1. Organization. Parts are Ends and Means.

2. Not merely mutually dependent.

3. Not merely mutually Cause and Effect.

4. Notion of End not derived from Facts.

5. This notion has regulated Physiology.

6. Notion of Design comes from within.
7- Design not understood by Savages.

8. Design opposed to Morphology.

9. Impression of Design when fresh.

10. Acknowledgement of an End by adverse Physiolo

gists.

1 1 . This included in the Notion of Disease.

12. It belongs to Organized Creatures only.

13. The term Final Cause-

14. Law and Design.

15. Final Causes and Morphology.

16. Expressions of physiological Ends.

17. The Conditions of Existence.

18. The asserted presumption of Teleology.

19. Final Causes in other subjects.

20. Transition to Palaetiology.

THE FIRST VOLUME. XXXI

PAGK

BOOK X.

THE PHILOSOPHY OF PAI^ETIOLOGY.

CHAP. I. OF PAL^ETIOLOGICAL SCIENCES IN GENERAL . 637

Art. 1. Description of Palaetiology.

2. Its Members.

3. Other Members.

4. Connexion of the whole subject.

5. We shall take Material Sciences only;

6. But these are connected with others.

CHAP. II. OF THE THREE MEMBERS OF A PALJETIOLOGICAL

SCIENCE . -. . . 642

Art. I. Divisions of such Sciences.

2. The Study of Causes.

3. ^Etiology.

4. Phenomenology requires Classification. Phenomenal

Geology.

5. Phenomenal Uranology.

6. Phenomenal Geography of Plants and Animals.
7- Phenomenal Glossology.

8. The Study of Phenomena leads to Theory.

9. No sound Theory without ^Etiology.

10. Causes in Palietiology.

11. Various kinds of Cause.

12. Hypothetical Order of Patatiological Causes.

13. Mode of Cultivating ^Etiology : In Geology :

14. In the Geography of Plants and Animals :

15. In Languages.

16. Construction of Theories.

17. No sound Palastiological Theory yet extant.
CHAP. III. OF THE DOCTRINE OF CATASTROPHES AND THE DOC
TRINE OF UNIFORMITY . . . . 665

Art. 1. Doctrine of Catastrophes.

2. Doctrine of Uniformity.

3. Is Uniformity probable a priori ?

4. Cycle of Uniformity indefinite.

5. Uniformitarian Arguments are Negative only.

6. Uniformity in the Organic World.

7- Origin of the present Organic World.

8. Nebular Origin of the Solar System.

9. Origin of Languages.

10. No Natural Origin discoverable.

XXX11 CONTENTS OF THE FIRST VOLUME.

PAGE

CHAP. IV. OF THE RELATION OF TRADITION TO PALJETIOLOGY 680

Art, 1. Importance of Tradition.

2. Connexion of Tradition and Science.

3. Natural and Providential History of the World.

4. The Sacred Narrative.

5. Difficulties in interpreting the Sacred Narrative.

6. Such Difficulties inevitable.

7. Science tells us nothing concerning Creation.

8. Scientific views, when familiar, do not disturb the

authority of Scripture.

9. When should Old Interpretations be given up?

10. In what Spirit should the Change be accepted ?

11. In what Spirit should the Change be urged?

12. Duty of Mutual Forbearance.

13. Case of Galileo.

CHAP. V. OF THE CONCEPTION OF A FIRST CAUSE /uu

Art. 1. The Origin of things is not naturally discoverable;

2. Yet has always been sought after.

3. There must be a First Cause.

4. This is an Axiom.

5. Involved in the Proof of a Deity.

6. The Mind is not satisfied without it.

7- The Whole Course of Nature must have a Cause.

8. Necessary Existence of God.

9. Forms of the Proof.

10. Idea of a First Cause is Necessary.

11. Conception of a First Cause.

12. The First Cause in all Sciences is the same.

13. We are thus led to Moral Subjects.
Conclusion of Part I.

THE

PHILOSOPHY

OF THE

INDUCTIVE SCIENCES.

PART I.

OF IDEAS.

VOL. I. W. P.

Quee adhuc inventa sunt in Scientiis, ea Imjusmodi sunt
ut Notionibus Vulgaribus fere subjaceant : lit vero ad inte-
riora et retnotiora Naturae penetretur, necesse est ut tarn
NOTIONES quam AXIOMATA magis certa et munita via a
particularibus abstrahantur ; atque omnino melior et certior
intellectus adoperaUo in usum veniat.

BACON, Nov. Org., Lib. i. Aphor. xviii.

BOOK I.

OF IDEAS IN GENERAL.

CHAPTER I.
INTRODUCTION.

THE PHILOSOPHY or SCIENCE, if the phrase were to be
understood in the comprehensive sense which most na
turally offers itself to our thoughts, would imply nothing
less than a complete insight into the essence and con
ditions of all real knowledge, and an exposition of the
best methods for the discovery of new truths. We must
narrow and lower this conception, in order to mould it
into a form in which we may make it the immediate
object of our labours with a good hope of success ; yet
still it may be a rational and useful undertaking, to
endeavour to make some advance towards such a Philo
sophy, even according to the most ample conception
of it which we can form. The present work has been
written with a view of contributing, in some measure,
however small it may be, towards such an undertaking.

But in this, as in every attempt to advance beyond
the position which we at present occupy, our hope of
success must depend mainly upon our being able to
profit, to the fullest extent, by the progress already
made. We may best hope to understand the nature and
conditions of real knowledge, by studying the nature
and conditions of the most certain and stable portions of
knowledge which we already possess : and we are most
likely to learn the best methods of discovering truth, by
VOL. i. \v. p. B

2 OF IDEAS IN GENERAL.

examining how truths, now universally recognized, have
really been discovered. Now there do exist among us
doctrines of solid and acknowledged certainty, and
truths of which the discovery has been received with
universal applause. These constitute what we com
monly term Sciences ; and of these bodies of exact and
enduring knowledge, we have within our reach so large
and raoied- a; collection, that we may examine them, and
the .history, of their formation, with a good prospect of
deriving froa i the study such instruction as we seek.
We may best hope to make some progress towards the
Philosophy of Science, by employing ourselves upon THE
PHILOSOPHY OF THE SCIENCES.

The Sciences to which the name is most commonly
and unhesitatingly given, are those which are concerned
about the material world ; whether they deal with the
celestial bodies, as the sun and stars, or the earth and
its products, or the elements ; whether they consider the
differences which prevail among such objects, or their
origin, or their mutual operation. And in all these
Sciences it is familiarly understood and assumed, that
their doctrines are obtained by a common process of
collecting general truths from particular observed facts,
which process is termed Induction. It is further assumed
that both in these and in other provinces of knowledge,
so long as this process is duly and legitimately per
formed, the results will be real substantial truth. And
although this process, with the conditions under which
it is legitimate, and the general laws of the formation of
Sciences, will hereafter be subjects of discussion in this
work, I shall at present so far adopt the assumption of
which I speak, as to give to the Sciences from which
our lessons are to be collected the name of Inductive
Sciences. And thus it is that I am led to designate my
work as THE PHILOSOPHY OF THE INDUCTIVE SCIENCES.

INTRODUCTION, 3

The views respecting the nature and progress of
knowledge, towards which we shall be directed by such
a course of inquiry as I have pointed out, though derived
from those portions of human knowledge which are
more peculiarly and technically termed Sciences, will by
no means be confined, in their bearing, to the domain of
such Sciences as deal with the material world, nor even
to the whole range of Sciences now existing. On the
contrary, we shall be led to believe that the nature of
truth is in all subjects the same, and that its discovery
involves, in all cases, the like conditions. On one sub
ject of human speculation after another, man s know
ledge assumes that exact and substantial character which
leads us to term it Science ; and in all these cases, whe
ther inert matter or living bodies, whether permanent
relations or successive occurrences, be the subject of our
attention, we can point out certain universal characters
which belong to truth, certain general laws which have
regulated its progress among men. And we naturally
expect that, even when we extend our range of specu
lation wider still, when we contemplate the world within
us as well as the world without us, when we consider
the thoughts and actions of men as well as the motions
and operations of unintelligent bodies, we shall still find
some general analogies which belong to the essence of
truth, and run through the whole intellectual universe.
Hence we have reason to trust that a just Philosophy of
the Sciences may throw light upon the nature and extent
of our knowledge in every department of human specu
lation. By considering what is the real import of our
acquisitions, where they are certain and definite, we may
learn something respecting the difference between true
knowledge and its precarious or illusory semblances ; by
examining the steps by which such acquisitions have
been made, we may discover the conditions under which

4 OF IDEAS IN GENERAL.

truth is to be obtained ; by tracing the boundary-line
between our knowledge and our ignorance, we may
ascertain in some measure the extent of the powers of
man s understanding.

But it may be said, in such a design there is nothing
new; these are objects at which inquiring men have
often before aimed. To determine the difference be
tween real and imaginary knowledge, the conditions
under which we arrive at truth, the range of the powers
of the human mind, has been a favourite employment of
speculative men from the earliest to the most recent
times. To inquire into the original, certainty, and com
pass of man s knowledge, the limits of his capacity, the
strength and weakness of his reason, has been the pro
fessed purpose of many of the most conspicuous and
valued labours of the philosophers of all periods up to
our own day. It may appear, therefore, that there is
little necessity to add one more to these numerous
essays ; and little hope that any new attempt will make
any very important addition to the stores of thought
upon such questions, which have been accumulated by
the profoundest and acutest thinkers of all ages.

To this I reply, that without at all disparaging the
value or importance of the labours of those who have
previously written respecting the foundations and con
ditions of human knowledge, it may still be possible to
add something to what they have done. The writings of
all great philosophers, up to our own time, form a series
which is not yet terminated. The books and systems of
philosophy which have, each in its own time, won the
admiration of men, and exercised a powerful influence
upon their thoughts, have had each its own part and
functions in the intellectual history of the world ; and
other labours which shall succeed these may also have
their proper office and useful effect. We may not be

INTRODUCTION, i)

able to do much, and yet still it may be in our power to
effect something. Perhaps the very advances made by
former inquirers may have made it possible for us, at
present, to advance still further. In the discovery of
truth, in the developement of man s mental powers and
privileges, each generation has its assigned part ; and it
is for us to endeavour to perform our portion of this
perpetual task of our species. Although the terms
which describe our undertaking may be the same which
have often been employed by previous writers to express
their purpose, yet our position is different from theirs,
and thus the result may be different too. We have, as
they had, to run our appropriate course of speculation
with the exertion of our best powers ; but our course
lies in a more advanced part of the great line along
which Philosophy travels from age to age. However
familiar and old, therefore, be the design of such a work
as this, the execution may have, and if it be performed
in a manner suitable to the time, will have, something
that is new and not unimportant.

Indeed, it appears to be absolutely necessary, in
order to check the prevalence of grave and pernicious
errour, that the doctrines which are taught concerning
the foundations of human knowledge and the powers of
the human mind, should be from time to time revised
and corrected or extended. Erroneous and partial views
are promulgated and accepted ; one portion of the truth
is insisted upon to the undue exclusion of another ; or
principles true in themselves are exaggerated till they
produce on men s minds the effect of falsehood. When
evils of this kind have grown to a serious height, a
Reform is requisite. The faults of the existing systems
must be remedied by correcting what is wrong, and sup
plying what is wanting. In such cases, all the merits
and excellencies of the labours of the preceding times do

6 OF IDEAS IN GENERAL.

not supersede the necessity of putting forth new views
suited to the emergency which has arrived. The new
form which errour has assumed makes it proper to
endeavour to give a new and corresponding form to
truth. Thus the mere progress of time, and the natural
growth of opinion from one stage to another, leads to
the production of new systems and forms of philosophy.
It will be found, I think, that some of the doctrines now
most widely prevalent respecting the foundations and
nature of truth are of such a kind that a Reform is
needed. The present age seems, by many indications, to
be called upon to seek a sounder Philosophy of Know
ledge than is now current among us. To contribute
towards such a Philosophy is the object of the present
work. The work is, therefore, like all works which
take into account the most recent forms of speculative
doctrine, invested with a certain degree of novelty in its
aspect and import, by the mere time and circumstances
of its appearance.

But, moreover, we can point out a very important
peculiarity by which this work is, in its design, distin
guished from preceding essays on like subjects ; and this
difference appears to be of such a kind as may well
entitle us to expect some substantial addition to our
knowledge as the result of our labours. The peculiarity
of which I speak has already been announced ; it is
this : that we purpose to collect our doctrines concerning
the nature of knowledge, and the best mode of acquiring
it, from a contemplation of the Structure and History of
those Sciences (the Material Sciences), which are univer
sally recognized as the clearest and surest examples of
knowledge and of discovery. It is by surveying and
studying the whole mass of such Sciences, and the
various steps of their progress, that we now hope to
approach to the true Philosophy of Science.

INTRODUCTION. 7

Now this, I venture to say, is a new method of pur
suing the philosophy of human knowledge. Those who
have hitherto endeavoured to explain the nature of
knowledge, and the process of discovery, have, it is true,
often illustrated their views by adducing special exam
ples of truths which they conceived to be established,
and by referring to the mode of their establishment.
But these examples have, for the most part, been taken
at random, not selected according to any principle or
system. Often they have involved doctrines so pre
carious or so vague that they confused rather than eluci
dated the subject ; and instead of a single difficulty,
What is the nature of Knowledge? these attempts at
illustration introduced two, What was the true analysis
of the Doctrines thus adduced? and, Whether they
might safely be taken as types of real Knowledge ?

This has usually been the case when there have
been adduced, as standard examples of the formation of
human knowledge, doctrines belonging to supposed sci
ences other than the material sciences; doctrines, for
example, of Political Economy, or Philology, or Morals,
or the Philosophy of the Fine Arts. I am very far from
thinking that, in regard to such subjects, there are no
important truths hitherto established : but it would seem
that those truths which have been obtained in these
provinces of knowledge, have not yet been fixed by
means of distinct and permanent phraseology, and sanc
tioned by universal reception, and formed into a con
nected system, and traced through the steps of their
gradual discovery and establishment, so as to make them
instructive examples of the nature and progress of truth
in general. Hereafter we trust to be able to show that
the progress of moral, and political, and philological,
and other knowledge, is governed by the same laws as
that of physical science. But since, at present, the

OF IDEAS IN GENERAL.

former class of subjects are full of controversy, doubt,
and obscurity, while the latter consist of undisputed
truths clearly understood and expressed, it may be con
sidered a wise procedure to make the latter class of
doctrines the basis of our speculations. And on the
having taken this course, is, in a great measure, my
hope founded, of obtaining valuable truths which have
escaped preceding inquirers.

But it may be said that many preceding writers on
the nature and progress of knowledge have taken their
examples abundantly from the Physical Sciences. It
would be easy to point out admirable works, which have
appeared during the present and former generations, in
which instances of discovery, borrowed from the Phy
sical Sciences, are introduced in a manner most happily
instructive. And to the works in which this has been
done, I gladly give my most cordial admiration. But at
the same time I may venture to remark that there still
remains a difference between my design and theirs : and
that I use the Physical Sciences as exemplifications of
the general progress of knowledge in a manner very
materially different from the course which is followed in
works such as are now referred to. For the conclusions
stated in the present work, respecting knowledge and
discovery, are drawn from a connected and systematic
survey of the whole range of Physical Science and its
History ; whereas, hitherto, philosophers have contented
themselves with adducing detached examples of scientific
doctrines, drawn from one or two departments of science.
So long as we select our examples in this arbitrary and
limited manner, we lose the best part of that philosophi
cal instruction, which the sciences are fitted to afford
when we consider them as all members of one series,
and as governed by rules which are the same for all.
Mathematical and chemical truths, physical and physio-

INTRODUCTION. 9

logical doctrines, the sciences of classification and of
causation, must alike be taken into our account, in order
that we may learn what are the general characters of
real knowledge. When our conclusions assume so com
prehensive a shape that they apply to a range of sub
jects so vast and varied as these, we may feel some con
fidence that they represent the genuine form of universal
and permanent truth. But if our exemplification is of a
narrower kind, it may easily cramp and disturb our phi
losophy. We may, for instance, render our views of
truth and its evidence so rigid and confined as to be
quite worthless, by founding them too much on the con
templation of mathematical truth. We may overlook
some of the most important steps in the general course
of discovery, by fixing our attention too exclusively
upon some one conspicuous group of discoveries, as, for
instance, those of Newton. We may misunderstand the
nature of physiological discoveries, by attempting to
force an analogy between them and discoveries of me
chanical laws, and by not attending to the intermediate
sciences which fill up the vast interval between these
extreme terms in the series of material sciences. In
these and in many other ways, a partial and arbitrary
reference to the material sciences in our inquiry into
human knowledge may mislead us ; or at least may fail
to give us those wider views, and that deeper insight,
which should result from a systematic study of the whole
range of sciences with this particular object.

The design of the following work, then, is to form a
Philosophy of Science, by analyzing the substance and
examining the progress of the existing body of the sci
ences. As a preliminary to this undertaking, a survey
of the history of the sciences was necessary. This,
accordingly, I have already performed ; and the result
of the labour thus undertaken has been laid before the
public as a History oftlie Inductive Sciences.

10 OF IDEAS IN GENERAL.

In that work I have endeavoured to trace the steps
by which men acquired each main portion of that know
ledge on which they now look with so much confidence
and satisfaction. The events which that History relates,
the speculations and controversies which are there de
scribed, and discussions of the same kind, far more
extensive, which are there omitted, must all be taken
into our account at present, as the prominent and
standard examples of the circumstances which attend
the progress of knowledge. With so much of real his
torical fact before us, we may hope to avoid such views
of the processes of the human mind as are too partial
and limited, or too vague and loose, or too abstract and
unsubstantial, to represent fitly the real forms of dis
covery and of truth.

Of former attempts, made with the same view of
tracing the conditions of the progress of knowledge, that
of Bacon is perhaps the most conspicuous : and his
labours on this subject were opened by his book on the
Advancement of Learning, which contains, among other
matter, a survey of the then existing state of knowledge.
But this review was undertaken rather with the object
of ascertaining in what quarters future advances were to
be hoped for, than of learning by what means they were
to be made. His examination of the domain of human
knowledge was conducted rather with the view of dis
covering what remained undone, than of finding out how
so much had been done. Bacon s survey was made for
the purpose of tracing the boundaries, rather than of
detecting the principles of knowledge. "I will now
attempt," he says*, "to make a general and faithful
perambulation of learning, with an inquiry what parts
thereof lie fresh and waste, and not improved and con
verted by the industry of man ; to the end that such a
plot made and recorded to memory, may both minister

* Advancement of Learning, b. i. p. 74.

INTRODUCTION. 11

light to any public designation, and also serve to excite
voluntary endeavours." Nor will it be foreign to our
scheme also hereafter to examine with a like purpose
the frontier-line of man s intellectual estate. But the
object of our perambulation in the first place, is not so
much to determine the extent of the field, as the sources
of its fertility. We would learn by what plan and rules
of culture, conspiring with the native forces of the boun
teous soil, those rich harvests have been produced which
fill our garners. Bacon s maxims, on the other hand,
respecting the mode in which he conceived that know
ledge was thenceforth to be cultivated, have little refer
ence to the failures, still less to the successes, which are
recorded in his Review of the learning of his time. His
precepts are connected with his historical views in a
slight and unessential manner. His Philosophy of the
Sciences is not collected from the Sciences which are
noticed in his survey. Nor, in truth, could this, at the
time when he wrote, have easily been otherwise. At
that period, scarce any branch of physics existed as a
science, except Astronomy. The rules which Bacon gives
for the conduct of scientific researches are obtained, as
it were, by divination, from the contemplation of sub
jects with regard to which no sciences as yet were. His
instances of steps rightly or wrongly made in this path,
are in a great measure cases of his own devising. He
could not have exemplified his Aphorisms by references
to treatises then extant, on the laws of nature ; for the
constant burden of his exhortation is, that men up to
his time had almost universally followed an erroneous
course. And however we may admire the sagacity with
which he pointed the way along a better path, we have
this great advantage over him ; that we can interrogate
the many travellers who since his time have journeyed
on this road. At the present day, when we have under

12 OF IDEAS IN GENERAL.

our notice so many sciences, of such wide extent, so well
established ; a Philosophy of the Sciences ought, it must
seem, to be founded, not upon conjecture, but upon an
examination of many instances; should not consist of
a few vague and unconnected maxims, difficult and
doubtful in their application, but should form a system
of which every part has been repeatedly confirmed and
verified.

This accordingly it is the purpose of the present
work to attempt. But I may further observe, that as
my hope of making any progress in this undertaking is
founded upon the design of keeping constantly in view
the whole result of the past history and present con
dition of science, I have also been led to draw my les
sons from my examples in a manner more systematic
and regular, as appears to me, than has been done by
preceding writers. Bacon, as I have just said, was led
to his maxims for the promotion of knowledge by the
sagacity of his own mind, w r ith little or no aid from
previous examples. Succeeding philosophers may often
have gathered useful instruction from the instances of
scientific truths and discoveries which they adduced, but
their conclusions were drawn from their instances casu
ally and arbitrarily. They took for their moral any
which the story might suggest. But such a proceeding
as this cannot suffice for us, whose aim is to obtain a
consistent body of philosophy from a contemplation of
the whole of Science and its History. For our purpose
it is necessary to resolve scientific truths into their con
ditions and ingredients, in order that we may see in
what manner each of these has been and is to be pro
vided, in the cases which we may have to consider. This
accordingly is necessarily the first part of our task : to
analyze Scientific Truth into its Elements. This attempt
will occupy the earlier portion of the present work ; and

INTRODUCTION. 1 3

will necessarily be somewhat long, and perhaps, in many
parts, abstruse and uninviting. The risk of such an
inconvenience is inevitable ; for the inquiry brings before
us many of the most dark and entangled questions in
which men have at any time busied themselves. And
even if these can now be made clearer and plainer than
of yore, still they can be made so only by means of men
tal discipline and mental effort. Moreover this analysis
of scientific truth into its elements contains much, both
in its principles and in its results, different from the
doctrines most generally prevalent among us in recent
times : but on that very account this analysis is an
essential part of the doctrines which I have now to lay
before the reader: and I must therefore crave his
indulgence towards any portion of it which may appear
to him obscure or repulsive.

There is another circumstance which may tend to
make the present work less pleasing than others on the
same subject, in the nature of the examples of human
knowledge to which I confine myself; all my instances
being, as I have said, taken from the material sciences.
For the truths belonging to these sciences are, for the
most part, neither so familiar nor so interesting to the
bulk of readers as those doctrines which belong to some
other subjects. Every general proposition concerning
politics or morals at once stirs up an interest in men s
bosoms, which makes them listen with curiosity to the
attempts to trace it to its origin and foundation. Every
rule of art or language brings before the mind of culti
vated men subjects of familiar and agreeable thought,
and is dwelt upon with pleasure for its own sake, as well
as on account of the philosophical lessons which it may
convey. But the curiosity which regards the truths of
physics or chemistry, or even of physiology and astro
nomy, is of a more limited and less animated kind.

14 OF IDEAS IN GENERAL.

Hence, in the mode of inquiry which I have prescribed
to myself, the examples which I have to adduce will not
amuse and relieve the reader s mind as much as they
might do, if I could allow myself to collect them from
the whole field of human knowledge. They will have in
them nothing to engage his fancy, or to warm his heart.
I am compelled to detain the listener in the chilly air
of the external world, in order that we may have the
advantage of full daylight.

But although I cannot avoid this inconvenience, so
far as it is one, I hope it will be recollected how great
are the advantages which we obtain by this restriction.
We are thus enabled to draw all our conclusions from
doctrines which are universally allowed to be eminently
certain, clear, and definite. The portions of knowledge
to which 1 refer are well known, and well established
among men, Their names are familiar, their assertions
uncontested. Astronomy and Geology, Mechanics and
Chemistry, Optics and Acoustics, Botany and Physiology,
are each recognized as large and substantial collections
of undoubted truths. Men are wont to dwell with pride
and triumph on the acquisitions of knowledge which
have been made in each of these provinces ; and to speak
with confidence of the certainty of their results. And all
can easily learn in what repositories these treasures of
human knowledge are to be found. When, therefore,
we begin our inquiry from such examples, we proceed
upon a solid foundation. With such a clear ground of
confidence, we shall not be met with general assertions
of the vagueness and uncertainty of human knowledge ;
with the question, What truth is, and How we are to
recognize it ; with complaints concerning the hopeless
ness and unprofitableness of such researches. We have,
at least, a definite problem before us. We have to
examine the structure and scheme, not of a shapeless

INTRODUCTION. 15

mass of incoherent materials, of which we doubt whether
it be a ruin or a natural wilderness, but of a fair and
lofty palace, still erect and tenanted, where hundreds of
different apartments belong to a common plan, where
every generation adds something to the extent and mag
nificence of the pile. The certainty and the constant
progress of science are things so unquestioned, that we
are at least engaged in an intelligible inquiry, when we
are examining the grounds and nature of that certainty,
the causes and laws of that progress.

To this enquiry, then, we now proceed. And in
entering upon this task, however our plan or our prin
ciples may differ from those of the eminent philosophers
who have endeavoured, in our own or in former times,
to illustrate or enforce the philosophy of science, we
most willingly acknowledge them as in many things our
leaders and teachers. Each reform must involve its own
peculiar principles, and the result of our attempts, so
far as they lead to a result, must be, in some respects,
different from those of former works. But we may still
share with the great writers who have treated this
subject before us, their spirit of hope and trust, their
reverence for the dignity of the subject, their belief in
the vast powers and boundless destiny of man. And we
may once more venture to use the words of hopeful
exhortation, with which the greatest of those who have
trodden this path encouraged himself and his followers
when he set out upon his way.

" Concerning ourselves we speak not ; but as touch
ing the matter which we have in hand, this we ask ;
that men deem it not to be the setting up an Opinion,
but the performing of a Work : and that they receive
this as a certainty; that we are not laying the founda
tions of any sect or doctrine, but of the profit and
dignity of mankind. Furthermore, that being well dis-

16 OF IDEAS IN GENERAL.

posed to what shall advantage themselves, and putting
off factions and prejudices, they take common counsel
with us, to the end that being by these our aids and
appliances freed and defended from wanderings and
impediments, they may lend their hands also to the
labours which remain to be performed : and yet further,
that they be of good hope ; neither imagine to them
selves this our Reform as something of infinite dimen
sion, and beyond the grasp of mortal man, when in truth
it is the end and true limit of infinite errour ; and is by
no means unmindful of the condition of mortality and
humanity, not confiding that such a thing can be carried
to its perfect close in the space of one single age, but
assigning it as a task to a succession of generations."

CHAPTER II.

OF THE FUNDAMENTAL ANTITHESIS OF
PHILOSOPHY.

SECT. 1. Thoughts and Things.

IN order that we may do something towards determining
the nature and conditions of human knowledge, (which
I have already stated as the purpose of this work,) I
shall have to refer to an antithesis or opposition, which
is familiar and generally recognized, and in which the
distinction of the things opposed to each other is com
monly considered very clear and plain. I shall have to
attempt to make this opposition sharper and stronger
than it is usually conceived, and yet to shew that the
distinction is far from being so clear and definite as it is
usually assumed to be : I shall have to point the con
trast, yet shew that the things which are contrasted

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 17

cannot be separated : I must explain that the anti
thesis is constant and essential, but yet that there is no
fixed and permanent line dividing its members. I may
thus appear, in different parts of my discussion, to be
proceeding in opposite directions, but I hope that the
reader who gives me a patient attention will see that
both steps lead to the point of view to which I wish to
lead him.

The antithesis or opposition of which I speak is
denoted, with various modifications, by various pairs of
terms : I shall endeavour to show the connexion of these
different modes of expression, and I will begin with that
form which is the simplest and most idiomatic.

The simplest and most idiomatic expression of the
antithesis to which I refer is that in which we oppose to
each other THINGS and THOUGHTS. The opposition is
familiar and plain. Our Thoughts are something which
belongs to ourselves; something which takes place
within us ; they are what me think ; they are actions of
our minds. Things, on the contrary, are something
different from ourselves and independent of us ; some
thing which is without us ; they are ; we see them,
touch them, and thus know that they exist ; but we do
not make them by seeing or touching them, as we make
our Thoughts by thinking them ; we are passive, and
Things act upon our organs of perception.

Now what I wish especially to remark is this : that
in all human KNOWLEDGE both Thoughts and Things are
concerned. In every part of my knowledge there must
be some thing about which I know, and an internal act
of me who know. Thus, to take simple yet definite parts
of our knowledge, if I know that a solar year consists of
365 days, or a lunar month of 30 days, I know some
thing about the sun or the moon ; namely, that those
objects perform certain revolutions and go through cer-

VOL. I. \V. P. C

18 OF IDEAS IN GENERAL.

tain changes, in those numbers of days; but I count
such numbers and conceive such revolutions and changes
by acts of my own thoughts. And both these elements
of my knowledge are indispensable. If there were not
such external Things as the sun and the moon I could
not have any knowledge of the progress of time as
marked by them. And however regular were the mo
tions of the sun and moon, if I could not count their
appearances and combine their changes into a cycle, or
if I could not understand this when done by other men,
I could not know anything about a year or a month. In
the former case I might be conceived as a human being,
possessing the human powers of thinking and reckoning,
but kept in a dark world with nothing to mark the pro
gress of existence. The latter is the case of brute ani
mals, which see the sun and moon, but do not know how
many days make a month or a year, because they have
not human powers of thinking and reckoning.

The two elements which are essential to our know
ledge in the above cases, are necessary to human know
ledge in all cases. In all cases, Knowledge implies a
combination of Thoughts and Things. Without this
combination, it would not be Knowledge. Without
Thoughts, there could be no connexion ; without Things,
there could be no reality. Thoughts and Things are so
intimately combined in our Knowledge, that we do not
look upon them as distinct. One single act of the mind
involves them both ; and their contrast disappears in
their union.

But though Knowledge requires the union of these
two elements, Philosophy requires the separation of
them, in order that the nature and structure of Know
ledge may be seen. Therefore I begin by considering
this separation. And I now proceed to speak of another
way of looking at the antithesis of which I have spoken ;

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 19

and which I may, for the reasons which I have just
mentioned, call the FUNDAMENTAL ANTITHESIS OF PHI
LOSOPHY.

SECT. 2. Necessary and Experiential Truths.

MOST persons are familiar with the distinction of ne
cessary and contingent truths. The former kind are
Truths which cannot but be true; as that 19 and 11
make 30 ; that parallelograms upon the same base and
between the same parallels are equal: that all the
angles in the same segment of a circle are equal. The
latter are Truths which it happens (contingit) are true ;
but which, for any thing which we can see, might have
been otherwise ; as that a lunar month contains 30 days,
or that the stars revolve in circles round the pole. The
latter kind of Truths are learnt by experience, and hence
we may call them Truths of Experience, or, for the sake
of convenience, Experiential Truths, in contrast with
Necessary Truths.

Geometrical propositions are the most manifest ex
amples of Necessary Truths. All persons who have read
and understood the elements of geometry, know that the
propositions above stated (that parallelograms upon the
same base and between the same parallels are equal ;
that all the angles in the same segment of a circle are
equal,) are necessarily true ; not only they are true, but
they must be true. The meaning of the terms being
understood, and the proof being gone through, the truth
of the propositions must be assented to. We learn these
propositions to be true by demonstrations deduced from
definitions and axioms ; and when we have thus learnt
them, we see that they could not be otherwise. In the
same manner, the truths which concern numbers are
necessary truths: 19 and 11 not only do make 30, but
must make that number, and cannot make anything else.

20 OF IDEAS IN GENERAL.

In the same manner, it is a necessary truth that half the
sum of two numbers added to half their difference is
equal to the greater number.

It is easy to find examples of Experiential Truths ;
propositions which we know to be true, but know by
experience only. We know, in this way, that salt will
dissolve in water ; that plants cannot live without light ;
in short, we know in this way all that we do know
in chemistry, physiology, and the material sciences in
general. I take the Sciences as my examples of human
knowledge, rather than the common truths of daily life,
or moral or political truths ; because, though the latter
are more generally interesting, the former are much
more definite and certain, and therefore better starting-
points for our speculations, as I have already said. And
we may take elementary astronomical truths as the most
familiar examples of Experiential Truths in the domain
of science.

With these examples, the distinction of Necessary
and Experiential Truths is, I hope, clear. The former
kind, we see to be true by thinking about them, and see
that they could not be otherwise. The latter kind, men
could never have discovered to be true without looking
at them ; and having so discovered them, still no one will
pretend to say they might not have been otherwise. For
aught we can see, the astronomical truths which express
the motions and periods of the sun, moon and stars,
might have been otherwise. If we had been placed in
another part of the solar system, our experiential truths
respecting days, years, and the motions of the heavenly
bodies, would have been other than they are, as we
know from astronomy itself.

It is evident that this distinction of Necessary and
Experiential Truths involves the same antithesis which
we have already considered ; the antithesis of Thoughts

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 21

and Things. Necessary Truths are derived from our own
Thoughts : Experiential Truths are derived from our
observation of Things about us. The opposition of
Necessary and Experiential Truths is another aspect of
the Fundamental Antithesis of Philosophy.

SECT. 3. Deduction and Induction.

I HAVE already stated that geometrical truths are
established by demonstrations deduced from definitions
and axioms. The term Deduction is specially applied
to such a course of demonstration of truths from defini
tions and axioms. In the case of the parallelograms
upon the same base and between the same parallels, we
prove certain triangles to be equal, by supposing them
placed so that their two bases have the same extremi
ties; and hence, referring to an Axiom respecting straight
lines, we infer that the bases coincide. We combine
these equal triangles with other equal spaces, and in this
way make up both the one and the other of the paral
lelograms, in such a manner as to shew that they are
equal. In this manner, going on step by step, deducing
the equality of the triangles from the axiom, and the
equality of the parallelograms from that of the triangles,
we travel to the conclusion. And this process of suc
cessive deduction is the scheme of all geometrical proof.
We begin with Definitions of the notions which we reason
about, and with Axioms, or self-evident truths, respecting
these notions; and we get, by reasoning from these, other
truths which are demonstratively evident; and from
these truths again, others of the same kind, and so on.
We begin with our own Thoughts, which supply us with
Axioms to start from; and we reason from these, till we
come to propositions which are applicable to the Things
about us; as for instance, the propositions respecting
circles and spheres are applicable to the motions of the

22 OF IDEAS IN GENERAL.

heavenly bodies. This is Deduction, or Deductive Rea
soning.

Experiential truths are acquired in a very different
way. In order to obtain such truths, we begin with
Things. In order to learn how many days there are in
a year, or in a lunar month, we must begin by observing
the sun and the moon. We must observe their changes
day by day, and try to make the cycle of change fit into
some notion of number which we supply from our own
Thoughts. We shall find that a cycle of 30 days nearly
will fit the changes of phase of the moon; that a cycle
of 365 days nearly will fit the changes of daily motion
of the sun. Or, to go on to experiential truths of
which the discovery comes within the limits of the his
tory of science we shall find (as Hipparchus found)
that the unequal motion of the sun among the stars,
such as observation shews it to be, may be fitly repre
sented by the notion of an eccentric; a circle in which
the sun has an equable annual motion, the spectator not
being in the center of the circle. Again, in the same
manner, at a later period, Kepler started from more
exact observations of the sun, and compared them with
a supposed motion in a certain ellipse; and was able to
shew that, not a circle about an eccentric point, but an
ellipse, supplied the mode of conception which truly
agreed with the motion of the sun about the earth ; or
rather, as Copernicus had already shewn, of the earth
about the sun. In such cases, in which truths are ob
tained by beginning from observation of external things
and by finding some notion with which the Things, as
observed, agree, the truths are said to be obtained by
Induction. The process is an Inductive Process.

The contrast of the Deductive and Inductive process
is obvious. In the former, we proceed at each step
from general truths to particular applications of them ;

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 23

in the latter, from particular observations to a general
truth which includes them. In the former case we
may be said to reason downwards, in the latter case,
upwards; for general notions are conceived as stand
ing above particulars. Necessary truths are proved,
like arithmetical sums, by adding together the portions
of which they consist. An inductive truth is proved,
like the guess which answers a riddle, by its agreeing
with the facts described. Demonstation is irresistible
in its effect on the belief, but does not produce surprize,
because all the steps to the conclusion are exhibited,
before we arrive at the conclusion. Inductive infer
ence is not demonstrative, but it is often more striking
than demonstrative reasoning, because the intermediate
links between the particulars and the inference are not
shown. Deductive truths are the results of relations
among our own Thoughts. Inductive Truths are re
lations which we discern among existing Things; and
thus, this opposition of Deduction and Induction is again
an aspect of the Fundamental Antithesis already spoken
of.

SECT. 4. Theories and Facts.

GENERAL experiential Truths, such as we have just
spoken of, are called Theories, and the particular
observations from which they are collected, and which
they include and explain, are called Facts. Thus Hip-
parchus s doctrine, that the sun moves in an eccentric
about the earth, is his Theory of the Sun, or the Eccen
tric Theory. The doctrine of Kepler, that the Earth
moves in an Ellipse about the Sun, is Kepler s Theory
of the Earth, the Elliptical Theory. Newton s doctrine
that this elliptical motion of the Earth about the Sun
is produced and governed by the Sun s attraction upon
the Earth, is the Newtonian theory, the Theory of
Attraction. Each of these Theories was accepted, be-

24 OF IDEAS IN GENERAL.

cause it included, connected and explained the Facts;
the Facts being, in the two former cases, the motions
of the Sun as observed; and in the other case, the ellip
tical motion of the Earth as known by Kepler s Theory.
This antithesis of Theory and Fact is included in what
has just been said of Inductive Propositions. A Theory
is an Inductive Proposition, and the Facts are the par
ticular observations from which, as I have said, such
Propositions are inferred by Induction. The Antithesis
of Theory and Fact implies the fundamental Antithesis
of Thoughts and Things; for a Theory (that is, a true
Theory) may be described as a Thought which is con
templated distinct from Things and seen to agree with
them; while a Fact is a combination of our Thoughts
with Things in so complete agreement that we do not
regard them as separate.

Thus the antithesis of Theory and Fact involves the
antithesis of Thoughts and Things, but is not identical
with it. Facts involve Thoughts, for we know Facts only
by thinking about them. The Fact that the year consists
of 365 days; the Fact that the month consists of 30 days,
cannot be known to us, except we have the Thoughts
of Time, Number and Recurrence. But these Thoughts
are so familiar, that we have the Fact in our mind
as a simple Thing without attending to the Thought
which it involves. When we mould our Thoughts into a
Theory, we consider the Thought as distinct from the
Facts; but yet, though distinct, not independent of them;
for it is a true Theory, only by including and agreeing
with the Facts.

SECT. 5. Ideas and Sensations.

WE have just seen that the antithesis of Theory and
Fact, although it involves the antithesis of Thoughts and
Things, is not identical with it. There are other modes

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 25

of expression also, which involve the same Fundamental
Antithesis, more or less modified. Of these, the pair of
words which in their relations appear to separate the
members of the antithesis most distinctly are Ideas and
Sensations. We see and hear and touch external things,
and thus perceive them by our senses; but in perceiving
them, we connect the impressions of sense according to
relations of space, time, number, likeness, cause, &c.
Now some at least of these kinds of connexion, as space,
time, number, may be contemplated distinct from the
things to which they are applied; and so contemplated,
I term them Ideas. And the other element, the impres
sions upon our senses which they connect, are called
Sensations.

I term space, time, cause, &c., Ideas, because they
are general relations among our sensations, apprehend
ed by an act of the mind, not by the senses simply.
These relations involve something beyond what the
senses alone could furnish. By the sense of sight we
see various shades and colours and shapes before us, but
the outlines by which they are separated into distinct
objects of definite forms, are the work of the mind itself.
And again, when we conceive visible things, not only as
surfaces of a certain form, but as solid bodies, placed at
various distances in space, we again exert an act of the
mind upon them. When we see a body move, we see
it move in a path or orbit, but this orbit is not itself
seen; it is constructed by the mind. In like manner
when we see the motions of a needle towards a mag
net, we do not see the attraction or force which pro
duces the effects; but we infer the force, by having in
our minds the Idea of Cause. Such acts of thought,
such Ideas, enter into our perceptions of external things.

But though our perceptions of external things in
volve some act of the mind, they must involve some-

26 OF IDEAS IN GENERAL.

thing else besides an act of the mind. If we must exer
cise an act of thought in order to see force exerted, or
orbits described by bodies in motion, or even in order
to see bodies existing in space, and to distinguish one
kind of object from another, still the act of thought
alone does not make the bodies. There must be some
thing besides, on which the thought is exerted. A
colour, a form, a sound, are not produced by the mind,
however they may be moulded, combined, and inter
preted by our mental acts. A philosophical poet has
spoken of

All the world

Of eye and ear, both what they half create,
And what perceive.

But it is clear, that though they half create, they do not
wholly create : there must be an external world of colour
and sound to give impressions to the eye and ear, as
well as internal powers by which we perceive what is
offered to our organs. The mind is in some way passive
as well as active: there are objects without as well as
faculties within; Sensations, as well as acts of Thought.
Indeed this is so far generally acknowledged, that
according to common apprehension, the mind is passive
rather than active in acquiring the knowledge which
it receives concerning the material world. Its sensa
tions are generally considered more distinct than its
operations. The world without is held to be more clearly
real than the faculties within. That there is some
thing different from ourselves, something external to us,
something independent of us, something which no act
of our minds can make or can destroy, is held by all
men to be at least as evident, as that our minds can
exert any effectual process in modifying and appreciating
the impressions made upon them. Most persons are
more likely to doubt whether the mind be always actively

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 27

applying Ideas to the objects which it perceives, than
whether it perceive them passively by means of Sen
sations.

But yet a little consideration will show us that an
activity of the mind, and an activity according to certain
Ideas, is requisite in all our knowledge of external
objects. We see objects, of various solid forms, and at
various distances from us. But we do not thus perceive
them by sensation alone. Our visual impressions can
not, of themselves, convey to us a knowledge of solid
form, or of distance from us. Such knowledge is inferred
from what we see : inferred by conceiving the objects
as existing in space, and by applying to them the Idea of
Space. Again : day after day passes, till they make up a
year : but we do not know that the days are 365, except
we count them; and thus apply to them our Idea of Num
ber. Again : we see a needle drawn to a magnet : but,
in truth, the drawing is what we cannot see. We see the
needle move, and infer the attraction, by applying to the
fact our Idea of Force, as the cause of motion. Again:
we see two trees of different kinds ; but we cannot know
that they are so, except by applying to them our Idea
of the resemblance and difference which makes kinds.
And thus Ideas, as well as Sensations, necessarily enter
into all our knowledge of objects : and these two words
express, perhaps more exactly than any of the pairs
before mentioned, that Fundamental Antithesis, in the
union of which, as I have said, all knowledge consists.

SECT 6. Reflexion and Sensation.

IT will hereafter be my business to show what the
Ideas are, which thus enter into our knowledge; and
how each Idea has been, as a matter of historical fact,
introduced into the Science to which it especially be
longs. But before I proceed to do this, I will notice

28 OF IDEAS IN GENERAL.

some other terms, besides the phrases already noticed,
which have a reference, more or less direct, to the Funda
mental Antithesis of Ideas and Sensations. I will mention
some of these, in order that if they should come under
the reader s notice, he may not be perplexed as to their
bearing upon the view here presented to him.

The celebrated doctrine of Locke, that all our
" Ideas," (that is, in his use of the word, all our objects
of thinking,) come from Sensation or Reflexion, will
naturally occur to the reader as connected with the
antithesis of which I have been speaking. But there is
a great difference between Locke s account of Sensation
and Reflexion, and our view of Sensation and Ideas. He
is speaking of the origin of our knowledge ; we, of its
nature and composition. He is content to say that all
the knowledge which we do not receive directly by
Sensation, we obtain by Reflex Acts of the mind, which
make up his Reflexion. But we hold that there is no
Sensation without an act of the mind, and that the
mind s activity is not only reflexly exerted upon itself,
but directly upon objects, so as to perceive in them con
nexions and relations which are not Sensations. He is
content to put together, under the name of Reflexion,
everything in our knowledge which is not Sensation : we
are to attempt to analyze all that is not Sensation ; not
only to say it consists of Ideas, but to point out what
those Ideas are, and to show the mode in which each of
them enters into our knowledge. His purpose was, to
prove that there are no Ideas, except the reflex acts of
the mind : our endeavour will be to show that the acts of
the mind, both direct and reflex, are governed by certain
Laws, which may be conveniently termed Ideas. His
procedure was, to deny that any knowledge could be
derived from the mind alone : our course will be, to
show that in every part of our most certain and exact

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 29

knowledge, those who have added to our knowledge in
every age have referred to principles which the mind
itself supplies. I do not say that my view is contrary to
his : but it is altogether different from his. If I grant
that all our knowledge comes from Sensation and Re
flexion, still my task then is only begun; for I want
further to determine, in each science, what portion
comes, not from mere Sensation, but from those Ideas
by the aid of which either Sensation or Reflexion can
lead to Science.

Locke s use of the word "idea" is, as the reader will
perceive, different from ours. He uses the word, as he
says, which " serves best to stand for whatsoever is the
object of the understanding when a man thinks." " I
have used it," he adds, " to express whatever is meant by
phantasm, notion, species, or whatever it is to which the
mind can be employed about in thinking." It might be
shown that this separation of the mind itself from the
ideal objects about which it is employed in thinking, may
lead to very erroneous results. But it may suffice to ob
serve that we use the word Ideas, in the manner already
explained, to express that element, supplied by the mind
itself, which must be combined with Sensation in order
to produce knowledge. For us, Ideas are not Objects of
Thought, but rather Laws of Thought. Ideas are not
synonymous with Notions; they are Principles which
give to our Notions whatever they contain of truth. But
our use of the term Idea will be more fully explained
hereafter.

SECT. 7 Subjective and Objective.

THE Fundamental Antithesis of Philosophy of which I
have to speak has been brought into great prominence
in the writings of modern German philosophers, and has
conspicuously formed the basis of their systems. They

30 OF IDEAS IN GENERAL.

have indicated this antithesis by the terms subjective and
objective. According to the technical language of old
writers, a thing and its qualities are described as subject
and attributes ; and thus a man s faculties and acts are
attributes of which he is the subject. The mind is the
subject in which ideas inhere. Moreover, the man s
faculties and acts are employed upon external objects;
and from objects all his sensations arise. Hence the
part of a man s knowledge which belongs to his own
mind, is subjective: that which flows in upon him from
the world external to him, is objective. And as in man s
contemplation of nature, there is always some act of
thought which depends upon himself, and some matter
of thought which is independent of him, there is, in every
part of his knowledge, a subjective and an objective
element. The combination of the two elements, the
subjective or ideal, and the objective or observed, is
necessary, in order to give us any insight into the laws of
nature. But different persons, according to their mental
habits and constitution, may be inclined to dwell by
preference upon the one or the other of these two
elements. It may perhaps interest the reader to see
this difference of intellectual character illustrated in two
eminent men of genius of modern times, Gothe and
Schiller.

Gothe himself gives us the account to which I refer,
in his history of the progress of his speculations con
cerning the Metamorphosis of Plants; a mode of viewing
their structure by which he explained, in a very striking
and beautiful manner, the relations of the different parts
of a plant to each other ; as has been narrated in the
History of the Inductive Sciences. Gothe felt a delight
in the passive contemplation of nature, unmingled with
the desire of reasoning and theorizing ; a delight such as
naturally belongs to those poets who merely embody the

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 31

images which a fertile genius suggests, and do not mix
with these pictures, judgments and reflexions of their
own. Schiller, on the other hand, both by his own
strong feeling of the value of a moral purpose in poetry,
and by his adoption of a system of metaphysics in which
the subjective element was made very prominent, was
well disposed to recognize fully the authority of ideas
over external impressions.

Gothe for a time felt a degree of estrangement
towards Schiller, arising from this contrariety in their
views and characters. But on one occasion they fell
into discussion on the study of natural history; and
Gothe endeavoured to impress upon his companion his
persuasion that nature was to be considered, not as com
posed of detached and incoherent parts, but as active
and alive, and unfolding herself in each portion, in
virtue of principles which pervade the whole. Schiller
objected that no such view of the objects of natural
history had been pointed out by observation, the only
guide which the natural historians recommended; and
was disposed on this account to think the whole of their
study narrow and shallow. "Upon this," says Gothe,
" I expounded to him, in as lively a way as I could, the
metamorphosis of plants, drawing on paper for him, as I
proceeded, a diagram to represent that general form of
a plant which shows itself in so many and so various
transformations. Schiller attended and understood; and,
accepting the explanation, he said, This is not observa
tion, but an idea. I replied," adds Gothe, " with some
degree of irritation ; for the point which separated us
was most luminously marked by this expression : but I
smothered my vexation, and merely said, I was happy
to find that I had got ideas without knowing it; nay,
that I saw them before my eyes. : Gothe then goes on
to say, that he had been grieved to the very soul by

32 OF IDEAS IN GENERAL.

maxims promulgated by Schiller, that no observed fact
ever could correspond with an idea. Since he himself
loved best to wander in the domain of external observa
tion, he had been led to look with repugnance and
hostility upon anything which professed to depend upon
ideas. "Yet," he observes, "it occurred to me that if
my Observation was identical with his Idea, there must
be some common ground on which we might meet."
They went on with their mutual explanations, and be
came intimate and lasting friends. "And thus," adds
the poet, " by means of that mighty and interminable
controversy between object and subject, we two concluded
an alliance which remained unbroken, and produced
much benefit to ourselves and others."

The general diagram of a plant, of which Gothe
here speaks, must have been a combination of lines and
marks expressing the relations of position and equiva
lence among the elements of vegetable forms, by which
so many of their resemblances and differences may be
explained. Such a symbol is not an Idea in that general
sense in which we propose to use the term, but is a
particular modification of the general Ideas of symmetry,
developement, and the like ; and we shall hereafter see,
according to the phraseology which we shall explain in
the next chapter, how such a diagram might express
the ideal conception of a plant.

The antithesis of subjective and objective is very
familiar in the philosophical literature of Germany and
France ; nor is it uncommon in any age of our own
literature. But though efforts have recently been made
to give currency among us to this phraseology, it has
not been cordially received, and has been much com
plained of as not of obvious meaning. Nor is the com
plaint without ground : for when we regard the mind as
the subject in which ideas inhere, it becomes for us an

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 33

object, and the antithesis vanishes. We are not so
much accustomed to use subject in this sense, as to
make it a proper contrast to object. The combination
"ideal and objective," would more readily convey to a
modern reader the opposition which is intended between
the ideas of the mind itself, and the objects which it
contemplates around it.

To the antitheses already noticed Thoughts and
Things ; Necessary and Experiential Truths ; Deduction
and Induction ; Theory and Fact ; Ideas and Sensations ;
Reflexion and Sensation ; Subjective and Objective ; we
may add others, by which distinctions depending more
or less upon the fundamental antithesis have been de
noted. Thus we speak of the internal and external
sources of our knowledge ; of the world within and the
world without us ; of Man and Nature. Some of the
more recent metaphysical writers of Germany have
divided the universe into the Me and the Not-me (Ich
and Nicht-ich). Upon such phraseology we may observe,
that to have the fundamental antithesis of which we
speak really understood, is of the highest consequence
to philosophy, but that little appears to be gained by
expressing it in any novel manner. The most weighty
part of the philosopher s task is to analyze the operations
of the mind ; and in this task, it can aid us but little to
call it, instead of the mind, the subject, or the me.

SECT. 8. Matter and Form.

THERE are some other ways of expressing, or rather
of illustrating, the fundamental antithesis, which I may
briefly notice. The antithesis has been at different times
presented by means of various images. One of the most
ancient of these, and one which is still very instructive,
is that which speaks of Sensations as the Matter, and
Ideas as the Form, of our knowledge ; just as ivory is
VOL. i. w. P. D

34 OF IDEAS IN GIONKRAL.

the matter, and a cube the form, of a die. This com
parison has the advantage of showing that two elements
of an antithesis which cannot be separated in fact, may
yet be advantageously separated in our reasonings. For
Matter and Form cannot by any means be detached
from each other. All matter must have some form ; all
form must be the form of some material thing. If the
ivory be not a cube, it must have a spherical or some
other form. And the cube, in order to be a cube, must
be of some material ; if not of ivory, of wood, or stone,
for instance. A figure without matter is merely a geo
metrical conception ; a modification of the idea of
space. Matter without figure is a mere abstract term ;
a supposed union of certain sensible qualities which,
so insulated from others, cannot exist. Yet the distinc
tion of Matter and Form is real ; and, as a subject of
contemplation, clear and plain. Nor is the distinction by
any means useless. The speculations which treat of the
two subjects, Matter and Figure, are very different.
Matter is the subject of the sciences of Mechanics and
Chemistry ; Figure, of Geometry. These two classes of
Sciences have quite different sets of principles. If we
refuse to consider the Matter and the Form of bodies
separately, because we cannot exhibit Matter and Form
separately, we shut the door to all philosophy on such
subjects. In like manner, though Sensations and Ideas
are necessarily united in all our knowledge, they can be
considered as distinct; and this distinction is the basis of
all philosophy concerning knowledge.

This illustration of the relation of Ideas and Sensa
tions may enable us to estimate a doctrine which has been
put forwards at various times. In a certain school of spe
culators there has existed a disposition to derive all our
Ideas from our Sensations, the term Idea being, in this
school, used in its wider sense, so as to include all modifi-

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 35

cations and limitations of our Fundamental Ideas. The
doctrines of this school have been summarily expressed
by saying that " Every Idea is a transformed Sensation."
Now, even supposing this assertion to be exactly true,
we easily see, from what has been said, how little we
are likely to answer the ends of philosophy by putting
forward such a maxim as one of primary importance.
For we might say, in like manner, that every statue is
but a transformed block of marble, or every edifice but
a collection of transformed stones. But what would
these assertions avail us, if our object were to trace the
rules of art by which beautiful statues were formed, or
great works of architecture erected ? The question
naturally occurs, What is the nature, the principle, the
law of this Transformation ? In what faculty resides the
transforming power? What train of ideas of beauty,
and symmetry, and stability, in the mind of the statuary
or the architect, has produced those great works which
mankind look upon as among their most valuable pos
sessions ; the Apollo of the Belvidere, the Parthenon,
the Cathedral of Cologne ? When this is what we want
to know, how are we helped by learning that the Apollo
is of Parian marble, or the Cathedral of basaltic stone ?
We must know much more than this, in order to acquire
any insight into the principles of statuary or of archi
tecture. In like manner, in order that we may make
any progress in the philosophy of knowledge, which is
our purpose, we must endeavour to learn something
further respecting ideas than that they are transformed
sensations, even if they were this.

But, in reality, the assertion that our ideas are trans
formed sensations, is erroneous as well as frivolous. For
it conveys, and is intended to convey, the opinion that
our sensations have one form which properly belongs to
them ; and that, in order to become ideas, they are con-

D 2

36 OF IDEAS IN GENERAL.

verted into some other form. But the truth is, that our
sensations, of themselves, without some act of the mind,
such as involves what we have termed an Idea, have no
form. We cannot see one object without the idea of
space ; we cannot see two without the idea of resem
blance or difference; and space and difference are not
sensations. Thus, if we are to employ the metaphor of
Matter and Form, which is implied in the expression to
which I have referred, our sensations, from their first
reception, have their Form not changed, but given by
our Ideas. Without the relations of thought which we
here term Ideas, the sensations are matter without form.
Matter without form cannot exist : and in like manner
sensations cannot become perceptions of objects, without
some formative power of the mind. By the very act of
being received as perceptions, they have a formative
power exercised upon them, the operation of which
might be expressed, by speaking of them, not as trans
formed, but simply as formed ; as invested with form,
instead of being the mere formless material of percep
tion. The word inform, according to its Latin etymo
logy, at first implied this process by which matter is
invested with form. Thus Virgil* speaks of the thunder
bolt as informed by the hands of Brontes, and Steropes,
and Pyracmon. And Dryden introduces the word in
another place :

Let others better mould the running mass
Of metals, or inform the breathing brass.

Even in this use of the word, the form is something
superior to the brute manner, and gives it a new signi
ficance and purpose. And hence the term is again used

* Ferrum exercebant vasto Cyclopes in Antro

Brontesque Steropesque et nudus membra Pyracmon ;
His informatum manibus, jam parte polita
Fulmen erat. Mn. viii. 424.

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 37

to denote the effect produced by an intelligent principle
of a still higher kind :

He informed

This ill-shaped body with a daring soul.

And finally even the soul itself, in its original condition,
is looked upon as matter, when viewed with reference
to education and knowledge, by which it is afterwards
moulded ; and hence these are, in our language, termed
information. If we confine ourselves to the first of
these three uses of the term, we may correct the erro
neous opinion of which we have just been speaking,
and retain the metaphor by which it is expressed, by
saying, that ideas are not transformed, but informed
sensations.

SECT. 9. Man the Interpreter of Nature.

THERE is another image by which writers have repre
sented the acts of thought through which knowledge is
obtained from the observation of the external world.
Nature is the Book, and Man is the Interpreter. The
facts of the external world are marks, in which man
discovers a meaning, and so reads them. Man is the
Interpreter of Nature, and Science is the right Interpre
tation. And this image also is, in many respects, instruc
tive. It exhibits to us the necessity of both elements ;
the marks which man has to look at, and the knowledge
of the alphabet and language which he must possess and
apply before he can find any meaning in what he sees.
Moreover this image presents to us, as the ideal element,
an activity of the mind of that very kind which we wish
to point out. Indeed the illustration is rather an
example than a comparison of the composition of our
knowledge. The letters and symbols which are pre
sented to the Interpreter are really objects of sensation :
the notion of letters as signs of words, the notion of

38 OF IDEAS IN GENERAL.

connexions among words by which they have meaning,
really are among our Ideas ; Signs and Meaning are
Ideas, supplied by the mind, and added to all that sensa
tion can disclose in any collection of visible marks. The
Sciences are not figuratively, but really, Interpretations
of Nature. But this image, whether taken as example or
comparison, may serve to show both the opposite charac
ter of the two elements of knowledge, and their neces
sary combination, in order that there may be knowledge.
This illustration may also serve to explain another
point in the conditions of human knowledge which we
shall have to notice : namely, the very different degrees
in which, in different cases, we are conscious of the
mental act by which our sensations are converted into
knowledge. For the same difference occurs in reading
an inscription. If the inscription were entire and plain,
in a language with which we were familiar, we should
be unconscious of any mental act in reading it. We
should seem to collect its meaning by the sight alone.
But if we had to decipher an ancient inscription, of
which only imperfect marks remained, with a few entire
letters among them, we should probably make several
suppositions as to the mode of reading it, before we
found any mode which was quite successful ; and thus,
our guesses, being separate from the observed facts, and
at first not fully in agreement with them, we should be
clearly aware that the conjectured meaning, on the one
hand, and the observed marks on the other, were dis
tinct things, though these two things would become
united as elements of one act of knowledge when we
had hit upon the right conjecture.

SECT. 10. The Fundamental Antithesis inseparable.

THE illustration just referred to, as well as other
ways of considering the subject, may help us to get over

FUNDAMENTAL ANTITHESIS OF J HILOSOPIl Y. 30

a difficulty which at first sight appears perplexing. We
have spoken of the common opposition of Theory and
Fact as important, and as involving what we have called
the Fundamental Antithesis of Philosophy. But after
all, it may be asked, Is this distinction of Theory and
Fact really tenable? Is it not often difficult to say
whether a special part of our knowledge is a Fact or
a Theory? Is it a Fact or a Theory that the stars
revolve round the pole? Is it a Fact or a Theory that
the earth is a globe revolving on its axis? Is it a Fact
or a Theory that the earth travels in an ellipse round
the sun? Is it a Fact or a Theory that the sun attracts
the earth? Is it a Fact or a Theory that the loadstone
attracts the needle? In all these cases, probably some
persons would answer one way, and some persons the
other. There are many persons by whom the doctrine
of the globular form of the earth, the doctrine of the
earth s elliptical orbit, the doctrine of the sun s attrac
tion on the earth, would be called theories, even if they
allowed them to be true theories. But yet if each of
these propositions be true, is it not &fact? And even
with regard to the simpler facts, as the motion of the
stars round the pole, although this may be a Fact to one
who has watched and measured the motions of the stars,
one who has not done this, and who has only carelessly
looked at these stars from time to time, may naturally
speak of the circles which the astronomer makes them
describe as Theories. It would seem, then, that we
cannot in such cases expect general assent, if we say,
This is a Fact and not a Theory, or, This is a Theory
and not a Fact. And the same is true in a vast range
of cases. It would seem, therefore, that we cannot rest
any reasoning upon this distinction of Theory and Fact:
and we cannot avoid asking whether there is any real
distinction in this antithesis, and if so, what it is.

40 OF IDEAS IN GENERAL.

To this I reply : the distinction between Theory
(that is, true Theory) and Fact, is this: that in Theory
the Ideas are considered as distinct from the Facts: in
Facts, though Ideas may be involved, they are not, in
our apprehension, separated from the sensations. In a
Fact, the Ideas are applied so readily and familiarly, and
incorporated with the sensations so entirely, that we
do not see them, we see through them. A person who
carefully notes the motion of a star all night, sees the
circle which it describes, as he sees the star, though
the circle is, in fact, a result of his own Ideas. A
person who has in his mind the measures of different
lines and countries on the earth s surface, and who can
put them together into one conception, finds that they
can make no figure but a globular one: to him, the
earth s globular form is a Fact, as much as the square
form of his chamber. A person to whom the grounds
of believing the earth to travel round the sun are as
familiar as the grounds for believing the movements
of the mail-coaches in this country, looks upon the
former event as a Fact, just as he looks upon the latter
events as Facts. And a person who, knowing the Fact
of the earth s annual motion, refers it distinctly to its
mechanical cause, conceives the sun s attraction as a
Fact, just as he conceives as a Fact, the action of the
wind which turns the sails of a mill. He cannot see
the force in either case ; he supplies it out of his own
Ideas. And thus, a true Theory is a Fact; a Fact is
a familiar Theory. That which is a Fact under one
aspect, is a Theory under another. The most recondite
Theories when firmly established are Facts: the sim
plest Facts involve something of the nature of Theory.
Theory and Fact correspond, in a certain degree, with
Ideas and Sensations, as to the nature of their opposi
tion. But the Facts are Facts, so far as the Ideas have

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 41

been combined with the Sensations and absorbed in
them: the Theories are Theories, so far as the Ideas
are kept distinct from the Sensations, and so far as it is
considered still a question whether those can be made
to agree with these.

We may, as I have said, illustrate this matter by
considering man as interpreting the phenomena which
he sees. He often interprets without being aware that
he does so. Thus when we see the needle move towards
the magnet, we assert that the magnet exercises an
attractive force on the needle. But it is only by an
interpretative act of our own minds that we ascribe
this motion to attraction. That, in this case, a force is
exerted something of the nature of the pull which we
could apply by our own volition is our interpretation
of the phenomena; although we may be conscious of the
act of interpretation, and may then regard the attrac
tion as a Fact.

Nor is it in such cases only that we interpret phe
nomena in our own way, without being conscious of
what we do. We see a tree at a distance, and judge it
to be a chestnut or a lime ; yet this is only an inference
from the colour or form of the mass according to pre
conceived classifications of our own. Our lives are full
of such unconscious interpretations. The farmer recog
nizes a good or a bad soil ; the artist a picture of a
favourite master ; the geologist a rock of a known local
ity, as we recognize the faces and voices of our friends ;
that is, by judgments formed on what we see and hear ;
but judgments in which we do not analyze the steps, or
distinguish the inference from the appearance. And in
these mixtures of observation and inference, we speak of
the judgment thus formed, as a Fact directly observed.

Even in the case in which our perceptions appear to
be most direct, and least to involve any interpretations

42 OF IDEAS IN GENERAL.

of our own, in the simple process of seeing, who does
not know how much we, by an act of the mind, add to
that which our senses receive ? Does any one fancy that
he sees a solid cube? It is easy to show that the solid
ity of the figure, the relative position of its faces and
edges to each other, are inferences of the spectator ; no
more conveyed to his conviction by the eye alone, than
they would be if he were looking at a painted represen
tation of a cube. The scene of nature is a picture with
out depth of substance, no less than the scene of art ;
and in the one case as in the other, it is the mind which,
by an act of its own, discovers that colour and shape
denote distance and solidity. Most men are unconscious
of this perpetual habit of reading the language of the
external world, and translating as they read. The
draughtsman, indeed, is compelled, for his purposes, to
return back in thought from the solid bodies which he
has inferred, to the shapes of surface which he really
sees. He knows that there is a mask of theory over the
whole face of nature, if it be theory to infer more than
we see. But other men, unaware of this masquerade,
hold it to be a fact that they see cubes and spheres, spa
cious apartments and winding avenues. And these things
are facts to them, because they are unconscious of the
mental operation by which they have penetrated nature s

disguise.

And thus, we still have an intelligible distinction of
Fact and Theory, if we consider Theory as a conscious, and
Fact as an unconscious inference, from the phenomena
which are presented to our senses.

But still, Theory and Fact, Inference and Perception,
Reasoning and Observation, are antitheses in none of
which can we separate the two members by any fixed
and definite line.

Even the simplest terms by which the antithesis is

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 43

expressed cannot be separated. Ideas and Sensations,
Thoughts and Things, Subject and Object, cannot in any
case be applied absolutely and exclusively. Our Sen
sations require Ideas to bind them together, namely,
Ideas of space, time, number, and the like. If not so
bound together, Sensations do not give us any appre
hension of Things or Objects. All Things, all Objects,
must exist in space and in time must be one or many.
Now space, time, number, are not Sensations or Things.
They are something different from, and opposed to Sen
sations and Things. We have termed them Ideas. It
may be said they are Relations of Things, or of Sensa
tions. But granting this form of expression, still a
Relation is not a Thing or a Sensation ; and therefore
we must still have another and opposite element, along
with our Sensations. And yet, though we have thus
these two elements in every act of perception, we cannot
designate any portion of the act as absolutely and exclu
sively belonging to one of the elements. Perception
involves Sensation, along with Ideas of time, space, and
the like ; or, if any one prefers the expression, we may
say, Perception involves Sensations along with the ap
prehension of Relations. Perception is Sensation, along
with such Ideas as make Sensation into an apprehension
of Things or Objects.

And as Perception of Objects implies Ideas, as Ob
servation implies Reasoning; so, on the other hand,
Ideas cannot exist where Sensation has not been ; Rea
soning cannot go on when there has riot been previous
Observation. This is evident from the necessary order
of developement of the human faculties. Sensation
necessarily exists from the first moments of our exist
ence, and is constantly at work. Observation begins
before we can suppose the existence of any Reasoning
which is not involved in Observation. Hence, at what-

44 OF IDEAS IN GENERAL.

ever period we consider our Ideas, we must consider
them as having been already engaged in connecting our
Sensations, and as having been modified by this employ
ment. By being so employed, our Ideas are unfolded
and defined ; and such developement and definition can
not be separated from the Ideas themselves. We cannot
conceive space, without boundaries or forms ; now Forms
involve Sensations. We cannot conceive time, without
events which mark the course of time ; but events involve
Sensations. We cannot conceive number, without con
ceiving things which are numbered ; and Things imply
sensations. And the forms, things, events, which are
thus implied in our Ideas, having been the objects of
Sensation constantly in every part of our life, have
modified, unfolded, and fixed our Ideas, to an extent
which we cannot estimate, but which we must suppose
to be essential to the processes which at present go on
in our minds. We cannot say that Objects create Ideas ;
for to perceive Objects we must already have Ideas.
But we may say, that Objects and the constant Perception
of Objects have so far modified our Ideas, that we cannot,
even in thought, separate our Ideas from the perception
of Objects.

We cannot say of any Ideas, as of the Idea of space,
or time, or number, that they are absolutely and exclu
sively Ideas. We cannot conceive what space, or time,
or number, would be in our minds, if we had never per
ceived any Thing or Things in space or time. We can
not conceive ourselves in such a condition as never to have
perceived any Thing or Things in space or time. But, on
the other hand, just as little can we conceive ourselves
becoming acquainted with space and time or numbers
as objects of Sensation. We cannot reason without
having the operations of our minds affected by previous
Sensations ; but we cannot conceive Reasoning to be

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 45

merely a series of Sensations. In order to be used in
Reasoning, Sensation must become Observation ; and, as
we have seen, Observation already involves Reasoning.
In order to be connected by our Ideas, Sensations must
be Things or Objects, and Things or Objects already in
clude Ideas. And thus, none of the terms by which the
fundamental antithesis is expressed can be absolutely
and exclusively applied.

I will make a remark suggested by the views which
have thus been presented. Since, as we have just seen,
none of the terms which express the fundamental anti
thesis can be applied absolutely and exclusively, the
absolute application of the antithesis in any particular
case can never be a conclusive or immoveable principle.
This remark is the more necessary to be borne in mind, as
the terms of this antithesis are often used in a vehement
and peremptory manner. Thus we are often told that
such a thing is a Fact; A FACT and not a Theory, with all
the emphasis which, in speaking or writing, tone or italics
or capitals can give. We see from what has been said,
that when this is urged, before we can estimate the
truth, or the value of the assertion, we must ask to
whom is it a Fact? what habits of thought, what pre
vious information, what Ideas does it imply, to conceive
the Fact as a Fact ? Does not the apprehension of the
Fact imply assumptions which may with equal justice
be called Theory, and which are perhaps false Theory ?
in which case, the Fact is no Fact. Did not the an
cients assert it as a Fact, that the earth stood still,
and the stars moved ? and can any Fact have stronger
apparent evidence to justify persons in asserting it em
phatically than this had ?

These remarks are by no means urged in order to
shew that no Fact can be certainly known to be true ;
but only, to shew that no Fact can be certainly shown

46 OF IDEAS IN GENERAL.

to be a Fact, merely by calling it a Fact, however
emphatically. There is by no means any ground of
general skepticism with regard to truth, involved in
the doctrine of the necessary combination of two ele
ments in all our knowledge. On the contrary, Ideas
are requisite to the essence, and Things to the reality
of our knowledge in every case. The proportions of
Geometry and Arithmetic are examples of knowledge
respecting our Ideas of space and number, with regard
to which there is no room for doubt. The doctrines of
Astronomy are examples of truths not less certain
respecting the Facts of the external world.

SECT. 11. Successive Generalization.

IN the preceding pages we have been led to the doctrine,
that though, in the Antithesis of Theory and Fact, there
is involved an essential opposition ; namely the opposition
of the thoughts within us and the phenomena without
us ; yet that we cannot distinguish and define the mem
bers of this antithesis separately. Theories become
Facts, by becoming certain and familiar : and thus, as
our knowledge becomes more sure and more extensive,
we are constantly transferring to the class of facts,
opinions which were at first regarded as theories.

Now we have further to remark, that in the progress
of human knowledge respecting any branch of specula
tion, there may be several such steps in succession, each
depending upon and including the preceding. The
theoretical views which one generation of discoverers
establishes, become the facts from which the next gene
ration advances to new theories. As men rise from the
particular to the general, so, in the same manner, they
rise from what is general to what is more general. Each
induction supplies the materials of fresh inductions ;
each generalization, with all that it embraces in its circle.

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 47

may be found to be but one of many circles, compre
hended within the circuit of some wider generalization.

This remark has already been made, and illustrated,
in the History of the Inductive Sciences* ; and, in truth,
the whole of the history of science is full of suggestions
and exemplifications of this course of things. It may be
convenient, however, to select a few instances which may
further explain and confirm this view of the progress of
scientific knowledge.

The most conspicuous instance of this succession is
to be found in that science which has been progressive
from the beginning of the world to our own times, and
which exhibits by far the richest collection of successive
discoveries : I mean Astronomy. It is easy to see that
each of these successive discoveries depended on those
antecedently made, and that in each, the truths which
were the highest point of the knowledge of one age
were the fundamental basis of the efforts of the age
which came next. Thus we find, in the days of Greek
discovery, Hipparchus and Ptolemy combining and ex
plaining the particular facts of the motion of the sun,
moon, and planets, by means of the theory of epicycles
and eccentrics ; a highly important step, which gave
an intelligible connexion and rule to the motions of each
of these luminaries. When these cycles and epicycles,
thus truly representing the apparent motions of the
heavenly bodies, had accumulated to an inconvenient
amount, by the discovery of many inequalities in the
observed motions, Copernicus showed that their effects
might all be more simply included, by making the sun
the center of motion of the planets, instead of the earth.
But in this new view, he still retained the epicycles and
eccentrics which governed the motion of each body.
Tycho Brahe s observations, and Kepler s calculations,

* Hist. Inductive Sciences, B. vn c. ii. Sect. a.

48 OF IDEAS IN GENERAL.

showed that, besides the vast number of facts which the
epicyclical theory could account for, there were some
which it would not exactly include, and Kepler was led
to the persuasion that the planets move in ellipses.
But this view of motion was at first conceived by Kepler
as a modification of the conception of epicycles. On one
occasion he blames himself for not sooner seeing that
such a modification was possible. " What an absurdity
on my part !" he cries* ; " as if libration in the diameter
of the epicycle might not come to the same thing as
motion in the ellipse." But again; Kepler s laws of the
elliptical motion of the planets were established; and
these laws immediately became the facts on which the
mathematicians had to found their mechanical theories.
From these facts, Newton, as we have related, proved
that the central force of the sun retains the planets in
their orbits, according to the law of the inverse square
of the distance. The same law was shown to prevail in
the gravitation of the earth. It was shown, too, by in
duction from the motions of Jupiter and Saturn, that
the planets attract each other ; by calculations from the
figure of the earth, that the parts of the earth attract
each other ; and, by considering the course of the tides,
that the sun and moon attract the waters of the ocean.
And all these curious discoveries being established as
facts, the subject was ready for another step of gene
ralization. By an unparalleled rapidity in the progress
of discovery in this case, not only were all the inductions
which we have first mentioned made by one individual,
but the new advance, the higher flight, the closing vic
tory, fell to the lot of the same extraordinary person.

The attraction of the sun upon the planets, of the
moon upon the earth, of the planets on each other, of the
parts of the earth on themselves, of the sun and moon

* Hif&gt;t. Inductive Sciences, B. v. c. iv. Sect. 3.

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 49

upon the ocean; all these truths, each of itself a great
discovery, were included by Newton in the higher gene
ralization^ of the universal gravitation of matter, by
which each particle is drawn to each other according to
the law of the inverse square : and thus this long ad
vance from discovery to discovery, from truths to truths,
each justly admired when new, and then rightly used as
old, was closed in a worthy and consistent manner, by
a truth which is the most worthy admiration, because it
includes all the researches of preceding ages of Astro
nomy.

We may take another example of a succession of this
kind from the history of a science, which, though it has
made wonderful advances, has not yet reached its goal,
as physical astronomy appears to have done, but seems to
have before it a long prospect of future progress. I now
refer to Chemistry, in which I shall try to point out how
the preceding discoveries afforded the materials of the
succeeding; although this subordination and connexion
is, in this case, less familiar to men s minds than in Astro
nomy, and is, perhaps, more difficult to present in a clear
and definite shape. Sylvius saw, in the facts which
occur, when an acid and an alkali are brought together,
the evidence that they neutralize each other. But cases
of neutralization, and acidification, and many other ef
fects of mixture of the ingredients of bodies, being thus
viewed as facts* had an aspect of unity and law given
them by Geoffroy and Bergman*, who introduced the con
ception of the Chemical Affinity or Elective Attraction,
by which certain elements select other elements, as if by
preference. That combustion, whether a chemical union
or a chemical separation of ingredients, is of the same
nature with acidification, was the doctrine of Boccher

* Hixf. Indue/ire Sciences, B. xiv. c. iii.
VOL. I. W. P. E

50 OF IDEAS IN GENERAL.

and Stahl, and was soon established as a truth which
must form a part of every succeeding physical theory.
That the rules of affinity and chemical composition may
include gaseous elements, was established by Black and
Cavendish. And all these truths, thus brought to light
by chemical discoverers, affinity, the identity of acidifi
cation and combustion, the importance of gaseous ele
ments, along with all the facts respecting the weight
of ingredients and compounds which the balance dis
closed, were taken up, connected, and included as
particulars in the oxygen theory of Lavoisier. Again,
the results of this theory, and the quantity of the several
ingredients which entered into each compound (such
results, for the most part, being now no longer mere
theoretical speculations, but recognized facts) were the
particulars from which Dalton derived that wide law of
chemical combination which we term the Atomic Theory.
And this law, soon generally accepted among chemists,
is already in its turn become one of the facts included
in Faraday s Theory of the identity of Chemical Affinity
and Electric Attraction.

It is unnecessary to give further exemplifications of
this constant ascent from one step to a higher; this
perpetual conversion of true theories into the materials
of other and wider theories. It will hereafter be our
business to exhibit, in a more full and formal manner,
the mode in which this principle determines the whole
scheme and structure of all the most exact sciences.
And thus, beginning with the facts of sense, we gradually
climb to the highest forms of human knowledge, and
obtain from experience and observation a vast collection
of the most wide and elevated truths.

There are, however, truths of a very different kind, to
which we must turn our attention, in order to pursue our

FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 51

researches respecting the nature and grounds of our
knowledge. But before we do this, we must notice one
more feature in that progress of science which we have
already in part described.

CHAPTER III.
OF TECHNICAL TERMS.

1 . IT has already been stated that we gather knowledge
from the external world, when we are able to apply, to
the facts which we observe, some ideal conception, which
gives unity and connexion to multiplied and separate
perceptions. We have also shown that our conceptions,
thus verified by facts, may themselves be united and con
nected by a new bond of the same nature ; and that man
may thus have to pursue his way from truth to truth
through a long progression of discoveries, each resting
on the preceding, and rising above it.

Each of these steps, in succession, is recorded, fixed,
and made available, by some peculiar form of words ;
and such words, thus rendered precise in their meaning,
and appropriated to the service of science, we may call
Technical Terms. It is in a great measure by inventing
such Terms that men not only best express the discoveries
they have made, but also enable their followers to become
so familiar with these discoveries, and to possess them
so thoroughly, that they can readily use them in ad
vancing to ulterior generalizations.

Most of our ideal conceptions are described by exact
and constant words or phrases, such as those of which we
here speak. We have already had occasion to employ
many of these. Thus we have had instances of technical
Terms expressing geometrical conceptions, as Ellipsis,

52 OF IDEAS IN GENERAL.

Radius Vector, Axis, Plane, the Proportion of the In
verse Square, and the like. Other Terms have described
mechanical conceptions, as Accelerating Force and
Attraction. Again, chemistry exhibits (as do all sciences)
a series of Terms which mark the steps of our progress.
The views of the first real founders of the science are
recorded by the Terms which are still in use, Neutral
Salts, Affinity, and the like. The establishment of Dai-
ton s theory has produced the use of the word Atom in
a peculiar sense, or of some other word, as Proportion,
in a sense equally technical. And Mr. Faraday has
found it necessary, in order to expound his electro-chemi
cal theory, to introduce such terms as Anode and Cathode,
Anion and Cathwn.

2. I need not adduce any further examples, for my
object at present is only to point out the use and influence
of such language : its rules and principles I shall here
after try, in some measure, to fix. But what we have
here to remark is, the extraordinary degree in which the
progress of science is facilitated, by thus investing each
new discovery with a compendious and steady form of
expression. These terms soon become part of the cur
rent language of all who take an interest in speculation.
However strange they may sound at first, they soon grow
familiar in our ears, and are used without any effort, or
any recollection of the difficulty they once involved. They
become as common as the phrases which express our
most frequent feelings and interests, while yet they have
incomparably more precision than belongs to any terms
which express feelings; and they carry with them, in
their import, the results of deep and laborious trains of
research. They convey the mental treasures of one
period to the generations that follow ; and laden with
this, their precious freight, they sail safely across gulfs
of time in which empires have suffered shipwreck, and

OF TECHNICAL TERMS. 53

the languages of common life have sunk into oblivion.
We have still in constant circulation among us the Terms
which belong to the geometry, the astronomy, the
zoology, the medicine of the Greeks, and the algebra
and chemistry of the Arabians. And we can in an in
stant, by means of a few words, call to our own recollec
tion, or convey to the apprehension of another person,
phenomena and relations of phenomena in optics, mine
ralogy, chemistry, which are so complex and abstruse,
that it might seem to require the utmost subtlety of the
human mind to grasp them, even if that were made the
sole object of its efforts. By this remarkable effect of
Technical Language, we have the results of all the
labours of past times not only always accessible, but so
prepared that we may (provided we are careful in the
use of our instrument) employ what is really useful and
efficacious for the purpose of further success, without
being in any way impeded or perplexed by the length
and weight of the chain of past connexions which we
drag along with us.

By such means, by the use of the Inductive Process,
and by the aid of Technical Terms, man has been con
stantly advancing in the path of scientific truth. In a
succeeding part of this work we shall endeavour to trace
the general rules of this advance, and to lay down the
maxims by which it may be most successfully guided
and forwarded. But in order that we may do this to
the best advantage, we must pursue still further the
analysis of knowledge into its elements ; and this will be
our employment in the first part of the work.

CHAPTER IV.
OF NECESSARY TRUTHS.

1. EVERY advance in human knowledge consists, as
we have seen, in adapting new ideal conceptions to ascer
tained facts, and thus in superinducing the Form upon
the Matter, the active upon the passive processes of our
minds. Every such step introduces into our knowledge
an additional portion of the ideal element, and of those
relations which flow from the nature of Ideas. It is,
therefore, important for our purpose to examine more
closely this element, and to learn what the relations are
which may thus come to form part of our knowledge.
An inquiry into those Ideas which form the foundations
of our sciences ; into the reality, independence, extent,
and principal heads of the knowledge which we thus ac
quire ; is a task on which we must now enter, and
which will employ us for several of the succeeding Books.

In this inquiry our object will be to pass in review all
the most important Fundamental Ideas which our
sciences involve ; and to prove more distinctly in refer
ence to each, what we have already asserted with regard
to all, that there are everywhere involved in our know
ledge acts of the mind as well as impressions of sense ;
and that our knowledge derives, from these acts, a gene
rality, certainty, and evidence which the senses could in
no degree have supplied. But before I proceed to do
this in particular cases, I will give some account of the
argument in its general form.

We have already considered the separation of our
knowledge into its two elements, Impressions of Sense
and Ideas, as evidently indicated by this ; that all know
ledge possesses characters which neither of these ele
ments alone could bestow. Without our ideas, our sen
sations could have no connexion ; without external

OF NECESSARY TRUTHS. 55

impressions, our ideas would have no reality ; and thus
both ingredients of our knowledge must exist.

2. There is another mode in which the distinction of
the two elements of knowledge appears, as I have already
said : (C. I. Sect. 2.) namely in the distinction of neces
sary and contingent or experiential truths. For of these
two classes of truths, the difference arises from this ;
that the one class derives its nature from the one, and
the other from the other, of the two elements of know
ledge. I have already stated briefly the difference of
these two kinds of truths : namely, that the former are
truths which, we see, must be true : the latter are true,
but so far as we can see, might be otherwise. The former
are true necessarily and universally : the latter are learnt
from experience and limited by experience. Now with
regard to the former kind of truths, I wish to show that
the universality and necessity which distinguish them
can by no means be derived from experience ; that these
characters do in reality flow from the ideas which these
truths involve ; and that when the necessity of the truth
is exhibited in the way of logical demonstration, it is
found to depend upon certain fundamental principles,
(Definitions and Axioms,) which may thus be considered
as expressing, in some measure, the essential characters
of our ideas. These fundamental principles I shall after
wards proceed to discuss and to exhibit in each of the
principal departments of science.

I shall begin by considering Necessary Truths more
fully than I have yet done. As I have already said,
necessary truths are those in which we not only learn
that the proposition is true, but see that it must be true ;
in which the negation of the truth is not only false, but
impossible; in which we cannot, even by an effort of
imagination, or in a supposition, conceive the reverse of
that which is asserted.

56 OF IDEAS IN GENERAL.

3. That there are such truths cannot be doubted.
We may take, for example, all relations of number.
Three and Two added together make Five. We cannot
conceive it to be otherwise. We cannot, by any freak
of thought, imagine Three and Two to make Seven.

It may be said that this assertion merely expresses
what we mean by our words ; that it is a matter of defi
nition ; that the proposition is an identical one.

But this is by no means so. The definition of Five
is not Three and Two, but Four and One. How does it
appear that Three and Two is the same number as Four
and One ? It is evident that it is so ; but why is it evi
dent ? not because the proposition is identical ; for if
that were the reason, all numerical propositions must be
evident for the same reason. If it be a matter of defi
nition that 3 and 2 make 5, it must be a matter of defi
nition that 39 and 27 make 66. But who will say that
the definition of 66 is 39 and 27 ? Yet the magnitude
of the numbers can make no difference in the ground of
the truth. How do we know that the product of 13 and
17 is 4 less than the product of 15 and 15? We see
that it is so, if we perform certain operations by the rules
of arithmetic ; but how do we know the truth of the
rules of arithmetic? If we divide 123375 by 987 ac
cording to the process taught us at school, how are we
assured that the result is correct, and that the number
125 thus obtained is really the number of times one
number is contained in the other ?

The correctness of the rule, it may be replied, can be
rigorously demonstrated. It can be shewn that the pro
cess must inevitably give the true quotient.

Certainly this can be shown to be the case. And
precisely because it can be shown that the result must be
true, we have here an example of a necessary truth ; and
this truth, it appears, is not therefore necessary because it

OF NECESSARY TRUTHS. 57

is itself evidently identical, however it may be possible to
prove it by reducing it to evidently identical propositions.
And the same is the case with all other numerical propo
sitions ; for, as we have said, the nature of all of them is
the same.

Here, then, we have instances of truths which are
not only true, but demonstrably and necessarily true.
Now such truths are, in this respect at least, altogether
different from truths, which, however certain they may
be, are learnt to be so only by the evidence of observa
tion, interpreted, as observation must be interpreted, by
our own mental faculties. There is no difficulty in find
ing examples of these merely observed truths. We find
that sugar dissolves in water, and forms a transparent
fluid, but no one will say that we can see any reason
beforehand why the result must be so. We find that all
animals which chew the cud have also the divided hoof;
but could any one have predicted that this would be
universally the case ? or supposing the truth of the rule
to be known, can any one say that he cannot conceive
the facts as occurring otherwise ? Water expands when
it crystallizes, some other substances contract in the same
circumstances ; but can any one know that this will be
so otherwise than by observation ? We have here propo
sitions rigorously true, (we will assume,) but can any
one say they are necessarily true ? These, and the great
mass of the doctrines established by induction, are actual,
but so far as we can see, accidental laws ; results deter
mined by some unknown selection, not demonstrable
consequences of the essence of things, inevitable and
perceived to be inevitable. According to the phrase
ology which has been frequently used by philosophical
writers, they are contingent, not necessary truths.

It is requisite to insist upon this opposition, because 1
no insight can be obtained into the true nature of

58 OF IDEAS IN GENERAL.

knowledge, and the mode of arriving at it, by any one
who does not clearly appreciate the distinction. The
separation of truths which are learnt by observation, and
truths which can be seen to be true by a pure act of
thought, is one of the first and most essential steps in
our examination of the nature of truth, and the mode of
its discovery. If any one does not clearly comprehend
this distinction of necessary and contingent truths, he
will not be able to go along with us in our researches
into the foundations of human knowledge ; nor, indeed,
to pursue with success any speculation on the subject.
But, in fact, this distinction is one that can hardly fail
to be at once understood. It is insisted upon by almost
all the best modern, as well as ancient, metaphysicians*,
as of primary importance. And if any person does not
fully apprehend, at first, the different kinds of truth thus
pointed out, let him study, to some extent, those sciences
which have necessary truth for their subject, as geometry,
or the properties of numbers, so as to obtain a familiar
acquaintance with such truth ; and he will then hardly
fail to see how different the evidence of the propositions
which occur in these sciences, is from the evidence of
the facts which are merely learnt from experience.
That the year goes through its course in 365 days, can
only be known by observation of the sun or stars : that
365 days is 52 weeks and a day, it requires no expe
rience, but only a little thought to perceive. That bees
build their cells in the form of hexagons, we cannot
know without looking at them ; that regular hexagons
may be arranged so as to fill space, may be proved with
the utmost rigour, even if there were not in existence
such a thing as a material hexagon.

4. As I have already said, one mode in which we
may express the difference of necessary truths and truths

* Aristotle, Dr. Whately, Dugald Stewart, &c.

OF NECESSARY TRUTHS. 59

of experience, is, that necessary truths are those of which
we cannot distinctly conceive the contrary. We can
very readily conceive the contrary of experiential truths.
We can conceive the stars moving about the pole or
across the sky in any kind of curves with any velocities ;
we can conceive the moon always appearing during the
whole month as a luminous disk, as she might do if her
light were inherent and not borrowed. But we cannot
conceive one of the parallelograms on the same base and
between the same parallels larger than the other; for
we find that, if we attempt to do this, when we separate
the parallelograms into parts, we have to conceive one
triangle larger than another, both having all their parts
equal ; which we cannot conceive at all, if we conceive
the triangles distinctly. We make this impossibility
more clear by conceiving the triangles to be placed so
that two sides of the one coincide with two sides of the
other ; and it is then seen, that in order to conceive the
triangles unequal, we must conceive the two bases which
have the same extremities both ways, to be different
lines, though both straight lines. This it is impossible
to conceive : we assent to the impossibility as an axiom,
when it is expressed by saying, that two straight lines
cannot inclose a space ; and thus we cannot distinctly
conceive the contrary of the proposition just mentioned
respecting parallelograms.

But it is necessarv. in annlvino- fVnc rKe+i nc tion, to

distinctly
For in a
\ the con-
hey erro-
e. Thus,
1 a means
lied, that
wo given

60 OF IDEAS IN GENERAL.

lines ; a problem which cannot be solved by plane
geometry. Hobbes not only proposed a construction for
this purpose, but obstinately maintained that it was
right, when it had been proved to be wrong. But then,
the discussion showed how indistinct the geometrical
conceptions of Hobbes were ; for when his critics had
proved that one of the lines in his diagram would not
meet the other in the point which his reasoning sup
posed, but in another point near to it ; he maintained, in
reply, that one of these points was large enough to
include the other, so that they might be considered as
the same point. Such a mode of conceiving the oppo
site of a geometrical truth, forms no exception to the
assertion, that this opposite cannot be distinctly con
ceived.

In like manner, the indistinct conceptions of children
and of rude savages do not invalidate the distinction of
necessary and experiential truths. Children and savages
make mistakes even with regard to numbers ; and might
easily happen to assert that 27 and 38 are equal to 63
or 64. But such mistakes cannot make arithmetical
truths cease to be necessary truths. When any person
conceives these numbers and their addition distinctly, by
resolving them into parts, or in any other way, he sees
that their sum is necessarily 65. If, on the ground of
the possibility of children and savages conceiving some
thing different, it be held that this is not a necessary
truth, it must be held on the same ground, that it is not
a necessary truth that 7 and 4 are equal to 11 ; for
children and savages might be found so unfamiliar with
numbers as not to reject the assertion that 7 and 4 are
10, or even that 4 and 3 are 6, or 8. But I suppose
that no persons would on such grounds hold that these
arithmetical truths are truths known only by experi
ence.

OF NECESSARY TRUTHS. 01

f&gt;. I have taken examples of necessary truths from
the properties of number and space; but such truths exist
no less in other subjects, although the discipline of
thought which is requisite to perceive them distinctly,
may not be so usual among men with regard to the
sciences of mechanics and hydrostatics, as it is with
regard to the sciences of geometry and arithmetic. Yet
every one may perceive that there are such truths in
mechanics. If I press the table with my hand, the
table presses my hand with an equal force : here is a
self-evident and necessary truth. In any machine,
constructed in whatever manner to increase the force
which I can exert, it is certain that what I gain in force
I must lose in the velocity which I communicate. This
is not a contingent truth, borrowed from and limited by
observation ; for a man of sound mechanical views applies
it with like confidence, however novel be the construc
tion of the machine. When I come to speak of the ideas
which are involved in our mechanical knowledge, I
may, perhaps, be able to bring more clearly into view
the necessary truth of general propositions on such
subjects. That reaction is equal and opposite to action,
is as necessarily true as that two straight lines cannot
inclose a space ; it is as impossible theoretically to make
a perpetual motion by mere mechanism as to make the
diagonal of a square commensurable with the side.

G. Necessary truths must be universal truths. If any
property belong to a right-angled triangle necessarily, it
must belong to all right-angled triangles. And it shall
be proved in the following Chapter, that truths possess
ing these two characters, of Necessity and Universality,
cannot possibly be the mere results of experience.

CHAPTER V.
OF EXPERIENCE.

1. I HERE employ the term Experience in a more defi
nite and limited sense than that which it possesses in
common usage ; for I restrict it to matters belonging to
the domain of science. In such cases, the knowledge
which we acquire, by means of experience, is of a clear
and precise nature ; and the passions and feelings and
interests, which make the lessons of experience in prac
tical matters so difficult to read aright, no longer disturb
and confuse us. We may, therefore, hope, by attending
to such cases, to learn what efficacy experience really
has, in the discovery of truth.

That from experience (including intentional expe
rience, or observation,} we obtain much knowledge which
is highly important, and which could not be procured
from any other source, is abundantly clear. We have
already taken several examples of such knowledge.
We know by experience that animals which ruminate
are cloven-hoofed ; and we know this in no other man
ner. We know, in like manner, that all the planets and
their satellites revolve round the sun from west to east.
It has been found by experience that all meteoric stones
contain chrome. Many similar portions of our know
ledge might be mentioned.

Now what we have here to remark is this ; that in
no case can experience prove a proposition to be neces
sarily or universally true. However many instances we
may have observed of the truth of a proposition, yet if it be
known merely by observation, there is nothing to assure
us that the next case shall not be an exception to the rule.
If it be strictly true that every ruminant animal yet
known has cloven hoofs, we still cannot be sure that

OF EXPERIENCE. 63

some creature will not hereafter be discovered which has
the first of these attributes without having the other.
When the planets and their satellites, as far as Saturn, had
been all found to move round the sun in one direction,
it was still possible that there might be other such bodies
not obeying this rule ; and, accordingly, when the satel
lites of Uranus were detected, they appeared to offer an
exception of this kind. Even in the mathematical sciences,
we have examples of such rules suggested by experience,
and also of their precariousness. However far they may
have been tested, we cannot depend upon their correct
ness, except we see some reason for the rule. For
instance, various rules have been given, for the purpose
of pointing out prime numbers; that is, those which can
not be divided by any other number. We may try, as
an example of such a rule, this one any odd power of
the number two, diminished by one. Thus the third
power of two, diminished by one, is seven; the fifth
power, diminished by one, is thirty-one; the seventh
power so diminished is one hundred and twenty-seven.
All these are prime numbers : and we might be led to
suppose that the rule is universal. But the next ex
ample shows us the fallaciousness of such a belief. The
ninth power of two, diminished by one, is five hundred
and eleven, which is not a prime, being divisible by seven.
Experience must always consist of a limited number
of observations. And, however numerous these may be,
they can show nothing with regard to the infinite
number of cases in which the experiment has not been
made. Experience being thus unable to prove a fact
to be universal, is, as will readily be seen, still more
incapable of proving a truth to be necessary. Expe
rience cannot, indeed, offer the smallest ground for the
necessity of a proposition. She can observe and record
what has happened ; but she cannot find, in any case, or

64 OF IDEAS IN GENERAL.

in any accumulation of cases, any reason for what wn$t
happen. She may see objects side by side ; but. she
cannot see a reason why they must ever be side by side.
She finds certain events to occur in succession ; but the
succession supplies, in its occurrence, no reason for its
recurrence. She contemplates external objects ; but she
cannot detect any internal bond, which indissolubly
connects the future with the past, the possible with the
real. To learn a proposition by experience, and to see
it to be necessarily true, are two altogether different pro
cesses of thought.

2. But it may be said, that we do learn by means
of observation and experience many universal truths;
indeed, all the general truths of which science consists.
Is not the doctrine of universal gravitation learnt by
experience ? Are not the laws of motion, the properties
of light, the general principles of chemistry, so learnt ?
How, with these examples before us, can we say that
experience teaches no universal truths ?

To this we reply, that these truths can only be
known to be general, not universal, if they depend upon
experience alone. Experience cannot bestow that uni
versality which she herself cannot have, and that necessity
of which she has no comprehension. If these doctrines
are universally true, this universality flows from the ideas
which we apply to our experience, and which are, as we
have seen, the real sources of necessary truth. How far
these ideas can communicate their universality and
necessity to the results of experience, it will hereafter
be our business to consider. It will then appear, that
when the mind collects from observation truths of a wide
and comprehensive kind, which approach to the sim
plicity and universality of the truths of pure science ;
she gives them this character by throwing upon them
the light of her own Fundamental Ideas.

OF EXPERIENCE. 65

But the truths which we discover by observation of
the external world, even when most strikingly simple
and universal, are not necessary truths. Is the doctrine
of universal gravitation necessarily true ? It was doubted
by Clairaut (so far as it refers to the moon), when the
progression of the apogee in fact appeared to be twice
as great as the theory admitted. It has been doubted,
even more recently, with respect to the planets, their
mutual perturbations appearing to indicate a deviation
from the law. It is doubted still, by some persons, with
respect to the double stars. But suppose all these
doubts to be banished, and the law to be universal ; is it
then proved to be necessary ? Manifestly not : the very
existence of these doubts proves that it is not so. For
the doubts were dissipated by reference to observation
and calculation, not by reasoning on the nature of the
law. Clairaut s difficulty was removed by a more exact
calculation of the effect of the sun s force on the motion
of the apogee. The suggestion of Bessel, that the in
tensity of gravitation might be different for different
planets, was found to be unnecessary, when Professor
Airy gave a more accurate determination of the mass of
Jupiter. And the question whether the extension of the
law of the inverse square to the double stars be true,
(one of the most remarkable questions now before the
scientific world,) must be answered, not by any specula
tions concerning what the laws of attraction must neces
sarily be, but by carefully determining the actual laws
of the motion of these curious objects, by means of the
observations such as those which Sir John Herschel has
collected for that purpose, by his unexampled survey of
both hemispheres of the sky. And since the extent of
this truth is thus to be determined by reference to ob
served facts, it is clear that no mere accumulation of
VOL. i. w. P. F

66 OF IDEAS IN GENERAL.

them can make its universality certain, or its necessity
apparent.

Thus no knowledge of the necessity of any truths
can result from the observation of what really happens.
This being clearly understood, we are led to an import
ant inquiry.

The characters of universality and necessity in the
truths which form part of our knowledge, can never
be derived from experience, by which so large a part
of our knowledge is obtained. But since, as we have
seen, we really do possess a large body of truths which
are necessary, and because necessary, therefore universal,
the question still recurs, from what source these charac
ters of universality and necessity are derived.

The answer to this question we will attempt to give
in the next chapter.

CHAPTER VI.
OF THE GROUNDS OF NECESSARY TRUTHS.

1 . To the question just stated, I reply, that the neces
sity and universality of the truths which form a part of
our knowledge, are derived from the Fundamental Ideas
which those truths involve. These ideas entirely shape
and circumscribe our knowledge ; they regulate the ac
tive operations of our minds, without which our passive
sensations do not become knowledge. They govern
these operations, according to rules which are not only
fixed and permanent, but which may be expressed in
plain and definite terms; and these rules, when thus
expressed, may be made the basis of demonstrations by
which the necessary relations imparted to our know
ledge by our Ideas may be traced to their consequences
in the most remote ramifications of scientific truth.

GROUNDS OF NECESSARY TRUTHS. 67

These enunciations of the necessary and evident con
ditions imposed upon our knowledge by the Fundamental
Ideas which it involves, are termed Axioms. Thus the
Axioms of Geometry express the necessary conditions
which result from the Idea of Space; the Axioms of
Mechanics express the necessary conditions which flow
from the Ideas of Force and Motion ; and so on.

2. It will be the office of several of the succeeding
Books of this work to establish and illustrate in detail
what I have thus stated in general terms. I shall there
pass in review many of the most important fundamental
ideas on which the existing body of our science depends ;
and I shall endeavour to show, for each such idea in
succession, that knowledge involves an active as well as
a passive element ; that it is not possible without an act
of the mind, regulated by certain laws. I shall further
attempt to enumerate some of the principal fundamental
relations which each idea thus introduces into our
thoughts, and to express them by means of definitions
and axioms, and other suitable forms.

I will only add a remark or two to illustrate further
this view of the ideal grounds of our knowledge.

3. To persons familiar with any of the demonstrative
sciences, it will be apparent that if we state all the
Definitions and Axioms which are employed in the
demonstrations, we state the whole basis on which those
reasonings rest. For the whole process of demonstrative
or deductive reasoning in any science, (as in geometry,
for instance,) consists entirely in combining some of these
first principles so as to obtain the simplest propositions
of the science ; then combining these so as to obtain
other propositions of greater complexity ; and so on, till
we advance to the most recondite demonstrable truths ;
these last, however, intricate and unexpected, still in
volving no principles except the original definitions and

F 2

68 OF IDEAS IN GENERAL.

axioms. Thus, by combining the Definition of a triangle,
and the Definitions of equal lines and equal angles,
namely, that they are such as when applied to each
other, coincide, with the Axiom respecting straight lines
(that two such lines cannot inclose a space,) we demon
strate the equality of triangles, under certain assumed
conditions. Again, by combining this result with the
Definition of parallelograms, and with the Axiom that if
equals be taken from equals the wholes are equal, we
prove the equality of parallelograms between the same
parallels and upon the same base. From this proposi
tion, again, we prove the equality of the square on the
hypotenuse of a triangle to the squares on the two sides
containing the right angle. But in all this there is
nothing contained which is not rigorously the result of
our geometrical Definitions and Axioms. All the rest
of our treatises of geometry consists only of terms and
phrases of reasoning, the object of which is to connect
those first principles, and to exhibit the effects of their
combination in the shape of demonstration.

4. This combination of first principles takes place
according to the forms and rules of Logic. All the
steps of the demonstration may be stated in the shape in
which logicians are accustomed to exhibit processes of
reasoning in order to show their conclusiveness, that is,
in Syllogisms. Thus our geometrical reasonings might
be resolved into such steps as the following :

All straight lines drawn from the centre of a circle
to its circumference are equal :

But the straight lines AB, AC, are drawn from the
centre of a circle to its circumference :

Therefore the straignt lines AB, AC, are equal.

Each step of geometrical, and all other demonstra
tive reasoning, may be resolved into three such clauses
as these ; and these three clauses are termed respectively,

GROUNDS OF NECESSARY TRUTHS. 69

the major premiss, the minor premiss, and the conclu
sion; or, more briefly, the major, the minor, and the
conclusion.

The principle which justifies the reasoning when
exhibited in this syllogistic form, is this : that a truth
which can be asserted as generally, or rather as univer
sally true, can be asserted as true also in each particular
case. The minor only asserts a certain particular case
to be an example of such conditions as are spoken of in
the major; and hence the conclusion, which is true of
the major by supposition, is true of the minor by conse
quence ; and thus we proceed from syllogism to syl
logism, in each one employing some general truth in
some particular instance. Any proof which occurs in
geometry, or any other science of demonstration, may
thus be reduced to a series of processes, in each of
which we pass from some general proposition to the
narrower and more special propositions which it in
cludes. And this process of deriving truths by the mere
combination of general principles, applied in particular
hypothetical cases, is called deduction; being opposed
to induction, in which, as we have seen, (Chap. i. Sect. 3.)
a new general principle is introduced at every step.

5. Now we have to remark that, this being so, how
ever far we follow such deductive reasoning, we can
never have, in our conclusion any truth which is not
virtually included in the original principles from which
the reasoning started. For since at any step we merely
take out of a general proposition something included in
it, while at the preceding step we have taken this ge
neral proposition out of one more general, and so on
perpetually, it is manifest that our last result was really
included in the principle or principles with which we
began. I say principles, because, although our logical
conclusion can only exhibit the legitimate issue of our

70 OF IDEAS IN GENERAL.

first principles, it may, nevertheless, contain the result
of the combination of several such principles, and may
thus assume a great degree of complexity, and may ap
pear so far removed from the parent truths, as to betray
at first sight hardly any relationship with them. Thus
the proposition which has already been quoted respect
ing the squares on the sides of a right-angled triangle,
contains the results of many elementary principles ; as,
the definitions of parallels, triangle, and square ; the
axioms respecting straight lines, and respecting paral
lels; and, perhaps, others. The conclusion is compli
cated by containing the effects of the combination of all
these elements ; but it contains nothing, and can contain
nothing, but such elements and their combinations.

This doctrine, that logical reasoning produces no new
truths, but only unfolds and brings into view those truths
which were, in effect, contained in the first principles of
the reasoning, is assented to by almost all who, in
modern times, have attended to the science of logic.
Such a view is admitted both by those who defend, and
by those who depreciate the value of logic. " Whatever
is established by reasoning, must have been contained
and virtually asserted in the premises""." "The only
truth which such propositions can possess consists in
conformity to the original principles."

In this manner the whole substance of our geometry
is reduced to the Definitions and Axioms which we
employ in our elementary reasonings ; and in like man
ner we reduce the demonstrative truths of any other
science to the definitions and axioms which we there
employ.

6. But in reference to this subject, it has sometimes
been said that demonstrative sciences do in reality depend
upon Definitions only; and that no additional kind of

* Whateley s Logic, pp. 237, 238.

GROUNDS OF NECESSARY TRUTHS. 71

principle, such as we have supposed Axioms to be, is
absolutely required. It has been asserted that in geo
metry, for example, the source of the necessary truth of
our propositions is this, that they depend upon definitions
alone, and consequently merely state the identity of the
same thing under different aspects.

That in the sciences which admit of demonstration,
as geometry, mechanics, and the like, Axioms as well as
Definitions are needed, in order to express the grounds
of our necessary convictions, must be shown hereafter
by an examination of each of these sciences in particular.
But that the propositions of these sciences, those of geo
metry for example, do not merely assert the identity of
the same thing, will, I think, be generally allowed, if we
consider the assertions which we are enabled to make.
When we declare that " a straight line is the shortest
distance between two points," is this merely an identical
proposition? the definition of a straight line in another
form ? Not so : the definition of a straight line involves
the notion of form only, and does not contain anything
about magnitude ; consequently, it cannot contain any
thing equivalent to " shortest." Thus the propositions
of geometry are not merely identical propositions; nor
have we in their general character anything to coun
tenance the assertion, that they are the results of defi
nitions alone. And when we come to examine this and
other sciences more closely, we shall find that axioms,
such as are usually in our treatises made the funda
mental principles of our demonstrations, neither have
ever been, nor can be, dispensed with. Axioms, as well
as Definitions, are in all cases requisite, in order pro
perly to exhibit the grounds of necessary truth.

7. Thus the real logical basis of every body of demon
strated truths are the Definitions and Axioms which are
the first principles of the reasonings. But when we are

72 OF IDEAS IN GENERAL.

arrived at this point, the question further occurs, what
is the ground of the truth of these Axioms? It is not
the logical, but the philosophical, not the formal, but the
real foundation of necessary truth, which we are seeking.
Hence this inquiry necessarily comes before us, What
is the ground of the Axioms of Geometry, of Mechanics,
and of any other demonstrable science ?

The answer which we are led to give, by the view
which we have taken of the nature of knowledge, has
already been stated. The ground of the axioms belong
ing to each science is the Idea which the axiom involves.
The ground of the Axioms of Geometry is the Idea of
Space: the ground of the Axioms of Mechanics is the
Idea of Force, of Action and Reaction, and the like. And
hence these Ideas are Fundamental Ideas ; and since they
are thus the foundations, not only of demonstration but
of truth, an examination into their real import and
nature is of the greatest consequence to our purpose.

8. Not only the Axioms, but the Definitions which
form the basis of our reasonings, depend upon our Fun
damental Ideas. And the Definitions are not arbitrary
definitions, but are determined by a necessity no less
rigorous than the Axioms themselves. We could not
think of geometrical truths without conceiving a circle ;
and we could not reason concerning such truths without
defining a circle in some mode equivalent to that which
is commonly adopted. The Definitions of parallels, of
right angles, and the like, are quite as necessarily pre
scribed by the nature of the case, as the Axioms which
these Definitions bring with them. Indeed we may
substitute one of these kinds of principles for another.
We cannot always put a Definition in the place of an
Axiom ; but we may always find an Axiom which shall
take the place of a Definition. If we assume a proper
Axiom respecting straight lines, we need no Definition

A GROUNDS OF NECESSARY TRUTHS. 73

a straight line. But in whatever shape the principle
jpear, as Definition or as Axiom, it has about it nothing
casual or arbitrary, but is determined to be what it is, as
to its import, by the most rigorous necessity, growing
out of the Idea of Space.

9. These principles, Definitions, and Axioms, thus
exhibiting the primary developements of a fundamental
idea, do in fact express the idea, so far as its expression
in words forms part of our science. They are different
views of the same body of truth ; and though each prin
ciple, by itself, exhibits only one aspect of this body,
taken together they convey a sufficient conception of it
for our purposes. The Idea itself cannot be fixed in
words ; but these various lines of truth proceeding from
it, suggest sufficiently to a fitly-prepared mind, the place
where the idea resides, its nature, and its efficacy.

It is true that these principles, our elementary Defi
nitions and Axioms, even taken altogether, express the
Idea incompletely. Thus the Definitions and Axioms of
Geometry, as they are stated in our elementary works,
do not fully express the Idea of Space as it exists in our
minds. For, in addition to these, other Axioms, inde
pendent of these, and no less evident, can be stated ; and
are in fact stated when we come to the Higher Geo
metry. Such, for instance, is the Axiom of Archimedes
that a curve line which joins two points is less than a
broken line which joins the same points and includes the
curve. And thus the Idea is disclosed but not fully re
vealed, imparted but not transfused, by the use we make
of it in science. When we have taken from the fountain
so much as serves our purpose, there still remains behind
a deep well of truth, which we have not exhausted, and
which we may easily believe to be inexhaustible.

CHAPTER VII.

THE FUNDAMENTAL IDEAS ARE NOT DERIVED
FROM EXPERIENCE.

1. BY the course of speculation contained in the last
three Chapters, we are again led to the conclusion which
we have already stated, that our knowledge contains an
ideal element, and that this element is not derived from
experience. For we have seen that there are proposi
tions which are known to be necessarily true ; and that
such knowledge is not, and cannot be, obtained by mere
observation of actual facts. It has been shown, also,
that these necessary truths are the results of certain fun
damental ideas, such as those of space, number, and the
like. Hence it follows inevitably that these ideas and
others of the same kind are not derived from experience.
For these ideas possess a power of infusing into their
developements that very necessity which experience can
in no way bestow. This power they do not borrow from
the external world, but possess by their own nature.
Thus we unfold out of the Idea of Space the propositions
of geometry, which are plainly truths of the most rigor
ous necessity and universality. But if the idea of space
were merely collected from observation of the external
world, it could never enable or entitle us to assert such
propositions : it could never authorize us to say that not
merely some lines, but all lines, not only have, but must
have, those properties which geometry teaches. Geo
metry in every proposition speaks a language which
experience never dares to utter; and indeed of which
she but half comprehends the meaning. Experience
sees that the assertions are true, but she sees not how
profound and absolute is their truth. She unhesitatingly
assents to the laws which geometry delivers, but she does

FUNDAMENTAL IDEAS NOT DERIVATIVE. 75

not pretend to see the origin of their obligation. She
is always ready to acknowledge the sway of pure scien
tific principles as a matter of fact, but she does not
dream of offering her opinion on their authority as a
matter of right ; still less can she justly claim to be her
self the source of that authority.

David Hume asserted 4 ", that we are incapable of
seeing in any of the appearances which the world pre
sents anything of necessary connexion ; and hence he
inferred that our knowledge cannot extend to any such
connexion. It will be seen from what we have said that
we assent to his remark as to the fact, but we differ from
him altogether in the consequence to be drawn from it.
Our inference from Hume s observation is, not the truth
of his conclusion, but the falsehood of his premises ;
not that, therefore, we can know nothing of natural con
nexion, but that, therefore, we have some other source of
knowledge than experience : not, that we can have no
idea of connexion or causation, because, in his language,
it cannot be the copy of an impression ; but that since
we have such an idea, our ideas are not the copies of
our impressions.

Since it thus appears that our fundamental ideas are
not acquired from the external world by our senses, but
have some separate and independent origin, it is im
portant for us to examine their nature and properties, as
they exist in themselves; and this it will be our business
to do through a portion of the following pages. But it
may be proper first to notice one or two objections
which may possibly occur to some readers.

2. It may be said that without the use of our senses,
of sight and touch, for instance, we should never have
any idea of space ; that this idea, therefore, may properly
be said to be derived from those senses. And to this I

* Essays, Vol. n. p. 70.

76 OF IDEAS IN GENERAL.

reply, by referring to a parallel instance. Without light
we should have no perception of visible figure ; yet the
power of perceiving visible figure cannot be said to be
derived from the light, but resides in the structure of the
eye. If we had never seen objects in the light, we
should be quite unaware that we possessed a power of
vision ; yet we should not possess it the less on that
account. If we had never exercised the senses of sight
and touch (if we can conceive such a state of human ex
istence) we know not that we should be conscious of an
idea of space. But the light reveals to us at the same
time the existence of external objects and our own power
of seeing. And in a very similar manner, the exercise
of our senses discloses to us, at the same time, the ex
ternal world, and our own ideas of space, time, and other
conditions, without which the external world can neither
be observed nor conceived. That light is necessary to
vision, does not, in any degree, supersede the importance
of a separate examination of the laws of our visual
powers, if we would understand the nature of our own
bodily faculties and the extent of the information they
can give us. In like manner, the fact that intercourse
with the external world is necessary for the conscious
employment of our ideas, does not make it the less es
sential for us to examine those ideas in their most inti
mate structure, in order that we may understand the
grounds and limits of our knowledge. Even before we
see a single object, we have a faculty of vision ; and in
like manner, if we can suppose a man who has never
contemplated an object in space or time, we must still
assume him to have the faculties of entertaining the ideas
of space and time, which faculties are called into play
on the very first occasion of the use of the senses.

3. In answer to such remarks as the above, it has
sometimes been said that to assume separate faculties in

FUNDAMENTAL IDEAS NOT DERIVATIVE. 77

the mind for so many different processes of thought, is to
give a mere verbal explanation, since we learn nothing
concerning our idea of space by being told that we have
a faculty of forming such an idea. It has been said that
this course of explanation leads to an endless multipli
cation of elements in man s nature, without any advan
tage to our knowledge of his true constitution. We
may, it is said, assert man to have a faculty of walking,
of standing, of breathing, of speaking ; but what, it is
asked, is gained by such assertions? To this I reply, that
we undoubtedly have such faculties as those just named;
that it is by no means unimportant to consider them; and
that the main question in such cases is, whether they are
separate and independent faculties, or complex and deri
vative ones ; and, if the latter be the case, what are the
simple and original faculties by the combination of which
the others are produced. In walking, standing, breath
ing, for instance, a great part of the operation can be
reduced to one single faculty ; the voluntary exercise of
our muscles. But in breathing this does not appear to
be the whole of the process. The operation is, in part at
least, involuntary ; and it has been held that there is a
certain sympathetic action of the nerves, in addition to
the voluntary agency which they transmit, which is essen
tial to the function. To determine whether or no this
sympathetic faculty is real and distinct, and if so, what
are its laws and limits, is certainly a highly philosophical
inquiry, and well deserving the attention which has been
bestowed upon it by eminent physiologists. And just of
the same nature are the inquiries with respect to man s
intellectual constitution, on which we propose to enter.
For instance, man has a faculty of apprehending time,
and a faculty of reckoning numbers: are these distinct, or
is one faculty derived from the other? To analyze the
various combinations of our ideas and observations into

78 OF IDEAS IN GENERAL.

the original faculties which they involve ; to show that
these faculties are original, and not capable of further
analysis : to point out the characters which mark these
faculties and lead to the most important features of our
knowledge; these are the kind of researches on which
we have now to enter, and these, we trust, will be found
to be far from idle or useless parts of our plan. If we
succeed in such attempts, it will appear that it is by
no means a frivolous or superfluous step to distinguish
separate faculties in the mind. If we do not learn much
by being told that we have a faculty of forming the idea
of space, we at least, by such a commencement, circum
scribe a certain portion of the field of our investigations,
which, we shall afterwards endeavour to show, requires
and rewards a special examination. And though we shall
thus have to separate the domain of our philosophy into
many provinces, these are, as we trust it will appear,
neither arbitrarily assigned, nor vague in their limits,
nor infinite in number.

CHAPTER VIII.
OF THE PHILOSOPHY OF THE SCIENCES.

WE proceed, in the ensuing Books, to the closer exami
nation of a considerable number of those Fundamental
Ideas on which the sciences, hitherto most successfully
cultivated, are founded. In this task, our objects will
be to explain and analyze such Ideas so as to bring into
view the Definitions and Axioms, or other forms, in
which we may clothe the conditions to which our specu
lative knowledge is subjected. I shall also try to prove,
for some of these Ideas in particular, what has been
already urged respecting them in general, that they are

PHILOSOPHY OF SCIENCES. 79

not derived from observation, but necessarily impose
their conditions upon that knowledge of which observa
tion supplies the materials. I shall further, in some
cases, endeavour to trace the history of these Ideas as
they have successively come into notice in the progress
of science; the gradual developement by which they have
arrived at their due purity and clearness; and, as a
necessary part of such a history, I shall give a view of
some of the principal controversies which have taken
place with regard to each portion of knowledge.

An exposition and discussion of the Fundamental
Ideas of each Science may, with great propriety, be
termed the PHILOSOPHY or such SCIENCE. These ideas
contain in themselves the elements of those truths which
the science discovers and enunciates; and in the progress
of the sciences, both in the world at large and in the
mind of each individual student, the most important
steps consist in apprehending these ideas clearly, and in
bringing them into accordance with the observed facts.
I shall, therefore, in a series of Books, treat of the Phi
losophy of the Pure Sciences, the Philosophy of the
Mechanical Sciences, the Philosophy of Chemistry, and
the like, and shall analyze and examine the ideas which
these sciences respectively involve.

In this undertaking, inevitably somewhat long, and
involving many deep and subtle discussions, I shall take,
as a chart of the country before me, by which my course
is to be guided, the scheme of the sciences which I was
led to form by travelling over the history of each in
order"". Each of the sciences of which I then narrated
the progress, depends upon several of the Fundamental
Ideas of which I have to speak : some of these Ideas are
peculiar to one field of speculation, others are common
to more. A previous enumeration of Ideas thus collected

* Hisiory of the Inductive Sciences.

80 OF IDEAS IN GENERAL.

may serve both to show the course and limits of this part
of our plan, and the variety of interest which it offers.

I shall, then, successively, have to speak of the Ideas
which are the foundation of Geometry and Arithmetic,
(and which also regulate all sciences depending upon
these, as Astronomy and Mechanics;) namely, the Ideas
of Space, Time, and Number :

Of the Ideas on which the Mechanical Sciences (as
Mechanics, Hydrostatics, Physical Astronomy) more pecu
liarly rest ; the ideas of Force and Matter, or rather the
idea of Cause, which is the basis of these :

Of the Ideas which the Secondary Mechanical Sciences
(Acoustics, Optics, and Thermotics) involve ; namely, the
Ideas of the Externality of objects, and of the Media
by which we perceive their qualities :

Of the Ideas which are the basis of Mechanico-che-
mical and Chemical Science; Polarity, Chemical Affinity,
and Substance ; and the Idea of Symmetry, a necessary
part of the Philosophy of Crystallography :

Of the Ideas on which the Classificatory Sciences
proceed (Mineralogy, Botany, and Zoology) ; namely, the
Ideas of Resemblance, and of its gradations, and of
Natural Affinity:

Finally, of those Ideas on which the Physiological
Sciences are founded ; the Ideas of separate Vital Powers,
such as Assimilation and Irritability ; and the Idea of
Final Cause.

We have, besides these, the Palsetiological Sciences,
which proceed mainly on the conception of Historical
Causation.

It is plain that when we have proceeded so far as
this, we have advanced to the verge of those speculations
which have to do with mind as well as body. The
extension of our philosophy to such a field, if it can be
justly so extended, will be one of the most important

PHILOSOPHY OF SCIENCES. 81

results of our researches; but on that very account we
must fully study the lessons which we learn in those
fields of speculation where our doctrines are most secure,
before we venture into a region where our principles will
appear to be more precarious, and where they are inevi
tably less precise.

We now proceed to the examination of the above
Ideas, and to such essays towards the philosophy of each
Science as this course of investigation may suggest.

VOL. i. w. p. G

BOOK II.

THE PHILOSOPHY OF THE PURE
SCIENCES.

CHAPTER I.
OF THE PUEE SCIENCES.

1. ALL external objects and events which we can con
template are viewed as having relations of Space, Time,
and Number ; and are subject to the general conditions
which these Ideas impose, as well as to the particular
laws which belong to each class of objects and occur
rences. The special laws of nature, considered under
the various aspects which constitute the different sciences,
are obtained by a mixed reference to experience and to
the fundamental ideas of each science. But besides the
sciences thus formed by the aid of special experience, the
conditions which flow from those more comprehensive
ideas first mentioned, Space, Time, and Number, consti
tute a body of science, applicable to objects and changes
of all kinds, and deduced without recurrence being had
to any observation in particular. These sciences, thus
unfolded out of ideas alone, unmixed with any reference
to the phenomena of matter, are hence termed Pure
Sciences. The principal sciences of this class are Geome
try, Theoretical Arithmetic, and Algebra considered in its
most general sense, as the investigation of the relations
of space and number by means of general symbols.

OF THE TURE SCIENCES. 83

2. These Pure Sciences were not included in our
survey of the history of the sciences, because they are
not inductive sciences. Their progress has not consisted
in collecting laws from phenomena, true theories from
observed facts, and more general from more limited laws ;
but in tracing the consequences of the ideas themselves,
and in detecting the most general and intimate analogies
and connexions which prevail among such conceptions as
are derivable from the ideas. These sciences have no
principles besides definitions and axioms, and no process
of proof but deduction ; this process, however, assuming
here a most remarkable character ; and exhibiting a com
bination of simplicity and complexity, of rigour and
generality, quite unparalleled in other subjects.

3. The universality of the truths, and the rigour of
the demonstrations of these pure sciences, attracted
attention in the earliest times ; and it was perceived that
they offered an exercise and a discipline of the intellec
tual faculties, in a form peculiarly free from admixture
of extraneous elements. They were strenuously culti
vated by the Greeks, both with a view to such a disci
pline, and from the love of speculative truth which pre
vailed among that people : and the name mathematics, by
which they are designated, indicates this their character
of disciplinal studies.

4. As has already been said, the ideas which these
sciences involve extend to all the objects and changes
which we observe in the external world ; and hence the
consideration of mathematical relations forms a large
portion of many of the sciences which treat of the phe
nomena and laws of external nature, as Astronomy,
Optics, and Mechanics. Such sciences are hence often
termed Mixed Mathematics, the relations of space and
number being, in these branches of knowledge, combined
with principles collected from special observation ;

G 2

84 PHILOSOPHY OF THE PURE SCIENCES.

while Geometry, Algebra, and the like subjects, which
involve no result of experience, are called Pure Mathe
matics.

5. Space, time, and number, may be conceived as
forms by which the knowledge derived from our sensa
tions is moulded, and which are independent of the dif
ferences in the matter of our knowledge, arising from the
sensations themselves. Hence the sciences which have
these ideas for their subject may be termed Formal
Sciences. In this point of view, they are distinguished
from sciences in which, besides these mere formal laws
by which appearances are corrected, we endeavour to
apply to the phenomena the idea of cause, or some of the
other ideas which penetrate further into the principles
of nature. We have thus, in the History, distinguished
Formal Astronomy and Formal Optics from Physical
Astronomy and Physical Optics.

We now proceed to our examination of the Ideas
which constitute the foundation of these formal or pure
mathematical sciences, beginning with the Idea of Space.

CHAPTER II.
OF THE IDEA OF SPACE.

1. BY speaking of space as an Idea, I intend to imply,
as has already been stated, that the apprehension of
objects as existing in space, and of the relations of posi
tion, &c., prevailing among them, is not a consequence
of experience, but a result of a peculiar constitution and
activity of the mind, which is independent of all expe
rience in its origin, though constantly combined with
experience in its exercise.

That the idea of space is thus independent of experi
ence, has already been pointed out in speaking of ideas

OF THE IDEA OF SPACE. 85

in general : but it may be useful to illustrate the doctrine
further in this particular case.

I assert, then, that space is not a notion obtained
by experience. Experience gives us information con
cerning things without us : but our apprehending them
as without us, takes for granted their existence in space.
Experience acquaints us what are the form, position,
magnitude of particular objects : but that they have form,
position, magnitude, presupposes that they are in space.
We cannot derive from appearances, by the way of
observation, the habit of representing things to ourselves
as in space ; for no single act of observation is possible
any otherwise than by beginning with such a representa
tion, and conceiving objects as already existing in space.

2. That our mode of representing space to ourselves
is not derived from experience, is clear also from this :
that through this mode of representation we arrive at
propositions which are rigorously universal and neces
sary. Propositions of such a kind could not possibly be
obtained from experience ; for experience can only teach
us by a limited number of examples, and therefore can
never securely establish a universal proposition : and
again, experience can only inform us that anything is so,
and can never prove that it must be so. That two sides
of a triangle are greater than the third is a universal
and necessary geometrical truth: it is true of all tri
angles ; it is true in such a way that the contrary cannot
be conceived. Experience could not prove such a propo
sition. And experience has not proved it ; for perhaps
no man ever made the trial as a means of removing
doubts : and no trial could, in feet, add in the smallest
degree to the certainty of this truth. To seek for proof
of geometrical propositions by an appeal to observation
proves nothing in reality, except that the person who
has recourse to such grounds has no due apprehension

86 PHILOSOPHY OF THE PURE SCIENCES.

of the nature of geometrical demonstration. We have
heard of persons who convinced themselves by measure
ment that the geometrical rule respecting the squares
on the sides of a right-angled triangle was true : but
these were persons whose minds had been engrossed by
practical habits, and in whom the speculative develope-
ment of the idea of space had been stifled by other em
ployments. The practical trial of the rule may illustrate,
but cannot prove it. The rule will of course be con
firmed by such trial, because what is true in general is
true in particular: but the rule cannot be proved from any
number of trials, for no accumulation of particular cases
makes up a universal case. To all persons who can see
the force of any proof, the geometrical rule above referred
to is as evident, and its evidence as independent of ex
perience, as the assertion that sixteen and nine make
twenty-five. At the same time, the truth of the geome
trical rule is quite independent of numerical truths, and
results from the relations of space alone. This could
not be if our apprehension of the relations of space were
the fruit of experience : for experience has no element
from which such truth and such proof could arise.

3. Thus the existence of necessary truths, such as
those of geometry, proves that the idea of space from
which they flow, is not derived from experience. Such
truths are inconceivable on the supposition of their being
collected from observation ; for the impressions of sense
include no evidence of necessity. But we can readily
understand the necessary character of such truths, if we
conceive that there are certain necessary conditions under
which alone the mind receives the impressions of sense.
Since these conditions reside in the constitution of the
mind, and apply to every perception of an object to
which the mind can attain, we easily see that their rules
must include, not only all that has been, but all that can

OF THE IDEA OF SPACE. 87

be, matter of experience. Our sensations can each con
vey no information except about itself; each can contain
no trace of another additional sensation ; and thus no
relation and connexion between two sensations can be
given by the sensations themselves. But the mode in
which the mind perceives these impressions as objects,
may and will introduce necessary relations among them :
and thus by conceiving the idea of space to be a con
dition of perception in the mind, we can conceive the
existence of necessary truths, which apply to all per
ceived objects.

4. If we consider the impressions of sense as the
mere materials of our experience, such materials may
be accumulated in any quantity and in any order. But
if we suppose that this matter has a certain form given
it, in the act of being accepted by the mind, we can
understand how it is that these materials are subject to
inevitable rules ; how nothing can be perceived exempt
from the relations which belong to such a form. And
since there are such truths applicable to our experience,
and arising from the nature of space, we may thus
consider space as a, form which the materials given by
experience necessarily assume in the mind; as an ar
rangement derived from the perceiving mind, and not
from the sensations alone.

5. Thus this phrase, that space is &form belonging
to our perceptive power, may be employed to express
that we cannot perceive objects as in space, without an
operation of the mind as well -* as of the senses without
active as well as passive faculties. This phrase, how
ever, is not necessary to the exposition of our doctrines.
Whether we call the conception of space a condition of
perception, a form of perception, or an idea, or by any
other term, it is something originally inherent in the
mind perceiving, and not in the objects perceived. And

88 PHILOSOPHY OF THE PURE SCIENCES.

it is because the apprehension of all objects is thus sub
jected to certain mental conditions, forms or ideas, that
our knowledge involves certain inviolable relations and
necessary truths. The principles of such truths, so far
as they regard space, are derived from the idea of space,
and we must endeavour to exhibit such principles in
their general form. But before we do this, we may
notice some of the conditions which belong, not to our
Ideas in general, but to this Idea of Space in parti
cular.

CHAPTER III.

OF SOME PECULABITIES OF THE IDEA OF

SPACE.

1. SOME of the Ideas which we shall have to examine
involve conceptions of certain relations of objects, as the
idea of Cause and of Likeness ; and may appear to be
suggested by experience, enabling us to abstract this
general relation from particular cases. But it will be
seen that Space is not such a general conception of a
relation. For we do not speak of Spaces as we speak of
Causes and Likenesses, but of Space. And when we
speak of spaces, we understand by the expression, parts
of one and the same identical every where -extended
Space. We conceive a Universal Space; which is not
made up of these partial spaces as its component parts,
for it would remain if these were taken away ; and these
cannot be conceived without presupposing absolute space.
Absolute Space is essentially one ; and the complication
which exists in it, and the conception of various spaces,
depends merely upon boundaries. Space must, there
fore, be, as we have said, not a general conception
abstracted from particulars, but a universal mode of
representation, altogether independent of experience.

PECULIARITIES OF THE IDEA OF SPACE. 89

2. Space is infinite. We represent it to ourselves as
an infinitely great magnitude. Such an idea as that of
Likeness or Cause, is, no doubt, found in an infinite
number of particular cases, and so far includes these
cases. But these ideas do not include an infinite number
of cases as parts of an infinite whole. When we say
that all bodies and partial spaces exist in infinite space,
we use an expression which is not applied in the same
sense to any cases except those of Space and Time.

3. What is here said may appear to be a denial of
the real existence of space. It must be observed, how
ever, that we do not deny, but distinctly assert, the
existence of space as a real and necessary condition of
all objects perceived ; and that we not only allow that
objects are seen external to us, but we found upon the
fact of their being so seen, our view of the nature of
space. If, however, it be said that we deny the reality
of space as an object or thing, this is true. Nor does it
appear easy to maintain that space exists as a thing,
when it is considered that this thing is infinite in all its
dimensions; and, moreover, that it is a thing, which,
being nothing in itself, exists only that other things may
exist in it. And those who maintain the real existence
of space, must also maintain the real existence of time in
the same sense. Now two infinite things, thus really
existing, and yet existing only as other things exist in
them, are notions so extravagant that we are driven to
some other mode of explaining the state of the matter.

4. Thus space is not an object of which we perceive
the properties, but a form of our perception; not a thing
which affects our senses, but an idea to which we con
form the impressions of sense. And its peculiarities ap
pear to depend upon this, that it is not only a form of
sensation, but of intuition ; that in reference to space,
we not only perceive but contemplate objects. We see

00 PHILOSOPHY OF THE PURE SCIENCES.

objects in space, side by side, exterior to each other;
space, and objects in so far as they occupy space, hare
parts exterior to other parts ; and have the whole thus
made up by the juxtaposition of parts. This mode of
apprehension belongs only to the ideas of space and
time. Space and Time are made up of parts, but Cause
and Likeness are not apprehended as made up of parts.
And the term intuition (in its rigorous sense) is appli
cable only to that mode of contemplation in which we
thus look at objects as made up of parts, and apprehend
the relations of those parts at the same time and by the
same act by which we apprehend the objects themselves.

5. As we have said, space limited by boundaries gives
rise to various conceptions which we have often to con
sider. Thus limited, space assumes form m figure; and
the variety of conceptions thus brought under our notice
is infinite. We have every possible form of line, straight
line, and curve ; and of curves an endless number ; cir
cles, parabolas, hyperbolas, spirals, helices. We have
plane surfaces of various shapes, parallelograms, poly
gons, ellipses ; and we have solid figures, cubes, cones,
cylinders, spheres, spheroids, and so on. All these have
their various properties, depending on the relations of
their boundaries ; and the investigation of their proper
ties forms the business of the science of Geometry.

6. Space has three dimensions, or directions in which
it may be measured ; it cannot have more or f&lt;3Ver. The
simplest measurement is that of a straight line, which
has length alone. A surface has both length and
breadth : and solid space has length, breadth, and thick
ness or depth. The origin of such a difference of dimen
sions will be seen if we reflect that each portion of space
has a boundary, and is extended both in the direction in
which its boundary extends, and also in a direction from
its boundary ; for otherwise it would not be a boundary.

PECULARITIES OF THE IDEA OF SPACE. 01

A point has no dimensions. A line has but one dimen
sion, the distance from its boundary, or its length. A
plane, bounded by a straight line, has the dimension
which belongs to this line, and also has another dimen
sion arising from the distance of its parts from this bound
ary line; and this may be called breadth. A solid,
bounded by a plane, has the dimensions which this plane
has ; and has also a third dimension, which we may call
height or depth, as we consider the solid extended above
or below the plane ; or thickness, if we omit all con
sideration of up and down. And no space can have any
dimensions which are not resoluble into these three.

We may now proceed to consider the mode in which
the idea of space is employed in the formation of
Geometry.

CHAPTER IV.

OF THE DEFINITIONS AND AXIOMS WHICH
RELATE TO SPACE.

1. THE relations of space have been apprehended
with peculiar distinctness and clearness from the very
first unfolding of man s speculative powers. This was a
consequence of the circumstance which we have just
noticed, that the simplest of these relations, and those on
which the others depend, are seen by intuition. Hence,
as soon as men were led to speculate concerning the
relations of space, they assumed just principles, and
obtained true results. It is said that the science of
geometry had its origin in Egypt, before the dawn of the
Greek philosophy : but the knowledge of the early
Egyptians (exclusive of their mythology) appears to have
been purely practical; and, probably, their geometry
consisted only in some maxims of land-measuring, which
is what the term implies. The Greeks of the time of

92 PHILOSOPHY OF THE PURE SCIENCES.

Plato, had, however, not only possessed themselves of
many of the most remarkable elementary theorems of
the science ; but had, in several instances, reached the
boundary of the science in its elementary form ; as when
they proposed to themselves the problems of doubling
the cube and squaring the circle.

But the deduction of these theorems by a systematic
process, and the primary exhibition of the simplest prin
ciples involved in the idea of space, which such a
deduction requires, did not take place, so far as we are
aware, till a period somewhat later. The Elements of
Geometry of Euclid, in which this task was performed,
are to this day the standard work on the subject: the
author of this work taught mathematics with great
applause at Alexandria, in the reign of Ptolemy Lagus,
about 280 years before Christ. The principles which
Euclid makes the basis of his system have been very
little simplified since his time ; and all the essays and
controversies which bear upon these principles, have
had a reference to the form in which they are stated
by him.

2. Definitions. The first principles of Euclid s geo
metry are, as the first principles of any system of
geometry must be, definitions and axioms respecting
the various ideal conceptions which he introduces; as
straight lines, parallel lines, angles, circles, and the like.
But it is to be observed that these definitions and
axioms are very far from being arbitrary hypotheses and
assumptions. They have their origin in the idea of
space, and are merely modes of exhibiting that idea in
such a manner as to make it afford grounds of deductive
reasoning. The axioms are necessary consequences of
the conceptions respecting which they are asserted ; and
the definitions are no less necessary limitations of con
ceptions ; not requisite in order to arrive at this or that

DEFINITIONS AND AXIOMS RELATING TO SPACE. 93

consequence ; but necessary in order that it may be
possible to draw any consequences, and to establish any
general truths.

For example, if we rest the end of one straight
staff upon the middle of another straight staif, and move
the first staff into various positions, we, by so doing,
alter the angles which the first staff makes with the
other to the right hand and to the left. But if we
place the staff in that special position in which these
two angles are equal, each of them is a right angle,
according to Euclid ; and this is the definition of a right
angle, except that Euclid employs the abstract con
ception of straight lines, instead of speaking, as we have
done, of staves. But this selection of the case in which
the two angles are equal is not a mere act of caprice ;
as it might have been if he had selected a case in which
these angles are unequal in any proportion. For the
consequences which can be drawn concerning the cases
of unequal angles, do not lead to general truths, without
some reference to that peculiar case in which the angles
are equal : and thus it becomes necessary to single out
and define that special case, marking it by a special
phrase. And this definition not only gives complete and
distinct knowledge what a right angle is, to any one
who can form the conception of an angle in general ; but
also supplies a principle from which all the properties of
right angles may be deduced.

3. Axioms. With regard to other conceptions also,
as circles, squares, and the like, it is possible to lay
down definitions which are a sufficient basis for our
reasoning, so far as such figures are concerned. But,
besides these definitions, it has been found necessary to
introduce certain axioms among the fundamental prin
ciples of geometry. These are of the simplest character ;
for instance, that two straight lines cannot cut each

94 PHILOSOPHY OF THE PURE SCIENCES,

other in more than one point, and an axiom concerning
parallel lines. Like the definitions, these axioms flow
from the Idea of Space, and present that idea under
various aspects. They are different from the definitions ;
nor can the definitions be made to take the place of the
axioms in the reasoning by which elementary geo
metrical properties are established. For example, the
definition of parallel straight lines is, that they are such
as, however far continued, can never meet : but, in order
to reason concerning such lines, we must further adopt
some axiom respecting them : for example, we may very
conveniently take this axiom; that two straight lines
which cut one another are not both of them parallel to
a third straight line*. The definition and the axiom are
seen to be inseparably connected by our intuition of the
properties of space; but the axiom cannot be proved
from the definition, by any rigorous deductive demon
stration. And if we were to take any other definition of
two parallel straight lines, (as that they are both per
pendicular to a third straight line,) we should still, at
some point or other of our progress, fall in with the
same difficulty of demonstratively establishing their pro
perties without some further assumption.

4. Thus the elementary properties of figures, which
are the basis of our geometry, are necessary results of
our Idea of Space ; and are connected with each other
by the nature of that idea, and not merely by our hypo
theses and constructions. Definitions and axioms must
be combined, in order to express this idea so far as
the purposes of demonstrative reasoning require. These
verbal enunciations of the results of the idea cannot be
made to depend on each other by logical consequence ;
but have a mutual dependence of a more intimate kind,

* This axiom is simpler and more convenient than that of Euclid.
It is employed by the late Professor Playfair in his Geometry.

DEFINITIONS AND AXIOMS RELATING TO SPACE. 95

which words cannot fully convey. It is not possible to
resolve these truths into certain hypotheses, of which all
the rest shall be the necessary logical consequence. The
necessity is not hypothetical, but intuitive. The axioms
require not to be granted, but to be seen. If any one
were to assent to them without seeing them to be true,
his assent would be of no avail for purposes of reason
ing: for he would be also unable to see in what cases
they might be applied. The clear possession of the
Idea of Space is the first requisite for all geometrical
reasoning ; and this clearness of idea may be tested by
examining whether the axioms offer themselves to the
mind as evident.

5. The necessity of ideas added to sensations, in
order to produce knowledge, has often been overlooked
or denied in modern times. The ground of necessary
truth which ideas supply being thus lost, it was con
ceived that there still remained a ground of necessity in
definitions; that we might have necessary truths, by
asserting especially what the definition implicity involved
in general. It was held, also, that this was the case in
geometry : that all the properties of a circle, for
instance, were implicitly contained in the definition of a
circle. That this alone is not the ground of the neces
sity of the truths which regard the circle, that we
could not in this way unfold a definition into propor
tions, without possessing an intuition of the relations to
which the definition led, has already been shown. But
the insufficiency of the above account of the grounds of
necessary geometrical truth appeared in another way
also. It was found impossible to lay down a system of
definitions out of which alone the whole of geometrical
truth could be evolved. It was found that axioms could
not be superseded. No definition of a straight line
could be given which rendered the axiom concerning

96 PHILOSOPHY OF THE PURE SCIENCES.

straight lines superfluous. And thus it appeared that
the source of geometrical truths was not definition
alone ; and we find in this result a confirmation of the
doctrine which we are here urging, that this source of
truth is to be found in the form or conditions of our
perception ; in the idea which we unavoidably combine
with the impressions of sense ; in the activity, and not
in the passivity of the mind"".

6. This will appear further when we come to con
sider the mode in which we exercise our observation
upon the relations of space. But we may, in the first
place, make a remark which tends to show the con
nexion between our conception of a straight line, and
the axiom which is made the foundation of our reason
ings concerning space. The axiom is this ; that two
straight lines, which have both their ends joined, cannot
have the intervening parts separated so as to inclose a
space. The necessity of this axiom is of exactly the
same kind as the necessity of the definition of a right
angle, of which we have already spoken. For as the line
standing on another makes right angles when it makes
the angles on the two sides of it equal ; so a line is a
straight line when it makes the two portions of space,
on the two sides of it, similar. And as there is only a
single position of the line first mentioned, which can
make the angles equal, so there is only a single form of
a line which can make the spaces near the line similar
on one side and on the other : and therefore there can
not be two straight lines, such as the axiom describes,

* I formerly stated views similar to these in some " Remarks"
appended to a work which I termed The Mechanical Euclid, pub
lished in 1837- These Remarks, so far as they bear upon the question
here discussed, were noticed and controverted in No. 135 of the Edin
burgh Review. As an examination of the reviewer s objections may
serve further to illustrate the subject, I shall annex to this chapter an
answer to the article to which I have referred.

DEFINITIONS AND AXIOMS RELATING TO SPACE. 97

which, between the same limits, give two different
boundaries to space thus separated. And thus we see a
reason for the axiom. Perhaps this view may be further
elucidated if we take a leaf of paper, double it, and
crease the folded edge. We shall thus obtain a straight
line at the folded edge ; and this line divides the surface
of the paper, as it was originally spread out, into two
similar spaces. And that these spaces are similar so far
as the fold which separates them is concerned, appears
from this; that these two parts coincide when the
paper is doubled. And thus a fold in a sheet of paper
at the same time illustrates the definition of a straight
line according to the above view, and confirms the
axiom that two such lines cannot enclose a space.

If the separation of the two parts of space were made
by any other than a straight line ; if, for instance, the
paper were cut by a concave line ; then, on turning one
of the parts over, it is easy to see that the edge of one
part being concave one way, and the edge of the other
part concave the other way, these two lines would
enclose a space. And each of them would divide the
whole space into two portions which were not similar ;
for one portion would have a concave edge, and the
other a convex edge. Between any two points, there
might be innumerable lines drawn, some, convex one
way, and some, convex the other way ; but the straight
line is the line which is not convex either one way or
the other ; it is the single medium standard from which
the others may deviate in opposite directions.

Such considerations as these show sufficiently that
the singleness of the straight line which connects any
two points is a result of our fundamental conceptions of
space. But yet the above conceptions of the similar
form of the two parts of space on the two sides of a line,
and of the form of a line which is intermediate among

VOL. i. w. p. H

98 PHILOSOPHY OF THE PURE SCIENCES.

all other forms, are of so vague a nature, that they can
not fitly be made the basis of our elementary geometry ;
and they are far more conveniently replaced, as they
have been in almost all treatises of geometry, by the
axiom, that two straight lines cannot inclose a space.

7. But we may remark that, in what precedes, we
have considered space only under one of its aspects : as
a plane. The sheet of paper which we assumed in order
to illustrate the nature of a straight line, was supposed
to be perfectly plane orflat: for otherwise, by folding it,
we might obtain a line not straight. Now this assump
tion of a plane appears to take for granted that very
conception of a straight line which the sheet was em
ployed to illustrate ; for the definition of a plane given
in the Elements of Geometry is, that it is a surface on
which lie all straight lines drawn from one point of the
surface to another. And thus the explanation above
given of the nature of a straight line, that it divides a
plane space into similar portions on each side, appears
to be imperfect or nugatory.

To this we reply, that the explanation must be ren
dered complete and valid by deriving the conception of
a plane from considerations of the same kind as those
which we employed for a straight line. Any portion of
solid space may be divided into two portions by surfaces
passing through any given line or boundaries. And
these surfaces may be convex either on one side or on
the other, and they admit of innumerable changes from
being convex on one side to being convex on the other
in any degree. So long as the surface is convex either
way, the two portions of space which it separates are not
similar, one having a convex and the other a concave
boundary. But there is a certain intermediate position of
the surface, in which position the two portions of space
which it divides have their boundaries exactly similar.

DEFINITIONS AND AXIOMS RELATING TO SPACE. 09

In this position, the surface is neither convex nor concave,
but plane. And thus a plane surface is determined by
this condition of its being that single surface which is
the intermediate form among all convex and concave
surfaces by which solid space can be divided, and of
its separating such space into two portions, of whiqh
the boundaries, though they are the same surface in
two opposite positions, are exactly similar.

Thus a plane is the simplest and most symmetrical
boundary by which a solid can be divided ; and a straight
line is the simplest and most symmetrical boundary by
which a plane can be separated. These conceptions are
obtained by considering the boundaries of an intermin
able space, capable of imaginary division in every direc
tion. And as a limited space may be separated into two
parts by a plane, and a plane again separated into two
parts by a straight line, so a line is divided into two por
tions by a point, which is the common boundary of the
t\vo portions ; the end of the one and the beginning of the
other portion having itself no magnitude, form, or parts.

8. The geometrical properties of planes and solids
are deducible from the first principles of the Elements,
without any new axioms ; the definition of a plane above
quoted, that all straight lines joining its points lie in
the plane, being a sufficient basis for all reasoning upon
these subjects. And thus, the views which we have pre
sented of the nature of space being verbally expressed
by means of certain definitions and axioms, become the
groundwork of a long series of deductive reasoning, by
which is established a very large and curious collection
of truths, namely, the whole science of Elementary
Plane and Solid Geometry.

This science is one of indispensable use and constant
reference, for every student of the laws of nature ; for the
relations of space and number are the alphabet in which

II 2

100 PHILOSOPHY OF THE PUKE SCIENCES.

those laws are written. But besides the interest and im
portance of this kind which geometry possesses, it has a
great and peculiar value for all who wish to understand
the foundations of human knowledge, and the methods
by which it is acquired. For the student of geometry
acquires, with a degree of insight and clearness which
the unmathematical reader can but feebly imagine, a
conviction that there are necessary truths, many of them
of a very complex and striking character; and that a
few of the most simple and self-evident truths which it is
possible for the mind of man to apprehend, may, by
systematic deduction, lead to the most remote and unex
pected results.

In pursuing such philosophical researches as that
in which we are now engaged, it is of great advantage
to the speculator to have cultivated to some extent the
study of geometry ; since by this study he may become
fully aware of such features in human knowledge as
those which we have mentioned. By the aid of the
lesson thus learned from the contemplation of geome
trical truths, we have been endeavouring to establish
those further doctrines; that these truths are but dif
ferent aspects of the same Fundamental Idea, and that
the grounds of the necessity which these truths possess
reside in the Idea from which they flow, this Idea not
being a derivative result of experience, but its primary
rule. When the reader has obtained a clear and satis
factory view of these doctrines, so far as they are appli
cable to our knowledge concerning space, he has, we may
trust, overcome the main difficulty which will occur in
following the course of the speculations now presented
to him. He is then prepared to go forwards with us ; to
see over how wide a field the same doctrines are appli
cable: and how rich and various a harvest of knowledge
springs from these seemingly scanty principles.

DEFINITIONS AND AXIOMS RELATING TO SPACE. 101

But before we quit the subject now under our con
sideration, we shall endeavour to answer some objections
which have been made to the views here presented; and
shall attempt to illustrate further the active powers which
we have ascribed to the mind.

CHAPTER V.

OF SOME OBJECTIONS WHICH HAVE BEEN
MADE TO THE DOCTRINES STATED
IN THE PREVIOUS CHAPTER-".

THE Edinburgh Review, No. cxxxv., contains a cri
tique on a work termed The Mechanical Euclid, in which
opinions were delivered to nearly the same effect as some
of those stated in the last chapter, and in Chapter xi.
of the First Book. Although I believe that there are no
arguments used by the reviewer to which the answers
will not suggest themselves in the mind of any one who
has read with attention what has been said in the pre
ceding chapters (except, perhaps, one or two remarks
which have reference to mechanical ideas), it may serve to

* In order to render the present chapter more intelligible, it may
be proper to state briefly the arguments which gave occasion to the
review. After noticing Stewart s assertions, that the certainty of mathe
matical reasoning arises from its depending upon definitions, and that
mathematical truth is hypothetical; I urged, that no one has yet
been able to construct a system of mathematical truths by the aid of
definitions alone ; that a definition would not be admissible or appli
cable except it agreed with a distinct conception in the mind ; that the
definitions which we employ in mathematics are not arbitrary or hypo
thetical, but necessary definitions; that if Stewart had taken as his
examples of axioms the peculiar geometrical axioms, his assertions
would have been obviously erroneous ; and that the real foundation of
the truths of mathematics is the Idea of Space, which may be expressed
(for purposes of demonstration) partly by definitions and partly by
axioms.

102 PHILOSOPHY OF THE PURE SCIENCES.

illustrate the subject if I reply to the objections directly,
taking them as the reviewer has stated them.

1. I had dissented from Stewart s assertion that
mathematical truth is hypothetical, or depends upon arbi
trary definitions ; since we understand by an hypothesis
a t supposition, not only which we may make, but may
abstain -from- ^making, or may replace by a different sup
position ;, whereas the definitions and hypotheses of geo
metry are -i&gt;ecessarily such as they are, and cannot be
altered or excluded. The reviewer (p. 84), informs us
that he understands Stewart, when he speaks of hypo
theses and definitions being the foundation of geometry,
to speak of the hypothesis that real objects correspond
to our geometrical definitions. " If a crystal be an exact
hexahedron, the geometrical properties of the hexahe
dron may be predicated of that crystal." To this I reply,
that such hypotheses as this are the grounds of our
applications of geometrical truths to real objects, but
can in no way be said to be the foundation of the truths
themselves; that I do not think that the sense which the
reviewer gives was Stewart s meaning; but that if it was,
this view of the use of mathematics does not at all affect
the question which both he and I proposed to discuss,
which was, the ground of mathematical certainty. I may
add, that whether a crystal be an exact hexahedron, is
a matter of observation and measurement, not of defini
tion. I think the reader can have no difficulty in seeing
how little my doctrine is affected by the connexion on
which the reviewer thus insists. I have asserted that the
proposition which affirms the square on the diagonal of
a rectangle to be equal to the squares on two sides, does
not rest upon arbitrary hypotheses; the objector answers,
that the proposition that the square on the diagonal of
this page is equal to the squares on the sides, depends
upon the arbitrary hypothesis that the page is a rect-

ANSWER TO OBJECTIONS. 103

angle. Even if this fact were a matter of arbitrary
hypothesis, what could it have to do with the general
geometrical proposition? How could a single fact, ob
served or hypothetical, affect a universal and necessary
truth, which would be equally true if the fact were false?
If there be nothing arbitrary or hypothetical in geometry
till we come to such steps in its application, it is plain
that the truths themselves are not hypothetical; which is
the question for us to decide.

2. The reviewer then (p. 85), considers the doctrine
that axioms as well as definitions are the foundations of
geometry; and here he strangely narrows and confuses
the discussion by making himself the advocate of Stewart,
instead of arguing the question itself. I had asserted
that some axioms are necessary as the foundations of
mathematical reasoning, in addition to the definitions.
If Stewart did not intend to discuss this question, I had
no concern with what he had said about axioms. But I
had every reason to believe that this was the question
which Stewart did intend to discuss. I conceive there is
no doubt that he intended to give an opinion upon the
grounds of mathematical reasoning in general. For he
begins his discussions (Elements, Vol. IL, p. 38) by contest
ing Reid s opinion on this subject, which is stated gene
rally; and he refers again to the same subject, asserting
in general terms, that the first principles of mathematics
are not axioms but definitions. If, then, afterwards, he
made his proof narrower than his assertion ; if having
declared that no axioms are necessary, he afterwards
limited himself to showing that seven out of twelve of
Euclid s axioms are barren truisms, it was no concern of
mine to contest this assertion, which left my thesis un
touched. I had asserted that the proper geometrical
axioms (that two straight lines cannot inclose a spa ce,
and the axiom about parallel lines) are indispensable in

104 PHILOSOPHY OF THE PURE SCIENCES.

geometry. What account the reviewer gives of these
axioms we shall soon see; but if Stewart allowed them to
be axioms necessary to geometrical reasoning, he over
turned his own assertion as to the foundations of such
reasoning ; and if he said nothing decisive about these
axioms, which are the points on which the battle must
turn, he left his assertion altogether unproved ; nor was
it necessary for me to pursue the war into a barren and
unimportant corner, when the metropolis was surrendered.
The reviewer s exultation that I have not contested the
first seven axioms is an amusing example of the self-
complacent zeal of advocacy.

3. But let us turn to the material point, the proper
geometrical axioms. What is the reviewer s account of
these? Which side of the alternative does he adopt?
Do they depend upon the definitions, and is he prepared
to show the dependence ? Or are they superfluous, and
can he erect the structure of geometry without their aid?
One of these two courses, it would seem, he must take.
For we both begin by asserting the excellence of geo
metry as an example of demonstrated truth. It is
precisely this attribute which gives an interest to our
present inquiry. How, then, does the reviewer explain
this excellence on his views ? How does he reckon the
foundation courses of the edifice which we agree in con
sidering as a perfect example of intellectual building ?

I presume I may take, as his answer to this question,
his hypothetical statement of what Stewart would have
said, (p. 87,) on the supposition that there had been,
among the foundations of geometry, self-evident indemon
strable truths : although it is certainly strange that the
reviewer should not venture to make up his mind as to
the truth or falsehood of this supposition. If there were
such truths they would be, he says, " legitimate filiations"
of the definitions. They would be involved in the defi-

ANSWER TO OBJECTIONS. 105

nitions. And again he speaks of the foundation of the
geometrical doctrine of parallels as a flaw, and as a
truth which requires, but has not received demonstration.
And yet again, he tells us that each of these supposed
axioms (Euclid s twelfth, for instance), is "merely an
indication of the point at which geometry fails to per
form that which it undertakes to perform" (p. 91); and
that in reality her truths are not yet demonstrated. The
amount of this is, that the geometrical axioms are to be
held to be legitimate filiations of the definitions, because
though certainly true, they cannot be proved from the
definitions; that they are involved in the definitions,
although they cannot be evolved out of them ; and that
rather than admit that they have any other origin than
the definitions, we are to proclaim that geometry has
failed to perform what she undertakes to perform.

To this I reply that I cannot understand what is
meant by "legitimate filiations" of principles, if the phrase
not mean consequences of such principles established by
rigorous and formal demonstrations ; that the reviewer,
if he claims any real signification for his phrase, must
substantiate the meaning of it by such a demonstration ;
he must establish his " legitimate filiation" by a genea
logical table in a satisfactory form. When this cannot
be done, to assert, notwithstanding, that the propositions
are involved in the definitions, is a mere begging the
question; and to excuse this defect by saying that geo
metry fails to perform what she has promised, is to calum
niate the character of that science which we profess to
make our standard, rather than abandon an arbitrary
and unproved assertion respecting the real grounds of
her excellence. I add, further, that if the doctrine of
parallel lines, or any other geometrical doctrine of which
we see the truth, with the most perfect insight of its
necessity, have not hitherto received demonstration to the

106 PHILOSOPHY OF THE PURE SCIENCES.

satisfaction of any school of reasoners, the defect must
arise from their erroneous views of the nature of demon
strations, and the grounds of mathematical certainty.

4. I conceive, then, that the reviewer has failed alto
gether to disprove the doctrine that the axioms of geo
metry are necessary as a part of the foundations of the
science. I had asserted further that these axioms supply
what the definitions leave deficient ; and that they, along
with definitions, serve to present the idea of space under
such aspects that we can reason logically concerning it.
To this the reviewer opposes (p. 96) the common opinion
that a perfect definition is a complete explanation of a
name, and that the test of its perfection is, that we
may substitute the definition for the name wherever
it occurs. I reply, that my doctrine, that a definition
expresses a part, but not the whole, of the essential cha
racters of an idea, is certainly at variance with an opinion
sometimes maintained, that a definition merely explains
a word, and should explain it so fully that it may always
replace it. The error of this common opinion may, I think,
be shown from considerations such as these ; that if we
undertake to explain one word by several, we may be
called upon, on the same ground, to explain each of these
several by others, and that in this way we can reach no
limit nor resting-place ; that in point of fact, it is not
found to lead to clearness, but to obscurity, when in the
discussion of general principles, we thus substitute defi
nitions for single terms ; that even if this be done, we
cannot reason without conceiving what the terms mean ;
and that, in doing this, the relations of our concep
tions, and not the arbitrary equivalence of two forms of
expression, are the foundations of our reasoning.

5. The reviewer conceives that some of the so-called
axioms are really definitions. The axiom, that " magni
tudes which coincide with each other, that is, which fill

ANSWER TO OBJECTIONS. 107

the same space, are equal," is a definition of geometrical
equality : the axiom, that " the whole is greater than its
part," is a definition of whole and part. But surely there
are very serious objections to this view. It would seem
more natural to say, if the former axiom is a definition
of the word equal, that the latter is a definition of the
word greater. And how can one short phrase define two
terms ? If I say, " the heat of summer is greater than
the heat of winter," does this assertion define anything,
though the proposition is perfectly intelligible and dis
tinct? I think, then, that this attempt to reduce these
axioms to definitions is quite untenable.

6. I have stated that a definition can be of no use,
except we can conceive the possibility and truth of the
property connected with it ; and that if we do conceive
this, we may rightly begin our reasonings by stating the
property as an axiom ; which Euclid does, in the case of
straight lines and of parallels. The reviewer inquires,
(p. 92,) whether I am prepared to extend this doctrine to
the case of circles, for which the reasoning is usually
rested upon the definition ; whether I would replace this
definition by an axiom, asserting the possibility of such a
circle. To this I might reply, that it is not at all incum
bent upon me to assent to such a change ; for I have all
along stated that it is indifferent whether the fundamen
tal properties from which we reason be exhibited as defi
nitions or as axioms, provided their necessity be clearly
seen. But I am ready to declare that I think the form
of our geometry would be not at all the worse, if, instead
of the usual definition of a circle, that it is a figure
contained by one line, which is called the circumference,
and which is such, that all straight lines drawn from a
certain point within the circumference are equal to one
another," we were to substitute an axiom and a defini
tion, as follows :

108 PHILOSOPHY OF THE PURE SCIENCES.

Axiom. If a line be drawn so as to be at every point
equally distant from a certain point, this line will return
into itself, or will be one line including a space.

Definition. The space is called a circle, the line the
circumference, and the point the center.

And this being done, it would be true, as the reviewer
remarks, that geometry cannot stir one step without
resting on an axiom. And I do not at all hesitate to say,
that the above axiom, expressed or understood, is no less
necessary than the definition, and is tacitly assumed in
every proposition into which circles enter.

7. I have, I think, now disposed of the principal
objections which bear upon the proper axioms of geo
metry. The principles which are stated as the first seven
axioms of Euclid s Elements, need not, as I have said, be
here discussed. They are principles which refer, not to
Space in particular, but to Quantity in general : such ?
for instance, as these ; " If equals be added to equals the
wholes are equal ;" " If equals be taken from equals
the remainders are equal." But I will make an obser
vation or two upon them before I proceed.

Both Locke and Stewart have spoken of these axioms
as barren truisms : as propositions from which it is not
possible to deduce a single inference : and the reviewer
asserts that they are not first principles, but laws of
thought, (p. 88.) To this last expression I am willing
to assent ; but I would add, that not only these, but all
the principles which express the fundamental conditions
of our knowledge, may with equal propriety be termed
laws of thought ; for these principles depend upon our
ideas, and regulate the active operations of the mind, by
which coherence and connexion are given to its passive
impressions. But the assertion that no conclusions can
be drawn from simple axioms, or laws of human thought,
which regard quantity, is by no means true. The whole.

ANSWER TO OBJECTIONS. 100

of arithmetic, for instance, the rules for the multiplica
tion and division of large numbers, for finding a common
measure, and, in short, a vast body of theory respecting
numbers, rests upon no other foundation than such
axioms as have been just noticed, that if equals be added
to equals the wholes will be equal. And even when
Locke s assertion, that from these axioms no truths can
be deduced, is modified by Stewart and the reviewer,
and limited to geometrical truths, it is hardly tenable
(although, in fact, it matters little to our argument
whether it is or no). For the greater part of the Seventh
Book of Euclid s Elements, (on Commensurable and In
commensurable Quantities,) and the Fifth Book, (on
Proportion,) depend upon these axioms, with the addi
tion only of the definition or axiom (for it may be stated
either way) which expresses the idea of proportionality
in numbers. So that the attempt to disprove the neces
sity and use of axioms, as principles of reasoning, fails
even when we take those instances which the opponents
consider as the more manifestly favourable to their
doctrine.

8. But perhaps the question may have already sug
gested itself to the reader s mind, of what use can it be
formally to state such principles as these, (for example,
that if equals be added to equals the wholes are equal,)
since, whether stated or no, they will be assumed in our
reasoning ? And how can such principles be said to be
necessary, when our proof proceeds equally well without
any reference to them ? And the answer is, that it is
precisely because these are the common principles of
reasoning, which we naturally employ without specially
contemplating them, that they require to be separated
from the other steps and formally stated, when we
analyze the demonstrations which we have obtained
In every mental process many principles are combined

110 PHILOSOPHY OF THE PURE SCIENCES.

and abbreviated, and thus in some measure concealed
and obscured. In analyzing these processes, the combi
nation must be resolved, and the abbreviation expanded,
and thus the appearance is presented of a pedantic and
superfluous formality. But that which is superfluous for
proof, is necessary for the analysis of proof. In order to
exhibit the conditions of demonstration distinctly, they
must be exhibited formally. In the same manner, in
demonstration we do not usually express every step in
the form of a syllogism, but we see the grounds of the
conclusiveness of a demonstration, by resolving it into
syllogisms. Neither axioms nor syllogisms are necessary
for conviction; but they are necessary to display the
conditions under which conviction becomes inevitable.
The application of a single one of the axioms just spoken
of is so minute a step in the proof, that it appears pe
dantic to give it a marked place ; but the very essence
of demonstration consists in this, that it is composed of
an indissoluble succession of such minute steps. The
admirable circumstance is, that by the accumulation of
such apparently imperceptible advances, we can in the
end make so vast and so sure a progress. The com
pleteness of the analysis of our knowledge appears in the
smallness of the elements into which it is thus resolved.
The minuteness of any of these elements of truth, of
axioms for instance, does not prevent their being as
essential as others which are more obvious. And any
attempt to assume one kind of element only, when the
course of our analysis brings before us two or more
kinds, is altogether unphilosophical. Axioms and defi
nitions are the proximate constituent principles of our
demonstrations; and the intimate bond which connects
together a definition and an axiom on the same subject
is not truly expressed by asserting the latter to be de
rived from the former. This bond of connexion exists

OF THE PERCEPTION OF SPACE. Ill

in the mind of the reasoner, in his conception of that to
which both definition and axiom refer, and consequently
in the general Fundamental Idea of which that concep
tion is a modification.

CHAPTER VI.
OF THE PERCEPTION OF SPACE.

1. ACCORDING to the views above explained, certain
of the impressions of our senses convey to us the per
ception of objects as existing in space ; inasmuch as by
the constitution of our minds we cannot receive those
impressions otherwise than in a certain form, involving
such a manner of existence. But the question deserves
to be asked, What are the impressions of sense by which
we thus become acquainted with space and its relations ?
And as we have seen that this idea of space implies an
act of the mind as well as an impression on the sense,
what manifestations do we find of this activity of the
mind, in our observation of the external world ?

It is evident that sight and touch are the senses by
which the relations of space are perceived, principally or
entirely. It does not appear that an odour, or a feeling
of warmth or cold, would, independently of experience,
suggest to us the conception of a space surrounding us.
But when we see objects, we see that they are extended
and occupy space; when we touch them, we feel that
they are in a space in which we also are. We have
before our eyes any object, for instance, a board covered
with geometrical diagrams ; and we distinctly perceive,
by vision, those lines of which the relations are the
subjects of our mathematical reasoning. Again, we see
before us a solid object, a cubical box for instance ; we
see that it is within reach ; we stretch out the hand and

112 PHILOSOPHY OE THE PURE SCIENCES.

perceive by the touch that it has sides, edges, corners,
which we had already perceived by vision.

2. Probably most persons do not generally appre
hend that there is any material difference in these two
cases ; that there are any different acts of mind con
cerned in perceiving by sight a mathematical diagram
upon paper, and a solid cube lying on a table. Yet it is
not difficult to show that, in the latter case at least, the
perception of the shape of the object is not immediate.
A very little attention teaches us that there is an act of
judgment as well as a mere impression of sense requisite,
in order that we may see any solid object. For there is
no visible appearance which is inseparably connected
with solidity. If a picture of a cube be rightly drawn in
perspective and skilfully shaded, the impression upon the
sense is the same as if it were a real cube. The picture
may be mistaken for a solid object. But it is clear that,
in this case, the solidity is given to the object by an act
of mental judgment. All that is seen is outline and
shade, figures and colours on a flat board. The solid
angles and edges, the relation of the faces of the figure
by which they form a cube, are matters of inference.
This, which is evident in the case of the pictured cube, is
true in all vision whatever. We see a scene before us
on which are various figures and colours, but the eye
cannot see more. It sees length and breadth, but no
third dimension. In order to know that there are solids,
we must infer as well as see. And this we do readily
and constantly; so familiarly, indeed, that we do not
perceive the operation. Yet we may detect this latent
process in many ways; for instance, by attending to
cases in which the habit of drawing such inferences mis
leads us. Most persons have experienced this delusion
in looking at a scene in a theatre, and especially that
kind of scene which is called a diorama, when the

OF THE PERCEPTION OF SPACE. 113

interior of a building is represented. In these cases,
the perspective representations of the various members
of the architecture and decoration impress us almost
irresistibly with the conviction that we have before us a
space of great extent and complex form, instead of a flat
painted canvass. Here, at least, the space is our own
creation, but yet here, it is manifestly created by the
same act of thought as if we were really in the palace or
the cathedral of which the halls and aisles thus seem to
inclose us. And the act by which we thus create space
of three dimensions out of visible extent of length and
breadth, is constantly and imperceptibly going on. We
are perpetually interpreting in this manner the language
of the visible world. From the appearances of things
which we directly see, we are constantly inferring that
which we cannot directly see, their distance from us,
and the position of their parts.

3. The characters which we thus interpret are
various. They are, for instance, the visible forms,
colours, and shades of the parts, understood according
to the maxims of perspective ; (for of perspective every
one has a practical knowledge, as every one has of
grammar ;) the effort by which we fix both our eyes on
the same object, and adjust each eye to distinct vision ;
and the like. The right interpretation of the informa
tion which such circumstances give us respecting the
true forms and distances of things, is gradually learned ;
the lesson being begun in our earliest infancy, and
inculcated upon us every hour during which we use our
eyes. The completeness with which the lesson is mas
tered is truly admirable ; for we forget that our con
clusion is obtained indirectly, and mistake a judgment
on evidence for an intuitive perception. We see the
breadth of the street, as clearly and readily as we see
the house on the other side of it ; and we see the house
VOL. i. w. P. I

114 PHILOSOPHY OF THE PURE SCIENCES.

to be square, however obliquely it be presented to us.
This, however, by no means throws any doubt or diffi
culty on the doctrine that in all these cases we do inter
pret and infer. The rapidity of the process, and the
unconsciousness of the effort, are not more remarkable
in this case than they are when we understand the
meaning of the speech which we hear, or of the book
which we read. In these latter cases we merely hear
noises or see black marks ; but we make, out of these
elements, thought and feeling, without being aware of
the act by which we do so. And by an exactly similar
process we see a variously-coloured expanse, and collect
from it a space occupied by solid objects. In both
cases the act of interpretation is become so habitual
that we can hardly stop short at the mere impression
of sense.

4. But yet there are various ways in which we may
satisfy ourselves that these two parts of the process of
seeing objects are distinct. To separate these operations
is precisely the task which the artist has to execute in
making a drawing of what he sees. He has to recover
the consciousness of his real and genuine sensations, and
to discern the lines of objects as they appear. This at
first he finds difficult ; for he is tempted to draw what
he knows of the forms of visible objects, and not what
he sees : but as he improves in his art, he learns to put
on paper what he sees only, separated from what he
infers, in order that thus the inference, and with it a
conception like that of the reality, may be left to the
spectator. And thus the natural process of vision is the
habit of seeing that which cannot be seen ; and the diffi
culty of the art of drawing consists in learning not to
see more than is visible.

5. But again ; even in the simplest drawing we
exhibit something which we do not see. However

OF THE PERCEPTION OF SPACE. 115

slight is our representation of objects, it contains some
thing which we create for ourselves. For we draw an
outline. Now an outline has no existence in nature.
There are no visible lines presented to the eye by a
group of figures. We separate each figure from the rest,
and the boundary by which we do this is the outline of
the figure ; and the like may be said of each member of
every figure. A painter of our own times has made this
remark in a work upon his art*. "The effect which
natural objects produce upon our sense of vision is that
of a number of parts, or distinct masses of form and
colour, and not of lines. But when we endeavour to
represent by painting the objects which are before us, or
which invention supplies to our minds, the first and the
simplest means we resort to is this picture, by which we
separate the form of each object from those that sur
round it, marking its boundary, the extreme extent of
its dimensions in every direction, as impressed on our
vision : and this is termed drawing its outline."

6. Again, there are other ways in which we see clear
manifestations of the act of thought by which we assign
to the parts of objects their relations in space, the im
pressions of sense being merely subservient to this act.
If we look at a medal through a glass which inverts it,
we see the figures upon it become concave depressions
instead of projecting convexities; for the light which
illuminates the nearer side of the convexity will be trans
ferred to the opposite side by the apparent inversion of
the medal, and will thus imply a hollow in which the
side nearest the light gathers the shade. Here our deci
sion as to which part is nearest to us, has reference to
the side from which the light comes. In other cases
the decision is more spontaneous. If we draw black
outlines, such as represent the edges of a cube seen

* Phillips On Faulting.

I 2

116 PHILOSOPHY OF THE PURE SCIENCES.

in perspective, certain of the lines will cross each other ;
and we may make this cube appear to assume two dif
ferent positions, by determining in our own mind that
the lines which belong to one end of the cube shall be
understood to be before or to be behind those which
they cross. Here an act of the will, operating upon the
same sensible image, gives us two cubes, occupying two
entirely different positions. Again, many persons may
have observed that when a windmill in motion at a dis
tance from us, (so that the outline of the sails only is
seen,) stands obliquely to the eye, we may, by an effort
of thought, make the obliquity assume one or the other
of two positions ; and as we do this, the sails, which in
one instance appear to turn from right to left, in the other
case turn from left to right. A person a little familiar
with this mental effort, can invert the motion as often as
he pleases, so long as the conditions of form and light
do not offer a manifest contradiction to either position.

Thus we have these abundant and various manifesta
tions of the activity of the mind, in the process by which
we collect from vision the relations of solid space of three
dimensions. But we must further make some remarks
on the process by which we perceive mere visible figure;
and also, on the mode in which we perceive the relations
of space by the touch ; and first, of the latter subject.

7. The opinion above illustrated, that our sight does
not give us a direct knowledge of the relations of solid
space, and that this knowledge is acquired only by an
inference of the mind, was first clearly taught by the
celebrated Bishop Berkeley"", and is a doctrine now
generally assented to by metaphysical speculators.

But does the sense of touch give us directly a know
ledge of space ? This is a question which has attracted
considerable notice in recent times; and new light has

* Theory of Vision.

OF THE PERCEPTION OF SPACE. 117

been thrown upon it in a degree which is very remark
able, when we consider that the philosophy of perception
has been a prominent subject of inquiry from the earliest
times. Two philosophers, advancing to this inquiry from
different sides, the one a metaphysician, the other a phy
siologist, have independently arrived at the conviction
that the long current opinion, according to which we
acquire a knowledge of space by the sense of touch, is
erroneous. And the doctrine which they teach instead
of the ancient errour, has a very important bearing upon
the principle which we are endeavouring to establish,
that our knowledge of space and its properties is derived
rather from the active operations than from the passive
impressions of the percipient mind.

Undoubtedly the persuasion that we acquire a know
ledge of form by the touch is very obviously suggested
by our common habits. If we wish to know the form of
any body in the dark, or to correct the impressions con
veyed by sight, when we suspect them to be false, we
have only, it seems to us, at least at first, to stretch forth
the hand and touch the object ; and we learn its shape
with no chance of error. In these cases, form appears
to be as immediate a perception of the sense of touch,
as colour is of the sense of sight.

8. But is this perception really the result of the
passive sense of touch merely ? Against such an opinion
Dr. Brown, the metaphysician of whom I speak, urges*
that the feeling of touch alone, when any object is ap
plied to the hand, or any other part of the body, can no
more convey the conception of form or extension, than
the sensation of an odour or a taste can do, except we
have already some knowledge of the relative position of
the parts of our bodies; that is, except we are already in
possession of an idea of space, and have, in our minds,

* Lectures, Vol. I. p. 459, (1824).

118 PHILOSOPHY OF THE PURE SCIENCES.

referred our limbs to their positions; which is to sup
pose the conception of form already acquired.

9. By what faculty then do we originally acquire our
conceptions of the relations of position ? Brown answers
by the muscular sense; that is, by the conscious exer
tions of the various muscles by which we move our limbs.
When we feel out the form and position of bodies by
the hand, our knowledge is acquired, not by the mere
touch of the body, but by perceiving the course the
fingers must take in order to follow the surface of the
body, or to pass from one body to another. We are
conscious of the slightest of the volitions by which we
thus feel out form and place ; we know whether we move
the finger to the right or left, up or down, to us or from
us, through a large or a small space ; and all these con
scious acts are bound together and regulated in our
minds by an idea of an extended space in which they are
performed. That this idea of space is not borrowed from
the sight, and transferred to the muscular feelings by
habit, is evident. For a man born blind can feel out his
way with his staff, and has his conceptions of position
determined by the conditions of space, no less than one
who has the use of his eyes. And the muscular con
sciousness which reveals to us the position of objects and
parts of objects, when we feel them out by means of the
hand, shews itself in a thousand other ways, and in all
our limbs: for our habits of standing, walking, and all
other attitudes and motions, are regulated by our feeling
of our position and that of surrounding objects. And
thus, we cannot touch any object without learning some
thing respecting its position ; not that the sense of
touch directly conveys such knowledge ; but we have
already learnt, from the muscular sense, constantly
exercised, the position of the limb which the object thus
touches.

OF THE PERCEPTION OF SPACE. 119

10. The justice of this distinction will, I think, be
assented to by all persons who attend steadily to the
process itself, and might be maintained by many forcible
reasons. Perhaps one of the most striking evidences in
its favour is that, as I have already intimated, it is the
opinion to which another distinguished philosopher, Sir
Charles Bell, has been led, reasoning entirely upon phy
siological principles. From his researches it resulted
that besides the nerves which convey the impulse of the
will from the brain to the muscle, by which every motion
of our limbs is produced, there is another set of nerves
which carry back to the brain $ sense of the condition
of the muscle, and thus regulate its activity ; and give us
the consciousness of our position and relation to sur
rounding objects. The motion of the hand and fingers,
or the consciousness of this motion, must be combined
with the sense of touch properly so called, in order to
make an inlet to the knowledge of such relations. This
consciousness of muscular exertion, which he has called a
sixth sense" ", is our guide, Sir C. Bell shows, in the com
mon practical government of our motions ; and he states
that having given this explanation of perception as a
physiological doctrine, he had afterwards with satisfac
tion seen it confirmed by Dr. Brown s speculations.

11. Thus it appears that our consciousness of the
relations of space is inseparably and fundamentally con
nected with our own actions in space. We perceive only
while we act ; our sensations require to be interpreted by
our volitions. The apprehension of extension and figure
is far from being a process in which we are inert arid
passive. We draw lines with our fingers ; we construct
surfaces by curving our hands; we generate spaces by the
motion of our arms. When the geometer bids us form
lines, or surfaces, or solids by motion, he intends his

* Bridgewater Treatise, p. 195. Phil. Trans. 1826, Pt. n., p. 167.

120 PHILOSOPHY OF THE PURE SCIENCES.

injunction to be taken as hypothetical only ; we need only
conceive such motions. But yet this hypothesis repre
sents truly the origin of our knowledge ; we perceive
spaces by motion at first, as we conceive spaces by motion
afterwards : or if not always by actual motion, at least
by potential. If we perceive the length of a staff by
holding its two ends in our two hands without running
the finger along it, this is because by habitual motion we
have already acquired a measure of the distance of our
hands in any attitude of which we are conscious. Even
in the simplest case, our perceptions are derived not from
the touch, but from the sixth sense ; and this sixth sense
at least, whatever may be the case with the other five,
implies an active mind along with the passive sense.

12. Upon attentive consideration, it will be clear
that a large portion of the perceptions respecting space
which appear at first to be obtained by sight alone, are,
in fact, acquired by means of this sixth sense. Thus we
consider the visible sky as a single surface surrounding
us and returning into itself, and thus forming a hemi
sphere. But such a mode of conceiving an object of vision
could never have occurred to us, if we had not been able
to turn our heads, to follow this surface, to pursue it till
we find it returning into itself. And when we have done
this, we necessarily present it to ourselves as a concave
inclosure within which we are. The sense of sight alone,
without the power of muscular motion, could not have
led us to view the sky as a vault or hemisphere. Under
such circumstances, we should have perceived only what
was presented to the eye in one position ; and if dif
ferent appearances had been presented in succession, we
could not have connected them as parts of the same
picture, for want of any perception of their relative posi
tion. They would have been so many detached and
incoherent visual sensations. The muscular sense con-

OF THE PERCEPTION OF SPACE. 121

nccts their parts into a whole, making them to be only
different portions of one universal scene 4 ".

13. These considerations point out the fallacy of a
very curious representation made by Dr. Reid, of the
convictions to which man would be led, if he possessed
vision without the sense of touch. To illustrate this sub
ject, Reid uses the fiction of a nation whom he terms the
Idomenians, who have no sense except that of sight. He
describes their notions of the relations of space as being
entirely different from ours. The axioms of their geome
try are quite contradictory to our axioms. For example,
it is held to be self-evident among them that two straight
lines which intersect each other once, must intersect a
second time; that the three angles of any triangle are
greater than two right angles; and the like. These
paradoxes are obtained by tracing the relations of lines
on the surface of a concave sphere, which surrounds the
spectator, and on which all visible appearances may be
supposed to be presented to him. But from what is said
above it appears that the notion of such a sphere, and
such a connexion of visible objects which are seen in dif
ferent directions, cannot be arrived at by sight alone.

* It has been objected to this view, that we might obtain a con
ception of the sky as a hemisphere, by being ourselves turned round, (as
on a music-stool, for instance,) and thus seeing in succession all parts of
the sky. But this assertion I conceive to be erroneous. By being thus
turned round, we should see a number of pictures which we should put
together as parts of a plane picture ; and when we came round to the
original point, we should have no possible means of deciding that it
was the same point : it would appear only as a repetition of the pic
ture. That sight, of itself, can give us only a plane picture, the doctrine
of Berkeley, appears to be indisputable ; and, no less so, the doctrine
that it is the consciousness of our own action in space which puts toge
ther these pictures so that they cover the surface of a solid body. We
can see length and breadth with our eyes, but we must thrust out our
arm towards the flat surface, in order that we may, in our thoughts,
combine a third dimension with the other two.

122 PHILOSOPHY OF THE PURE SCIENCES.

When the spectator combines in his conception the rela
tions of long-drawn lines and large figures, as he sees
them by turning his head to the right and to the left,
upwards and downwards, he ceases to be an Idomenian.
And thus our conceptions of the properties of space, de
rived through the exercise of one mode of perception,
are not at variance with those obtained in another way ;
but all such conceptions, however produced or suggested,
are in harmony with each other; being, as has already
been said, only different aspects of the same idea.

14. If our perceptions of the position of objects
around us do not depend on the sense of vision alone,
but on the muscular feeling brought into play when we
turn our head, it will obviously follow that the same is
true when we turn the eye instead of the head. And
thus we may learn the form of objects, not by looking
at them with a fixed gaze, but by following the boundary
of them with the eye. While the head is held perfectly
still, the eye can rove along the outlines of visible ob
jects, scrutinize each point in succession, arid leap from
one point to another ; each such act being accompanied
by a muscular consciousness which makes us aware of
the direction in which the look is travelling. And we
may thus gather information concerning the figures and
places which we trace out with the visual ray, as the
blind man learns the forms of things which he traces out
with his staff, being conscious of the motions of his hand.

15. This view of the mode in which the eye per
ceives position, which is thus supported by the analogy
of other members employed for the same purpose, is
further confirmed by Sir Charles Bell by physiological
reasons. He teaches us that* " when an object is seen we
employ two senses: there is an impression on the retina;
but we receive also the idea of position or relation in

* Phil. Trans., 1823. On the Motions of the Eye.

OF THE PERCEPTION OF SPACE. 123

space, which it is not the office of the retina to give, by
our consciousness of the efforts of the voluntary muscles
of the eye : and he has traced in detail the course of the
nerves by which these muscles convey their information.
The constant searching motion of the eye, as he terms
it*, is the means by which we become aware of the
position of objects about us.

16. It is not to our present purpose to follow the
physiology of this subject ; but we may notice that Sir
C. Bell has examined the special circumstances which
belong to this operation of the eye. We learn from him
that the particular point of the eye which thus traces the
forms of visible objects is a part of the retina which has
been termed the sensible spot; being that part which is
most distinctly sensible to the impressions of light and
colour. This part, indeed, is not a spot of definite size and
form, for it appears that proceeding from a certain point
of the retina, the distinct sensibility diminishes on every
side by degrees. And the searching motion of the eye
arises from the desire which we instinctively feel of re
ceiving upon the sensible spot the image of the object
to which the attention is directed. We are uneasy and

* Bridgewater Treatise, p. 282. I have adopted, in writing the
above, the views and expressions of Sir Charles Bell. The essential
part of the doctrine there presented is, that the eye constantly makes
efforts to turn, so that the image of an object to which our attention is
drawn, shall fall upon a certain particular point of the retina ; and that
when the image falls upon any other point, the eye turns away from
this oblique into the direct position. Other writers have maintained
that the eye thus turns, not because the point on which the image falls
in direct vision is the most sensible point, but that it is the point of
greatest distinctness of vision. They urge that a small star, which dis
appears when the eye is turned full upon it, may often be seen by
looking a little away from it : and hence, they infer that the parts of
the retina removed from the spot of direct vision, are more sensible than
it is. The facts are very curious, however they be explained, but they
do not disturb the doctrine delivered in the text.

124 PHILOSOPHY OF THE PURE SCIENCES.

impatient till the eye is turned so that this is effected.
And as our attention is transferred from point to point
of the scene before us, the eye, and this point of the eye
in particular, travel along with the thoughts ; and the
muscular sense, which tells us of these movements of
the organ of vision, conveys to us a knowledge of the
forms and places which we thus successively survey.

17. How much of activity there is in the process by
which we perceive the outlines of objects appears further
from the language by which we describe their forms.
We apply to them not merely adjectives of form, but
verbs of motion. An abrupt hill starts out of the plain ;
a beautiful figure has a gliding outline. We have

The windy summit, wild and high,
Roughly rushing on the sky.

These terms express the course of the eye as it follows
the lines by which such forms are bounded and marked.
In like manner another modern poet* says of Soracte,
that it

From out the plain

Heaves like a long-swept wave about to break,
And on the curl hangs pausing.

Thus the muscular sense, which is, inseparably con
nected with an act originating in our own mind, not only
gives us all that portion of our perceptions of space in
which we use the sense of touch, but also, at least in a
great measure, another large portion of such perceptions,
in which we employ the sense of sight. As we have
before seen that our knowledge of solid space and its
properties is not conceivable in any other way than as
the result of a mental act, governed by conditions depend
ing on its own nature ; so it now appears that our per
ceptions of visible figure are not obtained without an act
performed under the same conditions. The sensations
of touch and sight are subordinated to an idea which is
* Byron, Ch. Har. vi., st. 75.

OF THE PERCEPTION OF SPACE. 125

the basis of our speculative knowledge concerning space
and its relations ; and this same idea is disclosed to our
consciousness by its practically regulating our inter
course with the external world.

By considerations such as have been adduced and
referred to, it is proved beyond doubt, that in a great
number of cases our knowledge of form and position is
acquired from the muscular sense, and not from sight
directly: for instance, in all cases in which we have
before us objects so large and prospects so extensive
that we cannot see the whole of them in one position of
the eye*.

We now quit the consideration of the properties of
Space, and consider the Idea of Time.

CHAPTER VII.
OF THE IDEA OF TIME.

1. RESPECTING the Idea of Time, we may make
several of the same remarks which we made concerning

* The expression in the first edition was " large objects and exten
sive spaces." In the text as now given, I state a definite size and
extent, within which the sight by itself can judge of position and figure.

The doctrine that we require the assistance of the muscular sense to
enable us to perceive space of three dimensions, is not at all inconsistent
with this other doctrine, that within the space which is seen by the
fixed eye, we perceive the relative positions of points directly by vision,
and that, consequently, we have a perception of visible t figure.

Sir Charles Bell has said, (Phil. Trans. 1823, p. 181,) "It appears
to me that the utmost ingenuity will be at a loss to devise an explana
tion of that power by which the eye becomes acquainted with the
position and relation of objects, if the sense of muscular activity be
excluded which accompanies the motion of the eyeball." But surely we
should have no difficulty in perceiving the relation of the sides and
angles of a small triangle, placed before the eye, even if the muscles of
the eyeball were severed. This subject is resumed B. iv. c. ii. sect. 11.

126 PHILOSOPHY OF THE PURE SCIENCES.

the .idea of space, in order to shew that it is not bor
rowed from experience, but is a bond of connexion
among the impressions of sense, derived from a peculiar
activity of the mind, and forming a foundation both of
our experience and of our speculative knowledge.

Time is not a notion obtained by experience. Expe
rience, that is, the impressions of sense and our con
sciousness of our thoughts, gives us various percep
tions; and different successive perceptions considered
together exemplify the notion of change. But this very
connexion of different perceptions, this successiveness,
presupposes that the perceptions exist in time. That
things happen either together, or one after the other, is
intelligible only by assuming time as the condition under
which they are presented to us.

Thus time is a necessary condition in the presentation
of all occurrences to our minds. We cannot conceive
this condition to be taken away. We can conceive
time to go on while nothing happens in it ; but we can
not conceive anything to happen while time does not
go on.

It is clear from this that time is not an impression
derived from experience, in the same manner in which
we derive from experience our information concerning
the objects which exist, and the occurrences which take
place in time. The objects of experience can easily be
conceived to be, or not to be : to be absent as well as
present. Time always is, and always is present, and
even in our thoughts we cannot form the contrary sup
position.

2. Thus time is something distinct from the matter
or substance of our experience, and may be considered
as a necessary form which that matter (the experience of
change) must assume, in order to be an object of con
templation to the mind. Time is one of the necessary

OF THE IDEA OF TIME. 127

conditions under which we apprehend the information
which our senses and consciousness give us. By con
sidering time as a form which belongs to our power of
apprehending occurrences and changes, and under which
alone all such experience can be accepted by the mind,
we explain the necessity, which we find to exist, of con
ceiving all such changes as happening in time ; and we
thus see that time is not a property perceived as existing
in objects, or as conveyed to us by our senses ; but a con
dition impressed upon our knowledge by the constitution
of the mind itself; involving an act of thought as well as
an impression of sense.

3. We showed that space is an idea of the mind, or
form of our perceiving power, independent of experience,
by pointing out that we possess necessary and universal
truths concerning the relations of space, which could
never be given by means of experience ; but of which
the necessity is readily conceivable, if we suppose them
to have for their basis the constitution of the mind.
There exist also respecting number, many truths abso
lutely necessary, entirely independent of experience and
anterior to it ; and so far as the conception of number
depends upon the idea of time, the same argument might
be used to show that the idea of time is not derived from
experience, but is a result of the native activity of the
mind : but we shall defer all views of this kind till we
come to the consideration of Number.

4. Some persons have supposed that we obtain the
notion of time from the perception of motion. But it
is clear that the perception of motion, that is, change of
place, presupposes the conception of time, and is not
capable of being presented to the mind in any other way.
If we contemplate the same body as being in different
places at different times, and connect these observations,
we have the conception of motion, which thus presup-

128 PHILOSOPHY OF THE PURE SCIENCES.

poses the necessary conditions that existence in time
implies. And thus we see that it is possible there should
be necessary truths concerning all motion, and conse
quently, concerning those motions which are the objects
of experience ; but that the source of this necessity is the
Ideas of time and space, which, being universal conditions
of knowledge residing in the mind, afford a foundation
for necessary truths.

CHAPTER VIIL
OF SOME PECULIARITIES OF THE IDEA OF TIME.

1. THE Idea of Time, like the Idea of Space, offers to
our notice some characters which do not belong to our
fundamental ideas generally, but which are deserving of
remark. These characters are, in some respects, closely
similar with regard to time and to space, while, in other
respects, the peculiarities of these two ideas are widely
different. We shall point out some of these characters.

Time is not a general abstract notion collected from
experience ; as, for example, a certain general concep
tion of the relations of things. For we do not consider
particular times as examples of Time in general, (as we
consider particular causes to be examples of Cause,) but
we conceive all particular times to be parts of a single
and endless Time. This continually-flowing and endless
time is what offers itself to us when we contemplate any
series of occurrences. All actual and possible times
exist as Parts, in this original and general Time. And
since all particular times are considered as derivable
from time in general, it is manifest that the notion of
time in general cannot be derived from the notions of
particular times. The notion of time in general is there-

SOME PECULIARITIES OF THE IDEA OF TIME. 129

fore not a general conception gathered from experi
ence.

2. Time is infinite. Since all actual and possible
times exist in the general course of time, this general
time must be infinite. All limitation merely divides,
and does not terminate, the extent of absolute time.
Time has no beginning and no end ; but the beginning
and the end of every other existence takes place in it.

3. Time, like space, is not only a form of perception,
but of intuition. We contemplate events as taking
place in time. We consider its parts as added to one
another, and events as filling a larger or smaller extent
of such parts. The time which any event takes up is
the sum of all such parts, and the relation of the same
to time is fully understood when we can clearly see what
portions of time it occupies, and what it does not.
Thus the relation of known occurrences to time is
perceived by intuition ; and time is a form of intuition
of the external world.

4. Time is conceived as a quantity of one dimension ;
it has great analogy with a line, but none at all with a
surface or solid. Time may be considered as consisting
of a series of instants, which are before and after one
another ; and they have no other relation than this, of
before and after. Just the same would be the case with
a series of points taken along a line ; each would be
after those on one side of it, and before those on another.
Indeed the analogy between time, and space of one
dimension, is so close, that the same terms are applied to
both ideas, and we hardly know to which they originally
belong. Times and lines are alike called long and short ;
we speak of the beginning and end of a line ; of a point
of time, and of the limits of a portion of duration.

5. But, as has been said, there is nothing in time
which corresponds to more than one dimension in space,

VOL. i. w. p. K

130 PHILOSOPHY OF THE PURE SCIENCES.

and hence nothing which has any obvious analogy with
figure. Time resembles a line indefinitely extended both
ways ; all partial times are portions of this line ; and no
mode of conceiving time suggests to us a line making
any angle with the original line, or any other combina
tion which might give rise to figures of any kind. The
analogy between time and space, which in many circum
stances is so clear, here disappears altogether. Spaces
of two and of three dimensions, planes and solids, have
nothing to which we can compare them in the concep
tions arising out of time.

6. As figure is a conception solely appropriate to
space, there is also a conception which peculiarly belongs
to time, namely, the conception of recurrence of times
similarly marked; or, as it may be termed, rhythm,
using this word in a general sense. The term rhythm
is most commonly used to designate the recurrence of
times marked by the syllables of a verse, or the notes of
a melody : but it is easy to see that the general concep
tion of such a recurrence does not depend on the mode
in which it is impressed upon the sense. The forms of
such recurrence are innumerable. Thus in such a line as

Quddrupedante putrm sonitu quatit lingula campum,

we have alternately one long or forcible syllable, and
two short or light ones, recurring over and over. In
like manner in our own language, in the line

At the close of the day when the hamlet is still,

we have two light and one strong syllable repeated four
times over. Such repetition is the essence of versification.
The same kind of rhythm is one of the main elements of
music, with this difference only, that in music the forcible
syllables are made so for the purposes of rhythm by
their length only or principally ; for example, if either of
the above lines were imitated by a melody in the most

SOME PECULIARITIES OF THE IDEA OF TIME. 131

simple and obvious manner, each strong syllable would
occupy exactly twice as much time as two of the weaker
ones. Something very analogous to such rhythm may
be traced in other parts of poetry and art, which we need
not here dwell upon. But in reference to our present
subject, we may remark that by the introduction of such
rhythm, the flow of time, which appears otherwise so
perfectly simple and homogeneous, admits of an infinite
number of varied yet regular modes of progress. All
the kinds of versification which occur in all languages,
and the still more varied forms of recurrence of notes of
different lengths, which are heard in all the varied strains
of melodies, are only examples of such modifications, or
configurations as we may call them, of time. They in
volve relations of various portions of time, as figures
involve relations of various portions of space. But yet
the analogy between rhythm and figure is by no means
very close ; for in rhythm we have relations of quantity
alone in the parts of time, whereas in figure we have re
lations not only of quantity, but of a kind altogether
different, namely, of position. On the other hand, a
repetition of similar elements, which does not necessarily
occur in figures, is quite essential in order to impress
upon us that measured progress of time of which we here
speak. And thus the ideas of time and space have each
its peculiar and exclusive relations ; position and figure
belonging only to space, while repetition and rhythm are
appropriate to time.

7. One of the simplest forms of recurrence is alter
nation, as when we have alternate strong and slight syl
lables. For instance,

Awake, arise, or be for e"ver fdll n.

Or without any subordination, as when we reckon
numbers, and call them in succession, odd, even, odd,
even.

K 2

132 PHILOSOPHY OF THE PURE SCIENCES.

8. But the simplest of all forms of recurrence is that
which has no variety ; in which a series of units, each
considered as exactly similar to the rest, succeed each
other ; as one, one, one, and so on. In this case, how
ever, we are led to consider each unit with reference to
all that have preceded ; and thus the series one, one, one,
and so forth, becomes one, two, three, four, Jive, and so
on ; a series with which all are familiar, and which may
be continued without limit.

We thus collect from that repetition of which time
admits, the conception of Number.

9. The relations of position and figure are the sub
ject of the science of geometry ; and are, as we have
already said, traced into a very remarkable and extensive
body of truths, which rests for its foundations on axioms
involved in the Idea of Space. There is, in like manner,
a science of great complexity and extent, which has its
foundation in the Idea of Time. But this science, as it
is usually pursued, applies only to the conception of Num
ber, which is, as we have said, the simplest result of
repetition. This science is Theoretical Arithmetic, or
the speculative doctrine of the properties and relations
of numbers ; and we must say a few words concerning
the principles which it is requisite to assume as the basis
of this science.

CHAPTER IX.
OF THE AXIOMS WHICH RELATE TO NUMBER.

1. THE foundations of our speculative knowledge of
the relations and properties of Number, as well as of
Space, are contained in the mode in which we represent to
ourselves the magnitudes which are the subjects of our
reasonings. To express these foundations in axioms in the

OF THE AXIOMS WHICH RELATE TO NUMBER. 133

case of number, is a matter requiring some consideration,
for the same reason as in the case of geometry ; that is,
because these axioms are principles which we assume as
true, without being aware that we have made any assump
tion ; and we cannot, without careful scrutiny, determine
when we have stated, in the form of axioms, all that is
necessary for the formation of the science, and no more
than is necessary. We will, however, attempt to detect
the principles which really must form the basis of theo
retical arithmetic.

2. Why is it that three and two are equal to four and
one ? Because if we look at five things of any kind, we
see that it is so. The five are four and one ; they, are
also three and two. The truth of our assertion is in
volved in our being able to conceive the number five at
all. We perceive this truth by intuition, for we cannot
see, or imagine we see, five things, without perceiving
also that the assertion above stated is true.

But how do we state in words this fundamental prin
ciple of the doctrine of numbers ? Let us consider a
very simple case. If we wish to show that seven and
two are equal to four and five, we say that seven are four
and three, therefore seven and two are four and three
and two ; and because three and two are five, this is four
and five. Mathematical reasoners justify the first infer
ence (marked by the conjunctive word therefore), by
saying that " When equals are added to equals the
wholes are equal," and that thus, since seven is equal
to three and four, if we add two to both, seven and two
are equal to four and three and two.

3. Such axioms as this, that when equals are added
to equals the wholes are equal, are, in fact, expressions
of the general condition of intuition, by which a whole
is contemplated as made up of parts, and as identical
with the aggregate of the parts. And a yet more gene-

134 PHILOSOPHY OF THE PURE SCIENCES.

ral form in which we might more adequately express
this conditon of intuition would be this ; that " Two mag
nitudes are equal when they can be divided into parts
which are equal, each to each." Thus in the above ex
ample, seven and two are equal to four and five, because
each of the two sums can be divided into the parts, four,
three, and two.

4. In all these cases, a person who had never seen
such axioms enunciated in a verbal form would employ
the same reasoning as a practised mathematician, in order
to satisfy himself that the proposition was true. The
steps of the reasoning, being seen to be true by intuition,
would carry an entire conviction, whether or not the
argument were made verbally complete. Hence the
axioms may appear superfluous, and on this account
such axioms have often been spoken contemptuously of
as empty and barren assertions. In fact, however, al
though they cannot supply the deficiency of the clear in
tuition of number and space in the reasoner himself, and
although when he possesses such a faculty, he will reason
rightly if he have never heard of such axioms, they still
have their place properly at the beginning of our trea
tises on the science of quantity ; since they express, as
simply as words can express, those conditions of the
intuition of magnitudes on which all reasoning concern
ing quantity must be based ; and are necessary when we
want, not only to see the truth of the elementary reason
ings on these subjects, but to put such reasonings in a
formal and logical shape.

5. We have considered the above-mentioned axioms
as the basis of all arithmetical operations of the nature
of addition. But it is easily seen that the same prin
ciple may be carried into other cases ; as for instance,
multiplication, which is merely a repeated addition,
and admits of the same kind of evidence. Thus

OF THE AXIOMS WHICH RELATE TO NUMBER. 135

five times three are equal to three times five ; why
is this ? If we arrange fifteen things in five rows of
three, it is seen by looking, or by imaginary looking,
which is intuition, that they may also be taken as three
rows of five. And thus the principle that those wholes
are equal which can be resolved into the same partial
magnitudes, is immediately applicable in this as in the
other case.

6. We may proceed to higher numbers, and may find
ourselves obliged to use artificial nomenclature and
notation in order to represent and reckon them ; but the
reasoning in these cases also is still the same. And the
usual artifice by which our reasoning in such instances
is assisted is, that the number which is the root of our
scale of notation (which is ten in our usual system), is
alternately separated into parts and treated as a single
thing. Thus 47 and 35 are 82 ; for 47 is four tens and
seven ; 35 is three tens and five ; whence 47 and 35 are
seven tens and twelve ; that is, 7 tens, 1 ten, and 2 ;
which is 8 tens and 2, or 82. The like reasoning is
applicable in other cases. And since the most remote
and complex properties of numbers are obtained by a
prolongation of a course of reasoning exactly similar to
that by which we thus establish the most elementary
propositions, we have, in the principles just noticed, the
foundation of the whole of Theoretical Arithmetic.

CHAPTER X.
OF THE PERCEPTION OF TIME AND NUMBER,

I. OUR perception of the passage of time involves a
series of acts of memory. This is easily seen and assented
to, when large intervals of time and a complex train of
occurrences are concerned. But since memory is requi-

136 PHILOSOPHY OF THE PURE SCIENCES.

site in order to apprehend time in such cases, we cannot
doubt that the same faculty must be concerned in the
shortest and simplest cases of succession ; for it will
hardly be maintained that the process by which we con
template the progress of time is different when small
and when large intervals are concerned. If memory be
absolutely requisite to connect two events which begin
and end a day, and to perceive a tract of time between
them, it must be equally indispensable to connect the
beginning and end of a minute, or a second ; though in
this case the effort may be smaller, and consequently
more easily overlooked. In common cases, we are un
conscious of the act of thought by which we recollect
the preceding instant, though we perceive the effort when
we recollect some distant event. And this is analogous
to what happens in other instances. Thus, we walk
without being conscious of the volitions by which we
move our muscles ; but, in order to leap, a distinct and
manifest exertion of the same muscles is necessary. Yet
no one will doubt that we walk as well as leap by an
act of the will exerted through the muscles ; and in like
manner, our consciousness of small as well as large inter
vals of time involves something of the nature of an act
of memory.

2. But this constant and almost imperceptible kind
of memory, by which we connect the beginning and end
of each instant as it passes, may very fitly be distinguished
in common cases from manifest acts of recollection,
although it may be difficult or impossible to separate
the two operations in general. This perpetual and latent
kind of memory may be termed a sense of successive
ness ; and must be considered as an internal sense by
which we perceive ourselves existing in time, much in
the same way as by our external and muscular sense
we perceive ourselves existing in space. And both our

PERCEPTION OF TIME AND NUMBER. 137

internal thoughts and feelings, and the events which
take place around us, are apprehended as objects of this
internal sense, and thus as taking place in time.

3. In the same manner in which our interpretation
of the notices of the muscular sense implies the power of
moving our limbs, and of touching at will this object or
that ; our apprehension of the relations of time by means
of the internal sense of successiveness implies a power of
recalling what has past, and of retaining what is pass
ing. We are able to seize the occurrences which have
just taken place, and to hold them fast in our minds
so as mentally to measure their distance in time from
occurrences now present. And thus, this sense of suc
cessiveness, like the muscular sense with which we have
compared it, implies activity of the mind itself, and is
not a sense passively receiving impressions.

4. The conception of Number appears to require the
exercise of the same sense of succession. At first sight,
indeed, we seem to apprehend Number without any act
of memory, or any reference to time : for example, we
look at a horse, and see that his legs are four ; and this
we seem to do at once, without reckoning them. But it
is not difficult to see that this seeming instantaneousness
of the perception of small numbers is an illusion. This
resembles the many other cases in which we perform
short and easy acts so rapidly and familiarly that we are
unconscious of them ; as in the acts of seeing, and of arti
culating our words. And this is the more manifest, since
we begin our acquaintance with number by counting
even the smallest numbers. Children and very rude
savages must use an effort to reckon even their five
fingers, and find a difficulty in going further. And per
sons have been known who were able by habit, or by a
peculiar natural aptitude, to count by dozens as rapidly
as common persons can by units. We may conclude.

138 PHILOSOPHY OF THE PURE SCIENCES.

therefore, that when we appear to catch a small number
by a single glance of the eye, we do in fact count the
units of it in a regular, though very brief succession. To
count requires an act of memory. Of this we are sen
sible when we count very slowly, as when we reckon the
strokes of a church-clock ; for in such a case we may
forget in the intervals of the strokes, and miscount. Now
it will not be doubted that the nature of the process in
counting is the same whether we count fast or slow.
There is no definite speed of reckoning at which the
faculties which it requires are changed; and therefore
memory, which is requisite in some cases, must be so
in all*.

The act of counting, (one, two, three, and so on,) is
the foundation of all our knowledge of number. The
intuition of the relations of number involves this act of
counting; for, as we have just seen, the conception of
number cannot be obtained in any other way. And thus
the whole of theoretical arithmetic depends upon an act
of the mind, and upon the conditions which the exercise
of that act implies. These have been already explained
in the last chapter.

5. But if the apprehension of number be accompanied
by an act of the mind, the apprehension of rhythm is so
still more clearly. All the forms of versification and the
measures of melodies are the creations of man, who thus
realizes in words and sounds the forms of recurrence
which rise within his own mind. When we hear in a

* I have considered Number as involving the exercise of the sense
of succession, because I cannot draw any line between those cases of
large numbers, in which, the process of counting being performed, there
is a manifest apprehension of succession ; and those cases of small num
bers, in which we seem to see the number at one glance. But if any
one holds Number to be apprehended by a direct act of intuition, as
Space and Time are, this view will not disturb the other doctrines
delivered in the text.

PERCEPTION OF TIME AND NUMBER. 139

quiet scene any rapidly-repeated sound, as those made by
the hammer of the smith or the saw of the carpenter,
every one knows how insensibly we throw these noises
into a rhythmical form in our own apprehension. We
do this even without any suggestion from the sounds
themselves. For instance, if the beats of a clock or
watch be ever so exactly alike, we still reckon them
alternately tick-tack, tick-tack. That this is the case,
may be proved by taking a watch or clock of such a con
struction that the returning swing of the pendulum is
silent, and in which therefore all the beats are rigorously
alike : we shall find ourselves still reckoning its sounds
as tick-tack. In this instance it is manifest that the
rhythm is entirely of our own making. In melodies,
also, and in verses in which the rhythm is complex, ob
scure, and difficult, we perceive something is required
on our part ; for we are often incapable of contributing
our share, and thus lose the sense of the measure alto
gether. And when we consider such cases, and attend
to what passes within us when we catch the measure,
even of the simplest and best-known air, we shall no
longer doubt that an act of our own thoughts is requisite
in such cases, as well as impressions on the sense. And
thus the conception of this peculiar modification of time,
which we have called rhythm, like all the other views
which we have taken of the subject, shows that we must,
in order to form such conceptions, supply a certain idea
by our own thoughts, as well as merely receive by senses,
whether external or internal, the impressions of appear
ances and collections of appearances.

NOTE TO CHAPTER X.

I HAVE in the last ten chapters described Space, Time, and Number by
various expressions, all intended to point out their office as exemplifying
the Ideal Element of human knowledge. I have called them Funda-

140 PHILOSOPHY OF THE PURE SCIENCES-

mental Ideas ; Forms of Perception ; Forms of Intuition ; and per
haps other names. I might add yet other phrases. I might say that
the properties of Space, Time, and Number are Laws of the Mind s
Activity in apprehending what is. For the mind cannot apprehend any
thing or event except conformably to the properties of space, time, and
number. It is not only that it does not, but it can not : and this
impossibility shows that the law is a law of the mind, and not of
objects extraneous to the mind.

It is usual for some of those who reject the doctrines here presented
to say that the axioms of geometry, and of other sciences, are obtained
by Induction from facts constantly presented by experience. But I do
not see how Induction can prove that a proposition must be true. The
only intelligible usage of the word Induction appears to me to be, that in
which it is applied to a proposition which, being separable from tho
facts in our apprehension, and being compared with them, is seen to
agree with them. But in the cases now spoken of, the proposition is
not separable from the facts. We cannot infer by induction that two
straight lines cannot inclose a space, because we cannot contemplate
special cases of two lines inclosing a space, in which it remains to be
determined whether or not the proposition, that both are straight,
is true.

I do not deny that the activity of the mind by which it perceives
objects and events as related according to the laws of space, time, and
number, is awakened and developed by being constantly exercised ; and
that we cannot imagine a stage of human existence in which the powers
have not been awakened and developed by such exercise. In this way,
experience and observation are necessary conditions and prerequisites of
our apprehension of geometrical (and other) axioms. We cannot see
the truth of these axioms without some experience, because we cannot
see any thing, or be human beings, without some experience. This
might be expressed by saying that such truths are acquired necessarily
in the course of all experience ; but I think it is very undesirable to
apply, to such a case, the word Induction, of which it is so important
to us to keep the scientific meaning free from confusion. Induction
cannot give demonstrative proofs, as I have already stated in Book i.
C. ii. sect. 3, and therefore cannot be the ground of necessary truths.

Another expression which may be used to describe the Funda
mental Ideas here spoken of is suggested by the language of a very
profound and acute Review of the former edition. The Reviewer holds
that we pass from special experiences to universal truths in virtue of
" the inductive propensity the irresistible impulse of the mind to
generalize ad injinitum." I have already given reasons why I cannot
adopt the former expression ; but I do not see why space, time, number,

PERCEPTION OF TIME AND NUMBER. 141

cause, and the rest, may not be termed different forms of the impulse of
the mind to generalize. If we put together all the Fundamental Ideas
as results of the Generalizing Impulse, we must still separate them as
different modes of action of that Impulse, showing themselves in various
characteristic ways in the axioms and modes of reasoning which belong
to different sciences. The Generalizing Impulse in one case proceeds
according to the Idea of Space ; in another, according to the Idea of
Mechanical Cause ; and so in other subjects.

CHAPTER XL
OF MATHEMATICAL REASONING.

1. Discursive Reasoning. WE have thus seen that
our notions of space, time, and their modifications, neces
sarily involve a certain activity of the mind; and that
the conditions of this activity form the foundations of
those sciences which have the relations of space, time,
and number, for their object. Upon the fundamental
principles thus established, the various sciences which
are included in the term Pure Mathematics, (Geometry,
Algebra, Trigonometry, Conic Sections, and the rest of
the Higher Geometry, the Differential Calculus, and the
like,) are built up by a series of reasonings. These rea
sonings are subject to the rules of Logic, as we have
already remarked ; nor is it necessary here to dwell long
on the nature and rules of such processes. But we may
here notice that such processes are termed discursive,
in opposition to the operations by which we acquire our
fundamental principles, which are, as we have seen, intui
tive. This opposition was formerly very familiar to our
writers ; as Milton,

. . . Thus the soul reason receives,
Discursive or intuitive. Paradise Lost, v. 438.

For in such reasonings we obtain our conclusions, not
by looking at our conceptions steadily in one view, which

142 PHILOSOPHY OF THE PURE SCIENCES.

is intuition, but by passing from one view to another, like
those who run from place to place (discursus). Thus a
straight line may be at the same time a side of a triangle
and a radius of a circle : and in the first proposition of
Euclid a line is considered, first in one of these relations,
and then in the other, and thus the sides of a certain
triangle are proved to be equal. And by this " discourse
of reason," as by our older writers it was termed, we set
forth from those axioms which we perceive by intuition,
travel securely over a vast and varied region, and become
possessed of a copious store of mathematical truths.

2. Technical Terms of Reasoning. The reasoning of
mathematics, thus proceeding from a few simple princi
ples to many truths, is conducted according to the rules
of Logic. If it be necessary, mathematical proofs may be
reduced to logical forms, and expressed in Syllogisms,
consisting of major, minor, and conclusion. But in most
cases the syllogism is of that kind which is called by
logical writers an Enthymeme; a word which implies
something existing in the thoughts only, and which desig
nates a syllogism in which one of the premises is under
stood, and not expressed. Thus we say in a mathematical
proof, " because the point c is the center of the circle AB,
AC is equal to BC ;" not stating the major, that all lines
drawn from the center of a circle to the circumference
are equal; or introducing it only by a transient reference
to the definition of a circle. But the enthymeme is so
constantly used in all habitual forms of reasoning, that
it does not occur to us as being anything peculiar in
mathematical works.

The propositions which are proved to be generally
true are termed Theorems: but when any thing is required
to be done, as to draw a line or a circle under given
conditions, this proposition is a Problem. A theorem re
quires demonstration ; a problem, solution. And for both

OF MATHEMATICAL REASONING. 143

purposes the mathematician usually makes a Construe-
tion. He directs us to draw certain lines, circles, or other
curves, on which is to be founded his demonstration that
his theorem is true, or that his problem is solved. Some
times, too, he establishes some Lemma, or preparatory
proposition, before he proceeds to his main task ; and
often he deduces from his demonstration some conclusion
in addition to that which was the professed object of his
proposition ; and this is termed a Corollary.

These technical terms are noted here, not as being
very important, but in order that they may not sound
strange and unintelligible if we should have occasion to
use some of them. There is, however, one technical dis
tinction more peculiar, and more important.

3. Geometrical Analysis and Syntfiesis. In geome
trical reasoning such as we have described, we introduce
at every step some new consideration ; and it is by com
bining all these considerations, that we arrive at the
conclusion, that is, the demonstration of the proposition.
Each step tends to the final result, by exhibiting some
part of the figure under a new relation. To what we
have already proved, is added something more ; and hence
this process is called Synthesis, or putting together. The
proof flows on, receiving at every turn new contribu
tions from different quarters ; like a river fed and aug
mented by many tributary streams. And each of these
tributaries flows from some definition or axiom as its
fountain, or is itself formed by the union of smaller rivulets
which have sources of this kind. In descending along its
course, the synthetical proof gathers all these accessions
into one common trunk, the proposition finally proved.

But we may proceed in a different manner. We
may begin from the formed river, and ascend to its
sources. We may take the proposition of which we
require a proof, and may examine what the supposition

144 PHILOSOPHY OF THE PURE SCIENCES.

of its truth implies. If this be true, then something else
may be seen to be true ; and from this, something else,
and so on. We may often, in this way, discover of what
simpler propositions our theorem or solution is com
pounded, and may resolve these in succession, till we
come to some proposition which is obvious. This is geo
metrical Analysis. Having succeeded in this analytical
process, we may invert it ; and may descend again from
the simple and known propositions, to the proof of a
theorem, or the solution of a problem, which was our
starting-place.

This process resembles, as we have said, tracing a
river to its sources. As we ascend the stream, we per
petually meet with bifurcations; and some sagacity is
needed to enable us to see which, in each case, is the
main stream : but if we proceed in our research, we
exhaust the unexplored valleys, and finally obtain a clear
knowledge of the place whence the waters flow. Analy
tical is sometimes confounded with symbolical reasoning,
on which subject we shall make a remark in the next
chapter. The object of that chapter is to notice certain
other fundamental principles and ideas, not included in
those hitherto spoken of, which we find thrown in our
way as we proceed in our mathematical speculations.
It would detain us too long, and involve us in subtle and
technical disquisitions, to examine fully the grounds of
these principles ; but the Mathematics hold so important
a place in relation to the inductive sciences, that I shall
briefly notice the leading ideas which the ulterior pro
gress of the subject involves.

145

CHAPTER XII.

OF THE FOUNDATIONS OF THE HIGHER
MATHEMATICS,

1. The Idea of a Limit. THE general truths concern
ing relations of space which depend upon the axioms
and definitions contained in Euclid s Elements, and which
involve only properties of straight lines and circles, are
termed Elementary Geometry : all beyond this belongs to
the Higher Geometry. To this latter province appertain,
for example, all propositions respecting the lengths of any
portions of curve lines ; for these cannot be obtained by
means of the principles of the Elements alone. Here
then we must ask to what other principles the geometer
has recourse, and from what source these are drawn. Is
there any origin of geometrical truth which we have not
yet explored ?

The Idea of a Limit supplies a new mode of establish
ing mathematical truths. Thus with regard to the length
of any portion of a curve, a problem which we have just
mentioned ; a curve is not made up of straight lines, and
therefore we cannot by means of any of the doctrines of
elementary geometry measure the length of any curve.
But we may make up a figure nearly resembling any
curve by putting together many short straight lines, just
as a polygonal building of very many sides may nearly
resemble a circular room. And in order to approach
nearer and nearer to the curve, we may make the sides
more and more small, more and more numerous. We
may then possibly find some mode of measurement, some
relation of these small lines to other lines, which is not
disturbed by the multiplication of the sides, however far
it be carried. And thus, we may do what is equivalent to

VOL. i. w. P. L

146 PHILOSOPHY OF THE PURE SCIENCES.

measuring the curve itself; for by multiplying the sides
we may approach more and more closely to the curve till
no appreciable difference remains. The curve line is the
Limit of the polygon ; and in this process we proceed on
the Axiom,, that "What is true up to the limit is true at
the limit."

This mode of conceiving mathematical magnitudes is
of wide extent and use ; for every curve may be con
sidered as the limit of some polygon; every varied
magnitude, as the limit of some aggregate of simpler
forms ; and thus the relations of the elementary figures
enable us to advance to the properties of the most com
plex cases.

A Limit is a peculiar and fundamental conception, the
use of which in proving the propositions of the Higher
Geometry cannot be superseded by any combination of
other hypotheses and definitions*. The axiom just no
ticed, that what is true up to the limit is true at the limit,
is involved in the very conception of a limit : and this
principle, with its consequences, leads to all the results
which form the subject of the higher mathematics, whe-

* This assertion cannot be fully proved and illustrated without a
reference to mathematical reasonings which would not be generally
intelligible. I have shown the truth of the assertion in my Thoughts
on the Study of Mathematics^ annexed to the Principles of English
University Education. The proof is of this kind : The ultimate
equality of an arc of a curve and the corresponding periphery of a
polygon, when the sides of the polygon are indefinitely increased in
number, is evident. But this truth cannot be proved from any other
axiom. For if we take the supposed axiom, that a curve is always
less than the including broken line, this is not true, except with a con
dition ; and in tracing the import of this condition, we find its neces
sity becomes evident only when we introduce a reference to a Limit.
And the same is the case if we attempt to supersede the notion of a
Limit in proving any other simple and evident proposition in which
that notion is involved. Therefore these evident truths are ^//-evident,
in virtue of the Idea of a Limit,

THE FOUNDATIONS OF THE HIGHER MATHEMATICS. 147

ther proved by the consideration of evanescent triangles,
by the processes of the Differential Calculus, or in any
other way.

The ancients did not expressly introduce this con
ception of a Limit into their mathematical reasonings ;
although in the application of what is termed the
Method of Exhaustions, (in which they show how to
exhaust the difference between a polygon and a curve, or
the like,) they were in fact proceeding upon an obscure
apprehension of principles equivalent to those of the
Method of Limits. Yet the necessary fundamental prin
ciple not having, in their time, been clearly developed,
their reasonings were both needlessly intricate and im
perfectly satisfactory. Moreover they were led to put in
the place of axioms, assumptions which were by no means
self-evident ; as when Archimedes assumed, for the basis
of his measure of the circumference of the circle, the
proposition that a circular arch is necessarily less than
two lines which inclose it, joining its extremities. The
reasonings of the older mathematicians, which professed
to proceed upon such assumptions, led to true results
in reality, only because they were guided by a latent
reference to the limiting case of such assumptions. And
this latent employment of the conception of a Limit,
reappeared in various forms during the early period of
modern mathematics ; as for example, in the Method of
Indivisibles of Ca,v&\\eii, and the Characteristic Triangle
of Barrow ; till at last, Newton distinctly referred such
reasonings to the conception of a Limit, and established
the fundamental principles and processes which that
conception introduces, with a distinctness and exactness
which required little improvement to make it as unim
peachable as the demonstrations of geometry. And when
such processes as Newton thus deduced from the con
ception of a Limit are represented by means of general

148 PHILOSOPHY OF THE PURE SCIENCES.

algebraical symbols instead of geometrical diagrams, we
have then before us the Method of Fluocions, or the
Differential Calculus; a mode of treating mathematical
problems justly considered as the principal weapon by
which the splendid triumphs of modern mathematics
have been achieved.

2. The Use of General Symbols. The employment
of algebraical symbols, of which we have just spoken,
has been another of the main instruments to which the
successes of modern mathematics are owing. And here
again the processes by which we obtain our results de
pend for their evidence upon a fundamental conception,
the conception of arbitrary symbols as the Signs of
quantity and its relations ; and upon a corresponding
axiom, that " The interpretation of such symbols must
be perfectly general." In this case, as in the last, it was
only by degrees that mathematicians were led to a just
apprehension of the grounds of their reasoning. For
symbols were at first used only to represent numbers
considered with regard to their numerical properties;
and thus the science of Algebra was formed. But it was
found, even in cases belonging to common algebra, that
the symbols often admitted of an interpretation which
went beyond the limits of the problem, and which yet was
not unmeaning, since it pointed out a question closely
analogous to the question proposed. This was the case,
for example, when the answer was a negative quantity ;
for when Descartes had introduced the mode of repre
senting curves by means of algebraical relations among
the symbols of the co-ordinates, or distances of each of
their points from fixed lines, it was found that negative
quantities must be dealt with as not less truly significant
than positive ones. And as the researches of mathema
ticians proceeded, other cases also were found, in which
the symbols, although destitute of meaning according to

THE FOUNDATIONS OF THE HIGHER MATHEMATICS. 140

the original conventions of their institution, still pointed
out truths which could be verified in other ways ; as in
the cases in which what are called impossible quantities
occur. Such processes may usually be confirmed upon
other principles, and the truth in question may be esta
blished by means of a demonstration in which no such
seeeming fallacies defeat the reasoning. But it has also
been shown in many such cases, that the process in which
some of the steps appear to be without real meaning,
does in fact involve a valid proof of the proposition.
And what we have here to remark is, that this is not
true accidentally or partially only, but that the results
of systematic symbolical reasoning must always express
general truths, by their nature, and do not, for their
justification, require each of the steps of the process to
represent some definite operation upon quantity. The
absolute universality of the interpretation of symbols is
the fundamental principle of their use. This has been
shown very ably by Dr. Peacock in his Algebra. He
has there illustrated, in a variety of ways, this prin
ciple : that " If general symbols express an identity
when they are supposed to be of any special nature,
they must also express an identity when they are gene
ral in their nature." And thus, this universality of sym
bols is a principle in addition to those we have already
noticed; and is a principle of the greatest importance
in the formation of mathematical science, according to
the wide generality which such science has in modern
times assumed.

3. Connexion of Symbols and Analysis. Since in
our symbolical reasoning our symbols thus reason for us,
we do not necessarily here, as in geometrical reasoning,
go on adding carefully one known truth to another, till
we reach the desired result. On the contrary, if we have
a theorem to prove or a problem to solve which can be

150 PHILOSOPHY OF THE PURE SCIENCES.

brought under the domain of our symbols, we may at
once state the given but unproved truth, or the given
combination of unknown quantities, in its symbolical
form. After this first process, we may then proceed to
trace, by means of our symbols, what other truth is
involved in the one thus stated, or what the unknown
symbols must signify; resolving step by step the sym
bolical assertion with which we began, into others more
fitted for our purpose. The former process is a kind of
synthesis, the latter is termed analysis. And although
symbolical reasoning does not necessarily imply such
analysis; yet the connexion is so familiar, that the
term analysis is frequently used to designate symbolical
reasoning.

CHAPTER XIII.
THE DOCTRINE OF MOTION.

1. Pure Mechanism,. THE doctrine of Motion, of
which we have here to speak, is that in which motion is
considered quite independently of its cause, force; for
all consideration of force belongs to a class of ideas
entirely different from those with which we are here
concerned. In this view it may be termed the pure
doctrine of motion, since it has to do solely with space
and time, which are the subjects of pure mathematics.
(See C. i. of this Book.) Although the doctrine of
motion in connexion with force, which is the subject
of mechanics, is by far the most important form in
which the consideration of motion enters into the form
ation of our sciences, the Pure Doctrine of Motion,
which treats of space, time, and velocity, might be fol
lowed out so as to give rise to a very considerable and
curious body of science. Such a science is the science

THE DOCTRINE OF MOTION. 151

of Mechanism, independent of force, and considered as
the solution of a problem which may be thus enunciated:
" To communicate any given motion from a first mover
to a given body." The science which should have for its
object to solve all the various cases into which this pro
blem would ramify, might be termed Pure Mechanism,
in contradistinction to Mechanics Proper, or Machinery,
in which Force is taken into consideration. The greater
part of the machines which have been constructed for
use in manufactures have been practical solutions of some
of the cases of this problem. We have also important
contributions to such a science in the works of mathe
maticians; for example, the various investigations and
demonstrations which have been published respecting
the form of the Teeth of Wheels, and Mr. Babbage s
memoir"" on the Language of Machinery. There are
also several works which contain collections of the
mechanical contrivances which have been invented for
the purpose of transmitting and modifying motion, and
these works may be considered as treatises on the science
of Pure Mechanism. But this science has not yet been
reduced to the systematic simplicity which is desirable,
nor indeed generally recognized as a separate science. It
has been confounded, under the common name of Me
chanics, with the other science, Mechanics Proper, or
Machinery, which considers the effect of force transmitted
by mechanism from one part of a material combination
to another. For example, the Mechanical Powers, as
they are usually termed, (the Lever, the Wheel and
Axle, the Inclined Plane, the Wedge, and the Screw,)
have almost always been treated with reference to the
relation between the Power and the Weight, and not
primarily as a mode of changing the velocity and kind

* On a Method of expressing In) Signs the Action of Machinery.
Pliil. Trans., 1820, p. 250.

152 PHILOSOPHY OF THE PUKE SCIENCES.

of the motion. The science of pure motion has not
generally been separated from the science of motion
viewed with reference to its causes.

Recently, indeed, the necessity of such a separation
has been seen by those who have taken a philosophical
view of science. Thus this necessity has been urged by
M. Ampere, in his Essai sur la Philosophic des Sciences
(1834): "Long," he says, (p. 50), "before I employed
myself upon the present work, I had remarked that it is
usual to omit, in the beginning of all books treating of
sciences which regard motion and force, certain consi
derations which, duly developed, must constitute a special
science : of which science certain parts have been treated
of, either in memoirs or in special works ; such, for ex
ample, as that of Carnot upon Motion considered geome
trically, and the essay of Lanz and Betancourt upon the
Composition of Machines." He then proceeds to describe
this science nearly as we have done, and proposes to
term it Kinematics (Cinematique), from /aV^ua, motion.

2. Formal Astronomy. I shall not attempt here
further to develop the form which such a science must
assume. But I may notice one very large province which
belongs to it. When men had ascertained the apparent
motions of the sun, moon, and stars, to a moderate
degree of regularity and accuracy, they tried to conceive
in their minds some mechanism by which these motions
might be produced; and thus they in fact proposed to
themselves a very extensive problem in Kinematics.
This, indeed, was the view originally entertained of the
nature of the science of astronomy. Thus Plato in the
seventh Book of his Republic*, speaks of astronomy as
the doctrine of the motion of solids, meaning thereby,
spheres. And the same was a proper description of the
science till the time of Kepler, and even later: for

* P. 528.

THE DOCTRINE OF MOTION. 153

Kepler endeavoured in vain to conjoin with the know
ledge of the motions of the heavenly bodies, those true
mechanical conceptions which converted formal into
physical astronomy *.

The astronomy of the ancients admitted none but
uniform circular motions, and could therefore be com
pletely cultivated by the aid of their elementary geo
metry. But the pure science of motion might be
extended to all motions, however varied as to the speed
or the path of the moving body. In this form it must
depend upon the doctrine of limits ; and the funda
mental principle of its reasonings would be this : That
velocity is measured by the Limit of the space described,
considered with reference to the time in which it is
described. I shall not further pursue this subject ; and
in order to complete what I have to say respecting the
Pure Sciences, I have only a few words to add respect
ing their bearing on Inductive Science in general.

CHAPTER XIV.

OF THE APPLICATION OF MATHEMATICS TO
THE INDUCTIVE SCENCES.

1. ALL objects in the world which can be made the
subjects of our contemplation are subordinate to the
conditions of Space, Time, and Number; and on this
account, the doctrines of pure mathematics have most
numerous and extensive applications in every depart
ment of our investigations of nature. And there is a
peculiarity in these Ideas, which has caused the mathe
matical sciences to be, in all cases, the first successful
efforts of the awakening speculative powers of nations at

* Hist. Ind Sc. 9 ii. 130.

154 PHILOSOPHY OF THE PURE SCIENCES.

the commencement of their intellectual progress. Con
ceptions derived from these Ideas are, from the very
first, perfectly precise and clear, so as to be fit elements
of scientific truths. This is not the case with the other
conceptions which form the subjects of scientific in
quiries. The conception of statical force, for instance,
was never presented in a distinct form till the works of
Archimedes appeared : the conception of accelerating
force was confused, in the mind of Kepler and his con
temporaries, and only became clear enough for purposes
of sound scientific reasoning in the succeeding century :
the just conception of chemical composition of elements
gradually, in modern times, emerged from the erroneous
and vague notions of the ancients. If we take works
published on such subjects before the epoch when the
foundations of the true science were laid, we find the
knowledge not only small, but worthless. The writers
did not see any evidence in what we now consider as the
axioms of the science ; nor any inconsistency where we
now see self-contradiction. But this was never the case
with speculations concerning space and number. From
their first rise, these were true as far as they went.
The Geometry and Arithmetic of the Greeks and Indians,
even in their first and most scanty form, contained none
but true propositions. Men s intuitions upon these sub
jects never allowed them to slide into error and confu
sion ; and the truths to which they were led by the first
efforts of their faculties, so employed, form part of the
present stock of our mathematical knowledge.

2. But we are here not so much concerned with
mathematics in their pure form, as with their applica
tion to the phenomena and laws of nature. And here
also the very earliest history of civilization presents to
us some of the most remarkable examples of man s suc
cess in his attempts to attain to science. Space and

INDUCTIVE APPLICATION OF MATHEMATICS. 155

time, position and motion, govern all visible objects ;
but by far the most conspicuous examples of the rela
tions which arise out of such elements, are displayed by
the ever-moving luminaries of the sky, which measure
days, and months, and years, by their motions, and
man s place on the earth by their position. Hence the
sciences of space and number were from the first culti
vated with peculiar reference to Astronomy. I have
elsewhere* quoted Plato s remark, that it is absurd
to call the science of the relations of space geometry,
the measure of the earth, since its most important office
is to be found in its application to the heavens. And
on other occasions also it appears how strongly he, who
may be considered as the representative of the scientific
and speculative tendencies of his time and country, had
been impressed with the conviction, that the formation
of a science of the celestial motions must depend entirely
upon the progress of mathematics. In the Epilogue to
the Dialogue on the Laws\, he declares mathematical
knowledge to be the first and main requisite for the
astronomer, and describes the portions of it which he
holds necessary for astronomical speculators to culti
vate. These seem to be, Plane Geometry, Theoretical
Arithmetic, the Application of Arithmetic to planes
and to solids, and finally the doctrine of Harmonics.
Indeed the bias of Plato appears to be rather to con
sider mathematics as the essence of the science of
astronomy, than as its instrument; and he seems dis
posed, in this as in other things, to disparage observa
tion, and to aspire after a science founded upon demon
stration alone. " An astronomer," he says in the same
place, "must not be like Hesiod and persons of that
kind, whose astronomy consists in noting the settings
and risings of the stars; but he must be one who
* Hist. Ind. Sc., B. in. c . ii. t Epinomis, p. 900.

156 PHILOSOPHY OF THE PURE SCIENCES.

understands the revolutions of the celestial spheres, each
performing its proper cycle."

A large portion of the mathematics of the Greeks,
so long as their scientific activity continued, was directed
towards astronomy. Besides many curious propositions
of plane and solid Geometry, to which their astronomers
were led, their Arithmetic, though very inconvenient in
its fundamental assumptions, was cultivated to a great
extent ; and the science of Trigonometry, in which pro
blems concerning the relations of space were resolved by
means of tables of numerical results previously obtained,
was created. Menelaus of Alexandria wrote six Books
on Chords, probably containing methods of calculating
Tables of these quantities ; such Tables were familiarly
used by the later Greek astronomers. The same author
also wrote three Books on Spherical Trigonometry,
which are still extant.

3. The Greeks, however, in the first vigour of their
pursuit of mathematical truth, at the time of Plato and
soon after, had by no means confined themselves to
those propositions which had a visible bearing on the
phenomena of nature ; but had followed out many beau
tiful trains of research, concerning various kinds of
figures, for the sake of their beauty alone ; as for in
stance in their doctrine of Conic Sections, of which
curves they had discovered all the principal properties.
But it is curious to remark, that these investigations,
thus pursued at first as mere matters of curiosity and
intellectual gratification, were destined, two thousand
years later, to play a very important part in establishing
that system of the celestial motions which succeeded the
Platonic scheme of cycles and epicycles. If the proper
ties of the conic sections had not been demonstrated by
the Greeks, and thus rendered familiar to the mathe
maticians of succeeding ages, Kepler would probably

INDUCTIVE APPLICATION OF MATHEMATICS. 157

not have been able to discover those laws respecting the
orbits and motions of the planets which were the occa
sion of the greatest revolution that ever happened in
the history of science.

4. The Arabians, who, as I have elsewhere said,
added little of their own to the stores of science which
they received from the Greeks, did however make some
very important contributions in those portions of pure
mathematics which are subservient to astronomy. Their
adoption of the Indian mode of computation by means
of the Ten Digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, and by the
method of Local Values, instead of the cumbrous sexa
gesimal arithmetic of the Greeks, was an improvement
by which the convenience and facility of numerical cal
culations were immeasurably augmented. The Arabians
also rendered several of the processes of trigonometry
much more commodious, by using the Sine of an arc
instead of the Chord ; an improvement which Albateg-
nius appears to claim for himself"""; and by employing
also the Tangents of arcs, or, as they called themf,
upright shadows.

5. The constant application of mathematical know
ledge to the researches of Astronomy, and the mutual
influence of each science on the progress of the other,
has been still more conspicuous in modern times. New
ton s Method of Prime and Ultimate Ratios, which we
have already noticed as the first correct exposition of
the doctrine of a Limit, is stated in a series of Lemmas,
or preparatory theorems, prefixed to his Treatise on the
System of the World. Both the properties of curve
lines and the doctrines concerning force and motion,
which he had to establish, required that the common
mathematical methods should be methodized and ex
tended. If Newton had not been a most expert and in-

* Delambre, Art., M. A., p. 12. t Ibid., p. 17-

158 PHILOSOPHY OF THE PURE SCIENCES.

ventive mathematician, as well as a profound and philo
sophical thinker, he could never have made any one of
those vast strides in discovery of which the rapid succes
sion in his work strikes us with wonder"". And if we
see that the great task begun by him, goes on more
slowly in the hands of his immediate successors, and
lingers a little before its full completion, we perceive
that this arises, in a great measure, from the defect of
the mathematical methods then used. Newton s syn
thetical modes of investigation, as we have elsewhere
observed, were an instrument f, powerful indeed in his
mighty hand, but too ponderous for other persons to
employ with effect. The countrymen of Newton clung
to it the longest, out of veneration for their master ; and
English cultivators of physical astronomy were, on that
very account, left behind the progress of mathematical
science in France and Germany, by a wide interval,
which they have only recently recovered. On the Conti
nent, the advantages offered by a familiar use of symbols,
and by attention to their symmetry and other relations,
were accepted without reserve. In this manner the
Differential Calculus of Leibnitz, which was in its origin
and signification identical with the Method of Fluxions
of Newton, soon surpassed its rival in the extent and
generality of its application to problems. This Calculus
was applied to the science of mechanics, to which it,
along with the symmetrical use of co-ordinates, gave a
new form ; for it was soon seen that the most difficult
problems might in general be reduced to finding inte
grals, which is the reciprocal process of that by which
differentials are found ; so that all difficulties of physical
astronomy were reduced to difficulties of symbolical cal
culation, these, indeed, being often sufficiently stubborn.
Clairaut, Euler, and D Alembert employed the increased

* Hist. Ind. Sc., B. vn. c. ii. t Ib., p. 175.

INDUCTIVE APPLICATION OF MATHEMATICS. 159

resources of mathematical science upon the Theory of
the Moon, and other questions relative to the system of
the world ; and thus began to pursue such inquiries in
the course in which mathematicians are still labouring
up to the present day. This course was not without its
checks and perplexities. We have elsewhere quoted*
Clairaut s expression when he had obtained the very
complex differential equations which contain the solu
tion of the problem of the moon s motion : " Now inte
grate them who can !" But in no very long time they
were integrated, at least approximately ; and the methods
of approximation have since then been improved ; so
that now, with a due expenditure of labour, they may be
carried to any extent which is thought desirable. If
the methods of astronomical observation should here
after reach a higher degree of exactness than they now
profess, so that irregularities in the motions of the sun,
moon, and planets, shall be detected which at present
escape us, the mathematical part of the theory of univer
sal gravitation is in such a condition that it can soon be
brought into comparison with the newly-observed facts.
Indeed at present the mathematical theory is in advance
of such observations. It can venture to suggest what
may afterwards be detected, as well as to explain what
has already been observed. This has happened recently;
for Professor Airy has calculated the law and amount
of an inequality depending upon the mutual attraction of
the Earth and Venus ; of which inequality (so small is
it,) it remains to be determined whether its effect can be
traced in the series of astronomical observations.

6. As the influence of mathematics upon the progress
of astronomy is thus seen in the cases in which theory
and observation confirm each other, so this influence ap
pears in another way, in the very few cases in which the

* Hist. Ind. Sc., B. vi. c. vi. sect. 7*

160 PHILOSOPflY OF THE PURE SCIENCES.

facts have not been fully reduced to an agreement with
theory. The most conspicuous case of this kind is the
state of our knowledge of the Tides. This is a portion
of astronomy : for the Newtonian theory asserts these
curious phenomena to be the result of the attraction of
the sun and moon. Nor can there be any doubt that
this is true, as a general statement ; yet the subject is
up to the present time a blot on the perfection of the
theory of universal gravitation ; for we are very far from
being able in this, as in the other parts of astronomy, to
show that theory will exactly account for the time, and
magnitude, and all other circumstances of the pheno
menon at every place on the earth s surface. And what
is the portion of our mathematics which is connected
with this solitary signal defect in astronomy ? It is the
mathematics of the Motion of Fluids ; a portion in which
extremely little progress has been made, and in which all
the more general problems of the subject have hitherto
remained entirely insoluble. The attempts of the greatest
mathematicians, Newton, Maclaurin, Bernoulli, Clairaut,
Laplace, to master such questions, all involve some gra
tuitous assumption, which is introduced because the
problem cannot otherwise be mathematically dealt with :
these assumptions confessedly render the result defective,
and how defective, it is hard to say. And it was pro
bably precisely the absence of a theory which could be
reasonably expected to agree with the observations, which
made Observations of this very curious phenomenon, the
Tides, to be so much neglected as till very recently they
were. Of late years such observations have been pur
sued, and their results have been resolved into empirical
laws, so that the rules of the phenomena have been
ascertained, although the dependence of these rules upon
the lunar and solar forces has not been shown. Here
then we have a portion of our knowledge relating to

INDUCTIVE APPLICATION OF MATHEMATICS. lf&gt;l

facts undoubtedly dependent upon universal gravitation,
in which Observation has outstripped Theory in her pro
gress, and is compelled to wait till her usual companion
overtakes her. This is a position of which Mathematical
Theory has usually been very impatient, and we may
expect that she will be no less so in the present instance.
7. It would be easy to show from the history of
other sciences, for example, Mechanics and Optics, how
essential the cultivation of pure mathematics has been to
their progress. The parabola was already familiar among
mathematicians when Galileo discovered that it was the
theoretical path of a Projectile ; and the extension and
generalization of the Laws of Motion could never have
been effected, unless the Differential and Integral Cal
culus had been at hand, ready to trace the results of every
hypothesis which could be made. D Alembert s mode of
expressing the Third Law of Motion in its most general
form*, if it did not prove the law, at least reduced the
application of it to analytical processes which could be
performed in most of those cases in which they were
needed. In many instances the demands of mechanical
science suggested the extension of the methods of pure
analysis. The problem of Vibrating Strings gave rise to
the Calculus of Partial Differences, which was still fur
ther stimulated by its application to the motions of fluids
and other mechanical problems. And we have in the
writings of Lagrange and Laplace other instances equally
remarkable of new analytical methods, to which mecha
nical problems, and especially cosmical problems, have
given occasion.

8. The progress of Optics as a science has, in like
manner, been throughout dependent upon the progress
of pure mathematics. The first rise of geometry was fol-

* Hixt. I ml. Sci., B. vi. c. vi. sort. 7
VOL. I. W. P.

162 PHILOSOPHY OK THE PURE SCIENCES.

lowed by some advances, slight ones no doubt, in the
doctrine of Reflection and in Perspective. The law of
Refraction was traced to its consequences by means of
Trigonometry, which indeed was requisite to express the
law in a simple form. The steps made in Optical science
by Descartes, Newton, Euler, and Huyghens, required
the geometrical skill which those philosophers possessed.
And if Young and Fresnel had not been, each in his
peculiar way, persons of eminent mathematical endow
ments, they would not have been able to bring the
Theory of Undulations and Interferences into a condi
tion in which it could be tested by experiments. We
may see how unexpectedly recondite parts of pure mathe
matics may bear upon physical science, by calling to
mind a circumstance already noticed in the History of
Science* ; that Fresnel obtained one of the most curious
confirmations of the theory (the laws of Circular Polar
ization by reflection) through an interpretation of an
algebraical expression, which, according to the original
conventional meaning of the symbols, involved an im
possible quantity. We have already remarked, that in
virtue of the principle of the generality of symbolical
language, such an interpretation may often point out
some real and important analogy.

9. From this rapid sketch it may be seen how
important an office in promoting the progress of the
physical sciences belongs to mathematics. Indeed in
the progress of many sciences, every step has been so
intimately connected with some advance in mathematics,
that we can hardly be surprized if some persons have
considered mathematical reasoning to be the most essen
tial part of such sciences ; and have overlooked the other
elements which enter into their formation. How erro-

* Hist. Ind. Sci., B. ix. c. xiii. sect. 2.

INDUCTIVE APPLICATION OF MATHEMATICS. 163

neous this view is we shall best see by turning our
attention to the other Ideas besides those of space, num
ber, and motion, which enter into some of the most
conspicuous and admired portions of what is termed
exact science ; and by showing that the clear and distinct
developement of such Ideas is quite as necessary to the
progress of exact and real knowledge as an acquaintance
with arithmetic and geometry.

164

BOOK III.

THE PHILOSOPHY OF THE MECHANICAL
SCIENCES.

CHAPTER I.
OF THE MECHANICAL SCIENCES.

IN the History of the Sciences, that class of which we
here speak occupies a conspicuous and important place ;
coming into notice immediately after those parts of astro
nomy which require for their cultivation merely the
ideas of space, time, motion, and number. It appears
from our History, that certain truths concerning the equi
librium of bodies were established by Archimedes ; that,
after a long interval of inactivity, his principles were
extended and pursued further in modern times : and
that to these doctrines concerning equilibrium and the
forces which produce it, (which constitute the science
Statics,) were added many other doctrines concerning
the motions of bodies, considered also as produced by
forces, and thus the science of Dynamics was produced.
The assemblage of these sciences composes the province
of Mechanics. Moreover, philosophers have laboured to
make out the laws of the equilibrium of fluid as well as
solid bodies ; and hence has arisen the science of Hydro
statics. And the doctrines of Mechanics have been found
to have a most remarkable bearing upon the motions
of the heavenly bodies ; with reference to which, indeed,
they were at first principally studied. The explanation

OF THE MECHANICAL SCIENCES. 165

of those cosmical facts by means of mechanical principles
and their consequences, forms the science of Physical
Astronomy. These are the principal examples of mecha
nical science ; although some other portions of Physics,
as Magnetism and Electrodynamics, introduce mecha
nical doctrines very largely into their speculations.

Now in all these sciences we have to consider Forces.
In all mechanical reasonings forces enter, either as pro
ducing motion, or as prevented from doing so by other
forces. Thus force, in its most general sense, is the cause
of motion, or of tendency to motion ; and in order to
discover the principles on which the mechanical sciences
truly rest, we must examine the nature and origin of
our knowledge of Causes.

In these sciences, however, we have not to deal with
Cause in its more general acceptation, in which it applies
to all kinds of agency, material or immaterial ; to the
influence of thought and will, as well as of bodily pres
sure and attractive force. Our business at present is
only with such causes as immediately operate upon
matter. We shall nevertheless, in the first place, con
sider the nature of Cause in its most general form ; and
afterwards narrow our speculations so as to direct them
specially to the mechanical sciences.

CHAPTER II.
OF THE IDEA OF CAUSE.

1. WE see in the world around us a constant suc
cession of causes and effects connected with each other.
The laws of this connexion we learn in a great measure
from experience, by observation of the occurrences which
present themselves to our notice, succeeding one another.

166 PHILOSOPHY OF THE MECHANICAL SCIENCES.

But in doing this, and in attending to this succession of
appearances, of which we are aware by means of our
senses, we supply from our own minds the Idea of Cause.
This Idea, as we have already shown with respect to
other Ideas, is not derived from experience, but has its
origin in the mind itself; is introduced into our expe
rience by the active, and not by the passive part of our
nature.

By Cause we mean some quality, power, or efficacy,
by which a state of things produces a succeeding state.
Thus the motion of bodies from rest is produced by a
cause which we call Force : and in the particular case
in which bodies fall to the earth, this force is termed
Gravity. In these cases, the Conceptions of Force and
Gravity receive their meaning from the Idea of Cause
which they involve : for Force is conceived as the Gauge
of Motion. That this Idea of Cause is not derived from
experience, we prove (as in former cases) by this con
sideration : that we can make assertions, involving this
idea, which are rigorously necessary and universal ;
whereas knowledge derived from experience can only be
true as far as experience goes, and can never contain in
itself any evidence whatever of its necessity. We assert
that " Every event must have a cause :" and this proposi
tion we know to be true, not only probably, and gene
rally, and as far as we can see : but we cannot suppose
it to be false in any single instance. We are as certain
of it as of the truths of arithmetic or geometry. We
cannot doubt that it must apply to all events past and
future, in every part of the universe, just as truly as
to those occurrences which we have ourselves observed.
What causes produce what effects; what is the cause
of any particular event ; what will be the effect of any
peculiar process ; these are points on which experience
may enlighten us. Observation and experience may be

OF THE IDEA OF CAUSE. 167

requisite, to enable us to judge respecting such matters.
But that every event has some cause, Experience cannot
prove any more than she can disprove. She can add
nothing to the evidence of the truth, however often she
may exemplify it. This doctrine, then, cannot have been
acquired by her teaching ; and the Idea of Cause, which
the doctrine involves, and on which it depends, cannot
have come into our minds from the region of observa
tion.

2. That we do, in fact, apply the Idea of Cause in a
more extensive manner than could be justified, if it were
derived from experience only, is easily shown. For from
the principle that everything must have a cause, we not
only reason concerning the succession of the events which
occur in the progress of the world, and which form the
course of experience ; but we infer that the world itself
must have a cause ; that the chain of events connected
by common causation, must have a First Cause of a
nature different from the events themselves. This we
are entitled to do, if our Idea of Cause be independent of,
and superior to, experience : but if we have no Idea of
Cause except such as we gather from experience, this
reasoning is altogether baseless and unmeaning.

3. Again ; by the use of our powers of observation,
we are aware of a succession of appearances and events.
But none of our senses or powers of external observa
tion can detect in these appearances the power or quality
which we call Cause. Cause is that which connects one
event with another ; but no sense or perception discloses
to us, or can disclose, any connexion among the events
which we observe. We see that one occurrence follows
another, but we can never see anything which shows that
one occurrence must follow another. We have already
noticed* 5 ", that this truth has been urged by metaphy-

Book i., chap. xiii.

168 PHILOSOPHY OF THE MECHANICAL SCIENCES.

sicians in modern times, and generally assented to by
those who examine carefully the connexion of their own
thoughts. The arguments are, indeed, obvious enough.
One ball strikes another and causes it to move forwards.
But by what compulsion ? Where is the necessity ? If
the mind can see any circumstance in this case which
makes the result inevitable, let this circumstance be
pointed out. But, in fact, there is no such discoverable
necessity ; for we can conceive this event not to take
place at all. The struck ball may stand still, for aught
we can see. " But the laws of motion will not allow it
to do so." Doubtless they will not. But the laws of
motion are learnt from experience, and therefore can
prove no necessity. Why should not the laws of motion
be other than they are? Are they necessarily true?
That they are necessarily such as do actually regulate the
impact of bodies, is at least no obvious truth ; and there
fore this necessity cannot be, in common minds, the
ground of connecting the impact of one ball with the
motion of another. And assuredly, if this fail, no other
ground of such necessary connexion can be shown. In
this case, then, the events are not seen to be necessarily
connected. But if this case, where one ball moves another
by impulse, be not an instance of events exhibiting a
necessary connexion, we shall look in vain for any ex
ample of such a connexion. There is, then, no case in
which events can be observed to be necessarily con
nected : our idea of causation, which implies that the
event is necessarily connected with the cause, cannot be
derived from observation.

4. But it may be said, we have not any such Idea of
Cause, implying necessary connexion with effect, and a
quality by which this connexion is produced. We see
nothing but the succession of events; and by cause we
mean nothing but a certain succession of events; name-

OF THE IDEA OF CAUSE. 169

ly, a constant, unvarying succession. Cause and effect
are only two events of which the second invariably
follows the first. We delude ourselves when we ima
gine that our idea of causation involves anything more
than this.

To this I reply by asking, what then is the meaning
of the maxim above quoted, and allowed by all to be
universally and necessarily true, that every event must
have a cause ? Let us put this maxim into the language
of the explanation just noticed ; and it becomes this :
" Every event must have a certain other event invariably
preceding it." But why must it? Where is the neces
sity ? Why must like events always be preceded by like,
except so far as other events interfere? That there is
such a necessity, no one can doubt. All will allow that
if a stone ascend because it is thrown upwards in one
case, a stone which ascends in another case has also
been thrown upwards, or has undergone some equi
valent operation. All will allow that in this sense,
every kind of event must have some other specific kind
of event preceding it. But this turn of men s thoughts
shows that they see in events a connexion which is not
mere succession. They see in cause and effect, not
merely what does, often or always, precede and follow,
but what must precede and follow. The events are not
only conjoined, they are connected. The cause is more
than the prelude, the effect is more than the sequel, of
the fact. The cause is conceived not as a mere occa
sion ; it is a power, an efficacy, which has a real ope
ration.

5. Thus we have drawn from the maxim, that Every
Effect must have a Cause, arguments to show that we
have an Idea of Cause which is not borrowed from expe
rience, and which involves more than mere succession.
Similar arguments might be derived from any other

170 PHILOSOPHY OF THE MECHANICAL SCIENCES.

maxims of universal and necessary validity, which we
can obtain concerning Cause : as, for example, the max
ims that Causes are measured by their Effects, and that
Reaction is equal and opposite to Action. These maxims
we shall soon have to examine ; but we may observe here,
that the necessary truth which belongs to them, shows
that they, and the Ideas which they involve, are not the
mere fruits of observation; while their meaning, including,
as it does, something quite different from the mere con
ception of succession of events, proves that such a con
ception is far from containing the whole import and
signification of our Idea of Cause.

The progress of the opinions of philosophers on the
points discussed in this chapter, has been one of the
most remarkable parts of the history of Metaphysics in
modern times : and I shall therefore briefly notice some
of its features.

CHAPTER III.

MODERN OPINIONS RESPECTING THE IDEA
OF CAUSE.

1. TOWARDS the end of the seventeenth century there
existed in the minds of many of the most vigorous and
active speculators of the European literary world, a strong
tendency to ascribe the whole of our Knowledge to the
teaching of Experience. This tendency, with its conse
quences, including among them the reaction which was
produced when the tenet had been pushed to a length
manifestly absurd, has exercised a very powerful in
fluence upon the progress of metaphysical doctrines up
to the present time. I proceed to notice some of the
most prominent of the opinions which have thus ob-

OPINIONS RESPECTING THE IDEA OF CAUSE. 171

tained prevalence among philosophers, so far as the Idea
of Cause is concerned.

Locke was one of the metaphysicians who produced
the greatest effect in diffusing this opinion, of the exclu
sive dependence of our knowledge upon experience.
Agreeably to this general system, he taught* that our
ideas of Cause and Effect are got from observation of
the things about us. Yet notwithstanding this tenet of
his, he endeavoured still to employ these ideas in rea
soning on subjects which are far beyond all limits of
experience : for he professed to prove, from our idea of
Causation, the existence of the Deity f.

Hume noticed this obvious inconsistency; but declared
himself unable to discover any remedy for a defect so
fatal to the most important parts of our knowledge. He
could see, in our belief of the succession of cause and
effect, nothing but the habit of associating in our minds
what had often been associated in our experience. He
therefore maintained that we could not, with logical
propriety, extend our belief of such a succession to cases
entirely distinct from all those of which our experience
consisted. We see, he said, an actual conjunction of two
events ; but we can in no way detect a necessary con
nexion ; and therefore we . have no means of inferring
cause from effect, or effect from cause J. The only way
in which we recognize Cause and Effect in the field of
our experience, is as an unfailing Sequence : we look in
vain for anything which can assure us of an infallible
Consequence. And since experience is the only source
of our knowledge, we cannot with any justice assert
that the world in which we live must necessarily have
had a cause.

2. This doctrine, taken in conjunction with the known

* Essay on the Human Understanding, B. n. c xxvi. t B. iv. c. x.
t Hume s Phil, of the Human Mind, Vol. i. p. 94.

172 PHILOSOPHY OF THE MECHANICAL SCIENCES.

skepticism of its author on religious points, produced a
considerable fermentation in the speculative world. The
solution of the difficulty thus thrown before philosophers,
was by no means obvious. It was vain to endeavour to
find in experience any other property of a Cause, than a
constant sequence of the effect. Yet it was equally vain
to try to persuade men that they had no idea of Cause ;
or even to shake their belief in the cogency of the fami
liar arguments concerning the necessity of an original
cause of all that is and happens. Accordingly these
hostile and apparently irreconcilable doctrines, the in
dispensable necessity of a cause of every event, and the
impossibility of our knowing such a necessity, were at
last allowed to encamp side by side. Reid, Beattie, and
others, formed one party, who showed how widely and
constantly the idea of a cause pervades all the processes
of the human mind : while another sect, including Brown,
and apparently Stewart, maintained that this idea is
always capable of being resolved into a constant se
quence ; and these latter reasoners tried to obviate the
dangerous and shocking inferences which some persons
might try to draw from their opinion, by declaring the
maxim that "Every event must have a cause," to be an
instinctive law of belief, or a fundamental principle of
the human mind*.

3. While this series of discussions was going on in
Britain, a great metaphysical genius in Germany was
unravelling the perplexity in another way. Kant s spe
culations originated, as he informs us, in the trains of
thought to which Hume s writings gave rise ; and the
Kritik der Reinen Vernunft, or Examination of the
Pure Reason, was published in 1787, with the view of
showing the true nature of our knowledge.

* Stewart s Active Powers, Vol. i. p. 347- Brown s Lectures,
Vol. i. p. 115.

OPINIONS RESPECTING THE IDEA OF CAUSE. 173

Kant s solution of the difficulties just mentioned
differs materially from that above stated. According to
Brown" r % succession observed and cause inferred, the
memory of past conjunctions of events and the belief of
similar future conjunctions, are facts, independent, so
far as we can discover, but inseparably combined by a
law of our mental nature. According to Kant, causality
is an inseparable condition of our experience : a con
nexion in events is requisite to our apprehending them as
events. Future occurrences must be connected by causa
tion as the past have been, because we cannot think of
past, present, and future, without such connexion. We
cannot fix the mind upon occurrences, without including
these occurrences in a series of causes and effects. The
relation of Causation is a condition under which we
think of events, as the relations of space are a condition
under \vhich we see objects.

4. On a subject so abstruse, it is not easy to make
our distinctions very clear. Some of Brown s illustrations
appear to approach very near to the doctrine of Kant.
Thus he saysf, "The form of bodies is the relation of
their elements to each other in space, the power of
bodies is their relation to each other in time." Yet not
withstanding such approximations in expression, the
Kantian doctrine appears to be different from the views
of Stewart and Brown, as commonly understood. Ac
cording to the Scotch philosophers, the cause and the
effect are two things, connected in our minds by a law
of our nature. But this view requires us to suppose that
we can conceive the law to be absent, and the course of
events to be unconnected. If we can understand what is
the special force of this law, we must be able to imagine
what the case would be if the law were non-existing. We
must be able to conceive a mind which does not connect
* Led.. Vol. i. p. 114. t Led., i. p. 127.

174 PHILOSOPHY OF THE MECHANICAL SCIENCES.

effects with causes. The Kantian doctrine, on the other
hand, teaches that we cannot imagine events liberated
from the connexion of cause and effect : this connexion is
a condition of our conceiving any real occurrences : we
cannot think of a real sequence of things, except as in
volving the operation of causes. In the Scotch system,
the past and the future are in their nature independent,
but bound together by a rule ; in the German system,
they share in a common nature and mutual relation, by
the act of thought which makes them past and future.
In the former doctrine cause is a tie which binds ; in the
latter it is a character which pervades and shapes events.
The Scotch metaphysicians only assert the universality
of the relation ; the German attempts further to explain
its necessity.

This being the state of the case, such illustrations as
that of Dr. Brown quoted above, in which he represents
cause as a relation of the same kind with form, do not
appear exactly to fit his opinions. Can the relations of
figure be properly said to be connected with each other
by a law of our nature, or a tendency of our mental con
stitution ? Can we ascribe it to a law of our thoughts,
that we believe the three angles of a triangle to be equal
to two right angles? If so, we must give the same
reason for our belief that two straight lines cannot
inclose a space ; or that three and two are five. But
will any one refer us to an ultimate law of our consti
tution for the belief that three and two are five ? Do
we not see that they are so, as plainly as we see that
they are three and two ? Can we imagine laws of our
constitution abolished, so that three and two shall make
something different from five ; so that an inclosed space
shall lie between two straight lines ; so that the three
angles of a plane triangle shall be greater than two
right angles? We cannot conceive this. If the num-

OPINIONS RESPECTING THE IDEA OF CAUSE. 175

bers are three and two ; if the lines are straight ; if the
triangle is a rectilinear triangle, the consequences are
inevitable. We cannot even imagine the contrary. We
do not want a law to direct that things should be what
they are. The relation, then, of cause and effect, being
of the same kind as the necessary relations of figure and
number, is not properly spoken of as established in our
minds by a special law of our constitution : for we reject
that loose and inappropriate phraseology which speaks
of the relations of figure and number as " determined by
laws of belief."

5. In the present work, we accept and adopt,-as the
basis of our inquiry concerning our knowledge, the exist
ence of necessary truths concerning causes, as there exist
necessary truths concerning figure and number. We
find such truths universally established and assented to
among the cultivators of science, and among speculative
men in general. All mechanicians agree that reaction
is equal and opposite to action, both when one body
presses another, and when one body communicates mo
tion to another. All reasoners join in the assertion, not
only that every observed change of motion has had a
cause, but that every change of motion must have a
cause. Here we have certain portions of substantial
and undoubted knowledge. Now the essential point in
the view which we must take of the idea of cause is
this, that our view must be such as to form a solid
basis for our knowledge. We have, in the Mechanical
Sciences, certain universal and necessary truths on the
subject of causes. Now any view which refers our be
lief in causation to mere experience or habit, cannot
explain the possibility of such necessary truths, since
experience and habit can never lead to a perception of
necessary connexion. But a view which teaches us to
acknowledge axioms concerning cause, as we acknow-

176 PHILOSOPHY OF THE MECHANICAL SCIENCES.

ledge axioms concerning space, will lead us to look upon
the science of mechanics as equally certain and univer
sal with the science of geometry ; and will thus mate
rially affect our judgment concerning the nature and
claims of our scientific knowledge.

Axioms concerning Cause, or concerning Force,
which as we shall see, is a modification of Cause, will
flow from an Idea of Cause, just as axioms concerning
space and number flow from the ideas of space and num
ber or time. And thus the propositions which con
stitute the science of Mechanics prove that we possess
an idea of cause, in the same sense in which the propo
sitions of geometry and arithmetic prove our possession
of the ideas of space and of time or number.

6. The idea of cause, like the ideas of space and
time, is a part of the active powers of the mind. The
relation of cause and effect is a relation or condition
under which events are apprehended, which relation is
not given by observation, but supplied by the mind itself.
According to the views which explain our apprehension
of cause by reference to habit, or to a supposed law of
our mental nature, causal connexion is a consequence of
agencies which the mind passively obeys ; but according
to the view to which we are led, this connexion is a
result of faculties which the mind actively exercises.
And thus the relation of cause and effect is a condition
of our apprehending successive events, a part of the
mind s constant and universal activity, a source of neces
sary truths ; or, to sum all this in one phrase, a Funda
mental Idea.

177

CHAPTER IV.

OF THE AXIOMS WHICH RELATE TO THE IDEA
OF CAUSE.

1. Causes are abstract Conceptions. WE have now
to express, as well as we can, the fundamental character
of that Idea of Cause, of which we have just proved the
existence. This may be done, at least for purposes of
reasoning, in this as in former instances, by means of
axioms. I shall state the principal axioms which belong
to this subject, referring the reader to his own thoughts
for the axiomatic evidence which belongs to them.

But I must first observe, that in order to express
general and abstract truths concerning cause and effect,
these terms, cause and effect, must be understood in a
general and abstract manner. When one event gives rise
to another, the first event is, in common language, often
called the cause, and the second the effect. Thus the
meeting of two billiard balls may be said to be the
cause of one of them turning aside out of the path in
which it was moving. For our present purposes, how
ever, we must not apply the term cause to such occur
rences as this meeting and turning, but to a certain
conception, force, abstracted from all such special events,
and considered as a quality or property by which one
body affects the motion of the other. And in like man
ner in other cases, cause is to be conceived as some
abstract quality, power, or efficacy, by which change is
produced; a quality not identical with the events, but
disclosed by means of them. Not only is this abstract
mode of conceiving force and cause useful in expressing
the fundamental principles of science ; but it supplies us
with the only mode by which such principles can be
VOL. i. \v. p. N

178 PHILOSOPHY OF THE MECHANICAL SCIENCES.

stated in a general manner, and made to lead to sub
stantial truth and real knowledge.

Understanding cause, therefore, in this sense, we
proceed to our Axioms.

2. First Axiom. Nothing can take place without a
Cause.

Every event, of whatever kind, must have a Cause in
the sense of the term which we have just indicated ; and
that it must, is a universal and necessary proposition to
which we irresistibly assent as soon as it is understood.
We believe each appearance to come into existence,
we conceive every change to take place, not only with
something preceding it, but something by which it is made
to be what it is. An effect without a cause ; an event
without a preceding condition involving the efficacy by
which the event is produced ; are suppositions which we
cannot for a moment admit. That the connexion of effect
with cause is universal and necessary, is a universal and
constant conviction of mankind. It persists in the minds
of all men, undisturbed by all the assaults of sophistry
and skepticism; and, as we have seen in the last chapter,
remains unshaken, even when its foundations seem to be
ruined. This axiom expresses, to a certain extent, our
Idea of Cause ; and when that idea is clearly appre
hended, the axiom requires no proof, and indeed admits
of none which makes it more evident. That notwith
standing its simplicity, it is of use in our speculations, we
shall hereafter see ; but in the first place, we must con
sider the other axioms belonging to this subject.

3. Second Axiom. Effects are proportional to their
Causes, and Causes are measured ~by their Effects.

We have already said that cause is that quality or
power, in the circumstances of each case, by which the
effect is produced ; and this power, an abstract property
of the condition of things to which it belongs, can in

AXIOMS WHICH RELATE TO THE IDEA OF CAUSE. 1 70

no way fall directly under the cognizance of the senses.
Cause, of whatever kind, is not apprehended as including
objects and events which share its nature by being co-ex
tensive with certain portions of it, as space and time are.
It cannot therefore, like them, be measured by repeti
tion of its own parts, as space is measured by repetition
of inches, and time by repetition of minutes. Causes may
be greater or less ; as, for instance, the force of a man is
greater than the force of a child. But how much is the
one greater than the other ? How are we to compare
the abstract conception, force, in such cases as these ?

To this, the obvious and only answer is, that we must
compare causes by means of their effects ; that we must
compare force by something which force can do. The
child can lift one fagot; the man can lift ten such fagots:
we have here a means of comparison. And whether or
not the rule is to be applied in this manner, that is, by
the number of the things operated on, (a question which
we shall have to consider hereafter,) it is clear that this
form of rule, namely, a reference to some effect or other
as our measure, is the right, because the only possible
form. The cause determines the effect. The cause being
the same, the effect must be the same. The connexion
of the two is governed by a fixed and inviolable rule.
It admits of no ambiguity. Every degree of intensity
in the cause has some peculiar modification of the effect
corresponding to it. Hence the effect is an unfailing
index of the amount of the cause ; and if it be a mea
surable effect, gives a measure of the cause. We can
have no other measure ; but we need no other, for this
is exact, sufficient, and complete.

It may be said, that various effects are produced by
the same cause. The sun s heat melts wax and expands
quicksilver. The force of gravity causes bodies to move
downwards if they are free, and to press down upon their

180 PHILOSOPHY OF THE MECHANICAL SCIENCES.

supports if they are supported. Which of the effects is to
be taken as the measure of heat, or of gravity, in these
cases ? To this we reply, that if we had merely different
states of the same cause to compare, any of the effects
might be taken. The sun s heat on different days might
be measured by the expansion of quicksilver, or by the
quantity of wax melted. The force of gravity, if it were
different at different places, might be measured by the
spaces through which a given weight would bend an
elastic support, or by the spaces through which a body
would fall in a given time. All these measures are con
sistent with the general character of our idea of cause.

4. Limitation of the Second Axiom. But there may
be circumstances in the nature of the case which may
further determine the kind of effect which we must take
for the measure of the cause. For example, if causes
are conceived to be of such a nature as to be capable of
addition, the effects taken as their measure must conform
to this condition. This is the case with mechanical
causes. The weights of two bodies are the causes of the
pressure which they exert downwards ; and these weights
are capable of addition. The weight of the two is the
sum of the weight of each. We are therefore not at
liberty to say that weights shall be measured by the
spaces through which they bend a certain elastic support,
except we have first ascertained that the whole weight
bends it through a space equal to the sum of the inflec
tions produced by the separate weights. Without this
precaution, we might obtain inconsistent results. Two
weights, each of the magnitude 3 as measured by their
effects, might, if we took the inflections of a spring for
the effects, be together equal to 5 or to 7 by the same
kind of measurement. For the inflection produced by
two weights of 3 might, for aught we can see before
hand, be more or less than twice as great as the inflection

AXIOMS WHICH RELATE TO THE IDEA OF CAUSE. 181

produced by one weight of 3. That forces are capable of
addition, is a condition which limits, and, as we shall see,
in some cases rigorously fixes, the kind of effects which
are to be taken as their measures.

Causes which are thus capable of addition are to be
measured by the repeated addition of equal quantities.
Two such causes are equal to each other when they pro
duce exactly the same effect. So far our axiom is applied
directly. But these two causes can be added together ;
and being thus added, they are double of one of them ;
and the cause composed by addition of three such, is
three times as great as the first ; and so on for any mea
sure whatever. By this means, and by this means only,
we have a complete and consistent measure of those
causes which are so conceived as to be subject to this
condition of being added and multiplied.

Causes are, in the present chapter, to be understood
in the widest sense of the term ; and the axiom now
under our consideration applies to them, whenever they
are of such a nature as to admit of any measure at all.
But the cases which we have more particularly in view
are mechanical causes, the causes of the motion and of
the equilibrium of bodies. In these cases, forces are con
ceived as capable of addition ; and what has been said of
the measure of causes in such cases, applies peculiarly to
mechanical forces. Two weights, placed together, may
be considered as a single weight, equal to the sum of the
two. Two pressures, pushing a body in the same direc
tion at the same point, are identical in all respects with
some single pressure, their sum, pushing in like manner;
and this is true whether or not they put the body in
motion. In the cases of mechanical forces, therefore, we
take some certain effect, velocity generated or weight
supported, which may fix the unit of force : and we then
measure all other forces by the successive repetition of

182 PHILOSOPHY OF THE MECHANICAL SCIENCES.

this unit, as we measure all spaces by the successive
repetition of our unit of lineal measure.

But these steps in the formation of the science of
Mechanics will be further explained, when we come to
follow our axioms concerning cause into their application
in that science. At present we have, perhaps, suffi
ciently explained the axiom that causes are measured
by their effects, and we now proceed to a third axiom,
also of great importance.

5. Third Axiom. Reaction is equal and opposite to
Action.

In the case of mechanical forces, the action of a
cause often takes place by an operation of one body
upon another ; and in this case, the action is always and
inevitably accompanied by an opposite action. If I press
a stone with my hand, the stone presses my hand in
return. If one ball strike another and put it in motion,
the second ball diminishes the motion of the first. In
these cases the operation is mutual; the Action is ac
companied by a Reaction. And in all such cases the
Reaction is a force of exactly the same nature as the
Action, exerted in an opposite direction. A pressure
exerted upon a body at rest is resisted and balanced by
another pressure ; when the pressure of one body puts
another in motion, the body, though it yields to the force,
nevertheless exerts upon the pressing body a force like
that which it suffers.

Now the axiom asserts further, that this Reaction
is equal, as well as opposite, to the Action. For the
Reaction is an effect of the Action, and is determined by
it. And since the two, Action and Reaction, are forces
of the same nature, each may be considered as cause
and as effect ; and they must, therefore, determine each
other by a common rule. But this consideration leads
necessarily to their equality : for since the rule is mutual,

AXIOMS WHICH RELATE TO THE IDEA OF CAUSE. 183

if we could for an instant suppose the Reaction to be
less than the Action, we must, by the same rule, sup
pose the Action to be less than the Reaction. And thus
Action and Reaction, in every such case, are rigorously
equal to each other.

It is easily seen that this axiom is not a proposition
which is, or can be, proved by experience ; but that its
truth is anterior to special observation, and depends on
our conception of Action and Reaction. Like our other
axioms, this has its source in an Idea ; namely, the Idea
of Cause, under that particular condition in which cause
and effect are mutual. The necessary and universal
truth which we cannot help ascribing to the axiom, shows
that it is not derived from the stores of experience,
which can never contain truths of this character. Ac
cordingly, it was asserted with equal confidence and
generality by those who did not refer to experience for
their principles, and by those who did. Leonicus Tomseus,
a commentator of Aristotle, whose work was published
in 1552, and therefore at a period when no right opinions
concerning mechanical reaction were current, at least
in his school, says, in his remarks on the Author s Ques
tions concerning the communication of motion, that
" Reaction is equal and contrary to Action." The same
principle was taken for granted by all parties, in all the
controversies concerning the proper measure of force, of
which we shall have to speak : and would be rigorously
true, as a law of motion, whichever of the rival inter
pretations of the measure of the term * Action" we were
to take.

G. Extent of the Third Axiom. It may naturally be
asked whether this third Axiom respecting causation
extends to any other cases than those of mechanical
action, since the notion of Cause in general has certainly
a much wider extent. For instance, when a hot body

184 PHILOSOPHY OF THE MECHANICAL SCIENCES.

heats a cold one, is there necessarily an equal reaction
of the second body upon the first? Does the snowball
cool the boy s hand exactly as much as the hand heats
the snow ? To this we reply, that, in every case in which
one body acts upon another by its physical qualities, there
must be some reaction. No body can affect another
without being itself also affected. But in any physical
change the action exerted is an abstract term which may
be variously understood. The hot hand may melt a
cold body, or may warm it : which kind of effect is to
be taken as action ? This remains to be determined by
other considerations.

In all cases of physical change produced by one body
in another, it is generally possible to assume such a
meaning of action, that the reaction shall be of the same
nature as the action ; and when this is done, the third
axiom of causation, that reaction is equal to action, is
universally true. Thus if a hot body heat a cold one,
the change may be conceived as the transfer of a certain
substance, heat or caloric, from the first body to the
second. On this supposition, the first body loses just as
much heat as the other gains ; action and reaction are
equal. But if the reaction be of a different kind to the
action we can no longer apply the axiom. If a hot body
melt a cold one, the latter cools the former : here, then, is
reaction ; but so long as the action and reaction are stated
in this form, we cannot assert any equality between them.

In treating of the secondary mechanical sciences, we
shall see further in what way we may conceive the
physical action of one body upon another, so that the
same axioms which are the basis of the science of
Mechanics shall apply to changes not at first sight mani
festly mechanical.

The three axioms of causation which we have now
stated are the fundamental maxims of all reasoning con-

AXIOMS WHICH RELATE TO THE IDEA OF C^USE. 185

cerning causes as to their quantities; and it will be
shown in the sequel that these axioms form the basis of
the science of Mechanics, determining its form, extent,
and certainty. We must, however, in the first place,
consider how we acquire those conceptions upon which
the axioms now established are to be employed.

CHAPTER V.

OF THE ORIGIN OF OUR CONCEPTIONS OF
FORCE AND MATTER.

1. Force. WHEN the faculties of observation and
thought are developed in man, the idea of causation is
applied to those changes which we see and feel in the
state of rest and motion of bodies around us. And
when our abstract conceptions are thus formed and
named, we adopt the term Force, and use it to
denote that property which is the cause of motion pro
duced, changed, or prevented. This conception is, it
would seem, mainly and primarily suggested by our
consciousness of the exertions by which we put bodies
in motion. The Latin and Greek words for Force, Vis,
F*v, were probably, like all abstract terms, derived at
first from some sensible object. The original meaning
of the Greek word was a muscle or tendon. Its first
application as an abstract term is accordingly to muscu
lar force.

AevVe^os UVT AiYts TToAu jue/oi/a \ciav detpas
rJK tirttivtja-asy 7repi&lt;r Be FIN a.Tre\e6pov.

Then Ajax a far heavier stone upheaved,
He whirled it, and impressing Force intense
Upon the mass, dismist it.

The property by which bodies affect each other s
motions, was naturally likened to that energy which we

186 PHILOSOPHY OF THE MECHANICAL SCIENCES.

exert upon them with similar effect : and thus the labour
ing horse, the rushing torrent, the descending weight, the
elastic bow, Avere said to exert force. Homer* speaks
of the force of the river, F^ TrorajuoTo; and Hesiodf of
the force of the north wind, F&lt;? av^ov fiopeao.

Thus man s general notion of force was probably first
suggested by his muscular exertions, that is, by an act
depending upon that muscular sense, to which, as we
have already seen, the perception of space is mainly due.
And this being the case, it will be easily understood that
the Direction of the force thus exerted is perceived by
the muscular sense, at the same time that the force itself
is perceived ; and that the direction of any other force is
understood by comparison with force which man must
exert to produce the same effect, in the same manner as
force itself is so understood.

This abstract notion of Force long remained in a very
vague and obscure condition, as may be seen by referring
to the History for the failures of attempts at a science of
force and motion, made by the ancients and their com
mentators in the middle ages. By degrees, in modern
times, we see the scientific faculty revive. The concep
tion of Force becomes so far distinct and precise that it
can be reasoned upon in a consistent manner, with de
monstrated consequences ; and a genuine science of Me
chanics comes into existence. The foundations of this
science are to be found in the Axioms concerning causa
tion which we have already stated ; these axioms being
interpreted and fixed in their application by a constant
reference to observed facts, as we shall show. But we
must, in the first place, consider further those primary
processes of observation by which we acquire the first
materials of thought on such subjects.

2. Matter. The conception of Force, as we have said,

* //. xxi. t Op. et D.

ORIGIN OF CONCEPTIONS OF FORCE AND MATTER. 187

arises with our consciousness of our own muscular exer
tions. But we cannot imagine such exertions without
also imagining some bodily substance against which they
are exercised. If we press, we press something : if we
thrust or throw, there must be something to resist the
thrust or to receive the impulse. Without body, mus
cular force cannot be exerted and force in general is not
conceivable.

Thus Force cannot exist without Body on which it
acts. The two conceptions, Force and Matter, are co
existent and correlative. Force implies resistance ; and
the force is effective only when the resistance is called
into play. If we grasp a stone, we have no hold of it
till the closing of the hand is resisted by the solid tex
ture of the stone. If we push open a gate, we must
surmount the opposition which it exerts while turning
on its hinges. However slight the resistance be, there
must be some resistance, or there would be no force.
If we imagine a state of things in which objects do not
resist our touch, they must also cease to be influenced
by our strength. Such a state of things we sometimes
imagine in our dreams ; and such are the poetical pic
tures of the regions inhabited by disembodied spirits. In
these, the figures which appear are conspicuous to the
eye, but impalpable like shadow or smoke ; and as they
do not resist the corporeal impressions, so neither do
they obey them. The spectator tries in vain to strike
or to grasp them.

Et ni cana vates tenues sine corpore vitas
Admoneat volitare cava sub imagine formse,
Irruat ac frustra ferro diverberet umbras.

The Sibyl warns him that there round him fly
Bodiless things, but substance to the eye;
Else had he pierced those shapes with life-like face,
And smitten, fierce, the unresisting space.

188 PHILOSOPHY OF THE MECHANICAL SCIENCES.

Neque ilium *

Prensantem nequlcquam umbras et multa volentem
Dicore, preterea vidit.
He grasps her form, and clutches but the shade.

Such may be the circumstances of the unreal world of
dreams, or of poetical fancies approaching to dreams:
for in these worlds our imaginary perceptions are bound
by no rigid conditions of force and reaction. In such
cases, the mind casts off the empire of the idea of cause,
as it casts off even the still more familiar sway of the
ideas of space and time. But the character of the
material world in which we live when awake is, that we
have at every instant and at every place, force operating
on matter and matter resisting force.

3. Solidity. From our consciousness of muscular
exertion, we derive, as we have seen, the conception of
force, and with that also the conception of matter. We
have already shown, in a former chapter, that the same
part of our frame, the muscular system, is the organ by
which we perceive extension and the relations of space.
Thus the same organ gives us the perception of body as
resisting force, and as occupying space ; and by combin
ing these conditions we have the conception of solid
extended bodies. In reality, this resistance is inevitably
presented to our notice in the very facts from which we
collect the notion of extension. For the action of the
hand and arm by which we follow the forms of objects,
implies that we apply our fingers to their surface; and
we are stopped there by the resistance which the body
offers. This resistance is precisely that which is requisite
in order to make us conscious of our muscular effort*.
Neither touch, nor any other mere passive sensation,
could produce the perception of extent, as we have
already urged : nor could the muscular sense lead to such
* Brown s Lectures, i. 466.

ORIGIN OF CONCEPTIONS OF FORCE AND MATTER. 189

a perception, except the extension of the muscles were
felt to be resisted. And thus the perception of resistance
enters the mind along with the perception of extended
bodies. All the objects with which we have to do are
not only extended but solid.

This sense of the term solidity, (the general property
of all matter,) is different to that in which we oppose
solidity to fluidity. We may avoid ambiguity by op
posing rigid to fluid bodies. By solid bodies, as we now
speak of them, we mean only such as resist the pressure
which we exert, so long as their parts continue in their
places. By fluid bodies, we mean those whose parts are,
by a slight pressure, removed out of their places. A drop
of water ceases to prevent the contact of our two hands,
not by ceasing to have solidity in this sense, but by being
thrust out of the way. If it could remain in its place,
it could not cease to exercise its resistance to our pres
sure, except by ceasing to be matter altogether.

The perception of solidity, like the perception of
extension, implies an act of the mind, as well as an
impression of the senses : as the perception of extension
implies the idea of space, so the perception of solidity
implies the idea of action and reaction. That an Idea
is involved in our knowledge on this subject appears, as
in other instances, from this consideration, that the con
victions of persons, even of those who allow of no ground
of knowledge but experience, do in fact go far beyond the
possible limits of experience. Thus Locke says*, that
" the bodies which we daily handle hinder by an insur
mountable force the approach of the parts of our hands
that press them." Now it is manifest that our observa
tion can never go to this length. By our senses we can
only perceive that bodies resist the greatest actual forces
that we exert upon them. But our conception of force

* Essay, B. n. c. 4.

190 PHILOSOPHY OF THE MECHANICAL SCIENCES.

carries us further : and since, so long as the body is
there to receive the action of the force, it must suffer
the whole of that action, and must react as much as
it suffers : it is therefore true, that so long as the body
remains there, the force which is exerted upon it can
never surmount the resistance which the body exercises.
And thus this doctrine, that bodies resist the intrusion
of other bodies by an insurmountable force, is, in fact,
a consequence of the axiom that the reaction is always
equal to the action.

4. Inertia. But this principle of the equality of
action and reaction appears also in another way. Not
only when we exert force upon bodies at rest, but when,
by our exertions, we put them in motion, they react. If
we set a large stone in motion, the stone resists ; for the
operation requires an effort. By increasing the effort, we
can increase the effect, that is, the motion produced ; but
the resistance still remains. And the greater the stone
moved, the greater is the effort requisite to move it.
There is, in every case, a resistance to motion, which shows
itself, not in preventing the motion, but in a reciprocal
force, exerted backwards upon the agent by which the
motion is produced. And this resistance resides in
each portion of matter, for it is increased as we add
one portion of matter to another. We can push a light
boat rapidly through the water ; but we may go on
increasing its freight, till we are barely able to stir it.
This property of matter, then, by which it resists the
reception of motion, or rather by which it reacts and
requires an adequate force in order that any motion may
result, is called its inertness, or inertia. That matter has
such a property, is a conviction flowing from that idea of
a reaction equal and opposite to the action, which the
conception of all force involves. By what laws this
inertia depends on the magnitude, form, and material of

ORIGIN OF CONCEPTIONS OF FORCE AND MATTER. 191

the body, must be the subject of our consideration here
after. But that matter has this inertia, in virtue of
which, as the matter is greater, the velocity which the
same effort can communicate to it is less, is a principle
inseparable from the notion of matter itself.

Hermann says that Kepler first introduced this " most
significant word" inertia. Whether it is to be found in
earlier writers I know not ; Kepler certainly does use it
familiarly in those attempts to assign physical reasons
for the motions of the planets which were among the
main occasions of the discovery of the true laws of me
chanics. He assumes the slowness of the motions of the
planets to increase, (other causes remaining the same,)
as the inertia increases ; and though, even in this as
sumption, there is an errour involved, (if we adopt that
interpretation of the term inertia to which subsequent
researches led,) the introduction of such a word was one
step in determining and expressing those laws, of motion
which depend on the fundamental principle of the equality
of action and reaction.

5. We have thus seen, I trust in a satisfactory
manner, the origin of our conceptions of Force, Matter,
Solidity, and Inertness. It has appeared that the organ
by which we obtain such conceptions is that very mus
cular frame, which is the main instrument of our percep
tions of space ; but that, besides bodily sensations, these
ideal conceptions, like all the others which we have
hitherto considered, involve also an habitual activity of
the mind, giving to our sensations a meaning which they
could not otherwise possess. And among the ideas thus
brought into play, is an idea of action with an equal and
opposite reaction, which forms a foundation for univer
sal truths to be hereafter established respecting the
conceptions thus obtained.

We must now endeavour to trace in what manner

192 PHILOSOPHY OF THE MECHANICAL SCIENCES.

these fundamental principles and conceptions are un
folded by means of observation and reasoning, till they
become an extensive yet indisputable science.

CHAPTER VI.

OF THE ESTABLISHMENT OF THE PRINCIPLES
OF STATICS.

1. Object of the Chapter. IN the present and the
succeeding chapters we have to show how the general
axioms of Causation enable us to construct the science
of Mechanics. We have to consider these axioms as
moulding themselves, in the first place, into certain fun
damental mechanical principles, which are of evident
and necessary truth in virtue of their dependence upon
the general axioms of Causation ; and thus as forming a
foundation for the whole structure of the science ; a
system of truths no less necessary than the fundamen
tal principles, because derived from these by rigorous
demonstration.

This account of the construction of the science of
Mechanics, however generally treated, cannot be other
wise than technical in its details, and will probably be
imperfectly understood by any one not acquainted with
Mechanics as a mathematical science.

I cannot omit this portion of my survey without
rendering my work incomplete ; but I may remark that
the main purpose of it is to prove, in a more particular
manner, what I have already declared in general, that
there are, in Mechanics no less than in Geometry, funda
mental principles of axiomatic evidence and necessity ;
that these principles derive their axiomatic character
from the Idea which they involve, namely the Idea of

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 193

Cause ; and that through the combination of principles
of this kind, the whole science of Mechanics, including
its most complex and remote results, exists as a body of
solid and universal truths.

2. Statics and Dynamics. We must first turn our
attention to a technical distinction of Mechanics into
two portions, according as the forces about which we
reason produce rest, or motion; the former portion is
termed Statics, the latter Dynamics. If a stone fall,
or a weight put a machine in motion, the problem
belongs to Dynamics ; but if the stone rest upon the
ground, or a weight be merely supported by a machine,
without being raised higher, the question is one of
Statics.

3. Equilibrium. In Statics, forces balance each
other, or keep each other in equilibrium. And forces
which directly balance each other, or keep each other in
equilibrium, are necessarily and manifestly equal. If
we see two boys pull at two ends of a rope so that
neither of them in the smallest degree prevails over the
other, we have a case in which two forces are in equili
brium. The two forces are evidently equal, and are a
statical exemplification of action and reaction, such as are
spoken of in the third axiom concerning causes. Now
the same exemplification occurs in every case of equili
brium. No point or body can be kept at rest except in
virtue of opposing forces acting upon it ; and these forces
must always be equal in their opposite effect. When a
stone lies on the floor, the weight of the stone down
wards is opposed and balanced by an equal pressure of
the floor upwards. If the stone rests on a slope, its
tendency to slide is counteracted by some equal and
opposite force, arising, it may be, from the resistance
which the sloping ground opposes to any motion along
its surface. Every case of rest is a case of equilibrium :

VOL. i. AV. p.

194 PHILOSOPHY OF THE MECHANICAL SCIENCES.

every case of equilibrium is a case of equal and opposite
forces.

The most complex frame-work on which weights are
supported, as the roof of a building, or the cordage of a
machine, are still examples of equilibrium. In such
cases we may have many forces all combining to balance
each other ; and the equilibrium will depend on various
conditions of direction and magnitude among the forces.
And in order to understand what are these conditions,
we must ask, in the first place, what we understand by
the magnitude of such forces ; what is the measure of
statical forces.

4. Measure of Statical Forces. At first we might
expect, perhaps, that since statical forces come under the
general notion of Cause, the mode of measuring them
would be derived from the second axiom of Causation,
that causes are measured by their effects. But we find
that the application of this axiom is controlled by the
limitation which we noticed, after stating that axiom ;
namely, the condition that the causes shall be capable of
addition. Further, as we have seen, a statical force pro
duces no other effect than this, that it balances some
other statical force ; and hence the measure of statical
forces is necessarily dependent upon their balancing,
that is, upon the equality of action and reaction.

That statical forces are capable of addition is involved
in our conception of such forces. When two men pull
at a rope in the same direction, the forces which they
exert are added together. When two heavy bodies are
put into a basket suspended by a string, their weights
are added, and the sum is supported by the string.

Combining these considerations, it will appear that
the measure of statical forces is necessarily given at once
by the fundamental principle of the equality of action
and reaction. Since two opposite forces which balance

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 195

each other are equal, each force is measured by that
which it balances ; and since forces are capable of addi
tion, a force of any magnitude is measured by adding to
gether a proper number of such equal forces. Thus a
heavy body which, appended to some certain elastic
branch of a tree, would bend it down through one inch,
may be taken as a unit of weight. Then if we remove
this first body, and find a second heavy body which will
also bend the branch through the same space, this is also
a unit of weight ; and in like manner we might go on to
a third and a fourth equal body; and adding together
the two, or the three, or the four heavy bodies, we have
a force twice, or three times, or four times the unit of
weight. And with such a collection of heavy bodies, or
weights, we can readily measure all other forces ; for the
same principle of the equality of action and reaction
leads at once to this maxim, that any statical force is
measured by the weight which it would support.

As has been said, it might at first have been sup
posed that we should have to apply, in this case, the
axiom that causes are measured by their effects in an
other manner ; that thus, if that body were a unit of
weight which bent the bough of a tree through one inch,
that body would be two units which bent it through two
inches, and so on. But, as we have already stated, the
measures of weight must be subject to this condition,
that they are susceptible of being added : and therefore
we cannot take the deflexion of the bough for our mea
sure, till we have ascertained, that which experience
alone can teach us, that under the burden of two equal
weights, the deflexion will be twice as great as it is with
one weight, which is not true, or at least is neither ob
viously nor necessarily true. In this, as in all other cases,
although causes must be measured by their effects, we
learn from experience only how the effects are to be

O 2

196 PHILOSOPHY OF THE MECHANICAL SCIENCES.

interpreted, so as to give a true and consistent mea
sure.

With regard, however, to the measure of statical
force, and of weight, no difficulty really occurred to phi
losophers from the time when they first began to specu
late on such subjects ; for it was easily seen that if we
take any uniform material, as wood, or stone, or iron,
portions of this which are geometrically equal, must also
be equal in statical effect ; since this was implied in the
very hypothesis of a uniform material. And a body ten
times as large as another of the same substance, will be
of ten times the weight. But before men could esta
blish by reasoning the conditions under which weights
would be in equilibrium, some other principles were
needed in addition to the mere measure of forces. The
principles introduced for this purpose still resulted from
the conception of equal action and reaction ; but it re
quired no small clearness of thought to select them
rightly, and to employ them successfully. This, however,
was done, to a certain extent, by the Greeks; and the
treatise of Archimedes On the Center of Gravity, is
founded on principles which may still be considered as
the genuine basis of statical reasoning. I shall make a
few remarks on the most important principle among
those which Archimedes thus employs.

5. The Center of Gravity. The most important of
the principles which enter into the demonstration of
Archimedes is this : that " Every body has a center of
gravity ;" meaning by the center of gravity, a point at
which the whole matter of the body may be supposed to
be collected, to all intents and purposes of statical
reasoning. This principle has been put in various forms
by succeeding writers : for instance, it has been thought
sufficient to assume a case much simpler than the general
one ; and to assert that two equal bodies have their

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 197

center of gravity in the point midway between them. It
is to be observed, that this assertion not only implies
that the two bodies will balance upon a support placed
at that midway point, but also, that they will exercise,
upon such a support, a pressure equal to their sum ;
for this point being the center of gravity, the whole
matter of the two bodies may be conceived to be col
lected there, and therefore the whole weight will press
there. And thus the principle in question amounts to
this, that when two equal heavy bodies are supported on
the middle point between them, the pressure upon the
support is equal to the sum of the weights of the bodies.

A clear understanding of the nature and grounds of
this principle is of great consequence : for in it we have
the foundation of a large portion of the science of
Mechanics. And if this principle can be shown to be
necessarily true, in virtue of our Fundamental Ideas, we
can hardly doubt that there exist many other truths of
the same kind, and that no sound view of the evidence
and extent of human knowledge can be obtained, so long-
as we mistake the nature of these, its first principles.

The above principle, that the pressure on the support
is equal to the sum of the bodies supported, is often
stated as an axiom in the outset of books on Mechanics.
And this appears to be the true place and character of
this principle, in accordance with the reasonings which
we have already urged. The axiom depends upon our
conception of action and reaction. That the two weights
are supported, implies that the supporting force must be
equal to the force or weight supported.

In order further to show the foundation of this
principle, we may ask the question : If it be not an
axiom, deriving its truth from the fundamental concep
tion of equal action and reaction, which equilibrium
always implies, what is the origin of its certainty ? The

198 PHILOSOPHY OF THE MECHANICAL SCIENCES.

principle is never for an instant denied or questioned: it is
taken for granted, even before it is stated. No one will
doubt that it is not only true, but true with the same
rigour and universality as the axioms of Geometry. Will
it be said, that it is borrowed from experience ? Expe
rience could never prove a principle to be universally
and rigorously true. Moreover, when from experience
we prove a proposition to possess great exactness and
generality, we approach by degrees to this proof: the
conviction becomes stronger, the truth more secure, as
we accumulate trials. But nothing of this kind is the
case in the instance before us. There is no gradation
from less to greater certainty; no hesitation which
precedes confidence. From the first, we know that the
axiom is exactly and certainly true. In order to be
convinced of it, we do not require many trials, but
merely a clear understanding of the assertion itself.

But in fact, not only are trials not necessary to the
proof, but they do not strengthen it. Probably no
one ever made a trial for the purpose of showing that
the pressure upon the support is equal to the sum of the
two weights. Certainly no person with clear mechanical
conceptions ever wanted such a trial to convince him of
the truth ; or thought the truth clearer after the trial
had been made. If to such a person, an experiment
were shown which seemed to contradict the principle, his
conclusion would be, not that the principle was doubtful,
but that the apparatus was out of order. Nothing can
be less like collecting truth from experience than this.

We maintain, then, that this equality of mechanical
action and reaction, is one of the principles which do
not flow from, but regulate our experience. To this
principle, the facts which we observe must conform ;
and we cannot help interpreting them in such a manner
that they shall be exemplifications of the principle. A

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 199

mechanical pressure not accompanied by an equal and
opposite pressure, can no more be given by experience,
than two unequal right angles. With the supposition of
such inequalities, space ceases to be space, force ceases to
be force, matter ceases to be matter. And this equality
of action and reaction, considered in the case in which
two bodies are connected so as to act on a single support,
leads to the axiom which we have stated above, and
which is one of the main foundations of the science of
Mechanics.

6. Oblique Forces. By the aid of this axiom and
a few others, the Greeks made some progress in the
science of Statics. But after a short advance, they
arrived at another difficulty, that of Oblique Forces,
which they never overcame ; and which no mathematician
mastered till modern times. The unpublished manuscripts
of Leonardo da Vinci, written in the fifteenth century,
and the works of Stevinus and Galileo, in the sixteenth,
are the places in which we find the first solid grounds of
reasoning on the subject of forces acting obliquely to
each other. And mathematicians, having thus become
possessed of all the mechanical principles which are
requisite in problems respecting equilibrium, soon framed
a complete science of Statics. Succeeding writers pre
sented this science in forms variously modified ; for it
was found, in Mechanics as in Geometry, that various
propositions might be taken as the starting points ; and
that the collection of truths which it was the mecha
nician s business to include in his course, might thus be
traversed by various routes, each path offering a series
of satisfactory demonstrations. The fundamental con
ceptions of force and resistance, like those of space and
number, could be contemplated under different aspects,
each of which might be made the basis of axioms,
or of principles employed as axioms. Hence the

200 PHILOSOPHY OF THE MECHANICAL SCIENCES.

grounds of the truth of Statics may be stated in various
ways ; and it would be a task of some length to examine
all these completely, and to trace them to their Funda
mental Ideas. This I shall not undertake here to do ;
but the philosophical importance of the subject makes
it proper to offer a few remarks on some of the main
principles involved in the different modes of presenting
Statics as a rigorously demonstrated science.

7. A Force may be supposed to act at any Point of its
Direction. It has been stated in the history of Mecha
nics*, that Leonardo da Vinci and Galileo obtained the
true measure of the effect of oblique forces, by reason
ings which were, in substance, the same. The principle
of these reasonings is that expressed at the head of this
paragraph ; and when we have a little accustomed our
selves to contemplate our conceptions of force, and its
action on matter, in an abstract manner, we shall have
no difficulty in assenting to the principle in this general
form. But it may, perhaps, be more obvious at first in
a special case.

If we suppose a wheel, moveable about its axis, and
carrying with it in its motion a weight, (as, for example,
one of the wheels by means of which the large bells of a
church are rung,) this weight may be supported by means
of a rope (not passing along the circumference of the
wheel, as is usual in the case of bells,) but fastened to
one of the spokes of the wheel. Now the principle which
is enunciated above asserts, that if the rope pass in a
straight line across several of the spokes of the wheel, it
makes no difference in the mechanical effect of the force
applied, for the purpose of putting the bell in motion, to
which of these spokes the rope is fastened* In each case,
the fastening of the rope to the wheel merely serves to
enable the force to produce motion about the centre ;

* Hist. Tnd. Set., B. vi. c. i. sect. 2. and Note (A).

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 201

and so long as the force acts in the same line, the effect
is the same, at whatever point of the rope the line of
action finishes.

This axiom very readily aids us in estimating the
effect of oblique forces. For when a force acts on one of
the arms of a lever at any oblique angle, we suppose
another arm projecting from the centre of motion, like
another spoke of the same wheel, so situated that it is
perpendicular to the force. This arm we may, with
Leonardo, call the virtual lever ; for, by the axiom, we
may suppose the force to act where the line of its direc
tion meets this arm; and thus we reduce the case to
that in which the force acts perpendicularly on the arm.

The ground of this axiom is, that matter, in Statics,
is necessarily conceived as transmitting force. That force
can be transmitted from one place to another, by means
of matter ; that we can push with a rod, pull with a
rope, are suppositions implied in our conceptions of
force and matter. Matter is, as we have said, that which
receives the impression of force, and the modes just
mentioned, are the simplest ways in which that impres
sion operates. And since, in any of these cases, the force
might be resisted by a reaction equal to the force itself,
the reaction in each case would be equal, and, therefore,
the action in each case is necessarily equal ; and thus the
forces must be transmitted, from one point to another,
without increase or diminution.

This property of matter, of transmitting the action of
force, is of various kinds. We have the coherence of a
rope which enables us to pull, and the rigidity of a staff,
which enables us to push with it in the direction of its
length ; and again, the same staff has a rigidity of another
kind, in virtue of which we can use it as a lever ; that is, a
rigidity to resist flexure, and to transmit the force which
turns a body round a fulcrum. There is, further, the

202 PHILOSOPHY OF THE MECHANICAL SCIENCES.

rigidity by which a solid body resists twisting. Of these
kinds of rigidity, the first is that to which our axiom
refers ; but in order to complete the list of the ele
mentary principles of Statics, we ought also to lay down
axioms respecting the other kinds of rigidity*. These,
however, I shall not here state, as they do not involve
any new principle. Like the one just considered, they
form part of our fundamental conception of matter ; they
are not the results of any experience, but are the hypo
theses to which we are irresistibly led, when we would
liberate our reasonings concerning force and matter from
a dependence on the special results of experience. We
cannot even conceive (that is, if we have any clear
mechanical conceptions at all) the force exerted by the
point of a staff and resisting the force which we steadily
impress on the head of it, to be different from the
impressed force.

8. Forces may have equivalent Forces substituted for
them. The Parallelogram of Forces. It has already been
observed, that in order to prove the doctrines of Statics,
we may take various principles as our starting points,
and may still find a course of demonstration by which
the leading propositions belonging to the subject may
be established. Thus, instead of beginning our reason
ings, as in the last section we supposed them to
commence, with the case in which forces act upon
different points of the same body in the same line of
force, and counteract each other in virtue of the inter
vening matter by which the effect of force is transferred
from one point to another, we may suppose different
forces to act at the same point, and may thus commence
our reasonings with a case in which we have to con
template force, without having to take into our account

* Such axioms are given in a little work (The Mechanical Euclid}
which I published on the Elements of Mechanics.

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 203

the resistance or rigidity of matter. Two statical forces,
thus acting at a mathematical point, are equivalent, in
all respects, to some single force acting at the same point;
and would be kept in equilibrium by a force equal and
opposite to that single force. And the rule by which
the single force is derived from the two, is commonly
termed the parallelogram offerees; the proposition being
this, That if the two forces be represented in magnitude
and direction by the two sides of a parallelogram, the
resulting force will be represented in the same manner
by the diagonal of the parallelogram. This proposition
has very frequently been made, by modern writers, the
commencement of the science of Mechanics : a position
for which, by its simplicity, it is well suited ; although,
in order to deduce from it the other elementary proposi
tions of the science, as, for instance, those respecting the
lever, we require the axiom stated in the last section.

9. The Parallelogram of Forces is a necessary Truth.
In the series of discussions in which we are here
engaged, our main business is to ascertain the nature and
grounds of the certainty of scientific truths. We have,
therefore, to ask whether this proposition, the parallelo
gram of forces, be a necessary truth ; and if so, on what
grounds its necessity ultimately rests. We shall find
that this, like the other fundamental doctrines of Statics,
justly claims a demonstrative certainty. Daniel Ber
noulli, in 1726, gave the first proof of this important
proposition on pure statical principles; and thus, as he
says*, "proved that statical theorems are not less
necessarily true than geometrical are." If we examine
this proof of Bernoulli, in order to discover what are
the principles on which it rests, we shall find that the
reasoning employs in its progress such axioms as this ;
That if from forces which are in equilibrium at a point

* Comm. Pctrop. Vol. i.

204 PHILOSOPHY OF THE MECHANICAL SCIENCES.

be taken away other forces which are in equilibrium at
the same point, the remainder will be in equilibrium ;
and generally ; That if forces can be resolved into other
equivalent forces, these may be separated, grouped, and
recombined, in any new manner, and the result will still
be identical with what it was at first. Thus in Ber
noulli s proof, the two forces to be compounded are repre
sented by P and Q ; p is resolved into two other forces, x
and u ; and Q into two others, Y and v, under certain
conditions. It is then assumed that these forces may be
grouped into the pairs x, Y, and u, v : and when it has
been shown that x and Y are in equilibrium, they may, by
what has been said, be removed, and the forces, P, Q, are
equivalent to u, v; which, being in the same direction
by the course of the construction, have a result equal to
their sum.

It is clear that the principles here assumed are
genuine axioms, depending upon our conception of the
nature of equivalence of forces, and upon their being
capable of addition and composition. If the forces P, Q,
be equivalent to forces x, u, Y, v, they are equivalent to
these forces added and compounded in any order; just
as a geometrical figure is, by our conception of space,
equivalent to its parts added together in any order. The
apprehension of forces as having magnitude, as made
up of parts, as capable of composition, leads to such
axioms in Statics, in the same manner as the like
apprehension of space leads to the axioms of Geometry.
And thus the truths of Statics, resting upon such founda
tions, are independent of experience in the same manner
in which geometrical truths are so.

The proof of the parallelogram of forces thus given
by Daniel Bernoulli, as it was the first, is also one of
the most simple proofs of that .proposition which have
been devised up to the present day. Many other demon-

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 205

strations, however, have been given of the same proposi
tion. Jacobi, a German mathematician, has collected
and examined eighteen of these *. They all depend
either upon such principles as have just been stated ;
That forces may in every way be replaced by those which
are equivalent to them ; or else upon those previously
stated, the doctrine of the lever, and the transfer of a
force from one point to another of its direction. In
either case, they are necessary results of our statical con
ceptions, independent of any observed laws of motion,
and indeed, of the conception of actual motion altogether.
There is another class of alleged proofs of the paral
lelogram of forces, which involve the consideration of
the motion produced by the forces. But such reasonings
are, in fact, altogether irrelevant to the subject of Statics.
In that science, forces are not measured by the motion
which they produce, but by the forces which they will
balance, as we have already seen. The combination of
two forces employed in producing motion in the same
body, either simultaneously or successively, belongs to
that part of Mechanics which has motion for its subject,
and is to be considered in treating of the laws of motion.
The composition of motion, (as when a man moves in a
ship while the ship moves through the water,) has con
stantly been confounded with the composition of force.
But though it has been done by very eminent mathe
maticians, it is quite necessary for us to keep the two
subjects distinct, in order to see the real nature of the
evidence of truth in either case. The conditions of equi
librium of two forces on a lever, or of three forces at

* These are by the following mathematicians; D. Bernoulli
(1726); Lambert (1771); Scarella (1756); Yenini (1764); Araldi
(1806); Wachter (1815); Ka?stner ; Marini ; Eytelwein ; Salimbeni ;
Duchayla ; two different proofs by Foncenex (1760) ; three by
D Alembert; and those of Laplace and M. Poisson.

206 PHILOSOPHY OF THE MECHANICAL SCIENCES.

a point, can be established without any reference what
ever to any motions which the forces might, under other
circumstances, produce. And because this can be done,
to do so is the only scientific procedure. To prove such
propositions by any other course, would be to support
truth by extraneous and inconclusive reasons; which
would be foreign to our purpose, since we seek not only
knowledge, but the grounds of our knowledge.

10. The Center of gravity seeks the lowest place.
The principles which we have already mentioned afford
a sufficient basis for the science of Statics in its most
extensive and varied applications ; and the conditions of
equilibrium of the most complex combinations of ma
chinery may be deduced from these principles with a
rigour not inferior to that of geometry. But in some of
the more complex cases, the results of long trains of
reasoning may be foreseen, in virtue of certain maxims
which appear to us self-evident, although it may not be
easy to trace the exact dependence of these maxims upon
our fundamental conceptions of force and matter. Of
this nature is the maxim now stated ; That in any com
bination of matter any how supported, the Center of
Gravity will descend into the lowest position which the
connexion of the parts allows it to assume by descend
ing. It is easily seem that this maxim carries to a much
greater extent the principle which the Greek mathe
maticians assumed, that every body has a Center of
Gravity, that is, a point in which, if the whole matter of
the body be collected, the effect will remain unchanged.
For the Greeks asserted this of a single rigid mass only ;
whereas, in the maxim now under our notice, it is asserted
of any masses, connected by strings, rods, joints, or in
any manner. We have already seen that more modern
writers on mechanics, desirous of assuming as funda
mental no wider principles than are absolutely necessary,

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 207

have not adopted the Greek axiom in all its generality,
but have only asserted that two equal weights have a
center of gravity midway between them. Yet the prin
ciple that every body, however irregular, has a center of
gravity, and will be supported if that center is supported,
and not otherwise, is so far evident, that it might be
employed as a fundamental truth, if we could not resolve
it into any simpler truths : and, historically speaking, it
was assumed as evident by the Greeks. In like manner
the still wider principle, that a collection of bodies, as,
for instance, a flexible chain hanging upon one or more
supports, has a center of gravity ; and that this point
will descend to the lowest possible situation, as a single
body would do, has been adopted at various periods in
the history of mechanics ; and especially at conjunctures
when mathematical philosophers have had new and dif
ficult problems to contend with. For in almost every
instance it has only been by repeated struggles that phi
losophers have reduced the solution of such problems to
a clear dependence upon the most simple axioms.

11. Stevinuss Proof for Oblique Forces. We have
an example of this mode of dealing with problems, in
Stevinus s mode of reasoning concerning the Inclined
Plane ; which, as we have stated in the History of Me
chanics, was the first correct published solution of that
problem. Stevinus supposes a loop of chain, or a loop
of string loaded with a series of equal balls at equal dis
tances, to hang over the Inclined Plane ; and his reason
ing proceeds upon this assumption, That such a loop
so hanging will find a certain position in which it will
rest : for otherwise, says he*, its motion must go on for
ever, which is absurd. It may be asked how this absurd
ity of a perpetual motion appears ; and it will perhaps
be added, that although the impossibility of a machine

* Stevin. Staliquc, Livre i., prop. 19.

208 PHILOSOPHY OF THE MECHANICAL SCIENCES.

with such a condition may be proved as a remote result
of mechanical principles, this impossibility can hardly
be itself recognized as a self-evident truth. But to this
we may reply, that the impossibility is really evident in
the case contemplated by Stevinus ; for we cannot con
ceive a loop of chain to go on through all eternity, slid
ing round and round upon its support, by the effect of
its own weight. And the ground of our conviction that
this cannot be, seems to be this consideration; that when
the chain moves by the effect of its weight, we consider
its motion as the result of an effort to reach some certain
position, in which it can rest ; just as a single ball in
a bowl moves till it comes to rest at the lowest point
of the bowl. Such an effect of weight in the chain, we
may represent to ourselves by conceiving all the matter
of the chain to be collected in one single point, and this
single heavy point to hang from the support in some way
or other, so as fitly to represent the mode of support of
the chain. In whatever manner this heavy point (the
center of gravity of the chain) be supported and con
trolled in its movements, there will still be some position
of rest which it will seek and find. And thus there will
be some corresponding position of rest for the chain ; and
the interminable shifting from one position to another,
with no disposition to rest in any position, cannot exist.

Thus the demonstration of the property of the
Inclined Plane by Stevinus, depends upon a principle
which, though far from being the simplest of those to
which the case can be reduced, is still both true and
evident : and the evidence of this principle, depending
upon the assumption of a center of gravity, is of the
same nature as the evidence of the Greek statical demon
strations, the earliest real advances in the science.

12. Principle of Virtual Velocities. We have
referred above to an assertion often made, that we

ESTABLISHMENT OF THE PRINCIPLES OF STATICS, 209

may, from the simple principles of Mechanics, demon
strate the impossibility of a perpetual motion. In reality,
however, the simplest proof of that impossibility, in
a machine acted upon by weight only, arises from the
very maxim above stated, that the center of gravity seeks
and finds the lowest place ; or from some similar propo
sition. For if, as is done by many writers, we profess
to prove the impossibility of a perpetual motion by means
of that proposition which includes the conditions of equi
librium, and is called the Principle of Virtual Velocities*,
we are under the necessity of first proving in a general
manner that principle. And if this be done by a mere
enumeration of cases, (as by taking those five cases which
are called the Mechanical Powers,} there may remain
some doubts whether the enumeration of possible mecha
nical combinations be complete. Accordingly, some writers
have attempted independent and general proofs of the
Principle of Virtual Velocities; and these proofs rest
upon assumptions of the same nature as that now under
notice. This is, for example, the case with Lagrange s
proof, which depends upon what he calls the Principle
of Pulleys. For this principle is, That a weight any
how supported, as by a string passing round any number
of pulleys any how placed, will be at rest then only,
when it cannot get lower by any small motion of the
pulleys. And thus the maxim that a weight will descend
if it can, is assumed as the basis of this proof.

There is, as we have said, no need to assume such
principles as these for the foundation of our mechanical
science. But it is, on various accounts, useful to direct
our attention to those cases in which truths, apprehended
at first in a complex and derivative form, have after
wards been reduced to their simpler elements ; in which,
also, sagacious and inventive men have fixed upon those

* See Hist. Ind. Sci., B. vi. c. ii. sect. 4.
VOL. I. \V. I . P

210 PHILOSOPHY OF THE MECHANICAL SCIENCES.

truths as self-evident, which now appear to us only cer
tain in virtue of demonstration. In these cases we can
hardly doubt that such men were led to assert the
doctrines which they discovered, not by any capricious
conjecture or arbitrary selection, but by having a keener
and deeper insight than other persons into the relations
which were the object of their contemplation ; and in the
science now spoken of, they were led to their assump
tions by possessing clearly and distinctly the conceptions
of mechanical cause and effect, action and reaction.
force, and the nature of its operation.

13. Fluids press Equally in all Directions. The
doctrines which concern the equilibrium of fluids depend
on principles no less certain and simple than those which
refer to the equilibrium of solid bodies ; and the Greeks,
who, as we have seen, obtained a clear view of some of
the principles of Statics, also made a beginning in the
kindred subject of Hydrostatics. We still possess a trea
tise of Archimedes On Floating Bodies, which contains
correct solutions of several problems belonging to this
subject, and of some which are by no means easy. In
this treatise, the fundamental assumption is of this kind :
" Let it be assumed that the nature of a fluid is such,
that the parts which are less pressed yield to those which
are more pressed." In this assumption or axiom it is
implied that a pressure exerted upon a fluid in one direc
tion produces a pressure in another direction ; thus, the
weight of the fluid which arises from a downward force
produces a lateral pressure against the sides of the con
taining vessel. Not only does the pressure thus diverge
from its original direction into all other directions, but the
pressure, is in all directions exactly equal, an equal extent
of the fluid being taken. This principle, which was in
volved in the reasoning of Archimedes, is still to the
present day the basis of all hydrostatical treatises, and is

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 2 1 1

expressed, as above, by saying that fluids press equally
in all directions.

Concerning this, as concerning previously-noticed
principles, we have to ask whether it can rightly be said
to be derived from experience. And to this the answer
must still be, as in the former cases, that the proposition
is not one borrowed from experience in any usual or
exact sense of the phrase. I will endeavour to illustrate
this. There are many elementary propositions in phy
sics, our knowledge of which indisputably depends upon
experience ; and in these cases there is no difficulty in
seeing the evidence of this dependence. In such cases,
the experiments which prove the law are prominently
stated in treatises upon the subject : they are given with
exact measures, and with an account of the means by
which errors were avoided : the experiments of more
recent times have either rendered more certain the law
originally asserted, or have pointed out some correction
of it as requisite : and the names, both of the discoverers
of the law and of its subsequent reformers, are well
known. For instance, the proposition that " The elastic
force of air varies as the density," was first proved by
Boyle, by means of operations of which the detail is given
in his Defence of his Pneumatical Experiments* ; and
by Marriotte in his Traite de VEquilibre des Liquides,
from whom it has generally been termed Marriotte s law.
After being confirmed by many other experimenters,
this law was suspected to be slightly inaccurate, and a
commission of the French Academy of Sciences was
appointed, consisting of several distinguished philoso-
phersf, to ascertain the truth or falsehood of this suspicion.

* Shaw s Boyle, Vol. u. p. 671.

t The members were Prony, Arago, Ampere, Girard, and Dulong.
The experiments were extended to a pressure of twenty-seven atmo
spheres , nnd in no instance did the difference between the observed

P "2

212 PHILOSOPHY OF THE MECHANICAL SCIENCES.

The result of their investigations appeared to be, that
the law is exact, as nearly as the inevitable inaccuracies
of machinery and measures will allow us to judge. Here
we have an example of a law which is of the simplest
kind and form ; and which yet is not allowed to rest
upon its simplicity or apparent probability, but is rigor
ously tested by experience. In this case, the assertion,
that the law depends upon experience, contains a refer
ence to plain and notorious passages in the history of
science.

Now with regard to the principle that fluids press
equally in all directions, the case is altogether different.
It is, indeed, often asserted in works on hydrostatics,
that the principle is collected from experience, and some
times a few experiments are described as exhibiting its
effect ; but these are such as to illustrate and explain,
rather than to prove, the truth of the principle : they
are never related to have been made with that exact
ness of precaution and measurement, or that frequency
of repetition, which are necessary to establish a purely
experimental truth. Nor did such experiments occur as
important steps in the history of science. It does not
appear that Archimedes thought experiment necessary
to confirm the truth of the law as he employed it : on
the contrary, he states it in exactly the same shape as
the axioms which he employs in statics, and even in geo
metry ; namely, as an assumption. Nor does any intel
ligent student of the subject find any difficulty in assent
ing to this fundamental principle of hydrostatics as soon
as it is propounded to him. Experiment was not requi
site for its discovery ; experiment is not necessary for
its proof at present ; and we may add, that experiment,

and calculated elasticity amount to one-hundredth of the whole ; nor
did the difference appear to increase with the increase of pressure.
Fechner, Repertorium, i. 110.

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 213

though it may make the proposition more readily intelli
gible, can add nothing to our conviction of its truth
when it is once understood.

14. Foundation of the above Axiom. But it will
naturally be asked, What then is the ground of our
conviction of this doctrine of the equal pressure of a
fluid in all directions? And to this I reply, that the
reasons of this conviction are involved in our idea of a
fluid, which is considered as matter, and therefore as
capable of receiving, resisting, and transmitting force
according to the general conception of matter ; and which
is also considered as matter which has its parts perfectly
moveable among one another. For it follows from
these suppositions, that if the fluid be confined, a pres
sure which thrusts in one side of the containing vessel,
may cause any other side to bulge outwards, if there be
a part of the surface which has not strength to resist
this pressure from within. And that this pressure, when
thus transferred into a direction different from the ori
ginal one, is not altered in intensity, depends upon this
consideration ; that any difference in the two pressures
would be considered as a defect of perfect fluidity, since
the fluidity would be still more complete, if this entire
and undiminished transmission of pressure in all direc
tions were supposed. If, for instance, the lateral pres
sure were less than the vertical, this could be conceived
no other way than as indicating some rigidity or adhesion
of the parts of the fluid. When the fluidity is perfect,
the two pressures which act in the two different parts of
the fluid exactly balance each other : they are the action
and the reaction; and must hence be equal by the same
necessity as two directly opposite forces in statics.

But it may be urged, that even if we grant that this
conception of a perfect fluid, as a body which has its
parts perfectly moveable among each other, leads us

214 PHILOSOPHY OF THE MECHANICAL SCIENCES.

necessarily to the principle of the equality of hydrostatic
pressure in all directions, still this conception itself is
obtained from experience, or suggested by observation.
And to this we may reply, that the conception of a fluid,
as contemplated in mechanical theory, cannot be said to
be derived from experience, except in the same manner
as the conception of a solid and rigid body may be said
to be acquired by experience. For if we imagine a
vessel full of small, smooth spherical balls, such a collec
tion of balls would approach to the nature of a fluid, in
having its parts moveable among each other ; and would
approach to perfect fluidity, as the balls became
smoother and smaller. And such a collection of balls
would also possess the statical properties of a fluid ; for
it would transmit pressure out of a vertical into a lateral
(or any other) direction, in the same manner as a fluid
would do. And thus a collection of solid bodies has
the same property which a fluid has; and the science
of Hydrostatics borrows from experience no principles
beyond those which are involved in the science of
Statics respecting solids. And since in this latter por
tion of science, as we have already seen, none of the
principles depend for their evidence upon any special
experience, the doctrines of Hydrostatics also are not
proved by experience, but have a necessary truth bor
rowed from the relations of our ideas.

It is hardly to be expected that the above reasoning
will, at first sight, produce conviction in the mind of the
reader, except he have, to a certain extent, acquainted
himself with the elementary doctrines of the science of
Hydrostatics as usually delivered; and have followed,
with clear and steady apprehension, some of the trains
of reasoning by which the pressures of fluids are deter
mined ; as, for instance, the explanation of what is called
the Hydrostatic, Paradox. The necessity of such a dis-

ESTABLISHMENT OF THE PRINCIPLES OF STATICS. 21.5

cipline in order that the reader may enter fully into this
part of our speculations, naturally renders them less
popular ; but this disadvantage is inevitable in our plan.
We cannot expect to throw light upon philosophy by
means of the advances which have been made in the
mathematical and physical sciences, except we really
understand the doctrines which have been firmly esta
blished in those sciences. This preparation for philoso
phizing may be somewhat laborious ; but such labour is
necessary if we would pursue speculative truth with all
the advantages which the present condition of human
knoAvledge places within our reach.

We may add, that the consequences to which we are
directed by the preceding opinions, are of very great im
portance in their bearing upon our general views respect
ing human knowledge. I trust to be able to show, that
some important distinctions are illustrated, some per
plexing paradoxes solved, and some large anticipations
of the future extension of our knowledge suggested, by
means of the conclusions to which the preceding discus
sions have conducted us. But before I proceed to these
general topics, I must consider the foundations of some
of the remaining portions of Mechanics.

CHAPTER VII.

OF THE ESTABLISHMENT OF THE PRINCIPLES
OF DYNAMICS.

1. IN the History of Mechanics, I have traced the
steps by which the three Laws of Motion and the other
principles of mechanics were discovered, established, and
extended to the widest generality of form and applica
tion. We have, in these laws, examples of principles
which were, historically speaking, obtained by reference

216 PHILOSOPHY OF THE MECHANICAL SCIENCES.

to experience. Bearing in mind the object and the re
sult of the preceding discussions, we cannot but turn
with much interest to examine these portions of science ;
to inquire whether there be any real difference in the
grounds and nature between the knowledge thus ob
tained, and those truths which we have already contem
plated; and which, as we have seen, contain their own
evidence, and do not require proof from experiment.

2. The First Law of Motion. The first law of mo
tion is, that When a body moves not acted upon by any
force, it will go on perpetually in a straight line, and
with a uniform velocity. Now what is the real ground
of our assent to this proposition ? That it is not at first
sight a self-evident truth, appears to be clear ; since from
the time of Aristotle to that of Galileo the opposite
assertion was held to be true ; and it was believed that
all bodies in motion had, by their own nature, a constant
tendency to move more and more slowly, so as to stop at
last. This belief, indeed, is probably even now enter
tained by most persons, till their attention is fixed upon
the arguments by which the first law of motion is esta
blished. It is, however, not difficult to lead any person
of a speculative habit of thought to see that the retard
ation which constantly takes place in the motion of all
bodies when left to themselves, is, in reality, the effect
of extraneous forces which destroy the velocity. A top
ceases to spin because the friction against the ground
and the resistance of the air gradually diminish its mo
tion, and not because its motion has any internal prin
ciple of decay or fatigue. This may be shown, and was,
in fact, shown by Hooke before the Royal Society, at the
time when the laws of motion were still under discus
sion, by means of experiments in which the weight of
the top is increased, and the resistance to motion offered
by its support, is diminished ; for by such contrivances

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 217

its motion is made to continue much longer than it
would otherwise do. And by experiments of this nature,
although we can never remove the whole of the external
impediments to continued motion, and although, conse
quently, there will always be some retardation ; and an
end of the motion of a body left to itself, however long
it may be delayed, must at last come ; yet we can esta
blish a conviction that if all resistance could de removed,
there would be no diminution of velocity, and thus the
motion would go on for ever.

If we call to mind the axioms which we formerly
stated, as containing the most important conditions
involved in the idea of Cause, it will be seen that our
conviction in this case depends upon the first axiom of
Causation, that nothing can happen without a cause.
Every change in the velocity of the moving body must
have a cause ; and if the change can, in any manner, be
referred to the presence of other bodies, these are said
to exert force upon the moving body: and the conception
of force is thus evolved from the general idea of cause.
Force is any cause which has motion, or change of
motion, for its effect ; and thus, all the change of velocity
of a body which can be referred to extraneous bodies, as
the air which surrounds it, or the support on which it
rests, is considered as the effect of forces; and this
consideration is looked upon as explaining the difference
between the motion which really takes places in the expe
riment, and that motion which, as the law asserts, would
take place if the body were not acted on by any forces.

Thus the truth of the first law of motion depends
upon the axiom that no change can take place without a
cause; and follows from the definition of force, if we sup
pose that there can be none but an external cause of
change. But in order to establish the law, it was neces
sary further to be assured that there is no internal cause

218 PHILOSOPHY OF THE MECHANICAL SCIENCES.

of change of velocity belonging to all matter whatever,
and operating in such a manner that the mere progress
of time is sufficient to produce a diminution of velocity
in all moving bodies. It appears from the history of
mechanical science, that this latter step required a refer
ence to observation and experiment ; and that the first
law of motion is so far, historically at least, dependent
upon our experience.

But notwithstanding this historical evidence of the
need which we have of a reference to observed facts, in
order to place this first law of motion out of doubt, it has
been maintained by very eminent mathematicians and
philosophers, that the law is, in truth, evident of itself,
and does not really rest upon experimental proof. Such,
for example, is the opinion of D Alembert *, who offers
what is called an d priori proof of this law ; that is, a
demonstration derived from our ideas alone. When a
body is put in motion, either, he says, the cause which
puts it in motion at first, suffices to make it move one
foot, or the continued action of the cause during this foot
is requisite for the motion. In the first case, the same
reason which made the body proceed to the end of the
first foot will hold for its going on through a second,
a third, a fourth foot, and so on for any number. In
the second case, the same reason which made the force
continue to act during the first foot, will hold for its
acting, and therefore for the body moving during each
succeeding foot. And thus the body, once beginning to
move, must go on moving for ever.

It is obvious that we might reply to this argument,
that the reasons for the body proceeding during each
succeeding foot may not necessarily be all the same ; for
among these reasons may be the time which has elapsed ;
and thus the velocity may undergo a change as the time

* Dynamiqne.

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 210

proceeds : and we require observation to inform us that
it does not do so.

Professor Playfair has presented nearly the same
argument, although in a different and more mathematical
form*. If the velocity change, says he, it must change
according to some expression of calculation depending
upon the time, or, in mathematical language, must be a
function of the time. If the velocity diminish as the
time increases, this may be expressed by stating the velo
city in each case as a certain number, from which another
quantity, or term, increasing as the time increases, is
subtracted. But, Playfair adds, there is no condition
involved in the nature of the case, by which the coeffi
cients, or numbers which are to be employed, along with
the number representing the time, in calculating this
second term, can be determined to be of one magnitude
rather than of any other. Therefore he infers there can
be no such coefficients, and that the velocity is in each
case equal to some constant number, independent of the
time ; and is therefore the same for all times.

In reply to this we may observe, that the circum
stance of our not seeing in the nature of the case any
thing which determines for us the coefficients above
spoken off, cannot prove that they have not some certain
value in nature. We do not see in the nature of the
case anything which should determine a body to fall six
teen feet in a second of time, rather than one foot or one
hundred feet : yet in fact the space thus run through by
falling bodies is determined to a certain magnitude. It
would be easy to assign a mathematical expression for
the velocity of a body, implying that one-hundredth of the
velocity, or any other fraction, is lost in each second f:

* Outlines, &c., p. 26.

t This would be the case, if, / being the number of seconds elapsed,

220 PHILOSOPHY OF THE MECHANICAL SCIENCES.

and where is the absurdity of supposing such an expres
sion really to represent the velocity ?

Most modern writers on mechanics have embraced
the opposite opinion, and have ascribed our knowledge
of this first law of motion to experience. Thus M.
Poisson, one of the most eminent of the mathematicians
who have written on this subject, says*, " We cannot
affirm a priori that the velocity communicated to a body
will not become slower and slower of itself, and end by
being entirely extinguished. It is only by experience
and induction that this question can be decided."

Yet it cannot be denied that there is much force in
those arguments by which it is attempted to shew that
the First Law of Motion, such as we find it, is more
consonant to our conceptions than any other would be.
The Law, as it exists, is the most simple that we can
conceive. Instead of having to determine by experi
ments what is the law of the natural change of velocity,
we find the Law to be that it does not change at all. To a
certain extent, the Law depends upon the evident axiom,
that no change can take place without a cause. But
the question further occurs, whether the mere lapse of
time may not be a cause of change of velocity. In order
to ensure this, we have recourse to experiment ; and the
result is that time alone does not produce any such
change. In addition to the conditions of change which
we collect from our own Ideas, we ask of Experience what
other conditions and circumstances she has to offer ; and
the answer is, that she can point out none. When we
have removed the alterations which external causes, in

and C some constant quantity, the velocity were expressed by this
mathematical formula,

r /j#v

" Viooy

* Poisson, Dynamiquc. Ed. 2, Art. 113.

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 221

our very conception of them, occasion, there are no
longer any alterations. Instead of having to guide our
selves by experience, we learn that on this subject she
has nothing to tell us. Instead of having to take into
account a number of circumstances, we find that we have
only to reject all circumstances. The velocity of a body
remains unaltered by time alone, of whatever kind the
body itself be.

But the doctrine that time alone is not a cause of
change of velocity in any body is further recommended
to us by this consideration ; that time is conceived by
us not as a cause, but only as a condition of other causes
producing their effects. Causes operate in time ; but it
is only when the cause exists, that the lapse of time can
give rise to alterations. When therefore all external
causes of change of velocity are supposed to be removed,
the velocity must continue identical with itself, whatever
the time which elapses. An eternity of negation can
produce no positive result.

Thus, though the discovery of the First Law of
Motion was made, historically speaking, by means of
experiment, we have now attained a point of view in
which we see that it might have been certainly known
to be true independently of experience. This law in its
ultimate form, when completely simplified and steadily
contemplated, assumes the character of a self-evident
truth. We shall find the same process to take place in
other instances. And this feature in the progress of
science will hereafter be found to suggest very important
views with regard both to the nature and prospects of
our knowledge.

3. Gravity is a Uniform Force. We shall find
observations of the same kind offering themselves in a
manner more or less obvious, with regard to the other
principles of Dynamics. The determination of the laws

PHILOSOPHY OF THE MECHANICAL SCIENCES.

according to which bodies fall downwards by the com
mon action of gravity, has already been noticed in the
History of Mechanics*, as one of the earliest positive
advances in the doctrine of motion. These laws were
first rightly stated by Galileo, and established by rea
soning and by experiment, not without dissent and con
troversy. The amount of these doctrines is this : That
gravity is a uniform accelerating force ; such a uniform
force having this for its character, that it makes the
velocity increase in exact proportion to the time of
motion. The relation which the spaces described by the
body bear to the times in which they are described, is
obtained by mathematical deduction from this definition
of the force.

The clear Definition of a uniform accelerating force,
and the Proposition that gravity is such a force, were
co-ordinate and contemporary steps in this discovery.
In defining accelerating force, reference, tacit or ex
press, was necessarily made to the second of the general
axioms respecting causation, That causes are measured
by their effects. Force, in the cases now under our
notice, is conceived to be, as we have already stated,
(p. 217,) any cause which, acting from without, changes
the motion of a body. It must, therefore, in this accep
tation, be measured by the magnitude of the changes
which are produced. But in what manner the changes
of motion are to be employed as the measures of force, is
learnt from observation of the facts which we see taking
place in the world. Experience interprets the axiom of
causation, from which otherwise we could riot deduce
any real knowledge. We may assume, in virtue of our
general conceptions of force, that under the same cir
cumstances, a greater change of motion implies a greater
force producing it ; but what are we to expect when the

* Hist. Ind. Sci., B. vi. c. ii. sect. 2.

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 223

circumstances change ? The weight of a body makes it
fall from rest at first, and causes it to move more quickly
as it descends lower. We may express this by saying,
that gravity, the universal force which makes all terres
trial bodies fall when not supported, by its continuous
action first gives velocity to the body when it has none,
and afterwards adds velocity to that which the body
already has. But how is the velocity added propor
tioned to the velocity which already exists? Force
acting on a body at rest, and on a body in motion,
appears under very different conditions; how are the
effects related ? Let the force be conceived to be in both
cases the same, since force is conceived to depend upon
the extraneous bodies, and not upon the condition of the
moving mass itself. But the force being the same, the
effects may still be different. It is at first sight con
ceivable that the body, acted upon by the same gravity,
may receive a less addition of velocity when it is already
moving in the direction in which this gravity impels it ;
for if we ourselves push a body forwards, we can produce
little additional effect upon it when it is already moving
rapidly away from us. May it not be true, in like man
ner, that although gravity be always the same force, its
effect depends upon the velocity which the body under
its influence already possesses ?

Observation and reasoning combined, as we have
said, enabled Galileo to answer these questions. He as
serted and proved that we may consistently and properly
measure a force by the velocity which is by it generated
in a body, in some certain time, as one second ; and
further, that if we adopt this measure, gravity will be a
force of the same value under all circumstances of the
body which it affects; since it appeared that, in fact, a
falling body does receive equal increments of velocity
in equal times from first to last.

224 PHILOSOPHY OF THE MECHANICAL SCIENCES.

If it be asked whether we could have known, anterior
to, or independent of, experiment, that gravity is a uni
form force in the sense thus imposed upon the term ;
it appears clear that we must reply, that we could not
have attained to such knowledge, since other laws of the
motion of bodies downwards are easily conceivable, and
nothing but observation could inform us that one of
these laws does not prevail in fact. Indeed, we may add,
that the assertion that the force of gravity is uniform, is
so far from being self-evident, that it is not even true ;
for gravity varies according to the distance from the
center of the earth ; and although this variation is so
small as to be, in the case of falling bodies, imperceptible,
it negatives the rigorous uniformity of the force as com
pletely, though not to the same extent, as if the weight
of a body diminished in a marked degree, when it was
carried from the lower to the upper room of a house. It
cannot, then, be a truth independent of experience, that
gravity is uniform.

Yet, in fact, the assertion that gravity is uniform was
assented to, not only before it was proved, but even
before it was clearly understood. It was readily granted
by all, that bodies which fall freely are uniformly accele
rated ; but while some held the opinion just stated, that
uniformly accelerated motion is that in which the velocity
increases in proportion to the time, others maintained,
that that is uniformly accelerated motion, in which the
velocity increases in proportion to the space ; so that, for
example, a body in falling vertically through twenty feet
should acquire twice as great a velocity as one which
falls through ten feet.

These two opinions are both put forward by the
interlocutors of Galileo s Dialogue on this subject*. And
the latter supposition is rejected, the author showing,

* Din logo, in. p. 95.

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 225

not that it is inconsistent with experience, but that it is
impossible in itself: inasmuch as it would inevitably lead
to the conclusion, that the fall through a large and a
small vertical space would occupy exactly the same time.
Indeed, Galileo assumes his definition of uniformly
accelerated motion as one which is sufficiently recom
mended by its own simplicity. " If we attend carefully,"
he says, "we shall h nd that no mode of increase of velocity
is more simple than that which adds equal increments in
equal times. Which we may easily understand if we
consider the close affinity of time and motion : for as the
uniformity of motion is defined by the equality of spaces
described in equal times, so we may conceive the uni
formity of acceleration to exist when equal velocities are
added in equal times."

Galileo s mode of supporting his opinion, that bodies
falling by the action of gravity are thus uniformly acce
lerated, consists, in the first place, in adducing the
maxim that nature always employs the most simple
means*. But he is far from considering this a decisive
argument. " I," says one of his speakers, " as it would
be very unreasonable in me to gainsay this or any other
definition which any author may please to make, since
they are all arbitrary, may still, without offence, doubt
whether such a definition, conceived and admitted in the
abstract, fits, agrees, and is verified in that kind of
accelerated motion which bodies have when they descend
naturally."

The experimental proof that bodies, when they fall
downwards, are uniformly accelerated, is (by Galileo)
derived from the inclined plane ; and therefore assumes
the proposition, that if such uniform acceleration prevail
in vertical motion, it will also hold when a body is com
pelled to describe an oblique rectilinear path. This pro-

* Dialogo, in. p. 91.
VOL. I. \V. P. Q

226 PHILOSOPHY OF THE MECHANICAL SCIENCES.

position may be shown to be true, if (assuming by anti
cipation the Third Law of Motion, of which we shall
shortly have to speak,) we introduce the conception of
a uniform statical force as the cause of uniform acce
leration. For the force on the inclined plane bears
a constant proportion to the vertical force, and this
proportion is known from statical considerations. But
in the work of which we are speaking, Galileo does
not introduce this abstract conception of force as the
foundation of his doctrines. Instead of this, he pro
poses, as a postulate sufficiently evident to be made
the basis of his reasonings, That bodies which descend
down inclined planes of different inclinations, but of
the same vertical height, all acquire the same velocity*".
But when this postulate has been propounded by one
of the persons of the dialogue, another interlocutor says,
"You discourse very probably; but besides this like
lihood, I wish to augment the probability so far, that
it shall be almost as complete as a necessary demon
stration." He then proceeds to describe a very inge
nious and simple experiment, which shows that when a
body is made to swing upwards at the end of a string,
it attains to the same height, whatever is the path it
follows, so long as it starts from the lowest point with
the same velocity. And thus Galileo s postulate is ex
perimentally confirmed, so far as the force of gravity can
be taken as an example of the forces which the postulate
contemplates : and conversely, gravity is proved to be a
uniform force, so far as it can be considered clear that
the postulate is true of uniform forces.

When we have introduced the conception and defi
nition of accelerating force, Galileo s postulate, that
bodies descending down inclined planes of the same
vertical height, acquire the same velocity, may, by a

* Dialogo, in. p. 36.

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 227

few steps of reasoning, be demonstrated to be true of
uniform forces : and thus the proof that gravity, either in
vertical or oblique motion, is a uniform force, is con
firmed by the experiment above mentioned ; as it also is,
on like grounds, by many other experiments, made upon
inclined planes and pendulums.

Thus the propriety of Galileo s conception of a uni
form force, and the doctrine that gravity is a uniform
force, were confirmed by the same reasonings and experi
ments. We may make here two remarks ; First, that the
conception, when established and rightly stated, appears
so simple as hardly to require experimental proof; a
remark which we have already made with regard to the
First Law of Motion : and Second, that the discovery of
the real law of nature was made by assuming proposi
tions which, without further proof, we should consider as
very precarious, and as far less obvious, as well as less
evident, than the law of nature in its simple form.

4. The Second Law of Motion. When a body, instead
of falling downwards from rest, is thrown in any direc
tion, it describes a curve line, till its motion is stopped.
In this, and in all other cases in which a body describes
a curved path in free space, its motion is determined by
the Second Law of Motion. The law, in its general
form, is as follows: When a body is thus cast forth
and acted upon by a force in a direction transverse to its
motion, the result is, That there is combined with the
motion with which the body is thrown, another motion,
exactly the same as that which the same force would have
communicated to a body at rest.

It will readily be understood that the basis of this
law is the axiom already stated, that effects are measured
by their causes. In virtue of this axiom, the effect of
gravity acting upon a body in a direction transverse to its
motion, must measure the accelerative or deflective force

228 PHILOSOPHY OF THE MECHANICAL SCIENCES.

of gravity under those circumstances. If this effect vary
with the varying velocity and direction of the body thus
acted upon, the deflective force of gravity also will vary
with those circumstances. The more simple supposition
is, that the deflective force of gravity is the same, whatever
be the velocity and direction of the body which is sub
jected to its influence : and this is the supposition which
we find to be verified by facts. For example, a ball let
fall from the top of a ship s upright mast, when she is
sailing steadily forward, will fall at the foot of the mast,
just as if it were let fall while the ship were at rest ; thus
showing that the motion which gravity gives to the ball
is compounded with the horizontal motion which the ball
shares with the ship from the first. This general and
simple conception of motions as compounded with one
another, represents, it is proved, the manner in which
the motion produced by gravity modifies any other mo
tion which the body may previously have had.

The discussions which terminated in the general re
ception of this Second Law of Motion among mechanical
writers, were much mixed up with the arguments for and
against the Copernican system, which system represented
the earth as revolving upon its axis. For the obvious
argument against this system was, that if each point of the
earth s surface were thus in motion from west to east, a
stone dropt from the top of a tower would be left behind,
the tower moving away from it : and the answer was, that
by this law of motion, the stone would have the earth s
motion impressed upon it, as well as that motion which
would arise from its gravity to the earth ; and that the
motion of the stone relative to the tower would thus be
the same as if both earth and tower were at rest. Gali
leo further urged, as a presumption in favour of the opi
nion that the two motions, the circular motion arising
from the rotation of the earth, and the downward motion

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 229

arising from the gravity of the stone, would be com
pounded in the way we have described, (neither of them
disturbing or diminishing the other,) that the first
motion w r as in its own nature not liable to any change or
diminution"", as we learn from the First Law of Motion.
Nor was the subject lightly dismissed. The experiment
of the stone let fall from the top of the mast was made
in various forms by Gassendi ; and in his Epistle, De
Motu impresso a Motore translate, the rule now in ques
tion is supported by reference to these experiments. In
this manner, the general truth, the Second Law of
Motion, was established completely and beyond dispute.
But when this law had been proved to be true in a
general sense, with such accuracy as rude experiments,
like those of Galileo and Gassendi, would admit, it still
remained to be ascertained (supposing our knowledge of
the law to be the result of experience alone,) whether it
were true with that precise and rigorous exactness which
more refined modes of experimenting could test. We
so willingly believe in the simplicity of laws of nature,
that the rigorous accuracy of such a law, known to be at
least approximately true, was taken for granted, till some
ground for suspecting the contrary should appear. Yet
calculations have not been wanting which might confirm
the law as true to the last degree of accuracy. Laplace
relates (Syst. du Monde, livre iv., chap. 1 6,) that at one
time he had conceived it possible that the effect of
gravity upon the moon might be slightly modified by the
moon s direction and velocity; and that in this way an
explanation might be found for the moon s acceleration
(a deviation of her observed from her calculated place,
which long perplexed mathematicians). But it was after
some time discovered that this feature in the moon s
motion arose from another cause; and the second law of
* Dialoga, ii. |). 114.

230 PHILOSOPHY OF THE MECHANICAL SCIENCES.

motion was confirmed as true in the most rigorous
sense.

Thus we see that although there were arguments
which might be urged in favour of this law, founded
upon the necessary relations of ideas, men became con
vinced of its truth only when it was verified and con
firmed by actual experiment. But yet in this case
again, as in the former ones, when the law had been
established beyond doubt or question, men were very
ready to believe that it was not a mere result of observa
tion, that the truth which it contained was not derived
from experience, that it might have been assumed as
true in virtue of reasonings anterior to experience, and
that experiments served only to make the law more plain
and intelligible, as visible diagrams in geometry serve to
illustrate geometrical truths; our knowledge not being
(they deemed) in mechanics, any more than in geometry,
borrowed from the senses. It was thought by many to
be self-evident, that the effect of a force in any direction
cannot be increased or diminished by any motion trans
verse to the direction of the force which the body may
have at the same time : or, to express it otherwise, that
if the motion of the body be compounded of a horizontal
and vertical motion, the vertical motion alone will be
affected by the vertical force. This principle, indeed,
not only has appeared evident to many persons, but even
at the present day is assumed as an axiom by many of
the most eminent mathematicians. It is, for example,
so employed in the Mccanique Celeste of Laplace, which
may be looked upon as the standard of mathematical
mechanics in our time; and in the Mecanique Analy-
tique of Lagrange, the most consummate example which
has appeared of subtilty of thought on such subjects, as
well as of power of mathematical generalization*. And

* I may observe that the rule that we may compound motions, as

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 231

thus we have here another example of that circumstance
which we have already noticed in speaking of the First
Law of Motion, (Art. 2 of this Chapter,) and of the Law
that Gravity is a uniform Force, (Art. 3) ; namely, that
the law, though historically established by experiments,
appears, when once discovered and reduced to its most
simple and general form, to be self-evident. I am the
more desirous of drawing attention to this feature in
various portions of the history of science, inasmuch as it
will be found to lead to some very extensive and impor
tant views, hereafter to be considered.

5. The Third Law of Motion. We have, in the
definition of Accelerating Force, a measure of Forces, so
far as they are concerned in producing motion. We had
before, in speaking of the principles of statics, defined
the measure of Forces or Pressures, so far as they are
employed in producing equilibrium. But these two
aspects of Force are closely connected; and we require a
law which shall lay down the rule of their connexion.
By the same kind of muscular exertion by which we

the Law supposes, is involved in the step of resolving them ; which is
done in the passage to which I refer (Mec. Analyt. Ptie. i., sect. i. art. 3,
p. 225). " Si on con9oit que la mouvement d un corps et les forces
qui le sollicitent soient decomposes suivant trois lignes droites perpen-
diculaires entre elles, on pourra considerer separement les mouvemens
et les forces relatives a chacun a de ces trois directions. Car a cause de
la perpendicularite des directions il est visible que chacun de ces mouve
mens partiels pent etre regarde comme independant des deux autres,
et qu il ne peut recevoir d alteration que de la part de la force qui agit
dans la direction de ce mouvement ; Ton peut conclure que ces trois
mouvements doivent suivre, chacun en particulier, les lois des mouve
mens rectilignes acceleres oti retardes par les forces donnees." Laplace
makes the same assumption in effect, (Mec. Cel. P. i., liv. i., art. 7,)
by resolving the forces which act upon a point in three rectangular
directions, and reasoning separately concerning each direction. But in
his mode of treating the subject is involved a principle which belongs
to the Third Law of Motion, namely, the doctrine that the velocity is
its the force, of which we shall have to speak elsewhere.

232 PHILOSOPHY OF THE MECHANICAL SCIENCES.

can support a heavy stone, we can also put it in motion.
The question then occurs, how is the rate and manner
of its motion determined ? The answer to this question
is contained in the Third Law of Motion, and it is to
this effect : that the Momentum which any pressure pro
duces in the mass in a given time is proportional to the
pressure. By Momentum is meant the product of the
numbers which express the velocity and the mass of the
body : and hence, if the mass of the body be the same
in the instances which we compare, the rule is, That
the velocity is as the force which produces it ; and this is
one of the simplest ways of expressing the Third Law
of Motion.

In agreement with our general plan, we have to ask,
What is the ground of this rule ? What is the simplest
and most satisfactory form to which we can reduce the
proof of it ? Or, to take an instance ; if a double pres
sure be exerted against a given mass, so disposed as to
be capable of motion, why must it produce twice the
velocity in the same time ?

To answer this question, suppose the double pressure
to be resolved into two single pressures : one of these
will produce a certain velocity; and the question is, why
an equal pressure, acting upon the same mass, will pro
duce an equal velocity in addition to the former? Or,
stating the matter otherwise, the question is, why each
of the two forces will produce its separate effect, unal
tered by the simultaneous action of the other force ?

This statement of the case makes it seem to approach
very near to such cases as are included in the Second
Law of Motion, and therefore it might appear that this
Third Law has no grounds distinct from the Second.
But it must be recollected that the word force has a dif
ferent meaning in this case and in that ; in this place it
signifies pressure ; in the statement of the Second Law

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 233

its import was accelerative or deflective force, measured
by the velocity or deflexion generated. And thus the
Third Law of Motion, so far as our reasonings yet go,
appears to rest on a foundation different from the Second.

Accordingly, that part of the Third Law of Motion
which we are now considering, that the velocity gene
rated is as the force, was obtained, in fact, by a separate
train of research. The first exemplification of this law
which was studied by mathematicians, was the motion
of bodies upon inclined planes : for the force which urges
a body down an inclined plane is known by statics, and
hence the velocity of its descent was to be determined.
Galileo originally* in his attempts to solve this problem
of the descent of a body down an inclined plane, did not
proceed from the principle which we have stated, (the
determination of the force which acts down the inclined
plane from statical considerations,) obvious as it may
seem ; but assumed, as we have already seen, a propo
sition apparently far more precarious ; namely, that
a body sliding down a smooth inclined plane acquires
always the same velocity, so long as the vertical height
fallen through is the same. And this conjecture, (for
at first it was nothing more than a conjecture,) he
confirmed by an ingenious experiment ; in which bodies
acquired or lost the same velocity by descending or
ascending through the same height, although their paths
were different in other respects.

This was the form in which the doctrine of the mo
tion of bodies down inclined planes was at first presented
in Galileo s Dialogues on the Science of Motion. But
his disciple Viviani was dissatisfied with the assumption
thus introduced ; and in succeeding editions of the Dia
logues, the apparent chasm in the reasoning was much
narrowed, by making the proof depend upon a principle

* Dial, tlclla \c. \nm\ in., j&gt;. &lt;)&lt;;. Sot- Hist. Ind. Sci. B.vi. c. ii. sect. ."").

234 PHILOSOPHY OF THE MECHANICAL SCIENCES.

nearly identical with the third law of motion as we have
just stated it. In the proof thus added, " We are agreed,"
says the interlocutor"", "that in a moving body the
impetus, energy, momentum, or propension to motion, is
as great as is the force or least resistance which suffices
to sustain it ;" and the impetus or momentum, in the
course of the proof, being taken to be as the velocity
produced in a given time, it is manifest that the prin
ciple so stated amounts to this ; that the velocity pro
duced is as the statical force. And thus this law of
motion appears, in the school of Galileo, to have been
suggested and established at first by experiment, but
afterwards confirmed and demonstrated by a priori
considerations.

We see, in the above reasoning, a number of abstract
terms introduced which are not, at first at least, very
distinctly defined, as impetus, momentum, &c. Of
these, momentum has been selected, to express that
quantity which, in a moving body, measures the statical
force impressed upon the body. This quantity is, as we
have just seen, proportional to the velocity in a given
body. It is also, in different bodies, proportional to the
mass of the body. This part of the third law of motion
follows from our conception of matter in general as con
sisting of parts capable of addition. A double pressure
must be required to produce the same velocity in a
double mass ; for if the mass be halved, each half will
require an equal pressure ; and the addition, both of the
pressures and of the masses, will take place without dis
turbing the effects.

The measure of the quantity of matter of a body con
sidered as affecting the velocity which pressure produces
in the body, is termed its inertia, as we have already
stated, (p. 190.) Inertia is the property by which a
* Dialogo, p. 104.

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 235

large mass of matter requires .a greater force than a
small mass, to give it an equal velocity. It belongs to
each portion of matter; and portions of inertia are
added whenever portions of matter are added. Hence
inertia is as the quantity of matter ; which is only an
other way of expressing this third law of motion, so far
as quantity of matter is concerned.

But how do we know the quantity of matter of a
body ? We may reply, that we take the weight as the
measure of the quantity of matter : but we may then be
again asked, how it appears that the weight is propor
tional to the inertia ; which it must be, in order that the
quantity of matter may be proportional to both one and
the other. We answer, that this appears to be true
experimentally, because all bodies fall with equal veloci
ties by gravity, when the known causes of difference are
removed. The observations of falling bodies, indeed,
are not susceptible of much exactness : but experiments
leading to the same result, and capable of great precision,
were made upon pendulums by Newton ; as he relates in
his Principia, Book in., prop. 6. They all agreed, he
says, with perfect accuracy : and thus the weight and the
inertia are proportional in all cases, and therefore each
proportional to the quantity of matter as measured by
the other.

The conception of inertia, as we have already seen in
chapter v., involves the notion of action and reaction;
and thus the laws which involve inertia depend upon the
idea of mutual causation. The rule, that the velocity is
as the force, depends upon the principle of causation,
that the effect is proportional to the cause ; the effect
being here so estimated as to be consistent both with
the other laws of motion and with experiment.

But here, as in other cases, the question occurs
again ; Is experiment really requisite for the proof of

236 PHILOSOPHY OF THE MECHANICAL SCIENCES.

this law ? If we look to authorities, we shall be not a
little embarrassed to decide. D Alembert is against the
necessity of experimental proof. "Why," says he*,
" should we have recourse to this principle employed, at
the present day, by everybody, that the force is propor
tional to the velocity? ... a principle resting solely
upon this vague and obscure axiom, that the effect is
proportional to the cause. We shall not examine here,"
he adds, " if this principle is necessarily true ; we shall
only avow that the proofs which have hitherto been
adduced do not appear to us unexceptionable : nor shall
we, with some geometers, adopt it as a purely contingent
truth; which would be to ruin the certainty of me
chanics, and to reduce it to be nothing more than an
experimental science. We shall content ourselves with
observing," he proceeds, " that certain or doubtful, clear
or obscure, it is useless in mechanics, and consequently
ought to be banished from the science." Though
D Alembert rejects the third law of motion in this form,
he accepts one of equivalent import, which appears to
him to possess axiomatic certainty ; and this procedure
is in consistence with the course which he takes, of
claiming for the science of mechanics more than mere
experimental truth. On the contrary, Laplace considers
this third law as established by experiment. " Is the
force," he saysf, "proportioned to the velocity? This,"
he replies, " we cannot know a priori, seeing that we
are in ignorance of the nature of moving force : we must
therefore, for this purpose, recur to experience ; for all
which is not a necessary consequence of the few data we
have respecting the nature of things, is, for us, only a re
sult of observation." And again he saysj, "Here, then,
we have two laws of motion, the law of inertia [the first
law of motion], and the law of the force proportional to

* Dynamique, Pref. p. x. t Mec Cel. p. 15. J P. 18.

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 237

the velocity, which are given by observation. They
are the most natural and the most simple laws which we
can imagine, and without doubt they flow from the very
nature of matter ; but this nature being unknown, they
are, for us, only observed facts : the only ones, however,
which mechanics borrows from experience."

It will appear, I think, from the views given in this
and several other parts of the present work, that we can
not with justice say that we have very " few data respect
ing the nature of things," in speculating concerning the
laws of the universe ; since all the consequences which
flow from the relations of our fundamental ideas, neces
sarily regulate our knowledge of things, so far as we
have any such knowledge. Nor can we say that the na
ture of matter is unknown to us, in any sense in which
we can conceive knowledge as possible. The nature ot
matter is no more unknown than the nature of space or
of number. In our conception of matter, as of space
and of number, are involved certain relations, which are
the necessary groundwork of our knowledge ; and any
thing which is independent of these relations, is not un
known, but inconceivable.

It must be already clear to the reader, from the
phraseology employed by these two eminent mathema
ticians, that the question respecting the formation of the
third law of motion can only be solved by a careful con
sideration of what we mean by observation and experi
ence, nature and matter. But it will probably be gene
rally allowed, that, taking into account the explanations
already offered of the necessary conditions of experience
and of the conception of inertia, this law of motion, that
the inertia is as the quantity of matter, is almost or alto
gether self-evident.

6. Action and Reaction are Equal in Moving Bodies.
When we have to consider bodies as acting upon one

238 PHILOSOPHY OF THE MECHANICAL SCIENCES.

another, and influencing each other s motions, the third
law of motion is still applied ; but along with this, we
also employ the general principle that action and reaction
are equal and opposite. Action and reaction are here to
be understood as momentum produced and destroyed,
according to the measure of action established by the
Third Law of Motion : and the cases in which this prin
ciple is thus employed form so large a portion of those
in which the third law of motion is used, that some
writers (Newton at the head of them) have stated the
equality of action and reaction as the third law of motion.

The third law of motion being once established, the
equality of action and reaction, in the sense of mo
mentum gained and lost, necessarily follows. Thus, if
a weight hanging by a string over the edge of a smooth
level table draw another weight along the table, the
hanging weight moves more slowly than it would do if
not so connected, and thus loses velocity by the con
nexion ; while the other weight gains by the connexion
all the velocity which it has, for if left to itself it would
rest. And the pressures which restrain the descent of the
first body and accelerate the motion of the second, are
equal at all instants of time, for each of these pressures
is the tension of the string : and hence, by the third law
of motion, the momentum gained by the one body, and
the momentum lost by the other in virtue of the action
of this string, are equal. And similar reasoning may be
employed in any other case where bodies are connected.

The case where one body does not push or draw,
but strikes another, appeared at first to mechanical rea-
soners to be of a different nature from the others ; but a
little consideration was sufficient to show that a blow
is, in fact, only a short and violent pressure ; and that,
therefore, the general rule of the equality of momentum
lost and gained applies to this as well as to the other cases.

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 239

Thus, in order to determine the case of the direct
action of bodies upon one another, we require no new
law of motion. The equality of action and reaction,
which enters necessarily into every conception of me
chanical operation, combined with the measure of action
as given by the third law of motion, enables us to trace
the consequences of every case, whether of pressure or
of impact.

7. UAlemberfs Principle. But what will be the
result when bodies do not act directly upon each other,
but are indirectly connected in any way by levers, strings,
pulleys, or in any other manner, so that one part of the
system has a mechanical advantage over another? The
result must still be determined by the principle that
action and reaction balance each other. The action and
reaction, being pressures in one sense, must balance each
other by the laws of statics, for these laws determine
the equilibrium of pressure. Now action and reaction,
according to their measures in the Third Law of Motion,
are momentum gained and lost, when the action is di
rect ; and except the indirect action introduce some
modification of the law, they must have the same mea
sure still. But, in fact, we cannot well conceive any
modification of the law to take place in this case ; for
direct action is only one (the ultimate) case of indirect
action. Thus if two heavy bodies act at different points
of a lever, the action of each on the other is indirect ;
but if the two points come together, the action becomes
direct. Hence the rule must be that which we have
already stated ; for if the rule were false for indirect
action, it would also be false for direct action, for which
case we have shown it to be true. And thus we obtain
the general principle, that in any system of bodies which
act on each other, action and reaction, estimated by mo
mentum gained and lost, balance each other according

240 PHILOSOPHY OF THE MECHANICAL SCIENCES.

to the laws of equilibrium. This principle, which is so
general as to supply a key to the solution of all pos
sible mechanical problems, is commonly called UAlem-
berfs Principle. The experimental proofs which con
vinced men of the truth of the Third Law of Motion
were, many or most of them, proofs of the law in this
extended sense. And thus the proof of D Alembert s
Principle, both from the idea of mechanical action and
from experience, is included in the proof of the law
already stated.

8. Connexion of Dynamical and Statical Principles.
The principle of equilibrium of D Alembert just stated,
is the law which he would substitute for the Third Law
of Motion ; and he would thus remove the necessity for
an independent proof of that law. In like manner, the
Second Law of Motion is by some writers derived from
the principle of the composition of statical forces ; and
they would thus supersede the necessity of a reference to
experiment in that case. Laplace takes this course, and
thus, as we have seen, rests only the First and Third Law
of Motion upon experience. Newton, on the other hand,
recognizes the same connexion of propositions, but for
a different purpose ; for he derives the composition of
statical forces from the Second Law of Motion.

The close connexion of these three principles, the
composition of (statical) forces, the composition of (ac
celerating) forces with velocities, and the measure of
(moving) forces by velocities, cannot be denied; yet it
appears to be by no means easy to supersede the neces
sity of independent proofs of the two last of these prin
ciples. Both may be proved or illustrated by expe
riment : and the experiments which prove the one are
different from those which establish the other. For
example, it appears by easy calculations, that when we
apply our principles to the oscillations of a pendulum,

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 241

the Second Law is proved by the fact, that the oscilla
tions take place at the same rate in an east and west,
and in a north and south direction : under the same cir
cumstances, the Third Law is proved by our finding that
the time of a small oscillation is proportional to the
square root of the length of a pendulum ; and similar
differences might be pointed out in other experiments,
as to their bearing upon the one law or the other.

9. Mechanical Principles become gradually more
simple and more evident. I will again point out in
general two circumstances which I have already noticed
in particular cases of the laws of motion. Truths are
often at first assumed in a form which is far from being
the most obvious or simple ; and truths once discovered
are gradually simplified, so as to assume the appearance
of self-evident truths.

The former circumstance is exemplified in several of
the instances which we have had to consider. The
assumption that a perpetual motion is impossible pre
ceded the knowledge of the first law of motion. The
assumed equality of the velocities acquired down two in
clined planes of the same height, was afterwards reduced
to the third law of motion by Galileo himself. In the
History "% we have noted Huyghens s assumption of the
equality of the actual descent and potential ascent of the
center of gravity : this was afterwards reduced by Her
man and the Bernoullis, to the statical equivalence of the
solicitations of gravity and the vicarious solicitations of
the effective forces which act on each point ; and finally
to the principle of D Alembert, which asserts that the
motions gained and lost balance each other.

This assertion of principles which now appear neither
obvious nor self-evident, is not to be considered as a
groundless assumption on the part of the discoverers by

* B. vi. c. v. sect. 2.
VOL. I. W. P. R,

242 PHILOSOPHY OF THE MECHANICAL SCIENCES.

whom it was made. On the contrary, it is evidence of
the deep sagacity and clear thought which were requisite
in order to make such discoveries. For these results are
really rigorous consequences of the laws of motion in
their simplest form : and the evidence of them was pro
bably present, though undeveloped, in the minds of the
discoverers. We are told of geometrical students, who,
by a peculiar aptitude of mind, perceived the evidence of
some of the more advanced propositions of geometry
without going through the introductory steps. We must
suppose a similar aptitude for mechanical reasonings,
which, existing in the minds of Stevinus, Galileo, New
ton, and Huyghens, led them to make those assumptions
which finally resolved themselves into the laws of motion.
We may observe further, that the simplicity and evi
dence which the laws of mechanics have at length as
sumed, are much favoured by the usage of words among
the best writers on such subjects. Terms which origi
nally, and before the laws of motion were fully known,
were used in a very vague and fluctuating sense, were
afterwards limited and rendered precise, so that asser
tions which at first appear identical propositions become
distinct and important principles. Thus force, motion,
momentum, are terms which were employed, though in a
loose manner, from the very outset of mechanical specu
lation. And so long as these words retained the vagueness
of common language, it would have been a useless and
barren truism to say that " the momentum is proportional
to the force," or that " a body loses as much motion as
it communicates to another." But when " momentum "
and "quantity of motion" are defined to mean the pro
duct of mass and velocity, these two propositions imme
diately become distinct statements of the third law of
motion and its consequences. In like manner, the asser
tion that " gravity is a uniform force " was assented to,

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 243

before it was settled what a uniform force was ; but this
assertion only became significant and useful when that
point had been properly determined. The statement
that "when different motions are communicated to the
same body their effects are compounded," becomes the
second law of motion, when we define what composition
of motions is. And the same process may be observed
in other cases.

And thus we see how well the form which science
ultimately assumes is adapted to simplify knowledge.
The definitions which are adopted, and the terms which
become current in precise senses, produce a complete
harmony between the matter and the form of our know
ledge ; so that truths which were at first unexpected and
recondite, became familiar phrases, and after a few gene
rations sound, even to common ears, like identical pro
positions.

10. Controversy of the Measure of Force. In the
History of Mechanics*, we have given an account of the
controversy which, for some time, occupied the mathema
ticians of Europe, whether the forces of bodies in motion
should be reckoned proportional to the velocity, or to the
square of the velocity. We need not here recall the
events of this dispute ; but we may remark, that its his
tory, as a metaphysical controversy, is remarkable in this
respect, that it has been finally and completely settled ;
for it is now agreed among mathematicians that both
sides were right, and that the results of mechanical action
may be expressed with equal correctness by means of
momentum and of vis viva. It is, in one sense, as D Alem-
bert has saidf, a dispute about words; but we are not

* B. vi. c. v. sect. 2.

t D Alembert has also remarked (Dynamique, Pref. xxi.,) that this
controversy "shows how little justice and precision there is in the
pretended axiom that causes are proportional to their effects." But

244 PHILOSOPHY OF THE MECHANICAL SCIENCES.

to infer that, on that account, it was frivolous or useless ;
for such disputes are one principal means of reducing the
principles of our knowledge to their utmost simplicity
and clearness. The terms which are employed in the
science of mechanics are now liberated for ever, in the
minds of mathematicians, from that ambiguity which
was the battle-ground in the war of the vis viva.

But we may observe that the real reason of this con
troversy was exactly that tendency which we have been
noticing ; the disposition of man to assume in his specu
lations certain general propositions as true, and to fix the
sense of terms so that they shall fall in with this truth.
It was agreed, on all hands, that in the mutual action of
bodies the same quantity of force is always preserved;
and the question was, by which of the two measures this
rule could best be verified. We see, therefore, that the
dispute was not concerning a definition merely, but con
cerning a definition combined with a general proposition.
Such a question may be readily conceived to have been
by no means unimportant ; and we may remark, in pass
ing, that such controversies, although they are commonly
afterwards stigmatized as quarrels about words and defi
nitions, are, in reality, events of considerable conse
quence in the history of science ; since they dissipate all
ambiguity and vagueness in the use of terms, and bring
into view the conditions under which the fundamental
principles of our knowledge can be most clearly and
simply presented.

It is worth our while to pause for a moment on the
prospect that we have thus obtained, of the advance of

this reflection is by no means well founded. For since both measures
are true, it appears that causes may be justly measured by their effects,
even when very different kinds of effects are taken. That the axiom
does not point out one precise measure, till illustrated by experience or
by other considerations, we grant : but the same thing occurs in the
application of other axioms also.

ESTABLISHMENT OF THE PRINCIPLES OF DYNAMICS. 245

knowledge, as exemplified in the history of Mechanics.
The general transformation of our views from vague to
definite, from complex to simple, from unexpected dis
coveries to self-evident truths, from seeming contradic
tions to identical propositions, is very remarkable, but it
is by no means peculiar to our subject. The same cir
cumstances, more or less prominent, more or less deve
loped, appear in the history of other sciences, according
to the point of advance which each has reached. They
bear upon very important doctrines respecting the pro
spects, the limits, and the very nature of our knowledge.
And though these doctrines require to be considered with
reference to the whole body of science, yet the peculiar
manner in which they are illustrated by the survey of
the history of Mechanics, on which we have just been
engaged, appears to make this a convenient place for
introducing them to the reader.

CHAPTER VIII.

OF THE PARADOX OF UNIVERSAL PROPOSI
TIONS OBTAINED FROM EXPERIENCE.

1. IT was formerly stated" " that experience cannot
establish any universal or necessary truths. The number
of trials which we can make of any proposition is neces
sarily limited, and observation alone cannot give us any
ground of extending the inference to untried cases. Ob
served facts have no visible bond of necessary connexion,
and no exercise of our senses can enable us to discover
such connexion. We can never acquire from a mere
observation of facts, the right to assert that a proposition
is true in all cases, and that it could not be otherwise
than we find it to be.

* B. i., c. v. Of Experience.

246 PHILOSOPHY OF THE MECHANICAL SCIENCES.

Yet, as we have just seen in the history of the laws of
motion, we may go on collecting our knowledge from
observation, and enlarging and simplifying it, till it ap
proaches or attains to complete universality and seeming
necessity. Whether the laws of motion, as we now know
them, can be rigorously traced to an absolute necessity in
the nature of things, we have not ventured absolutely to
pronounce. But we have seen that some of the most
acute and profound mathematicians have believed that,
for these laws of motion, or some of them, there was
such a demonstrable necessity compelling them to be
such as they are, and no other. Most of those who have
carefully studied the principles of Mechanics will allow
that some at least of the primary laws of motion approach
very near to this character of necessary truth ; and will
confess that it would be difficult to imagine any other
consistent scheme of fundamental principles. And almost
all mathematicians will allow to these laws an absolute
universality ; so that we may apply them without scruple
or misgiving, in cases the most remote from those to
which our experience has extended. What astronomer
would fear to refer to the known laws of motion, in rea
soning concerning the double stars; although these objects
are at an immeasurably remote distance from that solar
system which has been the only field of our observation
of mechanical facts? What philosopher, in speculating
respecting a magnetic fluid, or a luminiferous ether, would
hesitate to apply to it the mechanical principles which
are applicable to fluids of known mechanical properties ?
When we assert that the quantity of motion in the world
cannot be increased or diminished by the mutual actions
of bodies, does not every mathematician feel convinced
that it would be an unphilosophical restriction to limit
this proposition to such modes of action as we have
tried?

PARADOX OF UNIVERSAL PROPOSITIONS. 247

Yet no one can doubt that, in historical fact, these
laws were collected from experience. That such is the
case, is no matter of conjecture. We know the time, the
persons, the circumstances, belonging to each step of each
discovery. I have, in the History, given an account of
these discoveries ; and in the previous chapters of the pre
sent work, I have further examined the nature and the
import of the principles which were thus brought to light.

Here, then, is an apparent contradiction. Experi
ence, it would seem, has done that which we had proved
that she cannot do. She has led men to propositions,
universal at least, and to principles which appear to some
persons necessary. What is the explanation of this con
tradiction, the solution of this paradox ? Is it true that
Experience can reveal to us universal and necessary
truths ? Does she possess some secret virtue, some un
suspected power, by which she can detect connexions
and consequences which we have declared to be out of
her sphere? Can she see more than mere appearances,
and observe more than mere facts ? Can she penetrate,
in some way, to the nature of things ? descend below the
surface of phenomena to their causes and origins, so as
to be able to say what can and what can not be ; what
occurrences are partial, and what universal ? If this be
so, we have indeed mistaken her character and powers ;
and the whole course of our reasoning becomes pre
carious and obscure. But, then, when we return upon
our path we cannot find the point at which we deviated,
we cannot detect the false step in our deduction. It
still seems that by experience, strictly so called, we
cannot discover necessary and universal truths. Our
senses can give us no evidence of a necessary connexion
in phenomena. Our observation must be limited, and
cannot testify concerning anything which is beyond its
limits. A general view of our faculties appears to prove

248 PHILOSOPHY OF THE MECHANICAL SCIENCES.

it to be impossible that men should do what the history
of the science of mechanics shows that they have done.

2. But in order to try to solve this Paradox, let us
again refer to the History of Mechanics. In the cases
belonging to that science, in which propositions of the
most unquestionable universality, and most approaching
to the character of necessary truths, (as, for instance, the
laws of motion,) have been arrived at, what is the source
of the axiomatic character which the propositions thus
assume ? The answer to this question will, we may hope,
throw some light on the perplexity in which we appear
to be involved.

Now the answer to this inquiry is, that the laws
of motion borrow their axiomatic character from their
being merely interpretations of the Axioms of Causation.
Those axioms, being exhibitions of the Idea of Cause
under various aspects, are of the most rigorous univer
sality and necessity. And so far as the laws of motion
are exemplifications of those axioms, these laws must be
no less universal and necessary. How these axioms are
to be understood ; in what sense cause and effect, action
and reaction, are to be taken, experience and observa
tion did, in fact, teach inquirers on this subject ; and
without this teaching, the laws of motion could never
have been distinctly known. If two forces act together,
each must produce its effect, by the axiom of causation ;
and, therefore, the effects of the separate forces must be
compounded. But a long course of discussion and expe
riment must instruct men of what kind this composition
of forces is. Again ; action and reaction must be equal ;
but much thought and some trial were needed to show
what action and reaction are. Those metaphysicians who
enunciated Laws of motion without reference to expe
rience, propounded only such laws as were vague and
inapplicable. But yet these persons manifested the

PARADOX OF UNIVERSAL PROPOSITIONS. 249

indestructible conviction, belonging to man s speculative
nature, that there exist Laws of motion, that is, uni
versal formulae, connecting the causes and effects when
motion takes place. Those mechanicians, again, who,
observed facts involving equilibrium and motion, and
stated some narrow rules, without attempting to ascend
to any universal and simple principle, obtained laws no
less barren and useless than the metaphysicians; for
they could not tell in what new cases, or whether in
any, their laws would be verified ; they needed a more
general rule, to show them the limits of the rule they
had discovered. They went wrong in each attempt to
solve a new problem, because their interpretation of
the terms of the axioms, though true, perhaps, in certain
cases, was not right in general.

Thus Pappus erred in attempting to interpret as a
case of the lever, the problem of supporting a weight
upon an inclined plane ; thus Aristotle erred in inter
preting the doctrine that the weight of bodies is the
cause of their fall ; thus Kepler erred in interpreting the
rule that the velocity of bodies depends upon the force;
thus Bernoulli "" erred in interpreting the equality of
action and reaction upon a lever in motion. In each
of these instances, true doctrines, already established,
(whether by experiment or otherwise,) were erroneously
applied. And the error was corrected by further reflec
tion, which pointed out that another mode of interpreta
tion was requisite, in order that the axiom which was
appealed to in each case might retain its force in the
most general sense. And in the reas